LMFDB - Embedded newform 2420.2.b.h.969.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2420,2,Mod(969,2420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2420, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2420.969");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2420 = 2^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2420.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:	$19.3237972891$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 27x^{10} + 273x^{8} + 1287x^{6} + 2856x^{4} + 2610x^{2} + 775 $ x^12 + 27x^10 + 273x^8 + 1287x^6 + 2856x^4 + 2610*x^2 + 775
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			969.8
Root			$1.04013i$ of defining polynomial
Character	$\chi$	$=$	2420.969
Dual form			2420.2.b.h.969.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.04013i q^{3} +(-0.0700917 - 2.23497i) q^{5} +2.26282i q^{7} +1.91812 q^{9} -0.268248i q^{13} +(2.32467 - 0.0729048i) q^{15} +5.92413i q^{17} -5.74484 q^{19} -2.35363 q^{21} -2.92836i q^{23} +(-4.99017 + 0.313306i) q^{25} +5.11551i q^{27} +6.48082 q^{29} +0.250047 q^{31} +(5.05732 - 0.158605i) q^{35} +10.3040i q^{37} +0.279014 q^{39} +2.36323 q^{41} +7.56763i q^{43} +(-0.134444 - 4.28694i) q^{45} +1.13257i q^{47} +1.87966 q^{49} -6.16189 q^{51} -10.2012i q^{53} -5.97541i q^{57} -11.4072 q^{59} +4.62196 q^{61} +4.34036i q^{63} +(-0.599526 + 0.0188020i) q^{65} -6.11229i q^{67} +3.04589 q^{69} +5.09021 q^{71} +8.15669i q^{73} +(-0.325880 - 5.19045i) q^{75} -8.30362 q^{79} +0.433552 q^{81} +16.4266i q^{83} +(13.2402 - 0.415232i) q^{85} +6.74092i q^{87} +5.73205 q^{89} +0.606996 q^{91} +0.260083i q^{93} +(0.402666 + 12.8396i) q^{95} +4.51296i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{5} - 18 q^{9} + 2 q^{15} - 14 q^{19} + 14 q^{21} - 9 q^{25} - 22 q^{29} + 12 q^{31} + 4 q^{35} - 26 q^{39} + 40 q^{41} - 20 q^{45} - 8 q^{49} + 26 q^{51} + 16 q^{59} + 30 q^{61} + 11 q^{65}+ \cdots + 14 q^{95}+O(q^{100}) $ 12 * q + 3 * q^5 - 18 * q^9 + 2 * q^15 - 14 * q^19 + 14 * q^21 - 9 * q^25 - 22 * q^29 + 12 * q^31 + 4 * q^35 - 26 * q^39 + 40 * q^41 - 20 * q^45 - 8 * q^49 + 26 * q^51 + 16 * q^59 + 30 * q^61 + 11 * q^65 + 36 * q^69 - 24 * q^71 + 39 * q^75 + 2 * q^79 - 12 * q^81 + 21 * q^85 + 36 * q^89 - 40 * q^91 + 14 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2420\mathbb{Z}\right)^\times$.

$n$	$1211$	$1937$	$2301$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.04013i			0.600522i	0.953857	+	0.300261i	$0.0970737\pi$
$3$			1.04013i			0.600522i	−0.953857	+	0.300261i	$0.902926\pi$
$4$		0			0
$5$	−0.0700917	−	2.23497i	−0.0313460	−	0.999509i
$5$	−0.0700917	−	2.23497i	−0.0313460	−	0.999509i
$6$		0			0
$7$			2.26282i			0.855264i	0.903953	+	0.427632i	$0.140652\pi$
$7$			2.26282i			0.855264i	−0.903953	+	0.427632i	$0.859348\pi$
$8$		0			0
$9$	1.91812			0.639374
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		−	0.268248i		−	0.0743986i	−0.999308	−	0.0371993i	$-0.988156\pi$
$13$		−	0.268248i		−	0.0743986i	0.999308	−	0.0371993i	$-0.0118436\pi$
$14$		0			0
$15$	2.32467	−	0.0729048i	0.600227	−	0.0188239i
$16$		0			0
$17$			5.92413i			1.43681i	0.695624	+	0.718406i	$0.255128\pi$
$17$			5.92413i			1.43681i	−0.695624	+	0.718406i	$0.744872\pi$
$18$		0			0
$19$	−5.74484			−1.31796			−0.658979	−	0.752161i	$-0.729011\pi$
$19$	−5.74484			−1.31796			−0.658979	+	0.752161i	$0.729011\pi$
$20$		0			0
$21$	−2.35363			−0.513605
$22$		0			0
$23$		−	2.92836i		−	0.610605i	−0.952255	−	0.305303i	$-0.901242\pi$
$23$		−	2.92836i		−	0.610605i	0.952255	−	0.305303i	$-0.0987576\pi$
$24$		0			0
$25$	−4.99017	+	0.313306i	−0.998035	+	0.0626611i
$26$		0			0
$27$			5.11551i			0.984479i
$28$		0			0
$29$	6.48082			1.20346			0.601729	−	0.798700i	$-0.294479\pi$
$29$	6.48082			1.20346			0.601729	+	0.798700i	$0.294479\pi$
$30$		0			0
$31$	0.250047			0.0449098			0.0224549	−	0.999748i	$-0.492852\pi$
$31$	0.250047			0.0449098			0.0224549	+	0.999748i	$0.492852\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	5.05732	−	0.158605i	0.854844	−	0.0268091i
$36$		0			0
$37$			10.3040i			1.69396i	0.531625	+	0.846980i	$0.321582\pi$
$37$			10.3040i			1.69396i	−0.531625	+	0.846980i	$0.678418\pi$
$38$		0			0
$39$	0.279014			0.0446780
$40$		0			0
$41$	2.36323			0.369074			0.184537	−	0.982826i	$-0.440921\pi$
$41$	2.36323			0.369074			0.184537	+	0.982826i	$0.440921\pi$
$42$		0			0
$43$			7.56763i			1.15405i	0.816725	+	0.577027i	$0.195787\pi$
$43$			7.56763i			1.15405i	−0.816725	+	0.577027i	$0.804213\pi$
$44$		0			0
$45$	−0.134444	−	4.28694i	−0.0200418	−	0.639059i
$46$		0			0
$47$			1.13257i			0.165202i	0.996583	+	0.0826009i	$0.0263227\pi$
$47$			1.13257i			0.165202i	−0.996583	+	0.0826009i	$0.973677\pi$
$48$		0			0
$49$	1.87966			0.268523
$50$		0			0
$51$	−6.16189			−0.862837
$52$		0			0
$53$		−	10.2012i		−	1.40124i	−0.713536	−	0.700619i	$-0.752907\pi$
$53$		−	10.2012i		−	1.40124i	0.713536	−	0.700619i	$-0.247093\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	5.97541i		−	0.791462i
$58$		0			0
$59$	−11.4072			−1.48509			−0.742546	−	0.669795i	$-0.766382\pi$
$59$	−11.4072			−1.48509			−0.742546	+	0.669795i	$0.766382\pi$
$60$		0			0
$61$	4.62196			0.591781			0.295890	−	0.955222i	$-0.404384\pi$
$61$	4.62196			0.591781			0.295890	+	0.955222i	$0.404384\pi$
$62$		0			0
$63$			4.34036i			0.546833i
$64$		0			0
$65$	−0.599526	+	0.0188020i	−0.0743621	+	0.00233210i
$66$		0			0
$67$		−	6.11229i		−	0.746735i	−0.927683	−	0.373368i	$-0.878203\pi$
$67$		−	6.11229i		−	0.746735i	0.927683	−	0.373368i	$-0.121797\pi$
$68$		0			0
$69$	3.04589			0.366682
$70$		0			0
$71$	5.09021			0.604097			0.302049	−	0.953293i	$-0.402330\pi$
$71$	5.09021			0.604097			0.302049	+	0.953293i	$0.402330\pi$
$72$		0			0
$73$			8.15669i			0.954668i	0.878722	+	0.477334i	$0.158397\pi$
$73$			8.15669i			0.954668i	−0.878722	+	0.477334i	$0.841603\pi$
$74$		0			0
$75$	−0.325880	−	5.19045i	−0.0376294	−	0.599342i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−8.30362			−0.934230			−0.467115	−	0.884197i	$-0.654707\pi$
$79$	−8.30362			−0.934230			−0.467115	+	0.884197i	$0.654707\pi$
$80$		0			0
$81$	0.433552			0.0481724
$82$		0			0
$83$			16.4266i			1.80305i	0.432723	+	0.901527i	$0.357553\pi$
$83$			16.4266i			1.80305i	−0.432723	+	0.901527i	$0.642447\pi$
$84$		0			0
$85$	13.2402	−	0.415232i	1.43611	−	0.0450383i
$86$		0			0
$87$			6.74092i			0.722702i
$88$		0			0
$89$	5.73205			0.607596			0.303798	−	0.952736i	$-0.401745\pi$
$89$	5.73205			0.607596			0.303798	+	0.952736i	$0.401745\pi$
$90$		0			0
$91$	0.606996			0.0636305
$92$		0			0
$93$			0.260083i			0.0269693i
$94$		0			0
$95$	0.402666	+	12.8396i	0.0413127	+	1.31731i
$96$		0			0
$97$			4.51296i			0.458221i	0.973400	+	0.229111i	$0.0735818\pi$
$97$			4.51296i			0.458221i	−0.973400	+	0.229111i	$0.926418\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.37275			0.932623			0.466312	−	0.884621i	$-0.345583\pi$
$101$	9.37275			0.932623			0.466312	+	0.884621i	$0.345583\pi$
$102$		0			0
$103$			6.57513i			0.647867i	0.946080	+	0.323934i	$0.105005\pi$
$103$			6.57513i			0.647867i	−0.946080	+	0.323934i	$0.894995\pi$
$104$		0			0
$105$	0.164970	+	5.26030i	0.0160994	+	0.513352i
$106$		0			0
$107$		−	6.93987i		−	0.670903i	−0.942058	−	0.335451i	$-0.891111\pi$
$107$		−	6.93987i		−	0.670903i	0.942058	−	0.335451i	$-0.108889\pi$
$108$		0			0
$109$	10.0676			0.964299			0.482150	−	0.876089i	$-0.339856\pi$
$109$	10.0676			0.964299			0.482150	+	0.876089i	$0.339856\pi$
$110$		0			0
$111$	−10.7175			−1.01726
$112$		0			0
$113$			18.7787i			1.76655i	0.468857	+	0.883274i	$0.344666\pi$
$113$			18.7787i			1.76655i	−0.468857	+	0.883274i	$0.655334\pi$
$114$		0			0
$115$	−6.54479	+	0.205254i	−0.610305	+	0.0191400i
$116$		0			0
$117$		−	0.514532i		−	0.0475685i
$118$		0			0
$119$	−13.4052			−1.22885
$120$		0			0
$121$		0			0
$122$		0			0
$123$			2.45807i			0.221637i
$124$		0			0
$125$	1.05000	+	11.1309i	0.0939147	+	0.995580i
$126$		0			0
$127$			19.0904i			1.69400i	0.531594	+	0.846999i	$0.321593\pi$
$127$			19.0904i			1.69400i	−0.531594	+	0.846999i	$0.678407\pi$
$128$		0			0
$129$	−7.87136			−0.693034
$130$		0			0
$131$	1.78426			0.155891			0.0779457	−	0.996958i	$-0.475164\pi$
$131$	1.78426			0.155891			0.0779457	+	0.996958i	$0.475164\pi$
$132$		0			0
$133$		−	12.9995i		−	1.12720i
$134$		0			0
$135$	11.4330	−	0.358555i	0.983996	−	0.0308595i
$136$		0			0
$137$		−	7.57205i		−	0.646924i	−0.946241	−	0.323462i	$-0.895153\pi$
$137$		−	7.57205i		−	0.646924i	0.946241	−	0.323462i	$-0.104847\pi$
$138$		0			0
$139$	−11.0664			−0.938638			−0.469319	−	0.883029i	$-0.655501\pi$
$139$	−11.0664			−0.938638			−0.469319	+	0.883029i	$0.655501\pi$
$140$		0			0
$141$	−1.17802			−0.0992073
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−0.454252	−	14.4844i	−0.0377235	−	1.20287i
$146$		0			0
$147$			1.95510i			0.161254i
$148$		0			0
$149$	−11.6600			−0.955224			−0.477612	−	0.878571i	$-0.658498\pi$
$149$	−11.6600			−0.955224			−0.477612	+	0.878571i	$0.658498\pi$
$150$		0			0
$151$	0.242619			0.0197440			0.00987202	−	0.999951i	$-0.496858\pi$
$151$	0.242619			0.0197440			0.00987202	+	0.999951i	$0.496858\pi$
$152$		0			0
$153$			11.3632i			0.918660i
$154$		0			0
$155$	−0.0175262	−	0.558848i	−0.00140774	−	0.0448878i
$156$		0			0
$157$		−	2.95189i		−	0.235586i	−0.993038	−	0.117793i	$-0.962418\pi$
$157$		−	2.95189i		−	0.235586i	0.993038	−	0.117793i	$-0.0375820\pi$
$158$		0			0
$159$	10.6106			0.841473
$160$		0			0
$161$	6.62634			0.522229
$162$		0			0
$163$		−	10.3949i		−	0.814191i	−0.913386	−	0.407095i	$-0.866542\pi$
$163$		−	10.3949i		−	0.814191i	0.913386	−	0.407095i	$-0.133458\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.88770i			0.765133i	0.923928	+	0.382567i	$0.124960\pi$
$167$			9.88770i			0.765133i	−0.923928	+	0.382567i	$0.875040\pi$
$168$		0			0
$169$	12.9280			0.994465
$170$		0			0
$171$	−11.0193			−0.842668
$172$		0			0
$173$			15.7914i			1.20060i	0.799776	+	0.600299i	$0.204952\pi$
$173$			15.7914i			1.20060i	−0.799776	+	0.600299i	$0.795048\pi$
$174$		0			0
$175$	−0.708953	−	11.2918i	−0.0535918	−	0.853583i
$176$		0			0
$177$		−	11.8650i		−	0.891830i
$178$		0			0
$179$	−5.86594			−0.438441			−0.219221	−	0.975675i	$-0.570351\pi$
$179$	−5.86594			−0.438441			−0.219221	+	0.975675i	$0.570351\pi$
$180$		0			0
$181$	−22.5943			−1.67942			−0.839710	−	0.543035i	$-0.817275\pi$
$181$	−22.5943			−1.67942			−0.839710	+	0.543035i	$0.817275\pi$
$182$		0			0
$183$			4.80746i			0.355377i
$184$		0			0
$185$	23.0290	−	0.722222i	1.69313	−	0.0530988i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−11.5754			−0.841990
$190$		0			0
$191$	11.0863			0.802176			0.401088	−	0.916040i	$-0.368632\pi$
$191$	11.0863			0.802176			0.401088	+	0.916040i	$0.368632\pi$
$192$		0			0
$193$		−	6.40935i		−	0.461355i	−0.973030	−	0.230677i	$-0.925906\pi$
$193$		−	6.40935i		−	0.461355i	0.973030	−	0.230677i	$-0.0740942\pi$
$194$		0			0
$195$	−0.0195566	−	0.623588i	−0.00140047	−	0.0446560i
$196$		0			0
$197$			9.15141i			0.652011i	0.945368	+	0.326005i	$0.105703\pi$
$197$			9.15141i			0.652011i	−0.945368	+	0.326005i	$0.894297\pi$
$198$		0			0
$199$	14.6653			1.03960			0.519798	−	0.854289i	$-0.326007\pi$
$199$	14.6653			1.03960			0.519798	+	0.854289i	$0.326007\pi$
$200$		0			0
$201$	6.35760			0.448431
$202$		0			0
$203$			14.6649i			1.02927i
$204$		0			0
$205$	−0.165643	−	5.28174i	−0.0115690	−	0.368893i
$206$		0			0
$207$		−	5.61695i		−	0.390405i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−12.4543			−0.857391			−0.428695	−	0.903449i	$-0.641027\pi$
$211$	−12.4543			−0.857391			−0.428695	+	0.903449i	$0.641027\pi$
$212$		0			0
$213$			5.29450i			0.362773i
$214$		0			0
$215$	16.9134	−	0.530429i	1.15349	−	0.0361749i
$216$		0			0
$217$			0.565811i			0.0384098i
$218$		0			0
$219$	−8.48405			−0.573299
$220$		0			0
$221$	1.58914			0.106897
$222$		0			0
$223$		−	12.1921i		−	0.816442i	−0.912883	−	0.408221i	$-0.866149\pi$
$223$		−	12.1921i		−	0.816442i	0.912883	−	0.408221i	$-0.133851\pi$
$224$		0			0
$225$	−9.57176	+	0.600958i	−0.638117	+	0.0400639i
$226$		0			0
$227$			7.70538i			0.511424i	0.966753	+	0.255712i	$0.0823099\pi$
$227$			7.70538i			0.511424i	−0.966753	+	0.255712i	$0.917690\pi$
$228$		0			0
$229$	−18.2312			−1.20475			−0.602377	−	0.798212i	$-0.705780\pi$
$229$	−18.2312			−1.20475			−0.602377	+	0.798212i	$0.705780\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4.94752i		−	0.324123i	−0.986781	−	0.162062i	$-0.948186\pi$
$233$		−	4.94752i		−	0.324123i	0.986781	−	0.162062i	$-0.0518143\pi$
$234$		0			0
$235$	2.53125	−	0.0793836i	0.165121	−	0.00517841i
$236$		0			0
$237$		−	8.63688i		−	0.561025i
$238$		0			0
$239$	−16.9847			−1.09865			−0.549324	−	0.835610i	$-0.685115\pi$
$239$	−16.9847			−1.09865			−0.549324	+	0.835610i	$0.685115\pi$
$240$		0			0
$241$	−6.72951			−0.433486			−0.216743	−	0.976229i	$-0.569543\pi$
$241$	−6.72951			−0.433486			−0.216743	+	0.976229i	$0.569543\pi$
$242$		0			0
$243$			15.7975i			1.01341i
$244$		0			0
$245$	−0.131749	−	4.20099i	−0.00841712	−	0.268391i
$246$		0			0
$247$			1.54104i			0.0980542i
$248$		0			0
$249$	−17.0859			−1.08277
$250$		0			0
$251$	19.7671			1.24769			0.623845	−	0.781548i	$-0.285569\pi$
$251$	19.7671			1.24769			0.623845	+	0.781548i	$0.285569\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	0.431897	+	13.7716i	0.0270465	+	0.862413i
$256$		0			0
$257$			6.39498i			0.398908i	0.979907	+	0.199454i	$0.0639169\pi$
$257$			6.39498i			0.398908i	−0.979907	+	0.199454i	$0.936083\pi$
$258$		0			0
$259$	−23.3160			−1.44878
$260$		0			0
$261$	12.4310			0.769459
$262$		0			0
$263$			14.6417i			0.902843i	0.892311	+	0.451422i	$0.149083\pi$
$263$			14.6417i			0.902843i	−0.892311	+	0.451422i	$0.850917\pi$
$264$		0			0
$265$	−22.7993	+	0.715017i	−1.40055	+	0.0439231i
$266$		0			0
$267$			5.96210i			0.364875i
$268$		0			0
$269$	19.8736			1.21172			0.605858	−	0.795573i	$-0.292830\pi$
$269$	19.8736			1.21172			0.605858	+	0.795573i	$0.292830\pi$
$270$		0			0
$271$	19.2126			1.16708			0.583541	−	0.812084i	$-0.301667\pi$
$271$	19.2126			1.16708			0.583541	+	0.812084i	$0.301667\pi$
$272$		0			0
$273$			0.631357i			0.0382115i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	23.8344i		−	1.43207i	−0.698063	−	0.716036i	$-0.745955\pi$
$277$		−	23.8344i		−	1.43207i	0.698063	−	0.716036i	$-0.254045\pi$
$278$		0			0
$279$	0.479621			0.0287142
$280$		0			0
$281$	17.9286			1.06953			0.534765	−	0.845001i	$-0.320400\pi$
$281$	17.9286			1.06953			0.534765	+	0.845001i	$0.320400\pi$
$282$		0			0
$283$		−	21.0267i		−	1.24991i	−0.780663	−	0.624953i	$-0.785118\pi$
$283$		−	21.0267i		−	1.24991i	0.780663	−	0.624953i	$-0.214882\pi$
$284$		0			0
$285$	−13.3549	+	0.418827i	−0.791073	+	0.0248092i
$286$		0			0
$287$			5.34755i			0.315656i
$288$		0			0
$289$	−18.0953			−1.06443
$290$		0			0
$291$	−4.69408			−0.275172
$292$		0			0
$293$		−	21.2681i		−	1.24249i	−0.783615	−	0.621247i	$-0.786626\pi$
$293$		−	21.2681i		−	1.24249i	0.783615	−	0.621247i	$-0.213374\pi$
$294$		0			0
$295$	0.799551	+	25.4947i	0.0465516	+	1.48436i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.785527			−0.0454282
$300$		0			0
$301$	−17.1242			−0.987021
$302$		0			0
$303$			9.74891i			0.560060i
$304$		0			0
$305$	−0.323961	−	10.3299i	−0.0185499	−	0.591490i
$306$		0			0
$307$		−	32.6823i		−	1.86528i	−0.360812	−	0.932639i	$-0.617500\pi$
$307$		−	32.6823i		−	1.86528i	0.360812	−	0.932639i	$-0.382500\pi$
$308$		0			0
$309$	−6.83902			−0.389058
$310$		0			0
$311$	−18.9259			−1.07319			−0.536594	−	0.843840i	$-0.680289\pi$
$311$	−18.9259			−1.07319			−0.536594	+	0.843840i	$0.680289\pi$
$312$		0			0
$313$		−	12.0306i		−	0.680007i	−0.940424	−	0.340004i	$-0.889572\pi$
$313$		−	12.0306i		−	0.680007i	0.940424	−	0.340004i	$-0.110428\pi$
$314$		0			0
$315$	9.70056	−	0.304223i	0.546565	−	0.0171410i
$316$		0			0
$317$		−	10.2008i		−	0.572931i	−0.958091	−	0.286466i	$-0.907520\pi$
$317$		−	10.2008i		−	0.572931i	0.958091	−	0.286466i	$-0.0924804\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	7.21840			0.402892
$322$		0			0
$323$		−	34.0332i		−	1.89366i
$324$		0			0
$325$	0.0840436	+	1.33860i	0.00466190	+	0.0742524i
$326$		0			0
$327$			10.4716i			0.579082i
$328$		0			0
$329$	−2.56279			−0.141291
$330$		0			0
$331$	11.8942			0.653763			0.326882	−	0.945065i	$-0.394002\pi$
$331$	11.8942			0.653763			0.326882	+	0.945065i	$0.394002\pi$
$332$		0			0
$333$			19.7642i			1.08307i
$334$		0			0
$335$	−13.6608	+	0.428421i	−0.746368	+	0.0234071i
$336$		0			0
$337$			10.0503i			0.547476i	0.961804	+	0.273738i	$0.0882601\pi$
$337$			10.0503i			0.547476i	−0.961804	+	0.273738i	$0.911740\pi$
$338$		0			0
$339$	−19.5323			−1.06085
$340$		0			0
$341$		0			0
$342$		0			0
$343$			20.0930i			1.08492i
$344$		0			0
$345$	−0.213491	−	6.80746i	−0.0114940	−	0.366502i
$346$		0			0
$347$		−	23.1992i		−	1.24540i	−0.782462	−	0.622698i	$-0.786037\pi$
$347$		−	23.1992i		−	1.24540i	0.782462	−	0.622698i	$-0.213963\pi$
$348$		0			0
$349$	−25.3369			−1.35625			−0.678127	−	0.734944i	$-0.737208\pi$
$349$	−25.3369			−1.35625			−0.678127	+	0.734944i	$0.737208\pi$
$350$		0			0
$351$	1.37222			0.0732439
$352$		0			0
$353$		−	1.08185i		−	0.0575813i	−0.999585	−	0.0287907i	$-0.990834\pi$
$353$		−	1.08185i		−	0.0575813i	0.999585	−	0.0287907i	$-0.00916562\pi$
$354$		0			0
$355$	−0.356782	−	11.3765i	−0.0189360	−	0.603800i
$356$		0			0
$357$		−	13.9432i		−	0.737954i
$358$		0			0
$359$	9.19690			0.485394			0.242697	−	0.970102i	$-0.421968\pi$
$359$	9.19690			0.485394			0.242697	+	0.970102i	$0.421968\pi$
$360$		0			0
$361$	14.0032			0.737013
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.2299	−	0.571716i	0.954199	−	0.0299250i
$366$		0			0
$367$		−	19.4223i		−	1.01384i	−0.861994	−	0.506918i	$-0.830785\pi$
$367$		−	19.4223i		−	1.01384i	0.861994	−	0.506918i	$-0.169215\pi$
$368$		0			0
$369$	4.53296			0.235976
$370$		0			0
$371$	23.0834			1.19843
$372$		0			0
$373$		−	25.8608i		−	1.33902i	−0.742803	−	0.669510i	$-0.766504\pi$
$373$		−	25.8608i		−	1.33902i	0.742803	−	0.669510i	$-0.233496\pi$
$374$		0			0
$375$	−11.5777	+	1.09214i	−0.597868	+	0.0563978i
$376$		0			0
$377$		−	1.73847i		−	0.0895356i
$378$		0			0
$379$	28.7118			1.47483			0.737413	−	0.675442i	$-0.236047\pi$
$379$	28.7118			1.47483			0.737413	+	0.675442i	$0.236047\pi$
$380$		0			0
$381$	−19.8566			−1.01728
$382$		0			0
$383$			11.1760i			0.571066i	0.958369	+	0.285533i	$0.0921706\pi$
$383$			11.1760i			0.571066i	−0.958369	+	0.285533i	$0.907829\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			14.5156i			0.737871i
$388$		0			0
$389$	−30.1763			−1.53000			−0.764999	−	0.644031i	$-0.777261\pi$
$389$	−30.1763			−1.53000			−0.764999	+	0.644031i	$0.777261\pi$
$390$		0			0
$391$	17.3480			0.877325
$392$		0			0
$393$			1.85587i			0.0936162i
$394$		0			0
$395$	0.582015	+	18.5583i	0.0292843	+	0.933771i
$396$		0			0
$397$			28.3518i			1.42293i	0.702719	+	0.711467i	$0.251969\pi$
$397$			28.3518i			1.42293i	−0.702719	+	0.711467i	$0.748031\pi$
$398$		0			0
$399$	13.5213			0.676909
$400$		0			0
$401$	13.9491			0.696583			0.348292	−	0.937386i	$-0.386762\pi$
$401$	13.9491			0.696583			0.348292	+	0.937386i	$0.386762\pi$
$402$		0			0
$403$		−	0.0670747i		−	0.00334123i
$404$		0			0
$405$	−0.0303884	−	0.968975i	−0.00151001	−	0.0481487i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−12.9388			−0.639782			−0.319891	−	0.947454i	$-0.603646\pi$
$409$	−12.9388			−0.639782			−0.319891	+	0.947454i	$0.603646\pi$
$410$		0			0
$411$	7.87595			0.388492
$412$		0			0
$413$		−	25.8124i		−	1.27015i
$414$		0			0
$415$	36.7130	−	1.15137i	1.80217	−	0.0565185i
$416$		0			0
$417$		−	11.5105i		−	0.563673i
$418$		0			0
$419$	26.6889			1.30384			0.651920	−	0.758288i	$-0.273964\pi$
$419$	26.6889			1.30384			0.651920	+	0.758288i	$0.273964\pi$
$420$		0			0
$421$	26.7183			1.30217			0.651085	−	0.759005i	$-0.274314\pi$
$421$	26.7183			1.30217			0.651085	+	0.759005i	$0.274314\pi$
$422$		0			0
$423$			2.17240i			0.105626i
$424$		0			0
$425$	−1.85606	−	29.5624i	−0.0900323	−	1.43399i
$426$		0			0
$427$			10.4586i			0.506129i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	14.9263			0.718972			0.359486	−	0.933150i	$-0.382952\pi$
$431$	14.9263			0.718972			0.359486	+	0.933150i	$0.382952\pi$
$432$		0			0
$433$			19.8201i			0.952493i	0.879312	+	0.476247i	$0.158003\pi$
$433$			19.8201i			0.952493i	−0.879312	+	0.476247i	$0.841997\pi$
$434$		0			0
$435$	15.0657	−	0.472483i	0.722347	−	0.0226538i
$436$		0			0
$437$			16.8230i			0.804752i
$438$		0			0
$439$	34.4507			1.64424			0.822122	−	0.569312i	$-0.192790\pi$
$439$	34.4507			1.64424			0.822122	+	0.569312i	$0.192790\pi$
$440$		0			0
$441$	3.60542			0.171687
$442$		0			0
$443$		−	27.1532i		−	1.29009i	−0.764145	−	0.645044i	$-0.776839\pi$
$443$		−	27.1532i		−	1.29009i	0.764145	−	0.645044i	$-0.223161\pi$
$444$		0			0
$445$	−0.401769	−	12.8110i	−0.0190457	−	0.607298i
$446$		0			0
$447$		−	12.1280i		−	0.573633i
$448$		0			0
$449$	−27.3664			−1.29150			−0.645750	−	0.763549i	$-0.723455\pi$
$449$	−27.3664			−1.29150			−0.645750	+	0.763549i	$0.723455\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			0.252356i			0.0118567i
$454$		0			0
$455$	−0.0425454	−	1.35662i	−0.00199456	−	0.0635992i
$456$		0			0
$457$		−	38.5598i		−	1.80375i	−0.431996	−	0.901875i	$-0.642191\pi$
$457$		−	38.5598i		−	1.80375i	0.431996	−	0.901875i	$-0.357809\pi$
$458$		0			0
$459$	−30.3049			−1.41451
$460$		0			0
$461$	27.7494			1.29242			0.646208	−	0.763161i	$-0.276354\pi$
$461$	27.7494			1.29242			0.646208	+	0.763161i	$0.276354\pi$
$462$		0			0
$463$			10.9962i			0.511035i	0.966804	+	0.255518i	$0.0822459\pi$
$463$			10.9962i			0.511035i	−0.966804	+	0.255518i	$0.917754\pi$
$464$		0			0
$465$	0.581277	−	0.0182296i	0.0269561	−	0.000845380i
$466$		0			0
$467$			32.0412i			1.48269i	0.671123	+	0.741346i	$0.265812\pi$
$467$			32.0412i			1.48269i	−0.671123	+	0.741346i	$0.734188\pi$
$468$		0			0
$469$	13.8310			0.638656
$470$		0			0
$471$	3.07036			0.141475
$472$		0			0
$473$		0			0
$474$		0			0
$475$	28.6678	−	1.79989i	1.31537	−	0.0825847i
$476$		0			0
$477$		−	19.5671i		−	0.895914i
$478$		0			0
$479$	−15.4154			−0.704350			−0.352175	−	0.935934i	$-0.614558\pi$
$479$	−15.4154			−0.704350			−0.352175	+	0.935934i	$0.614558\pi$
$480$		0			0
$481$	2.76402			0.126028
$482$		0			0
$483$			6.89228i			0.313610i
$484$		0			0
$485$	10.0863	−	0.316321i	0.457996	−	0.0143634i
$486$		0			0
$487$		−	26.1849i		−	1.18655i	−0.805000	−	0.593275i	$-0.797834\pi$
$487$		−	26.1849i		−	1.18655i	0.805000	−	0.593275i	$-0.202166\pi$
$488$		0			0
$489$	10.8121			0.488939
$490$		0			0
$491$	6.47538			0.292230			0.146115	−	0.989268i	$-0.453323\pi$
$491$	6.47538			0.292230			0.146115	+	0.989268i	$0.453323\pi$
$492$		0			0
$493$			38.3932i			1.72914i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			11.5182i			0.516663i
$498$		0			0
$499$	−23.0274			−1.03085			−0.515424	−	0.856935i	$-0.672366\pi$
$499$	−23.0274			−1.03085			−0.515424	+	0.856935i	$0.672366\pi$
$500$		0			0
$501$	−10.2845			−0.459479
$502$		0			0
$503$			27.5506i			1.22842i	0.789142	+	0.614211i	$0.210525\pi$
$503$			27.5506i			1.22842i	−0.789142	+	0.614211i	$0.789475\pi$
$504$		0			0
$505$	−0.656952	−	20.9478i	−0.0292340	−	0.932165i
$506$		0			0
$507$			13.4469i			0.597198i
$508$		0			0
$509$	9.75473			0.432371			0.216185	−	0.976352i	$-0.430638\pi$
$509$	9.75473			0.432371			0.216185	+	0.976352i	$0.430638\pi$
$510$		0			0
$511$	−18.4571			−0.816493
$512$		0			0
$513$		−	29.3878i		−	1.29750i
$514$		0			0
$515$	14.6952	−	0.460862i	0.647549	−	0.0203080i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−16.4252			−0.720985
$520$		0			0
$521$	20.2020			0.885068			0.442534	−	0.896752i	$-0.354080\pi$
$521$	20.2020			0.885068			0.442534	+	0.896752i	$0.354080\pi$
$522$		0			0
$523$		−	22.9063i		−	1.00162i	−0.865557	−	0.500811i	$-0.833035\pi$
$523$		−	22.9063i		−	1.00162i	0.865557	−	0.500811i	$-0.166965\pi$
$524$		0			0
$525$	11.7450	−	0.737406i	0.512595	−	0.0321831i
$526$		0			0
$527$			1.48131i			0.0645270i
$528$		0			0
$529$	14.4247			0.627161
$530$		0			0
$531$	−21.8804			−0.949528
$532$		0			0
$533$		−	0.633931i		−	0.0274586i
$534$		0			0
$535$	−15.5104	+	0.486428i	−0.670573	+	0.0210301i
$536$		0			0
$537$		−	6.10137i		−	0.263293i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	35.8978			1.54337			0.771683	−	0.636008i	$-0.219415\pi$
$541$	35.8978			1.54337			0.771683	+	0.636008i	$0.219415\pi$
$542$		0			0
$543$		−	23.5011i		−	1.00853i
$544$		0			0
$545$	−0.705654	−	22.5007i	−0.0302269	−	0.963825i
$546$		0			0
$547$		−	38.5078i		−	1.64648i	−0.567697	−	0.823238i	$-0.692165\pi$
$547$		−	38.5078i		−	1.64648i	0.567697	−	0.823238i	$-0.307835\pi$
$548$		0			0
$549$	8.86547			0.378369
$550$		0			0
$551$	−37.2313			−1.58611
$552$		0			0
$553$		−	18.7896i		−	0.799014i
$554$		0			0
$555$	0.751208	+	23.9533i	0.0318870	+	1.01676i
$556$		0			0
$557$		−	15.9714i		−	0.676729i	−0.941015	−	0.338365i	$-0.890126\pi$
$557$		−	15.9714i		−	0.676729i	0.941015	−	0.338365i	$-0.109874\pi$
$558$		0			0
$559$	2.03000			0.0858600
$560$		0			0
$561$		0			0
$562$		0			0
$563$			0.570350i			0.0240374i	0.999928	+	0.0120187i	$0.00382576\pi$
$563$			0.570350i			0.0240374i	−0.999928	+	0.0120187i	$0.996174\pi$
$564$		0			0
$565$	41.9697	−	1.31623i	1.76568	−	0.0553742i
$566$		0			0
$567$			0.981048i			0.0412001i
$568$		0			0
$569$	−46.9936			−1.97007			−0.985037	−	0.172343i	$-0.944866\pi$
$569$	−46.9936			−1.97007			−0.985037	+	0.172343i	$0.944866\pi$
$570$		0			0
$571$	−39.8308			−1.66687			−0.833434	−	0.552619i	$-0.813628\pi$
$571$	−39.8308			−1.66687			−0.833434	+	0.552619i	$0.813628\pi$
$572$		0			0
$573$			11.5312i			0.481724i
$574$		0			0
$575$	0.917472	+	14.6130i	0.0382612	+	0.609405i
$576$		0			0
$577$		−	34.3595i		−	1.43040i	−0.698918	−	0.715202i	$-0.746335\pi$
$577$		−	34.3595i		−	1.43040i	0.698918	−	0.715202i	$-0.253665\pi$
$578$		0			0
$579$	6.66658			0.277054
$580$		0			0
$581$	−37.1704			−1.54209
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.14996	+	0.0360644i	−0.0475451	+	0.00149108i
$586$		0			0
$587$		−	32.4360i		−	1.33878i	−0.742912	−	0.669389i	$-0.766556\pi$
$587$		−	32.4360i		−	1.33878i	0.742912	−	0.669389i	$-0.233444\pi$
$588$		0			0
$589$	−1.43648			−0.0591892
$590$		0			0
$591$	−9.51869			−0.391547
$592$		0			0
$593$		−	19.9594i		−	0.819635i	−0.912168	−	0.409817i	$-0.865592\pi$
$593$		−	19.9594i		−	0.819635i	0.912168	−	0.409817i	$-0.134408\pi$
$594$		0			0
$595$	0.939595	+	29.9602i	0.0385196	+	1.22825i
$596$		0			0
$597$			15.2539i			0.624300i
$598$		0			0
$599$	−11.7718			−0.480982			−0.240491	−	0.970651i	$-0.577309\pi$
$599$	−11.7718			−0.480982			−0.240491	+	0.970651i	$0.577309\pi$
$600$		0			0
$601$	5.55080			0.226422			0.113211	−	0.993571i	$-0.463886\pi$
$601$	5.55080			0.226422			0.113211	+	0.993571i	$0.463886\pi$
$602$		0			0
$603$		−	11.7241i		−	0.477443i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			14.2128i			0.576882i	0.957498	+	0.288441i	$0.0931369\pi$
$607$			14.2128i			0.576882i	−0.957498	+	0.288441i	$0.906863\pi$
$608$		0			0
$609$	−15.2535			−0.618102
$610$		0			0
$611$	0.303809			0.0122908
$612$		0			0
$613$		−	43.7273i		−	1.76613i	−0.469250	−	0.883065i	$-0.655476\pi$
$613$		−	43.7273i		−	1.76613i	0.469250	−	0.883065i	$-0.344524\pi$
$614$		0			0
$615$	5.49372	−	0.172291i	0.221528	−	0.00694743i
$616$		0			0
$617$			0.0295713i			0.00119050i	1.00000		0.000595249i	$0.000189474\pi$
$617$			0.0295713i			0.00119050i	−1.00000		0.000595249i	$0.999811\pi$
$618$		0			0
$619$	9.45479			0.380020			0.190010	−	0.981782i	$-0.439148\pi$
$619$	9.45479			0.380020			0.190010	+	0.981782i	$0.439148\pi$
$620$		0			0
$621$	14.9800			0.601128
$622$		0			0
$623$			12.9706i			0.519655i
$624$		0			0
$625$	24.8037	−	3.12690i	0.992147	−	0.125076i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−61.0420			−2.43390
$630$		0			0
$631$	2.30973			0.0919489			0.0459744	−	0.998943i	$-0.485361\pi$
$631$	2.30973			0.0919489			0.0459744	+	0.998943i	$0.485361\pi$
$632$		0			0
$633$		−	12.9542i		−	0.514882i
$634$		0			0
$635$	42.6664	−	1.33808i	1.69317	−	0.0531000i
$636$		0			0
$637$		−	0.504216i		−	0.0199778i
$638$		0			0
$639$	9.76364			0.386244
$640$		0			0
$641$	1.58431			0.0625764			0.0312882	−	0.999510i	$-0.490039\pi$
$641$	1.58431			0.0625764			0.0312882	+	0.999510i	$0.490039\pi$
$642$		0			0
$643$		−	11.2380i		−	0.443182i	−0.975140	−	0.221591i	$-0.928875\pi$
$643$		−	11.2380i		−	0.443182i	0.975140	−	0.221591i	$-0.0711249\pi$
$644$		0			0
$645$	0.551717	+	17.5922i	0.0217238	+	0.692694i
$646$		0			0
$647$			11.4256i			0.449186i	0.974453	+	0.224593i	$0.0721052\pi$
$647$			11.4256i			0.449186i	−0.974453	+	0.224593i	$0.927895\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−0.588519			−0.0230659
$652$		0			0
$653$		−	14.7783i		−	0.578320i	−0.957281	−	0.289160i	$-0.906624\pi$
$653$		−	14.7783i		−	0.578320i	0.957281	−	0.289160i	$-0.0933759\pi$
$654$		0			0
$655$	−0.125062	−	3.98776i	−0.00488657	−	0.155815i
$656$		0			0
$657$			15.6455i			0.610390i
$658$		0			0
$659$	14.3223			0.557918			0.278959	−	0.960303i	$-0.410011\pi$
$659$	14.3223			0.557918			0.278959	+	0.960303i	$0.410011\pi$
$660$		0			0
$661$	−17.1886			−0.668561			−0.334280	−	0.942474i	$-0.608493\pi$
$661$	−17.1886			−0.668561			−0.334280	+	0.942474i	$0.608493\pi$
$662$		0			0
$663$			1.65291i			0.0641939i
$664$		0			0
$665$	−29.0535	+	0.911159i	−1.12665	+	0.0353332i
$666$		0			0
$667$		−	18.9782i		−	0.734838i
$668$		0			0
$669$	12.6814			0.490291
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	34.4696i		−	1.32871i	−0.747419	−	0.664353i	$-0.768707\pi$
$673$		−	34.4696i		−	1.32871i	0.747419	−	0.664353i	$-0.231293\pi$
$674$		0			0
$675$	−1.60272	−	25.5273i	−0.0616886	−	0.982545i
$676$		0			0
$677$		−	1.93721i		−	0.0744532i	−0.999307	−	0.0372266i	$-0.988148\pi$
$677$		−	1.93721i		−	0.0744532i	0.999307	−	0.0372266i	$-0.0118523\pi$
$678$		0			0
$679$	−10.2120			−0.391900
$680$		0			0
$681$	−8.01463			−0.307121
$682$		0			0
$683$			19.1445i			0.732543i	0.930508	+	0.366271i	$0.119366\pi$
$683$			19.1445i			0.732543i	−0.930508	+	0.366271i	$0.880634\pi$
$684$		0			0
$685$	−16.9233	+	0.530738i	−0.646606	+	0.0202785i
$686$		0			0
$687$		−	18.9629i		−	0.723481i
$688$		0			0
$689$	−2.73644			−0.104250
$690$		0			0
$691$	21.8944			0.832903			0.416451	−	0.909158i	$-0.363274\pi$
$691$	21.8944			0.832903			0.416451	+	0.909158i	$0.363274\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	0.775662	+	24.7330i	0.0294225	+	0.938177i
$696$		0			0
$697$			14.0001i			0.530290i
$698$		0			0
$699$	5.14609			0.194643
$700$		0			0
$701$	27.5159			1.03926			0.519630	−	0.854391i	$-0.326070\pi$
$701$	27.5159			1.03926			0.519630	+	0.854391i	$0.326070\pi$
$702$		0			0
$703$		−	59.1946i		−	2.23257i
$704$		0			0
$705$	0.0825696	+	2.63284i	0.00310975	+	0.0991586i
$706$		0			0
$707$			21.2088i			0.797639i
$708$		0			0
$709$	28.2730			1.06181			0.530907	−	0.847430i	$-0.321851\pi$
$709$	28.2730			1.06181			0.530907	+	0.847430i	$0.321851\pi$
$710$		0			0
$711$	−15.9273			−0.597322
$712$		0			0
$713$		−	0.732229i		−	0.0274222i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	17.6663i		−	0.659762i
$718$		0			0
$719$	10.9532			0.408484			0.204242	−	0.978920i	$-0.434527\pi$
$719$	10.9532			0.408484			0.204242	+	0.978920i	$0.434527\pi$
$720$		0			0
$721$	−14.8783			−0.554098
$722$		0			0
$723$		−	6.99960i		−	0.260318i
$724$		0			0
$725$	−32.3404	+	2.03048i	−1.20109	+	0.0754100i
$726$		0			0
$727$		−	26.1233i		−	0.968860i	−0.874830	−	0.484430i	$-0.839027\pi$
$727$		−	26.1233i		−	0.968860i	0.874830	−	0.484430i	$-0.160973\pi$
$728$		0			0
$729$	−15.1308			−0.560401
$730$		0			0
$731$	−44.8317			−1.65816
$732$		0			0
$733$			2.75923i			0.101914i	0.998701	+	0.0509572i	$0.0162272\pi$
$733$			2.75923i			0.101914i	−0.998701	+	0.0509572i	$0.983773\pi$
$734$		0			0
$735$	4.36959	−	0.137036i	0.161175	−	0.00505466i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−19.3095			−0.710312			−0.355156	−	0.934807i	$-0.615572\pi$
$739$	−19.3095			−0.710312			−0.355156	+	0.934807i	$0.615572\pi$
$740$		0			0
$741$	−1.60289			−0.0588837
$742$		0			0
$743$			3.00740i			0.110331i	0.998477	+	0.0551655i	$0.0175686\pi$
$743$			3.00740i			0.110331i	−0.998477	+	0.0551655i	$0.982431\pi$
$744$		0			0
$745$	0.817269	+	26.0597i	0.0299424	+	0.954755i
$746$		0			0
$747$			31.5082i			1.15283i
$748$		0			0
$749$	15.7037			0.573799
$750$		0			0
$751$	9.26094			0.337936			0.168968	−	0.985622i	$-0.445956\pi$
$751$	9.26094			0.337936			0.168968	+	0.985622i	$0.445956\pi$
$752$		0			0
$753$			20.5605i			0.749265i
$754$		0			0
$755$	−0.0170056	−	0.542245i	−0.000618896	−	0.0197343i
$756$		0			0
$757$			42.0908i			1.52982i	0.644140	+	0.764908i	$0.277216\pi$
$757$			42.0908i			1.52982i	−0.644140	+	0.764908i	$0.722784\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−45.1465			−1.63656			−0.818280	−	0.574820i	$-0.805072\pi$
$761$	−45.1465			−1.63656			−0.818280	+	0.574820i	$0.805072\pi$
$762$		0			0
$763$			22.7811i			0.824730i
$764$		0			0
$765$	25.3964	−	0.796466i	0.918209	−	0.0287963i
$766$		0			0
$767$			3.05996i			0.110489i
$768$		0			0
$769$	−45.9031			−1.65531			−0.827655	−	0.561237i	$-0.810325\pi$
$769$	−45.9031			−1.65531			−0.827655	+	0.561237i	$0.810325\pi$
$770$		0			0
$771$	−6.65164			−0.239553
$772$		0			0
$773$		−	8.49306i		−	0.305474i	−0.988267	−	0.152737i	$-0.951191\pi$
$773$		−	8.49306i		−	0.305474i	0.988267	−	0.152737i	$-0.0488087\pi$
$774$		0			0
$775$	−1.24778	+	0.0783412i	−0.0448216	+	0.00281410i
$776$		0			0
$777$		−	24.2517i		−	0.870026i
$778$		0			0
$779$	−13.5764			−0.486424
$780$		0			0
$781$		0			0
$782$		0			0
$783$			33.1527i			1.18478i
$784$		0			0
$785$	−6.59738	+	0.206903i	−0.235470	+	0.00738468i
$786$		0			0
$787$			16.1729i			0.576502i	0.957555	+	0.288251i	$0.0930738\pi$
$787$			16.1729i			0.576502i	−0.957555	+	0.288251i	$0.906926\pi$
$788$		0			0
$789$	−15.2293			−0.542177
$790$		0			0
$791$	−42.4927			−1.51087
$792$		0			0
$793$		−	1.23983i		−	0.0440277i
$794$		0			0
$795$	−0.743713	−	23.7143i	−0.0263768	−	0.841060i
$796$		0			0
$797$			6.40651i			0.226930i	0.993542	+	0.113465i	$0.0361950\pi$
$797$			6.40651i			0.226930i	−0.993542	+	0.113465i	$0.963805\pi$
$798$		0			0
$799$	−6.70947			−0.237364
$800$		0			0
$801$	10.9948			0.388481
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−0.464452	−	14.8097i	−0.0163698	−	0.521972i
$806$		0			0
$807$			20.6712i			0.727662i
$808$		0			0
$809$	−47.2818			−1.66234			−0.831169	−	0.556020i	$-0.812328\pi$
$809$	−47.2818			−1.66234			−0.831169	+	0.556020i	$0.812328\pi$
$810$		0			0
$811$	−22.6707			−0.796077			−0.398038	−	0.917369i	$-0.630309\pi$
$811$	−22.6707			−0.796077			−0.398038	+	0.917369i	$0.630309\pi$
$812$		0			0
$813$			19.9837i			0.700858i
$814$		0			0
$815$	−23.2323	+	0.728596i	−0.813790	+	0.0255216i
$816$		0			0
$817$		−	43.4749i		−	1.52099i
$818$		0			0
$819$	1.16429			0.0406836
$820$		0			0
$821$	−20.4587			−0.714012			−0.357006	−	0.934102i	$-0.616202\pi$
$821$	−20.4587			−0.714012			−0.357006	+	0.934102i	$0.616202\pi$
$822$		0			0
$823$			26.9963i			0.941032i	0.882391	+	0.470516i	$0.155932\pi$
$823$			26.9963i			0.941032i	−0.882391	+	0.470516i	$0.844068\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			18.5457i			0.644896i	0.946587	+	0.322448i	$0.104506\pi$
$827$			18.5457i			0.644896i	−0.946587	+	0.322448i	$0.895494\pi$
$828$		0			0
$829$	−6.99533			−0.242958			−0.121479	−	0.992594i	$-0.538764\pi$
$829$	−6.99533			−0.242958			−0.121479	+	0.992594i	$0.538764\pi$
$830$		0			0
$831$	24.7910			0.859990
$832$		0			0
$833$			11.1354i			0.385817i
$834$		0			0
$835$	22.0987	−	0.693046i	0.764757	−	0.0239838i
$836$		0			0
$837$			1.27912i			0.0442128i
$838$		0			0
$839$	−13.6764			−0.472163			−0.236081	−	0.971733i	$-0.575863\pi$
$839$	−13.6764			−0.472163			−0.236081	+	0.971733i	$0.575863\pi$
$840$		0			0
$841$	13.0010			0.448310
$842$		0			0
$843$			18.6481i			0.642276i
$844$		0			0
$845$	−0.906149	−	28.8938i	−0.0311725	−	0.993976i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	21.8705			0.750595
$850$		0			0
$851$	30.1737			1.03434
$852$		0			0
$853$		−	3.31642i		−	0.113552i	−0.998387	−	0.0567760i	$-0.981918\pi$
$853$		−	3.31642i		−	0.113552i	0.998387	−	0.0567760i	$-0.0180821\pi$
$854$		0			0
$855$	0.772362	+	24.6278i	0.0264142	+	0.842253i
$856$		0			0
$857$			29.2111i			0.997831i	0.866651	+	0.498915i	$0.166268\pi$
$857$			29.2111i			0.997831i	−0.866651	+	0.498915i	$0.833732\pi$
$858$		0			0
$859$	−32.2591			−1.10067			−0.550334	−	0.834945i	$-0.685499\pi$
$859$	−32.2591			−1.10067			−0.550334	+	0.834945i	$0.685499\pi$
$860$		0			0
$861$	−5.56217			−0.189558
$862$		0			0
$863$		−	27.6382i		−	0.940816i	−0.882449	−	0.470408i	$-0.844107\pi$
$863$		−	27.6382i		−	0.940816i	0.882449	−	0.470408i	$-0.155893\pi$
$864$		0			0
$865$	35.2933	−	1.10685i	1.20001	−	0.0376339i
$866$		0			0
$867$		−	18.8216i		−	0.639214i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−1.63961			−0.0555561
$872$		0			0
$873$			8.65640i			0.292975i
$874$		0			0
$875$	−25.1872	+	2.37595i	−0.851484	+	0.0803219i
$876$		0			0
$877$		−	0.854494i		−	0.0288542i	−0.999896	−	0.0144271i	$-0.995408\pi$
$877$		−	0.854494i		−	0.0288542i	0.999896	−	0.0144271i	$-0.00459245\pi$
$878$		0			0
$879$	22.1216			0.746145
$880$		0			0
$881$	−7.22437			−0.243395			−0.121698	−	0.992567i	$-0.538834\pi$
$881$	−7.22437			−0.243395			−0.121698	+	0.992567i	$0.538834\pi$
$882$		0			0
$883$		−	11.3730i		−	0.382733i	−0.981519	−	0.191366i	$-0.938708\pi$
$883$		−	11.3730i		−	0.382733i	0.981519	−	0.191366i	$-0.0612919\pi$
$884$		0			0
$885$	−26.5180	+	0.831640i	−0.891391	+	0.0279553i
$886$		0			0
$887$		−	7.65420i		−	0.257003i	−0.991709	−	0.128502i	$-0.958983\pi$
$887$		−	7.65420i		−	0.257003i	0.991709	−	0.128502i	$-0.0410167\pi$
$888$		0			0
$889$	−43.1980			−1.44882
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	6.50642i		−	0.217729i
$894$		0			0
$895$	0.411154	+	13.1102i	0.0137434	+	0.438226i
$896$		0			0
$897$		−	0.817053i		−	0.0272806i
$898$		0			0
$899$	1.62051			0.0540471
$900$		0			0
$901$	60.4330			2.01332
$902$		0			0
$903$		−	17.8114i		−	0.592727i
$904$		0			0
$905$	1.58367	+	50.4975i	0.0526430	+	1.67859i
$906$		0			0
$907$			54.3238i			1.80379i	0.431952	+	0.901897i	$0.357825\pi$
$907$			54.3238i			1.80379i	−0.431952	+	0.901897i	$0.642175\pi$
$908$		0			0
$909$	17.9781			0.596295
$910$		0			0
$911$	29.8776			0.989888			0.494944	−	0.868925i	$-0.335189\pi$
$911$	29.8776			0.989888			0.494944	+	0.868925i	$0.335189\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	10.7445	−	0.336963i	0.355203	−	0.0111396i
$916$		0			0
$917$			4.03745i			0.133328i
$918$		0			0
$919$	−17.6787			−0.583168			−0.291584	−	0.956545i	$-0.594182\pi$
$919$	−17.6787			−0.583168			−0.291584	+	0.956545i	$0.594182\pi$
$920$		0			0
$921$	33.9940			1.12014
$922$		0			0
$923$		−	1.36544i		−	0.0449440i
$924$		0			0
$925$	−3.22829	−	51.4185i	−0.106145	−	1.69063i
$926$		0			0
$927$			12.6119i			0.414229i
$928$		0			0
$929$	0.905778			0.0297176			0.0148588	−	0.999890i	$-0.495270\pi$
$929$	0.905778			0.0297176			0.0148588	+	0.999890i	$0.495270\pi$
$930$		0			0
$931$	−10.7984			−0.353902
$932$		0			0
$933$		−	19.6855i		−	0.644473i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			12.8792i			0.420744i	0.977621	+	0.210372i	$0.0674675\pi$
$937$			12.8792i			0.420744i	−0.977621	+	0.210372i	$0.932532\pi$
$938$		0			0
$939$	12.5134			0.408359
$940$		0			0
$941$	30.7949			1.00389			0.501943	−	0.864901i	$-0.332619\pi$
$941$	30.7949			1.00389			0.501943	+	0.864901i	$0.332619\pi$
$942$		0			0
$943$		−	6.92038i		−	0.225359i
$944$		0			0
$945$	0.811343	+	25.8708i	0.0263930	+	0.841576i
$946$		0			0
$947$			27.4841i			0.893112i	0.894756	+	0.446556i	$0.147350\pi$
$947$			27.4841i			0.893112i	−0.894756	+	0.446556i	$0.852650\pi$
$948$		0			0
$949$	2.18802			0.0710260
$950$		0			0
$951$	10.6101			0.344058
$952$		0			0
$953$			13.7716i			0.446105i	0.974806	+	0.223052i	$0.0716021\pi$
$953$			13.7716i			0.446105i	−0.974806	+	0.223052i	$0.928398\pi$
$954$		0			0
$955$	−0.777057	−	24.7775i	−0.0251450	−	0.801782i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	17.1342			0.553291
$960$		0			0
$961$	−30.9375			−0.997983
$962$		0			0
$963$		−	13.3115i		−	0.428958i
$964$		0			0
$965$	−14.3247	+	0.449242i	−0.461128	+	0.0144616i
$966$		0			0
$967$		−	45.5732i		−	1.46554i	−0.680478	−	0.732768i	$-0.738228\pi$
$967$		−	45.5732i		−	1.46554i	0.680478	−	0.732768i	$-0.261772\pi$
$968$		0			0
$969$	35.3991			1.13718
$970$		0			0
$971$	45.6069			1.46360			0.731798	−	0.681522i	$-0.238682\pi$
$971$	45.6069			1.46360			0.731798	+	0.681522i	$0.238682\pi$
$972$		0			0
$973$		−	25.0412i		−	0.802784i
$974$		0			0
$975$	−1.39233	+	0.0874166i	−0.0445902	+	0.00279957i
$976$		0			0
$977$		−	16.9438i		−	0.542080i	−0.962568	−	0.271040i	$-0.912632\pi$
$977$		−	16.9438i		−	0.542080i	0.962568	−	0.271040i	$-0.0873676\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	19.3108			0.616547
$982$		0			0
$983$			16.0075i			0.510561i	0.966867	+	0.255280i	$0.0821677\pi$
$983$			16.0075i			0.510561i	−0.966867	+	0.255280i	$0.917832\pi$
$984$		0			0
$985$	20.4531	−	0.641438i	0.651691	−	0.0204379i
$986$		0			0
$987$		−	2.66565i		−	0.0848485i
$988$		0			0
$989$	22.1608			0.704671
$990$		0			0
$991$	57.8601			1.83799			0.918993	−	0.394274i	$-0.129004\pi$
$991$	57.8601			1.83799			0.918993	+	0.394274i	$0.129004\pi$
$992$		0			0
$993$			12.3715i			0.392599i
$994$		0			0
$995$	−1.02792	−	32.7765i	−0.0325871	−	1.03908i
$996$		0			0
$997$			46.5621i			1.47464i	0.675545	+	0.737319i	$0.263908\pi$
$997$			46.5621i			1.47464i	−0.675545	+	0.737319i	$0.736092\pi$
$998$		0			0
$999$	−52.7099			−1.66767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2420.2.b.h.969.8		12
5.4	even	2	inner	2420.2.b.h.969.5		12
11.2	odd	10		220.2.t.a.169.4	yes	24
11.6	odd	10		220.2.t.a.69.3	✓	24
11.10	odd	2		2420.2.b.i.969.8		12
44.35	even	10		880.2.cd.d.609.3		24
44.39	even	10		880.2.cd.d.289.4		24
55.2	even	20		1100.2.n.f.301.4		24
55.13	even	20		1100.2.n.f.301.3		24
55.17	even	20		1100.2.n.f.201.4		24
55.24	odd	10		220.2.t.a.169.3	yes	24
55.28	even	20		1100.2.n.f.201.3		24
55.39	odd	10		220.2.t.a.69.4	yes	24
55.54	odd	2		2420.2.b.i.969.5		12
220.39	even	10		880.2.cd.d.289.3		24
220.79	even	10		880.2.cd.d.609.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.t.a.69.3	✓	24	11.6	odd	10
220.2.t.a.69.4	yes	24	55.39	odd	10
220.2.t.a.169.3	yes	24	55.24	odd	10
220.2.t.a.169.4	yes	24	11.2	odd	10
880.2.cd.d.289.3		24	220.39	even	10
880.2.cd.d.289.4		24	44.39	even	10
880.2.cd.d.609.3		24	44.35	even	10
880.2.cd.d.609.4		24	220.79	even	10
1100.2.n.f.201.3		24	55.28	even	20
1100.2.n.f.201.4		24	55.17	even	20
1100.2.n.f.301.3		24	55.13	even	20
1100.2.n.f.301.4		24	55.2	even	20
2420.2.b.h.969.5		12	5.4	even	2	inner
2420.2.b.h.969.8		12	1.1	even	1	trivial
2420.2.b.i.969.5		12	55.54	odd	2
2420.2.b.i.969.8		12	11.10	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:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2420.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.04013i q^{3} +(-0.0700917 - 2.23497i) q^{5} +2.26282i q^{7} +1.91812 q^{9} -0.268248i q^{13} +(2.32467 - 0.0729048i) q^{15} +5.92413i q^{17} -5.74484 q^{19} -2.35363 q^{21} -2.92836i q^{23} +(-4.99017 + 0.313306i) q^{25} +5.11551i q^{27} +6.48082 q^{29} +0.250047 q^{31} +(5.05732 - 0.158605i) q^{35} +10.3040i q^{37} +0.279014 q^{39} +2.36323 q^{41} +7.56763i q^{43} +(-0.134444 - 4.28694i) q^{45} +1.13257i q^{47} +1.87966 q^{49} -6.16189 q^{51} -10.2012i q^{53} -5.97541i q^{57} -11.4072 q^{59} +4.62196 q^{61} +4.34036i q^{63} +(-0.599526 + 0.0188020i) q^{65} -6.11229i q^{67} +3.04589 q^{69} +5.09021 q^{71} +8.15669i q^{73} +(-0.325880 - 5.19045i) q^{75} -8.30362 q^{79} +0.433552 q^{81} +16.4266i q^{83} +(13.2402 - 0.415232i) q^{85} +6.74092i q^{87} +5.73205 q^{89} +0.606996 q^{91} +0.260083i q^{93} +(0.402666 + 12.8396i) q^{95} +4.51296i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{5} - 18 q^{9} + 2 q^{15} - 14 q^{19} + 14 q^{21} - 9 q^{25} - 22 q^{29} + 12 q^{31} + 4 q^{35} - 26 q^{39} + 40 q^{41} - 20 q^{45} - 8 q^{49} + 26 q^{51} + 16 q^{59} + 30 q^{61} + 11 q^{65}+ \cdots + 14 q^{95}+O(q^{100}) \) 12 * q + 3 * q^5 - 18 * q^9 + 2 * q^15 - 14 * q^19 + 14 * q^21 - 9 * q^25 - 22 * q^29 + 12 * q^31 + 4 * q^35 - 26 * q^39 + 40 * q^41 - 20 * q^45 - 8 * q^49 + 26 * q^51 + 16 * q^59 + 30 * q^61 + 11 * q^65 + 36 * q^69 - 24 * q^71 + 39 * q^75 + 2 * q^79 - 12 * q^81 + 21 * q^85 + 36 * q^89 - 40 * q^91 + 14 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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