LMFDB - Embedded newform 2106.2.b.c.649.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2106,2,Mod(649,2106)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2106, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2106.649"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2106 = 2 \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2106.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,-14,0,0,0,0,0,0,0,0,-2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.8164946657$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 34x^{12} + 435x^{10} + 2617x^{8} + 7651x^{6} + 10260x^{4} + 5589x^{2} + 729 $ x^14 + 34x^12 + 435x^10 + 2617x^8 + 7651x^6 + 10260x^4 + 5589x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$1.04629i$ of defining polynomial
Character	$\chi$	$=$	2106.649
Dual form			2106.2.b.c.649.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} -0.484256i q^{5} -5.05438i q^{7} +1.00000i q^{8} -0.484256 q^{10} +3.21153i q^{11} +(3.42035 + 1.14069i) q^{13} -5.05438 q^{14} +1.00000 q^{16} +4.20245 q^{17} +3.21153i q^{19} +0.484256i q^{20} +3.21153 q^{22} +6.27217 q^{23} +4.76550 q^{25} +(1.14069 - 3.42035i) q^{26} +5.05438i q^{28} +4.59254 q^{29} -6.48369i q^{31} -1.00000i q^{32} -4.20245i q^{34} -2.44761 q^{35} -2.08429i q^{37} +3.21153 q^{38} +0.484256 q^{40} -11.0540i q^{41} -9.46735 q^{43} -3.21153i q^{44} -6.27217i q^{46} +5.28124i q^{47} -18.5467 q^{49} -4.76550i q^{50} +(-3.42035 - 1.14069i) q^{52} +6.41990 q^{53} +1.55520 q^{55} +5.05438 q^{56} -4.59254i q^{58} +3.62311i q^{59} -1.00173 q^{61} -6.48369 q^{62} -1.00000 q^{64} +(0.552388 - 1.65633i) q^{65} +1.08194i q^{67} -4.20245 q^{68} +2.44761i q^{70} +4.63041i q^{71} -0.325525i q^{73} -2.08429 q^{74} -3.21153i q^{76} +16.2323 q^{77} -7.83637 q^{79} -0.484256i q^{80} -11.0540 q^{82} +5.86613i q^{83} -2.03506i q^{85} +9.46735i q^{86} -3.21153 q^{88} -8.42912i q^{89} +(5.76550 - 17.2878i) q^{91} -6.27217 q^{92} +5.28124 q^{94} +1.55520 q^{95} -13.0505i q^{97} +18.5467i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{4} - 2 q^{13} - 8 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{23} - 14 q^{25} + 4 q^{26} + 16 q^{29} - 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} - 60 q^{53} + 8 q^{56} - 28 q^{61} + 34 q^{62} - 14 q^{64}+ \cdots - 8 q^{92}+O(q^{100}) $ 14 * q - 14 * q^4 - 2 * q^13 - 8 * q^14 + 14 * q^16 + 8 * q^17 + 8 * q^23 - 14 * q^25 + 4 * q^26 + 16 * q^29 - 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 - 60 * q^53 + 8 * q^56 - 28 * q^61 + 34 * q^62 - 14 * q^64 + 8 * q^65 - 8 * q^68 - 16 * q^74 + 24 * q^77 - 28 * q^79 - 24 * q^82 - 8 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times$.

$n$	$1379$	$1783$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	0.484256i		−	0.216566i	−0.994120	−	0.108283i	$-0.965465\pi$
$5$		−	0.484256i		−	0.216566i	0.994120	−	0.108283i	$-0.0345352\pi$
$6$		0			0
$7$		−	5.05438i		−	1.91038i	−0.296001	−	0.955188i	$-0.595653\pi$
$7$		−	5.05438i		−	1.91038i	0.296001	−	0.955188i	$-0.404347\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−0.484256			−0.153135
$11$			3.21153i			0.968312i	0.874982	+	0.484156i	$0.160873\pi$
$11$			3.21153i			0.968312i	−0.874982	+	0.484156i	$0.839127\pi$
$12$		0			0
$13$	3.42035	+	1.14069i	0.948635	+	0.316371i
$13$	3.42035	+	1.14069i	0.948635	+	0.316371i
$14$	−5.05438			−1.35084
$15$		0			0
$16$	1.00000			0.250000
$17$	4.20245			1.01924			0.509622	−	0.860398i	$-0.329785\pi$
$17$	4.20245			1.01924			0.509622	+	0.860398i	$0.329785\pi$
$18$		0			0
$19$			3.21153i			0.736775i	0.929672	+	0.368388i	$0.120090\pi$
$19$			3.21153i			0.736775i	−0.929672	+	0.368388i	$0.879910\pi$
$20$			0.484256i			0.108283i
$21$		0			0
$22$	3.21153			0.684700
$23$	6.27217			1.30784			0.653919	−	0.756565i	$-0.273124\pi$
$23$	6.27217			1.30784			0.653919	+	0.756565i	$0.273124\pi$
$24$		0			0
$25$	4.76550			0.953099
$26$	1.14069	−	3.42035i	0.223708	−	0.670787i
$27$		0			0
$28$			5.05438i			0.955188i
$29$	4.59254			0.852812			0.426406	−	0.904532i	$-0.359779\pi$
$29$	4.59254			0.852812			0.426406	+	0.904532i	$0.359779\pi$
$30$		0			0
$31$		−	6.48369i		−	1.16451i	−0.813008	−	0.582253i	$-0.802171\pi$
$31$		−	6.48369i		−	1.16451i	0.813008	−	0.582253i	$-0.197829\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	4.20245i		−	0.720715i
$35$	−2.44761			−0.413722
$36$		0			0
$37$		−	2.08429i		−	0.342654i	−0.985214	−	0.171327i	$-0.945194\pi$
$37$		−	2.08429i		−	0.342654i	0.985214	−	0.171327i	$-0.0548055\pi$
$38$	3.21153			0.520979
$39$		0			0
$40$	0.484256			0.0765676
$41$		−	11.0540i		−	1.72634i	−0.504914	−	0.863169i	$-0.668476\pi$
$41$		−	11.0540i		−	1.72634i	0.504914	−	0.863169i	$-0.331524\pi$
$42$		0			0
$43$	−9.46735			−1.44376			−0.721879	−	0.692020i	$-0.756721\pi$
$43$	−9.46735			−1.44376			−0.721879	+	0.692020i	$0.756721\pi$
$44$		−	3.21153i		−	0.484156i
$45$		0			0
$46$		−	6.27217i		−	0.924780i
$47$			5.28124i			0.770348i	0.922844	+	0.385174i	$0.125859\pi$
$47$			5.28124i			0.770348i	−0.922844	+	0.385174i	$0.874141\pi$
$48$		0			0
$49$	−18.5467			−2.64953
$50$		−	4.76550i		−	0.673943i
$51$		0			0
$52$	−3.42035	−	1.14069i	−0.474318	−	0.158186i
$53$	6.41990			0.881841			0.440921	−	0.897546i	$-0.354652\pi$
$53$	6.41990			0.881841			0.440921	+	0.897546i	$0.354652\pi$
$54$		0			0
$55$	1.55520			0.209703
$56$	5.05438			0.675420
$57$		0			0
$58$		−	4.59254i		−	0.603029i
$59$			3.62311i			0.471689i	0.971791	+	0.235844i	$0.0757856\pi$
$59$			3.62311i			0.471689i	−0.971791	+	0.235844i	$0.924214\pi$
$60$		0			0
$61$	−1.00173			−0.128258			−0.0641290	−	0.997942i	$-0.520427\pi$
$61$	−1.00173			−0.128258			−0.0641290	+	0.997942i	$0.520427\pi$
$62$	−6.48369			−0.823430
$63$		0			0
$64$	−1.00000			−0.125000
$65$	0.552388	−	1.65633i	0.0685153	−	0.205442i
$66$		0			0
$67$			1.08194i			0.132180i	0.997814	+	0.0660900i	$0.0210524\pi$
$67$			1.08194i			0.132180i	−0.997814	+	0.0660900i	$0.978948\pi$
$68$	−4.20245			−0.509622
$69$		0			0
$70$			2.44761i			0.292546i
$71$			4.63041i			0.549529i	0.961512	+	0.274764i	$0.0885999\pi$
$71$			4.63041i			0.549529i	−0.961512	+	0.274764i	$0.911400\pi$
$72$		0			0
$73$		−	0.325525i		−	0.0380998i	−0.999819	−	0.0190499i	$-0.993936\pi$
$73$		−	0.325525i		−	0.0380998i	0.999819	−	0.0190499i	$-0.00606414\pi$
$74$	−2.08429			−0.242293
$75$		0			0
$76$		−	3.21153i		−	0.368388i
$77$	16.2323			1.84984
$78$		0			0
$79$	−7.83637			−0.881660			−0.440830	−	0.897591i	$-0.645316\pi$
$79$	−7.83637			−0.881660			−0.440830	+	0.897591i	$0.645316\pi$
$80$		−	0.484256i		−	0.0541415i
$81$		0			0
$82$	−11.0540			−1.22071
$83$			5.86613i			0.643891i	0.946758	+	0.321946i	$0.104337\pi$
$83$			5.86613i			0.643891i	−0.946758	+	0.321946i	$0.895663\pi$
$84$		0			0
$85$		−	2.03506i		−	0.220734i
$86$			9.46735i			1.02089i
$87$		0			0
$88$	−3.21153			−0.342350
$89$		−	8.42912i		−	0.893485i	−0.894662	−	0.446743i	$-0.852584\pi$
$89$		−	8.42912i		−	0.893485i	0.894662	−	0.446743i	$-0.147416\pi$
$90$		0			0
$91$	5.76550	−	17.2878i	0.604388	−	1.81225i
$92$	−6.27217			−0.653919
$93$		0			0
$94$	5.28124			0.544718
$95$	1.55520			0.159560
$96$		0			0
$97$		−	13.0505i		−	1.32508i	−0.749027	−	0.662539i	$-0.769479\pi$
$97$		−	13.0505i		−	1.32508i	0.749027	−	0.662539i	$-0.230521\pi$
$98$			18.5467i			1.87350i
$99$		0			0
$100$	−4.76550			−0.476550
$101$	−6.84702			−0.681304			−0.340652	−	0.940189i	$-0.610648\pi$
$101$	−6.84702			−0.681304			−0.340652	+	0.940189i	$0.610648\pi$
$102$		0			0
$103$	3.55520			0.350304			0.175152	−	0.984541i	$-0.443958\pi$
$103$	3.55520			0.350304			0.175152	+	0.984541i	$0.443958\pi$
$104$	−1.14069	+	3.42035i	−0.111854	+	0.335393i
$105$		0			0
$106$		−	6.41990i		−	0.623556i
$107$	1.41392			0.136689			0.0683445	−	0.997662i	$-0.478228\pi$
$107$	1.41392			0.136689			0.0683445	+	0.997662i	$0.478228\pi$
$108$		0			0
$109$		−	4.97525i		−	0.476543i	−0.971199	−	0.238271i	$-0.923419\pi$
$109$		−	4.97525i		−	0.476543i	0.971199	−	0.238271i	$-0.0765808\pi$
$110$		−	1.55520i		−	0.148283i
$111$		0			0
$112$		−	5.05438i		−	0.477594i
$113$	−13.0883			−1.23124			−0.615622	−	0.788041i	$-0.711095\pi$
$113$	−13.0883			−1.23124			−0.615622	+	0.788041i	$0.711095\pi$
$114$		0			0
$115$		−	3.03733i		−	0.283233i
$116$	−4.59254			−0.426406
$117$		0			0
$118$	3.62311			0.333534
$119$		−	21.2408i		−	1.94714i
$120$		0			0
$121$	0.686087			0.0623716
$122$			1.00173i			0.0906921i
$123$		0			0
$124$			6.48369i			0.582253i
$125$		−	4.72900i		−	0.422975i
$126$		0			0
$127$	−4.56300			−0.404901			−0.202450	−	0.979293i	$-0.564890\pi$
$127$	−4.56300			−0.404901			−0.202450	+	0.979293i	$0.564890\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−1.65633	−	0.552388i	−0.145269	−	0.0484476i
$131$	2.46946			0.215758			0.107879	−	0.994164i	$-0.465594\pi$
$131$	2.46946			0.215758			0.107879	+	0.994164i	$0.465594\pi$
$132$		0			0
$133$	16.2323			1.40752
$134$	1.08194			0.0934654
$135$		0			0
$136$			4.20245i			0.360357i
$137$		−	19.6104i		−	1.67543i	−0.546110	−	0.837714i	$-0.683892\pi$
$137$		−	19.6104i		−	1.67543i	0.546110	−	0.837714i	$-0.316108\pi$
$138$		0			0
$139$	−0.387396			−0.0328585			−0.0164293	−	0.999865i	$-0.505230\pi$
$139$	−0.387396			−0.0328585			−0.0164293	+	0.999865i	$0.505230\pi$
$140$	2.44761			0.206861
$141$		0			0
$142$	4.63041			0.388576
$143$	−3.66337	+	10.9846i	−0.306346	+	0.918575i
$144$		0			0
$145$		−	2.22396i		−	0.184690i
$146$	−0.325525			−0.0269406
$147$		0			0
$148$			2.08429i			0.171327i
$149$			9.45874i			0.774890i	0.921893	+	0.387445i	$0.126642\pi$
$149$			9.45874i			0.774890i	−0.921893	+	0.387445i	$0.873358\pi$
$150$		0			0
$151$		−	1.33164i		−	0.108367i	−0.998531	−	0.0541836i	$-0.982744\pi$
$151$		−	1.33164i		−	0.108367i	0.998531	−	0.0541836i	$-0.0172556\pi$
$152$	−3.21153			−0.260489
$153$		0			0
$154$		−	16.2323i		−	1.30803i
$155$	−3.13977			−0.252192
$156$		0			0
$157$	−5.84243			−0.466277			−0.233139	−	0.972444i	$-0.574900\pi$
$157$	−5.84243			−0.466277			−0.233139	+	0.972444i	$0.574900\pi$
$158$			7.83637i			0.623428i
$159$		0			0
$160$	−0.484256			−0.0382838
$161$		−	31.7019i		−	2.49846i
$162$		0			0
$163$		−	15.1340i		−	1.18538i	−0.805430	−	0.592691i	$-0.798065\pi$
$163$		−	15.1340i		−	1.18538i	0.805430	−	0.592691i	$-0.201935\pi$
$164$			11.0540i			0.863169i
$165$		0			0
$166$	5.86613			0.455300
$167$			12.0268i			0.930663i	0.885137	+	0.465331i	$0.154065\pi$
$167$			12.0268i			0.930663i	−0.885137	+	0.465331i	$0.845935\pi$
$168$		0			0
$169$	10.3976	+	7.80315i	0.799818	+	0.600242i
$170$	−2.03506			−0.156082
$171$		0			0
$172$	9.46735			0.721879
$173$	8.85863			0.673509			0.336755	−	0.941592i	$-0.390671\pi$
$173$	8.85863			0.673509			0.336755	+	0.941592i	$0.390671\pi$
$174$		0			0
$175$		−	24.0866i		−	1.82078i
$176$			3.21153i			0.242078i
$177$		0			0
$178$	−8.42912			−0.631790
$179$	10.3704			0.775121			0.387561	−	0.921844i	$-0.373318\pi$
$179$	10.3704			0.775121			0.387561	+	0.921844i	$0.373318\pi$
$180$		0			0
$181$	10.8407			0.805783			0.402892	−	0.915248i	$-0.368005\pi$
$181$	10.8407			0.805783			0.402892	+	0.915248i	$0.368005\pi$
$182$	−17.2878	−	5.76550i	−1.28145	−	0.427367i
$183$		0			0
$184$			6.27217i			0.462390i
$185$	−1.00933			−0.0742072
$186$		0			0
$187$			13.4963i			0.986947i
$188$		−	5.28124i		−	0.385174i
$189$		0			0
$190$		−	1.55520i		−	0.112826i
$191$	11.2580			0.814600			0.407300	−	0.913294i	$-0.366470\pi$
$191$	11.2580			0.814600			0.407300	+	0.913294i	$0.366470\pi$
$192$		0			0
$193$			7.62566i			0.548907i	0.961600	+	0.274453i	$0.0884969\pi$
$193$			7.62566i			0.548907i	−0.961600	+	0.274453i	$0.911503\pi$
$194$	−13.0505			−0.936972
$195$		0			0
$196$	18.5467			1.32477
$197$			24.5170i			1.74677i	0.487035	+	0.873383i	$0.338079\pi$
$197$			24.5170i			1.74677i	−0.487035	+	0.873383i	$0.661921\pi$
$198$		0			0
$199$	−13.5310			−0.959187			−0.479593	−	0.877491i	$-0.659216\pi$
$199$	−13.5310			−0.959187			−0.479593	+	0.877491i	$0.659216\pi$
$200$			4.76550i			0.336971i
$201$		0			0
$202$			6.84702i			0.481755i
$203$		−	23.2124i		−	1.62919i
$204$		0			0
$205$	−5.35295			−0.373866
$206$		−	3.55520i		−	0.247703i
$207$		0			0
$208$	3.42035	+	1.14069i	0.237159	+	0.0790929i
$209$	−10.3139			−0.713428
$210$		0			0
$211$	23.8692			1.64322			0.821610	−	0.570050i	$-0.193076\pi$
$211$	23.8692			1.64322			0.821610	+	0.570050i	$0.193076\pi$
$212$	−6.41990			−0.440921
$213$		0			0
$214$		−	1.41392i		−	0.0966537i
$215$			4.58462i			0.312668i
$216$		0			0
$217$	−32.7710			−2.22464
$218$	−4.97525			−0.336967
$219$		0			0
$220$	−1.55520			−0.104852
$221$	14.3739	+	4.79371i	0.966892	+	0.322460i
$222$		0			0
$223$			12.8698i			0.861825i	0.902394	+	0.430913i	$0.141808\pi$
$223$			12.8698i			0.861825i	−0.902394	+	0.430913i	$0.858192\pi$
$224$	−5.05438			−0.337710
$225$		0			0
$226$			13.0883i			0.870621i
$227$			18.0552i			1.19836i	0.800613	+	0.599182i	$0.204507\pi$
$227$			18.0552i			1.19836i	−0.800613	+	0.599182i	$0.795493\pi$
$228$		0			0
$229$		−	21.1737i		−	1.39920i	−0.714536	−	0.699599i	$-0.753362\pi$
$229$		−	21.1737i		−	1.39920i	0.714536	−	0.699599i	$-0.246638\pi$
$230$	−3.03733			−0.200276
$231$		0			0
$232$			4.59254i			0.301515i
$233$	29.0610			1.90385			0.951923	−	0.306337i	$-0.0991033\pi$
$233$	29.0610			1.90385			0.951923	+	0.306337i	$0.0991033\pi$
$234$		0			0
$235$	2.55747			0.166831
$236$		−	3.62311i		−	0.235844i
$237$		0			0
$238$	−21.2408			−1.37684
$239$			7.26902i			0.470194i	0.971972	+	0.235097i	$0.0755408\pi$
$239$			7.26902i			0.470194i	−0.971972	+	0.235097i	$0.924459\pi$
$240$		0			0
$241$		−	17.9104i		−	1.15371i	−0.816846	−	0.576856i	$-0.804279\pi$
$241$		−	17.9104i		−	1.15371i	0.816846	−	0.576856i	$-0.195721\pi$
$242$		−	0.686087i		−	0.0441034i
$243$		0			0
$244$	1.00173			0.0641290
$245$			8.98136i			0.573798i
$246$		0			0
$247$	−3.66337	+	10.9846i	−0.233095	+	0.698931i
$248$	6.48369			0.411715
$249$		0			0
$250$	−4.72900			−0.299088
$251$	−3.84702			−0.242822			−0.121411	−	0.992602i	$-0.538742\pi$
$251$	−3.84702			−0.242822			−0.121411	+	0.992602i	$0.538742\pi$
$252$		0			0
$253$			20.1432i			1.26639i
$254$			4.56300i			0.286308i
$255$		0			0
$256$	1.00000			0.0625000
$257$	−14.7675			−0.921172			−0.460586	−	0.887615i	$-0.652361\pi$
$257$	−14.7675			−0.921172			−0.460586	+	0.887615i	$0.652361\pi$
$258$		0			0
$259$	−10.5348			−0.654598
$260$	−0.552388	+	1.65633i	−0.0342576	+	0.102721i
$261$		0			0
$262$		−	2.46946i		−	0.152564i
$263$	8.76248			0.540318			0.270159	−	0.962816i	$-0.412924\pi$
$263$	8.76248			0.540318			0.270159	+	0.962816i	$0.412924\pi$
$264$		0			0
$265$		−	3.10888i		−	0.190977i
$266$		−	16.2323i		−	0.995265i
$267$		0			0
$268$		−	1.08194i		−	0.0660900i
$269$	−8.90736			−0.543091			−0.271546	−	0.962426i	$-0.587535\pi$
$269$	−8.90736			−0.543091			−0.271546	+	0.962426i	$0.587535\pi$
$270$		0			0
$271$			3.58551i			0.217804i	0.994052	+	0.108902i	$0.0347335\pi$
$271$			3.58551i			0.217804i	−0.994052	+	0.108902i	$0.965266\pi$
$272$	4.20245			0.254811
$273$		0			0
$274$	−19.6104			−1.18471
$275$			15.3045i			0.922898i
$276$		0			0
$277$	−14.8793			−0.894012			−0.447006	−	0.894531i	$-0.647510\pi$
$277$	−14.8793			−0.894012			−0.447006	+	0.894531i	$0.647510\pi$
$278$			0.387396i			0.0232345i
$279$		0			0
$280$		−	2.44761i		−	0.146273i
$281$		−	16.5116i		−	0.985002i	−0.870312	−	0.492501i	$-0.836083\pi$
$281$		−	16.5116i		−	0.985002i	0.870312	−	0.492501i	$-0.163917\pi$
$282$		0			0
$283$	−0.535666			−0.0318420			−0.0159210	−	0.999873i	$-0.505068\pi$
$283$	−0.535666			−0.0318420			−0.0159210	+	0.999873i	$0.505068\pi$
$284$		−	4.63041i		−	0.274764i
$285$		0			0
$286$	10.9846	+	3.66337i	0.649531	+	0.216620i
$287$	−55.8709			−3.29795
$288$		0			0
$289$	0.660618			0.0388599
$290$	−2.22396			−0.130596
$291$		0			0
$292$			0.325525i			0.0190499i
$293$		−	19.5896i		−	1.14444i	−0.820100	−	0.572220i	$-0.806082\pi$
$293$		−	19.5896i		−	1.14444i	0.820100	−	0.572220i	$-0.193918\pi$
$294$		0			0
$295$	1.75451			0.102152
$296$	2.08429			0.121147
$297$		0			0
$298$	9.45874			0.547930
$299$	21.4530	+	7.15462i	1.24066	+	0.413762i
$300$		0			0
$301$			47.8516i			2.75812i
$302$	−1.33164			−0.0766272
$303$		0			0
$304$			3.21153i			0.184194i
$305$			0.485092i			0.0277763i
$306$		0			0
$307$		−	1.26064i		−	0.0719485i	−0.999353	−	0.0359743i	$-0.988547\pi$
$307$		−	1.26064i		−	0.0719485i	0.999353	−	0.0359743i	$-0.0114534\pi$
$308$	−16.2323			−0.924920
$309$		0			0
$310$			3.13977i			0.178327i
$311$	−17.5844			−0.997122			−0.498561	−	0.866855i	$-0.666138\pi$
$311$	−17.5844			−0.997122			−0.498561	+	0.866855i	$0.666138\pi$
$312$		0			0
$313$	0.205411			0.0116105			0.00580526	−	0.999983i	$-0.498152\pi$
$313$	0.205411			0.0116105			0.00580526	+	0.999983i	$0.498152\pi$
$314$			5.84243i			0.329708i
$315$		0			0
$316$	7.83637			0.440830
$317$			18.0806i			1.01551i	0.861502	+	0.507754i	$0.169524\pi$
$317$			18.0806i			1.01551i	−0.861502	+	0.507754i	$0.830476\pi$
$318$		0			0
$319$			14.7491i			0.825789i
$320$			0.484256i			0.0270707i
$321$		0			0
$322$	−31.7019			−1.76668
$323$			13.4963i			0.750954i
$324$		0			0
$325$	16.2997	+	5.43597i	0.904144	+	0.301533i
$326$	−15.1340			−0.838192
$327$		0			0
$328$	11.0540			0.610353
$329$	26.6934			1.47165
$330$		0			0
$331$		−	4.19502i		−	0.230579i	−0.993332	−	0.115290i	$-0.963220\pi$
$331$		−	4.19502i		−	0.230579i	0.993332	−	0.115290i	$-0.0367796\pi$
$332$		−	5.86613i		−	0.321946i
$333$		0			0
$334$	12.0268			0.658078
$335$	0.523936			0.0286257
$336$		0			0
$337$	13.7608			0.749600			0.374800	−	0.927106i	$-0.377711\pi$
$337$	13.7608			0.749600			0.374800	+	0.927106i	$0.377711\pi$
$338$	7.80315	−	10.3976i	0.424435	−	0.565557i
$339$		0			0
$340$			2.03506i			0.110367i
$341$	20.8226			1.12761
$342$		0			0
$343$			58.3615i			3.15123i
$344$		−	9.46735i		−	0.510445i
$345$		0			0
$346$		−	8.85863i		−	0.476243i
$347$	−23.8702			−1.28142			−0.640710	−	0.767783i	$-0.721360\pi$
$347$	−23.8702			−1.28142			−0.640710	+	0.767783i	$0.721360\pi$
$348$		0			0
$349$			19.3478i			1.03567i	0.855482	+	0.517833i	$0.173261\pi$
$349$			19.3478i			1.03567i	−0.855482	+	0.517833i	$0.826739\pi$
$350$	−24.0866			−1.28748
$351$		0			0
$352$	3.21153			0.171175
$353$			27.7828i			1.47873i	0.673305	+	0.739365i	$0.264874\pi$
$353$			27.7828i			1.47873i	−0.673305	+	0.739365i	$0.735126\pi$
$354$		0			0
$355$	2.24230			0.119009
$356$			8.42912i			0.446743i
$357$		0			0
$358$		−	10.3704i		−	0.548093i
$359$		−	23.8304i		−	1.25772i	−0.777517	−	0.628861i	$-0.783521\pi$
$359$		−	23.8304i		−	1.25772i	0.777517	−	0.628861i	$-0.216479\pi$
$360$		0			0
$361$	8.68609			0.457162
$362$		−	10.8407i		−	0.569775i
$363$		0			0
$364$	−5.76550	+	17.2878i	−0.302194	+	0.906125i
$365$	−0.157637			−0.00825112
$366$		0			0
$367$	−10.0929			−0.526843			−0.263421	−	0.964681i	$-0.584851\pi$
$367$	−10.0929			−0.526843			−0.263421	+	0.964681i	$0.584851\pi$
$368$	6.27217			0.326959
$369$		0			0
$370$			1.00933i			0.0524724i
$371$		−	32.4486i		−	1.68465i
$372$		0			0
$373$	−8.54741			−0.442568			−0.221284	−	0.975209i	$-0.571025\pi$
$373$	−8.54741			−0.442568			−0.221284	+	0.975209i	$0.571025\pi$
$374$	13.4963			0.697877
$375$		0			0
$376$	−5.28124			−0.272359
$377$	15.7081	+	5.23868i	0.809008	+	0.269806i
$378$		0			0
$379$			30.7125i			1.57760i	0.614653	+	0.788798i	$0.289296\pi$
$379$			30.7125i			1.57760i	−0.614653	+	0.788798i	$0.710704\pi$
$380$	−1.55520			−0.0797802
$381$		0			0
$382$		−	11.2580i		−	0.576009i
$383$		−	2.88551i		−	0.147442i	−0.997279	−	0.0737212i	$-0.976512\pi$
$383$		−	2.88551i		−	0.147442i	0.997279	−	0.0737212i	$-0.0234875\pi$
$384$		0			0
$385$		−	7.86058i		−	0.400612i
$386$	7.62566			0.388136
$387$		0			0
$388$			13.0505i			0.662539i
$389$	−11.8955			−0.603127			−0.301564	−	0.953446i	$-0.597509\pi$
$389$	−11.8955			−0.603127			−0.301564	+	0.953446i	$0.597509\pi$
$390$		0			0
$391$	26.3585			1.33301
$392$		−	18.5467i		−	0.936751i
$393$		0			0
$394$	24.5170			1.23515
$395$			3.79481i			0.190937i
$396$		0			0
$397$		−	23.8021i		−	1.19459i	−0.802020	−	0.597297i	$-0.796241\pi$
$397$		−	23.8021i		−	1.19459i	0.802020	−	0.597297i	$-0.203759\pi$
$398$			13.5310i			0.678247i
$399$		0			0
$400$	4.76550			0.238275
$401$		−	8.99705i		−	0.449291i	−0.974441	−	0.224646i	$-0.927878\pi$
$401$		−	8.99705i		−	0.449291i	0.974441	−	0.224646i	$-0.0721225\pi$
$402$		0			0
$403$	7.39591	−	22.1765i	0.368416	−	1.10469i
$404$	6.84702			0.340652
$405$		0			0
$406$	−23.2124			−1.15201
$407$	6.69374			0.331796
$408$		0			0
$409$		−	32.1404i		−	1.58924i	−0.607107	−	0.794620i	$-0.707670\pi$
$409$		−	32.1404i		−	1.58924i	0.607107	−	0.794620i	$-0.292330\pi$
$410$			5.35295i			0.264363i
$411$		0			0
$412$	−3.55520			−0.175152
$413$	18.3126			0.901103
$414$		0			0
$415$	2.84071			0.139445
$416$	1.14069	−	3.42035i	0.0559271	−	0.167697i
$417$		0			0
$418$			10.3139i			0.504470i
$419$	3.37377			0.164820			0.0824098	−	0.996599i	$-0.473738\pi$
$419$	3.37377			0.164820			0.0824098	+	0.996599i	$0.473738\pi$
$420$		0			0
$421$			2.61685i			0.127537i	0.997965	+	0.0637687i	$0.0203120\pi$
$421$			2.61685i			0.127537i	−0.997965	+	0.0637687i	$0.979688\pi$
$422$		−	23.8692i		−	1.16193i
$423$		0			0
$424$			6.41990i			0.311778i
$425$	20.0268			0.971441
$426$		0			0
$427$			5.06311i			0.245021i
$428$	−1.41392			−0.0683445
$429$		0			0
$430$	4.58462			0.221090
$431$		−	1.25113i		−	0.0602647i	−0.999546	−	0.0301324i	$-0.990407\pi$
$431$		−	1.25113i		−	0.0602647i	0.999546	−	0.0301324i	$-0.00959288\pi$
$432$		0			0
$433$	−4.77117			−0.229288			−0.114644	−	0.993407i	$-0.536573\pi$
$433$	−4.77117			−0.229288			−0.114644	+	0.993407i	$0.536573\pi$
$434$			32.7710i			1.57306i
$435$		0			0
$436$			4.97525i			0.238271i
$437$			20.1432i			0.963582i
$438$		0			0
$439$	33.7019			1.60850			0.804252	−	0.594289i	$-0.202566\pi$
$439$	33.7019			1.60850			0.804252	+	0.594289i	$0.202566\pi$
$440$			1.55520i			0.0741413i
$441$		0			0
$442$	4.79371	−	14.3739i	0.228014	−	0.683696i
$443$	−5.37751			−0.255493			−0.127747	−	0.991807i	$-0.540774\pi$
$443$	−5.37751			−0.255493			−0.127747	+	0.991807i	$0.540774\pi$
$444$		0			0
$445$	−4.08185			−0.193498
$446$	12.8698			0.609403
$447$		0			0
$448$			5.05438i			0.238797i
$449$			16.7813i			0.791958i	0.918260	+	0.395979i	$0.129595\pi$
$449$			16.7813i			0.791958i	−0.918260	+	0.395979i	$0.870405\pi$
$450$		0			0
$451$	35.5001			1.67163
$452$	13.0883			0.615622
$453$		0			0
$454$	18.0552			0.847371
$455$	−8.37170	−	2.79198i	−0.392471	−	0.130890i
$456$		0			0
$457$			23.6791i			1.10766i	0.832629	+	0.553831i	$0.186835\pi$
$457$			23.6791i			1.10766i	−0.832629	+	0.553831i	$0.813165\pi$
$458$	−21.1737			−0.989383
$459$		0			0
$460$			3.03733i			0.141616i
$461$			34.4622i			1.60507i	0.596608	+	0.802533i	$0.296515\pi$
$461$			34.4622i			1.60507i	−0.596608	+	0.802533i	$0.703485\pi$
$462$		0			0
$463$			22.4773i			1.04461i	0.852759	+	0.522304i	$0.174927\pi$
$463$			22.4773i			1.04461i	−0.852759	+	0.522304i	$0.825073\pi$
$464$	4.59254			0.213203
$465$		0			0
$466$		−	29.0610i		−	1.34622i
$467$	−4.43632			−0.205288			−0.102644	−	0.994718i	$-0.532730\pi$
$467$	−4.43632			−0.205288			−0.102644	+	0.994718i	$0.532730\pi$
$468$		0			0
$469$	5.46853			0.252513
$470$		−	2.55747i		−	0.117967i
$471$		0			0
$472$	−3.62311			−0.166767
$473$		−	30.4047i		−	1.39801i
$474$		0			0
$475$			15.3045i			0.702220i
$476$			21.2408i			0.973570i
$477$		0			0
$478$	7.26902			0.332477
$479$		−	11.8165i		−	0.539912i	−0.962873	−	0.269956i	$-0.912991\pi$
$479$		−	11.8165i		−	0.539912i	0.962873	−	0.269956i	$-0.0870091\pi$
$480$		0			0
$481$	2.37753	−	7.12899i	0.108406	−	0.325054i
$482$	−17.9104			−0.815798
$483$		0			0
$484$	−0.686087			−0.0311858
$485$	−6.31979			−0.286967
$486$		0			0
$487$		−	25.8373i		−	1.17080i	−0.810745	−	0.585400i	$-0.800938\pi$
$487$		−	25.8373i		−	1.17080i	0.810745	−	0.585400i	$-0.199062\pi$
$488$		−	1.00173i		−	0.0453461i
$489$		0			0
$490$	8.98136			0.405737
$491$	11.1999			0.505444			0.252722	−	0.967539i	$-0.418674\pi$
$491$	11.1999			0.505444			0.252722	+	0.967539i	$0.418674\pi$
$492$		0			0
$493$	19.2999			0.869225
$494$	10.9846	+	3.66337i	0.494219	+	0.164823i
$495$		0			0
$496$		−	6.48369i		−	0.291126i
$497$	23.4039			1.04981
$498$		0			0
$499$		−	15.4596i		−	0.692069i	−0.938222	−	0.346034i	$-0.887528\pi$
$499$		−	15.4596i		−	0.692069i	0.938222	−	0.346034i	$-0.112472\pi$
$500$			4.72900i			0.211487i
$501$		0			0
$502$			3.84702i			0.171701i
$503$	−6.00330			−0.267674			−0.133837	−	0.991003i	$-0.542730\pi$
$503$	−6.00330			−0.267674			−0.133837	+	0.991003i	$0.542730\pi$
$504$		0			0
$505$			3.31571i			0.147547i
$506$	20.1432			0.895476
$507$		0			0
$508$	4.56300			0.202450
$509$			21.5699i			0.956068i	0.878341	+	0.478034i	$0.158650\pi$
$509$			21.5699i			0.956068i	−0.878341	+	0.478034i	$0.841350\pi$
$510$		0			0
$511$	−1.64533			−0.0727849
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$			14.7675i			0.651367i
$515$		−	1.72163i		−	0.0758640i
$516$		0			0
$517$	−16.9609			−0.745937
$518$			10.5348i			0.462871i
$519$		0			0
$520$	1.65633	+	0.552388i	0.0726347	+	0.0242238i
$521$	24.0543			1.05384			0.526918	−	0.849916i	$-0.323347\pi$
$521$	24.0543			1.05384			0.526918	+	0.849916i	$0.323347\pi$
$522$		0			0
$523$	−10.8449			−0.474215			−0.237107	−	0.971483i	$-0.576199\pi$
$523$	−10.8449			−0.474215			−0.237107	+	0.971483i	$0.576199\pi$
$524$	−2.46946			−0.107879
$525$		0			0
$526$		−	8.76248i		−	0.382062i
$527$		−	27.2474i		−	1.18692i
$528$		0			0
$529$	16.3401			0.710438
$530$	−3.10888			−0.135041
$531$		0			0
$532$	−16.2323			−0.703758
$533$	12.6092	−	37.8085i	0.546164	−	1.63767i
$534$		0			0
$535$		−	0.684700i		−	0.0296022i
$536$	−1.08194			−0.0467327
$537$		0			0
$538$			8.90736i			0.384024i
$539$		−	59.5633i		−	2.56558i
$540$		0			0
$541$			12.2998i			0.528808i	0.964412	+	0.264404i	$0.0851753\pi$
$541$			12.2998i			0.528808i	−0.964412	+	0.264404i	$0.914825\pi$
$542$	3.58551			0.154011
$543$		0			0
$544$		−	4.20245i		−	0.180179i
$545$	−2.40930			−0.103203
$546$		0			0
$547$	−20.6903			−0.884655			−0.442327	−	0.896854i	$-0.645847\pi$
$547$	−20.6903			−0.884655			−0.442327	+	0.896854i	$0.645847\pi$
$548$			19.6104i			0.837714i
$549$		0			0
$550$	15.3045			0.652587
$551$			14.7491i			0.628331i
$552$		0			0
$553$			39.6080i			1.68430i
$554$			14.8793i			0.632162i
$555$		0			0
$556$	0.387396			0.0164293
$557$			14.2838i			0.605226i	0.953114	+	0.302613i	$0.0978589\pi$
$557$			14.2838i			0.605226i	−0.953114	+	0.302613i	$0.902141\pi$
$558$		0			0
$559$	−32.3817	−	10.7993i	−1.36960	−	0.456764i
$560$	−2.44761			−0.103430
$561$		0			0
$562$	−16.5116			−0.696501
$563$	−35.9346			−1.51446			−0.757231	−	0.653147i	$-0.773448\pi$
$563$	−35.9346			−1.51446			−0.757231	+	0.653147i	$0.773448\pi$
$564$		0			0
$565$			6.33809i			0.266646i
$566$			0.535666i			0.0225157i
$567$		0			0
$568$	−4.63041			−0.194288
$569$	−37.0631			−1.55376			−0.776882	−	0.629646i	$-0.783200\pi$
$569$	−37.0631			−1.55376			−0.776882	+	0.629646i	$0.783200\pi$
$570$		0			0
$571$	−46.3464			−1.93954			−0.969769	−	0.244026i	$-0.921532\pi$
$571$	−46.3464			−1.93954			−0.969769	+	0.244026i	$0.921532\pi$
$572$	3.66337	−	10.9846i	0.153173	−	0.459288i
$573$		0			0
$574$			55.8709i			2.33201i
$575$	29.8900			1.24650
$576$		0			0
$577$			31.4447i			1.30906i	0.756036	+	0.654530i	$0.227134\pi$
$577$			31.4447i			1.30906i	−0.756036	+	0.654530i	$0.772866\pi$
$578$		−	0.660618i		−	0.0274781i
$579$		0			0
$580$			2.22396i			0.0923450i
$581$	29.6496			1.23007
$582$		0			0
$583$			20.6177i			0.853898i
$584$	0.325525			0.0134703
$585$		0			0
$586$	−19.5896			−0.809241
$587$		−	31.2506i		−	1.28985i	−0.764247	−	0.644924i	$-0.776889\pi$
$587$		−	31.2506i		−	1.28985i	0.764247	−	0.644924i	$-0.223111\pi$
$588$		0			0
$589$	20.8226			0.857979
$590$		−	1.75451i		−	0.0722322i
$591$		0			0
$592$		−	2.08429i		−	0.0856636i
$593$		−	26.3978i		−	1.08403i	−0.840370	−	0.542013i	$-0.817662\pi$
$593$		−	26.3978i		−	1.08403i	0.840370	−	0.542013i	$-0.182338\pi$
$594$		0			0
$595$	−10.2860			−0.421684
$596$		−	9.45874i		−	0.387445i
$597$		0			0
$598$	7.15462	−	21.4530i	0.292574	−	0.877279i
$599$	−6.81049			−0.278269			−0.139135	−	0.990273i	$-0.544432\pi$
$599$	−6.81049			−0.278269			−0.139135	+	0.990273i	$0.544432\pi$
$600$		0			0
$601$	9.65032			0.393645			0.196822	−	0.980439i	$-0.436938\pi$
$601$	9.65032			0.393645			0.196822	+	0.980439i	$0.436938\pi$
$602$	47.8516			1.95028
$603$		0			0
$604$			1.33164i			0.0541836i
$605$		−	0.332242i		−	0.0135075i
$606$		0			0
$607$	27.5792			1.11940			0.559702	−	0.828694i	$-0.310916\pi$
$607$	27.5792			1.11940			0.559702	+	0.828694i	$0.310916\pi$
$608$	3.21153			0.130245
$609$		0			0
$610$	0.485092			0.0196408
$611$	−6.02428	+	18.0637i	−0.243716	+	0.730779i
$612$		0			0
$613$		−	16.0314i		−	0.647502i	−0.946142	−	0.323751i	$-0.895056\pi$
$613$		−	16.0314i		−	0.647502i	0.946142	−	0.323751i	$-0.104944\pi$
$614$	−1.26064			−0.0508753
$615$		0			0
$616$			16.2323i			0.654017i
$617$			13.2054i			0.531628i	0.964024	+	0.265814i	$0.0856408\pi$
$617$			13.2054i			0.531628i	−0.964024	+	0.265814i	$0.914359\pi$
$618$		0			0
$619$		−	1.77418i		−	0.0713104i	−0.999364	−	0.0356552i	$-0.988648\pi$
$619$		−	1.77418i		−	0.0713104i	0.999364	−	0.0356552i	$-0.0113518\pi$
$620$	3.13977			0.126096
$621$		0			0
$622$			17.5844i			0.705072i
$623$	−42.6040			−1.70689
$624$		0			0
$625$	21.5374			0.861497
$626$		−	0.205411i		−	0.00820988i
$627$		0			0
$628$	5.84243			0.233139
$629$		−	8.75911i		−	0.349249i
$630$		0			0
$631$		−	30.2211i		−	1.20308i	−0.798841	−	0.601542i	$-0.794553\pi$
$631$		−	30.2211i		−	1.20308i	0.798841	−	0.601542i	$-0.205447\pi$
$632$		−	7.83637i		−	0.311714i
$633$		0			0
$634$	18.0806			0.718072
$635$			2.20966i			0.0876876i
$636$		0			0
$637$	−63.4364	−	21.1561i	−2.51344	−	0.838237i
$638$	14.7491			0.583921
$639$		0			0
$640$	0.484256			0.0191419
$641$	−0.603345			−0.0238307			−0.0119154	−	0.999929i	$-0.503793\pi$
$641$	−0.603345			−0.0238307			−0.0119154	+	0.999929i	$0.503793\pi$
$642$		0			0
$643$			31.2057i			1.23063i	0.788280	+	0.615317i	$0.210972\pi$
$643$			31.2057i			1.23063i	−0.788280	+	0.615317i	$0.789028\pi$
$644$			31.7019i			1.24923i
$645$		0			0
$646$	13.4963			0.531005
$647$	−48.8937			−1.92221			−0.961105	−	0.276182i	$-0.910931\pi$
$647$	−48.8937			−1.92221			−0.961105	+	0.276182i	$0.910931\pi$
$648$		0			0
$649$	−11.6357			−0.456742
$650$	5.43597	−	16.2997i	0.213216	−	0.639326i
$651$		0			0
$652$			15.1340i			0.592691i
$653$	38.3888			1.50227			0.751135	−	0.660149i	$-0.229507\pi$
$653$	38.3888			1.50227			0.751135	+	0.660149i	$0.229507\pi$
$654$		0			0
$655$		−	1.19585i		−	0.0467258i
$656$		−	11.0540i		−	0.431585i
$657$		0			0
$658$		−	26.6934i		−	1.04062i
$659$	−41.9136			−1.63272			−0.816362	−	0.577541i	$-0.804012\pi$
$659$	−41.9136			−1.63272			−0.816362	+	0.577541i	$0.804012\pi$
$660$		0			0
$661$			12.1484i			0.472516i	0.971690	+	0.236258i	$0.0759211\pi$
$661$			12.1484i			0.472516i	−0.971690	+	0.236258i	$0.924079\pi$
$662$	−4.19502			−0.163044
$663$		0			0
$664$	−5.86613			−0.227650
$665$		−	7.86058i		−	0.304820i
$666$		0			0
$667$	28.8051			1.11534
$668$		−	12.0268i		−	0.465331i
$669$		0			0
$670$		−	0.523936i		−	0.0202414i
$671$		−	3.21708i		−	0.124194i
$672$		0			0
$673$	36.6505			1.41277			0.706386	−	0.707827i	$-0.250324\pi$
$673$	36.6505			1.41277			0.706386	+	0.707827i	$0.250324\pi$
$674$		−	13.7608i		−	0.530047i
$675$		0			0
$676$	−10.3976	−	7.80315i	−0.399909	−	0.300121i
$677$	38.5927			1.48324			0.741620	−	0.670820i	$-0.234058\pi$
$677$	38.5927			1.48324			0.741620	+	0.670820i	$0.234058\pi$
$678$		0			0
$679$	−65.9622			−2.53140
$680$	2.03506			0.0780411
$681$		0			0
$682$		−	20.8226i		−	0.797337i
$683$			31.4038i			1.20163i	0.799387	+	0.600816i	$0.205158\pi$
$683$			31.4038i			1.20163i	−0.799387	+	0.600816i	$0.794842\pi$
$684$		0			0
$685$	−9.49644			−0.362840
$686$	58.3615			2.22825
$687$		0			0
$688$	−9.46735			−0.360939
$689$	21.9583	+	7.32314i	0.836546	+	0.278990i
$690$		0			0
$691$			19.2588i			0.732640i	0.930489	+	0.366320i	$0.119382\pi$
$691$			19.2588i			0.732640i	−0.930489	+	0.366320i	$0.880618\pi$
$692$	−8.85863			−0.336755
$693$		0			0
$694$			23.8702i			0.906100i
$695$			0.187599i			0.00711603i
$696$		0			0
$697$		−	46.4538i		−	1.75956i
$698$	19.3478			0.732326
$699$		0			0
$700$			24.0866i			0.910389i
$701$	−26.7781			−1.01139			−0.505697	−	0.862711i	$-0.668765\pi$
$701$	−26.7781			−1.01139			−0.505697	+	0.862711i	$0.668765\pi$
$702$		0			0
$703$	6.69374			0.252459
$704$		−	3.21153i		−	0.121039i
$705$		0			0
$706$	27.7828			1.04562
$707$			34.6074i			1.30155i
$708$		0			0
$709$			46.4696i			1.74520i	0.488433	+	0.872602i	$0.337569\pi$
$709$			46.4696i			1.74520i	−0.488433	+	0.872602i	$0.662431\pi$
$710$		−	2.24230i		−	0.0841522i
$711$		0			0
$712$	8.42912			0.315895
$713$		−	40.6668i		−	1.52298i
$714$		0			0
$715$	5.31934	+	1.77401i	0.198932	+	0.0663442i
$716$	−10.3704			−0.387561
$717$		0			0
$718$	−23.8304			−0.889344
$719$	−4.81378			−0.179524			−0.0897619	−	0.995963i	$-0.528611\pi$
$719$	−4.81378			−0.179524			−0.0897619	+	0.995963i	$0.528611\pi$
$720$		0			0
$721$		−	17.9693i		−	0.669213i
$722$		−	8.68609i		−	0.323263i
$723$		0			0
$724$	−10.8407			−0.402892
$725$	21.8857			0.812815
$726$		0			0
$727$	12.7183			0.471696			0.235848	−	0.971790i	$-0.424213\pi$
$727$	12.7183			0.471696			0.235848	+	0.971790i	$0.424213\pi$
$728$	17.2878	+	5.76550i	0.640727	+	0.213684i
$729$		0			0
$730$			0.157637i			0.00583442i
$731$	−39.7861			−1.47154
$732$		0			0
$733$			4.25487i			0.157157i	0.996908	+	0.0785787i	$0.0250382\pi$
$733$			4.25487i			0.157157i	−0.996908	+	0.0785787i	$0.974962\pi$
$734$			10.0929i			0.372534i
$735$		0			0
$736$		−	6.27217i		−	0.231195i
$737$	−3.47468			−0.127991
$738$		0			0
$739$			9.71484i			0.357366i	0.983907	+	0.178683i	$0.0571837\pi$
$739$			9.71484i			0.357366i	−0.983907	+	0.178683i	$0.942816\pi$
$740$	1.00933			0.0371036
$741$		0			0
$742$	−32.4486			−1.19123
$743$		−	12.3252i		−	0.452168i	−0.974108	−	0.226084i	$-0.927408\pi$
$743$		−	12.3252i		−	0.452168i	0.974108	−	0.226084i	$-0.0725924\pi$
$744$		0			0
$745$	4.58045			0.167815
$746$			8.54741i			0.312943i
$747$		0			0
$748$		−	13.4963i		−	0.493474i
$749$		−	7.14649i		−	0.261127i
$750$		0			0
$751$	−2.86981			−0.104721			−0.0523603	−	0.998628i	$-0.516674\pi$
$751$	−2.86981			−0.104721			−0.0523603	+	0.998628i	$0.516674\pi$
$752$			5.28124i			0.192587i
$753$		0			0
$754$	5.23868	−	15.7081i	0.190781	−	0.572055i
$755$	−0.644854			−0.0234686
$756$		0			0
$757$	33.1438			1.20463			0.602316	−	0.798258i	$-0.294245\pi$
$757$	33.1438			1.20463			0.602316	+	0.798258i	$0.294245\pi$
$758$	30.7125			1.11553
$759$		0			0
$760$			1.55520i			0.0564131i
$761$			42.7544i			1.54985i	0.632056	+	0.774923i	$0.282211\pi$
$761$			42.7544i			1.54985i	−0.632056	+	0.774923i	$0.717789\pi$
$762$		0			0
$763$	−25.1468			−0.910375
$764$	−11.2580			−0.407300
$765$		0			0
$766$	−2.88551			−0.104258
$767$	−4.13286	+	12.3923i	−0.149229	+	0.447461i
$768$		0			0
$769$			48.1276i			1.73552i	0.496980	+	0.867762i	$0.334442\pi$
$769$			48.1276i			1.73552i	−0.496980	+	0.867762i	$0.665558\pi$
$770$	−7.86058			−0.283275
$771$		0			0
$772$		−	7.62566i		−	0.274453i
$773$			21.3661i			0.768484i	0.923232	+	0.384242i	$0.125537\pi$
$773$			21.3661i			0.768484i	−0.923232	+	0.384242i	$0.874463\pi$
$774$		0			0
$775$		−	30.8980i		−	1.10989i
$776$	13.0505			0.468486
$777$		0			0
$778$			11.8955i			0.426475i
$779$	35.5001			1.27192
$780$		0			0
$781$	−14.8707			−0.532115
$782$		−	26.3585i		−	0.942578i
$783$		0			0
$784$	−18.5467			−0.662383
$785$			2.82923i			0.100980i
$786$		0			0
$787$		−	34.1708i		−	1.21806i	−0.793148	−	0.609029i	$-0.791559\pi$
$787$		−	34.1708i		−	1.21806i	0.793148	−	0.609029i	$-0.208441\pi$
$788$		−	24.5170i		−	0.873383i
$789$		0			0
$790$	3.79481			0.135013
$791$			66.1533i			2.35214i
$792$		0			0
$793$	−3.42626	−	1.14266i	−0.121670	−	0.0405772i
$794$	−23.8021			−0.844706
$795$		0			0
$796$	13.5310			0.479593
$797$	12.5383			0.444131			0.222066	−	0.975032i	$-0.428720\pi$
$797$	12.5383			0.444131			0.222066	+	0.975032i	$0.428720\pi$
$798$		0			0
$799$			22.1942i			0.785173i
$800$		−	4.76550i		−	0.168486i
$801$		0			0
$802$	−8.99705			−0.317697
$803$	1.04543			0.0368925
$804$		0			0
$805$	−15.3518			−0.541081
$806$	−22.1765	−	7.39591i	−0.781135	−	0.260510i
$807$		0			0
$808$		−	6.84702i		−	0.240877i
$809$	−23.7653			−0.835543			−0.417772	−	0.908552i	$-0.637189\pi$
$809$	−23.7653			−0.835543			−0.417772	+	0.908552i	$0.637189\pi$
$810$		0			0
$811$			12.1844i			0.427853i	0.976850	+	0.213926i	$0.0686253\pi$
$811$			12.1844i			0.427853i	−0.976850	+	0.213926i	$0.931375\pi$
$812$			23.2124i			0.814596i
$813$		0			0
$814$		−	6.69374i		−	0.234615i
$815$	−7.32871			−0.256713
$816$		0			0
$817$		−	30.4047i		−	1.06372i
$818$	−32.1404			−1.12376
$819$		0			0
$820$	5.35295			0.186933
$821$			17.1510i			0.598573i	0.954163	+	0.299287i	$0.0967486\pi$
$821$			17.1510i			0.598573i	−0.954163	+	0.299287i	$0.903251\pi$
$822$		0			0
$823$	38.1966			1.33145			0.665725	−	0.746197i	$-0.268122\pi$
$823$	38.1966			1.33145			0.665725	+	0.746197i	$0.268122\pi$
$824$			3.55520i			0.123851i
$825$		0			0
$826$		−	18.3126i		−	0.637176i
$827$			16.3182i			0.567441i	0.958907	+	0.283721i	$0.0915688\pi$
$827$			16.3182i			0.567441i	−0.958907	+	0.283721i	$0.908431\pi$
$828$		0			0
$829$	13.6052			0.472529			0.236265	−	0.971689i	$-0.424077\pi$
$829$	13.6052			0.472529			0.236265	+	0.971689i	$0.424077\pi$
$830$		−	2.84071i		−	0.0986024i
$831$		0			0
$832$	−3.42035	−	1.14069i	−0.118579	−	0.0395464i
$833$	−77.9418			−2.70052
$834$		0			0
$835$	5.82406			0.201550
$836$	10.3139			0.356714
$837$		0			0
$838$		−	3.37377i		−	0.116545i
$839$			50.7828i			1.75322i	0.481204	+	0.876609i	$0.340200\pi$
$839$			50.7828i			1.75322i	−0.481204	+	0.876609i	$0.659800\pi$
$840$		0			0
$841$	−7.90862			−0.272711
$842$	2.61685			0.0901826
$843$		0			0
$844$	−23.8692			−0.821610
$845$	3.77872	−	5.03512i	0.129992	−	0.173213i
$846$		0			0
$847$		−	3.46774i		−	0.119153i
$848$	6.41990			0.220460
$849$		0			0
$850$		−	20.0268i		−	0.686913i
$851$		−	13.0730i		−	0.448136i
$852$		0			0
$853$		−	35.9527i		−	1.23100i	−0.788138	−	0.615499i	$-0.788955\pi$
$853$		−	35.9527i		−	1.23100i	0.788138	−	0.615499i	$-0.211045\pi$
$854$	5.06311			0.173256
$855$		0			0
$856$			1.41392i			0.0483268i
$857$	−4.57183			−0.156171			−0.0780853	−	0.996947i	$-0.524881\pi$
$857$	−4.57183			−0.156171			−0.0780853	+	0.996947i	$0.524881\pi$
$858$		0			0
$859$	11.6490			0.397459			0.198729	−	0.980054i	$-0.436319\pi$
$859$	11.6490			0.397459			0.198729	+	0.980054i	$0.436319\pi$
$860$		−	4.58462i		−	0.156334i
$861$		0			0
$862$	−1.25113			−0.0426136
$863$		−	26.2523i		−	0.893640i	−0.894624	−	0.446820i	$-0.852556\pi$
$863$		−	26.2523i		−	0.893640i	0.894624	−	0.446820i	$-0.147444\pi$
$864$		0			0
$865$		−	4.28985i		−	0.145859i
$866$			4.77117i			0.162131i
$867$		0			0
$868$	32.7710			1.11232
$869$		−	25.1667i		−	0.853722i
$870$		0			0
$871$	−1.23416	+	3.70062i	−0.0418180	+	0.125391i
$872$	4.97525			0.168483
$873$		0			0
$874$	20.1432			0.681355
$875$	−23.9021			−0.808040
$876$		0			0
$877$		−	46.7822i		−	1.57972i	−0.613284	−	0.789862i	$-0.710152\pi$
$877$		−	46.7822i		−	1.57972i	0.613284	−	0.789862i	$-0.289848\pi$
$878$		−	33.7019i		−	1.13738i
$879$		0			0
$880$	1.55520			0.0524258
$881$	52.3698			1.76438			0.882192	−	0.470889i	$-0.156067\pi$
$881$	52.3698			1.76438			0.882192	+	0.470889i	$0.156067\pi$
$882$		0			0
$883$	−28.7233			−0.966617			−0.483309	−	0.875450i	$-0.660565\pi$
$883$	−28.7233			−0.966617			−0.483309	+	0.875450i	$0.660565\pi$
$884$	−14.3739	−	4.79371i	−0.483446	−	0.161230i
$885$		0			0
$886$			5.37751i			0.180661i
$887$	12.0238			0.403719			0.201859	−	0.979415i	$-0.435302\pi$
$887$	12.0238			0.403719			0.201859	+	0.979415i	$0.435302\pi$
$888$		0			0
$889$			23.0631i			0.773512i
$890$			4.08185i			0.136824i
$891$		0			0
$892$		−	12.8698i		−	0.430913i
$893$	−16.9609			−0.567573
$894$		0			0
$895$		−	5.02193i		−	0.167865i
$896$	5.05438			0.168855
$897$		0			0
$898$	16.7813			0.559999
$899$		−	29.7766i		−	0.993105i
$900$		0			0
$901$	26.9793			0.898812
$902$		−	35.5001i		−	1.18202i
$903$		0			0
$904$		−	13.0883i		−	0.435311i
$905$		−	5.24968i		−	0.174505i
$906$		0			0
$907$	27.7948			0.922912			0.461456	−	0.887163i	$-0.347327\pi$
$907$	27.7948			0.922912			0.461456	+	0.887163i	$0.347327\pi$
$908$		−	18.0552i		−	0.599182i
$909$		0			0
$910$	−2.79198	+	8.37170i	−0.0925531	+	0.277519i
$911$	−20.6592			−0.684470			−0.342235	−	0.939614i	$-0.611184\pi$
$911$	−20.6592			−0.684470			−0.342235	+	0.939614i	$0.611184\pi$
$912$		0			0
$913$	−18.8392			−0.623488
$914$	23.6791			0.783235
$915$		0			0
$916$			21.1737i			0.699599i
$917$		−	12.4816i		−	0.412179i
$918$		0			0
$919$	56.3658			1.85933			0.929667	−	0.368400i	$-0.120094\pi$
$919$	56.3658			1.85933			0.929667	+	0.368400i	$0.120094\pi$
$920$	3.03733			0.100138
$921$		0			0
$922$	34.4622			1.13495
$923$	−5.28188	+	15.8376i	−0.173855	+	0.521303i
$924$		0			0
$925$		−	9.93265i		−	0.326584i
$926$	22.4773			0.738649
$927$		0			0
$928$		−	4.59254i		−	0.150757i
$929$			2.02201i			0.0663399i	0.999450	+	0.0331699i	$0.0105603\pi$
$929$			2.02201i			0.0663399i	−0.999450	+	0.0331699i	$0.989440\pi$
$930$		0			0
$931$		−	59.5633i		−	1.95211i
$932$	−29.0610			−0.951923
$933$		0			0
$934$			4.43632i			0.145161i
$935$	6.53566			0.213739
$936$		0			0
$937$	−15.3243			−0.500622			−0.250311	−	0.968166i	$-0.580533\pi$
$937$	−15.3243			−0.500622			−0.250311	+	0.968166i	$0.580533\pi$
$938$		−	5.46853i		−	0.178554i
$939$		0			0
$940$	−2.55747			−0.0834155
$941$		−	41.4208i		−	1.35028i	−0.737691	−	0.675139i	$-0.764084\pi$
$941$		−	41.4208i		−	1.35028i	0.737691	−	0.675139i	$-0.235916\pi$
$942$		0			0
$943$		−	69.3323i		−	2.25777i
$944$			3.62311i			0.117922i
$945$		0			0
$946$	−30.4047			−0.988541
$947$			56.8875i			1.84859i	0.381673	+	0.924297i	$0.375348\pi$
$947$			56.8875i			1.84859i	−0.381673	+	0.924297i	$0.624652\pi$
$948$		0			0
$949$	0.371324	−	1.11341i	0.0120537	−	0.0361428i
$950$	15.3045			0.496544
$951$		0			0
$952$	21.2408			0.688418
$953$	−40.5931			−1.31494			−0.657470	−	0.753481i	$-0.728373\pi$
$953$	−40.5931			−1.31494			−0.657470	+	0.753481i	$0.728373\pi$
$954$		0			0
$955$		−	5.45175i		−	0.176415i
$956$		−	7.26902i		−	0.235097i
$957$		0			0
$958$	−11.8165			−0.381775
$959$	−99.1182			−3.20069
$960$		0			0
$961$	−11.0383			−0.356074
$962$	−7.12899	−	2.37753i	−0.229848	−	0.0766547i
$963$		0			0
$964$			17.9104i			0.576856i
$965$	3.69277			0.118874
$966$		0			0
$967$			49.9936i			1.60769i	0.594841	+	0.803844i	$0.297215\pi$
$967$			49.9936i			1.60769i	−0.594841	+	0.803844i	$0.702785\pi$
$968$			0.686087i			0.0220517i
$969$		0			0
$970$			6.31979i			0.202916i
$971$	−31.1467			−0.999546			−0.499773	−	0.866156i	$-0.666583\pi$
$971$	−31.1467			−0.999546			−0.499773	+	0.866156i	$0.666583\pi$
$972$		0			0
$973$			1.95805i			0.0627721i
$974$	−25.8373			−0.827880
$975$		0			0
$976$	−1.00173			−0.0320645
$977$		−	6.90009i		−	0.220753i	−0.993890	−	0.110377i	$-0.964794\pi$
$977$		−	6.90009i		−	0.220753i	0.993890	−	0.110377i	$-0.0352057\pi$
$978$		0			0
$979$	27.0704			0.865173
$980$		−	8.98136i		−	0.286899i
$981$		0			0
$982$		−	11.1999i		−	0.357403i
$983$			52.4312i			1.67229i	0.548505	+	0.836147i	$0.315197\pi$
$983$			52.4312i			1.67229i	−0.548505	+	0.836147i	$0.684803\pi$
$984$		0			0
$985$	11.8725			0.378290
$986$		−	19.2999i		−	0.614635i
$987$		0			0
$988$	3.66337	−	10.9846i	0.116547	−	0.349465i
$989$	−59.3808			−1.88820
$990$		0			0
$991$	−26.0277			−0.826799			−0.413399	−	0.910550i	$-0.635659\pi$
$991$	−26.0277			−0.826799			−0.413399	+	0.910550i	$0.635659\pi$
$992$	−6.48369			−0.205857
$993$		0			0
$994$		−	23.4039i		−	0.742325i
$995$			6.55246i			0.207727i
$996$		0			0
$997$	27.2475			0.862936			0.431468	−	0.902128i	$-0.357996\pi$
$997$	27.2475			0.862936			0.431468	+	0.902128i	$0.357996\pi$
$998$	−15.4596			−0.489367
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.b.c.649.4		14
3.2	odd	2		2106.2.b.d.649.11		14
9.2	odd	6		702.2.t.a.415.11		28
9.4	even	3		234.2.t.a.25.12	yes	28
9.5	odd	6		702.2.t.a.181.4		28
9.7	even	3		234.2.t.a.103.5	yes	28
13.12	even	2	inner	2106.2.b.c.649.11		14
39.38	odd	2		2106.2.b.d.649.4		14
117.25	even	6		234.2.t.a.103.12	yes	28
117.38	odd	6		702.2.t.a.415.4		28
117.77	odd	6		702.2.t.a.181.11		28
117.103	even	6		234.2.t.a.25.5	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.t.a.25.5	✓	28	117.103	even	6
234.2.t.a.25.12	yes	28	9.4	even	3
234.2.t.a.103.5	yes	28	9.7	even	3
234.2.t.a.103.12	yes	28	117.25	even	6
702.2.t.a.181.4		28	9.5	odd	6
702.2.t.a.181.11		28	117.77	odd	6
702.2.t.a.415.4		28	117.38	odd	6
702.2.t.a.415.11		28	9.2	odd	6
2106.2.b.c.649.4		14	1.1	even	1	trivial
2106.2.b.c.649.11		14	13.12	even	2	inner
2106.2.b.d.649.4		14	39.38	odd	2
2106.2.b.d.649.11		14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2106.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -0.484256i q^{5} -5.05438i q^{7} +1.00000i q^{8} -0.484256 q^{10} +3.21153i q^{11} +(3.42035 + 1.14069i) q^{13} -5.05438 q^{14} +1.00000 q^{16} +4.20245 q^{17} +3.21153i q^{19} +0.484256i q^{20} +3.21153 q^{22} +6.27217 q^{23} +4.76550 q^{25} +(1.14069 - 3.42035i) q^{26} +5.05438i q^{28} +4.59254 q^{29} -6.48369i q^{31} -1.00000i q^{32} -4.20245i q^{34} -2.44761 q^{35} -2.08429i q^{37} +3.21153 q^{38} +0.484256 q^{40} -11.0540i q^{41} -9.46735 q^{43} -3.21153i q^{44} -6.27217i q^{46} +5.28124i q^{47} -18.5467 q^{49} -4.76550i q^{50} +(-3.42035 - 1.14069i) q^{52} +6.41990 q^{53} +1.55520 q^{55} +5.05438 q^{56} -4.59254i q^{58} +3.62311i q^{59} -1.00173 q^{61} -6.48369 q^{62} -1.00000 q^{64} +(0.552388 - 1.65633i) q^{65} +1.08194i q^{67} -4.20245 q^{68} +2.44761i q^{70} +4.63041i q^{71} -0.325525i q^{73} -2.08429 q^{74} -3.21153i q^{76} +16.2323 q^{77} -7.83637 q^{79} -0.484256i q^{80} -11.0540 q^{82} +5.86613i q^{83} -2.03506i q^{85} +9.46735i q^{86} -3.21153 q^{88} -8.42912i q^{89} +(5.76550 - 17.2878i) q^{91} -6.27217 q^{92} +5.28124 q^{94} +1.55520 q^{95} -13.0505i q^{97} +18.5467i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{4} - 2 q^{13} - 8 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{23} - 14 q^{25} + 4 q^{26} + 16 q^{29} - 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} - 60 q^{53} + 8 q^{56} - 28 q^{61} + 34 q^{62} - 14 q^{64}+ \cdots - 8 q^{92}+O(q^{100}) \) 14 * q - 14 * q^4 - 2 * q^13 - 8 * q^14 + 14 * q^16 + 8 * q^17 + 8 * q^23 - 14 * q^25 + 4 * q^26 + 16 * q^29 - 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 - 60 * q^53 + 8 * q^56 - 28 * q^61 + 34 * q^62 - 14 * q^64 + 8 * q^65 - 8 * q^68 - 16 * q^74 + 24 * q^77 - 28 * q^79 - 24 * q^82 - 8 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more