LMFDB - Embedded newform 2100.2.d.j.1301.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,2,Mod(1301,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1301");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2100.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$16.7685844245$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.121550625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 $ x^8 - x^7 - 4x^6 - 9x^5 + 23x^4 + 18x^3 - 16x^2 + 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1301.2
Root			$0.553538 - 0.676408i$ of defining polynomial
Character	$\chi$	$=$	2100.1301
Dual form			2100.2.d.j.1301.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70466 + 0.306808i) q^{3} -2.64575i q^{7} +(2.81174 - 1.04601i) q^{9} +O(q^{10})$	$q+(-1.70466 + 0.306808i) q^{3} -2.64575i q^{7} +(2.81174 - 1.04601i) q^{9} -5.55612i q^{11} +7.13235i q^{13} +5.75583 q^{17} +(0.811738 + 4.51011i) q^{21} +(-4.47214 + 2.64575i) q^{27} -4.83619i q^{29} +(1.70466 + 9.47129i) q^{33} +(-2.18826 - 12.1582i) q^{39} -1.28369 q^{47} -7.00000 q^{49} +(-9.81174 + 1.76593i) q^{51} +(-2.76748 - 7.43916i) q^{63} -11.8322i q^{71} -10.5830i q^{73} -14.7001 q^{77} +14.8704 q^{79} +(6.81174 - 5.88220i) q^{81} -8.94427 q^{83} +(1.48378 + 8.24406i) q^{87} +18.8704 q^{91} -3.45065i q^{97} +(-5.81174 - 15.6223i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{9}+O(q^{10}) $ 8 * q + 2 * q^9	$ 8 q + 2 q^{9} - 14 q^{21} - 38 q^{39} - 56 q^{49} - 58 q^{51} - 4 q^{79} + 34 q^{81} + 28 q^{91} - 26 q^{99}+O(q^{100}) $ 8 * q + 2 * q^9 - 14 * q^21 - 38 * q^39 - 56 * q^49 - 58 * q^51 - 4 * q^79 + 34 * q^81 + 28 * q^91 - 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$-1$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.70466	+	0.306808i	−0.984186	+	0.177136i
$3$	−1.70466	+	0.306808i	−0.984186	+	0.177136i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	2.64575i		−	1.00000i
$7$		−	2.64575i		−	1.00000i
$8$		0			0
$9$	2.81174	−	1.04601i	0.937246	−	0.348669i
$10$		0			0
$11$		−	5.55612i		−	1.67523i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$		−	5.55612i		−	1.67523i	0.546259	−	0.837616i	$-0.316051\pi$
$12$		0			0
$13$			7.13235i			1.97816i	0.147386	+	0.989079i	$0.452914\pi$
$13$			7.13235i			1.97816i	−0.147386	+	0.989079i	$0.547086\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.75583			1.39599			0.697997	−	0.716101i	$-0.254075\pi$
$17$	5.75583			1.39599			0.697997	+	0.716101i	$0.254075\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	0.811738	+	4.51011i	0.177136	+	0.984186i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.47214	+	2.64575i	−0.860663	+	0.509175i
$28$		0			0
$29$		−	4.83619i		−	0.898058i	−0.893517	−	0.449029i	$-0.851770\pi$
$29$		−	4.83619i		−	0.898058i	0.893517	−	0.449029i	$-0.148230\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	1.70466	+	9.47129i	0.296743	+	1.64874i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$	−2.18826	−	12.1582i	−0.350402	−	1.94688i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.28369			−0.187246			−0.0936230	−	0.995608i	$-0.529845\pi$
$47$	−1.28369			−0.187246			−0.0936230	+	0.995608i	$0.529845\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$	−9.81174	+	1.76593i	−1.37392	+	0.247280i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	−2.76748	−	7.43916i	−0.348669	−	0.937246i
$64$		0			0
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.8322i		−	1.40422i	−0.712069	−	0.702109i	$-0.752242\pi$
$71$		−	11.8322i		−	1.40422i	0.712069	−	0.702109i	$-0.247758\pi$
$72$		0			0
$73$		−	10.5830i		−	1.23865i	−0.785136	−	0.619324i	$-0.787407\pi$
$73$		−	10.5830i		−	1.23865i	0.785136	−	0.619324i	$-0.212593\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−14.7001			−1.67523
$78$		0			0
$79$	14.8704			1.67305			0.836527	−	0.547926i	$-0.184582\pi$
$79$	14.8704			1.67305			0.836527	+	0.547926i	$0.184582\pi$
$80$		0			0
$81$	6.81174	−	5.88220i	0.756860	−	0.653577i
$82$		0			0
$83$	−8.94427			−0.981761			−0.490881	−	0.871227i	$-0.663325\pi$
$83$	−8.94427			−0.981761			−0.490881	+	0.871227i	$0.663325\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.48378	+	8.24406i	0.159078	+	0.883856i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	18.8704			1.97816
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	3.45065i		−	0.350361i	−0.984536	−	0.175180i	$-0.943949\pi$
$97$		−	3.45065i		−	0.350361i	0.984536	−	0.175180i	$-0.0560509\pi$
$98$		0			0
$99$	−5.81174	−	15.6223i	−0.584102	−	1.57010i
$100$		0			0
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		−	16.1055i		−	1.58693i	−0.608618	−	0.793463i	$-0.708276\pi$
$103$		−	16.1055i		−	1.58693i	0.608618	−	0.793463i	$-0.291724\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	20.8704			1.99902			0.999512	−	0.0312328i	$-0.00994332\pi$
$109$	20.8704			1.99902			0.999512	+	0.0312328i	$0.00994332\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.46049	+	20.0543i	0.689723	+	1.85402i
$118$		0			0
$119$		−	15.2285i		−	1.39599i
$120$		0			0
$121$	−19.8704			−1.80640
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	2.18826	−	0.393847i	0.184285	−	0.0331679i
$142$		0			0
$143$	39.6282			3.31387
$144$		0			0
$145$		0			0
$146$		0			0
$147$	11.9326	−	2.14766i	0.984186	−	0.177136i
$148$		0			0
$149$		−	23.6643i		−	1.93866i	−0.245770	−	0.969328i	$-0.579041\pi$
$149$		−	23.6643i		−	1.93866i	0.245770	−	0.969328i	$-0.420959\pi$
$150$		0			0
$151$	6.87043			0.559107			0.279554	−	0.960130i	$-0.409814\pi$
$151$	6.87043			0.559107			0.279554	+	0.960130i	$0.409814\pi$
$152$		0			0
$153$	16.1839	−	6.02064i	1.30839	−	0.486740i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	21.1660i		−	1.68923i	−0.535373	−	0.844616i	$-0.679829\pi$
$157$		−	21.1660i		−	1.68923i	0.535373	−	0.844616i	$-0.320171\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	19.1722			1.48359			0.741796	−	0.670625i	$-0.233974\pi$
$167$	19.1722			1.48359			0.741796	+	0.670625i	$0.233974\pi$
$168$		0			0
$169$	−37.8704			−2.91311
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−26.2118			−1.99284			−0.996422	−	0.0845218i	$-0.973064\pi$
$173$	−26.2118			−1.99284			−0.996422	+	0.0845218i	$0.973064\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$			11.8322i			0.884377i	0.896922	+	0.442189i	$0.145798\pi$
$179$			11.8322i			0.884377i	−0.896922	+	0.442189i	$0.854202\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	31.9801i		−	2.33861i
$188$		0			0
$189$	7.00000	+	11.8322i	0.509175	+	0.860663i
$190$		0			0
$191$		−	26.3407i		−	1.90595i	−0.303052	−	0.952974i	$-0.598006\pi$
$191$		−	26.3407i		−	1.90595i	0.303052	−	0.952974i	$-0.401994\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.7954			−0.898058
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−26.8704			−1.84984			−0.924918	−	0.380166i	$-0.875867\pi$
$211$	−26.8704			−1.84984			−0.924918	+	0.380166i	$0.875867\pi$
$212$		0			0
$213$	3.63020	+	20.1698i	0.248737	+	1.38201i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	3.24695	+	18.0404i	0.219409	+	1.21906i
$220$		0			0
$221$			41.0526i			2.76150i
$222$		0			0
$223$		−	8.74216i		−	0.585418i	−0.956202	−	0.292709i	$-0.905443\pi$
$223$		−	8.74216i		−	0.585418i	0.956202	−	0.292709i	$-0.0945567\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.66058			−0.508450			−0.254225	−	0.967145i	$-0.581820\pi$
$227$	−7.66058			−0.508450			−0.254225	+	0.967145i	$0.581820\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$	25.0587	−	4.51011i	1.64874	−	0.296743i
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−25.3490	+	4.56237i	−1.64660	+	0.296358i
$238$		0			0
$239$		−	6.99597i		−	0.452532i	−0.974066	−	0.226266i	$-0.927348\pi$
$239$		−	6.99597i		−	0.452532i	0.974066	−	0.226266i	$-0.0726518\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−9.80700	+	12.1170i	−0.629119	+	0.777309i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	15.2470	−	2.74417i	0.966236	−	0.173905i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−4.47214			−0.278964			−0.139482	−	0.990225i	$-0.544544\pi$
$257$	−4.47214			−0.278964			−0.139482	+	0.990225i	$0.544544\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−5.05869	−	13.5981i	−0.313125	−	0.841701i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	−32.1677	+	5.78960i	−1.94688	+	0.350402i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			25.6208i			1.52841i	0.644974	+	0.764204i	$0.276868\pi$
$281$			25.6208i			1.52841i	−0.644974	+	0.764204i	$0.723132\pi$
$282$		0			0
$283$			30.3703i			1.80532i	0.430350	+	0.902662i	$0.358390\pi$
$283$			30.3703i			1.80532i	−0.430350	+	0.902662i	$0.641610\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.1296			0.948798
$290$		0			0
$291$	1.05869	+	5.88220i	0.0620614	+	0.344820i
$292$		0			0
$293$	8.32322			0.486248			0.243124	−	0.969995i	$-0.421828\pi$
$293$	8.32322			0.486248			0.243124	+	0.969995i	$0.421828\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	14.7001	+	24.8477i	0.852986	+	1.44181i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	23.0069i		−	1.31307i	−0.754295	−	0.656535i	$-0.772021\pi$
$307$		−	23.0069i		−	1.31307i	0.754295	−	0.656535i	$-0.227979\pi$
$308$		0			0
$309$	4.94131	+	27.4545i	0.281101	+	1.56183i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	31.9801i		−	1.80762i	−0.427934	−	0.903810i	$-0.640759\pi$
$313$		−	31.9801i		−	1.80762i	0.427934	−	0.903810i	$-0.359241\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$	−26.8704			−1.50446
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−35.5770	+	6.40321i	−1.96741	+	0.354099i
$328$		0			0
$329$			3.39633i			0.187246i
$330$		0			0
$331$	8.00000			0.439720			0.219860	−	0.975531i	$-0.429440\pi$
$331$	8.00000			0.439720			0.219860	+	0.975531i	$0.429440\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			18.5203i			1.00000i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	−18.8704	−	31.8968i	−1.00723	−	1.70253i
$352$		0			0
$353$	35.1560			1.87117			0.935583	−	0.353106i	$-0.114874\pi$
$353$	35.1560			1.87117			0.935583	+	0.353106i	$0.114874\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	4.67222	+	25.9594i	0.247280	+	1.37392i
$358$		0			0
$359$			11.8322i			0.624477i	0.950004	+	0.312239i	$0.101079\pi$
$359$			11.8322i			0.624477i	−0.950004	+	0.312239i	$0.898921\pi$
$360$		0			0
$361$	19.0000			1.00000
$362$		0			0
$363$	33.8723	−	6.09641i	1.77784	−	0.319978i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			19.7872i			1.03289i	0.856322	+	0.516443i	$0.172744\pi$
$367$			19.7872i			1.03289i	−0.856322	+	0.516443i	$0.827256\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	34.4934			1.77650
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−35.7771			−1.82812			−0.914062	−	0.405575i	$-0.867071\pi$
$383$	−35.7771			−1.82812			−0.914062	+	0.405575i	$0.867071\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			36.7330i			1.86244i	0.364459	+	0.931219i	$0.381254\pi$
$389$			36.7330i			1.86244i	−0.364459	+	0.931219i	$0.618746\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			0.231042i			0.0115956i	0.999983	+	0.00579782i	$0.00184551\pi$
$397$			0.231042i			0.0115956i	−0.999983	+	0.00579782i	$0.998154\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			38.1729i			1.90626i	0.302556	+	0.953131i	$0.402160\pi$
$401$			38.1729i			1.90626i	−0.302556	+	0.953131i	$0.597840\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	3.12957			0.152526			0.0762630	−	0.997088i	$-0.475701\pi$
$421$	3.12957			0.152526			0.0762630	+	0.997088i	$0.475701\pi$
$422$		0			0
$423$	−3.60941	+	1.34275i	−0.175496	+	0.0652869i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−67.5526	+	12.1582i	−3.26147	+	0.587005i
$430$		0			0
$431$		−	18.1082i		−	0.872241i	−0.899888	−	0.436121i	$-0.856352\pi$
$431$		−	18.1082i		−	0.872241i	0.899888	−	0.436121i	$-0.143648\pi$
$432$		0			0
$433$		−	10.5830i		−	0.508587i	−0.967127	−	0.254293i	$-0.918157\pi$
$433$		−	10.5830i		−	0.508587i	0.967127	−	0.254293i	$-0.0818429\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−19.6822	+	7.32205i	−0.937246	+	0.348669i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	7.26040	+	40.3396i	0.343405	+	1.90800i
$448$		0			0
$449$			28.5005i			1.34502i	0.740087	+	0.672511i	$0.234784\pi$
$449$			28.5005i			1.34502i	−0.740087	+	0.672511i	$0.765216\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−11.7117	+	2.10790i	−0.550266	+	0.0990379i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$	−25.7409	+	15.2285i	−1.20148	+	0.710805i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−28.1165			−1.30108			−0.650538	−	0.759473i	$-0.725457\pi$
$467$	−28.1165			−1.30108			−0.650538	+	0.759473i	$0.725457\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	6.49390	+	36.0809i	0.299223	+	1.66252i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	13.7886i		−	0.622273i	−0.950365	−	0.311136i	$-0.899290\pi$
$491$		−	13.7886i		−	0.622273i	0.950365	−	0.311136i	$-0.100710\pi$
$492$		0			0
$493$		−	27.8363i		−	1.25368i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−31.3050			−1.40422
$498$		0			0
$499$	5.12957			0.229631			0.114816	−	0.993387i	$-0.463372\pi$
$499$	5.12957			0.229631			0.114816	+	0.993387i	$0.463372\pi$
$500$		0			0
$501$	−32.6822	+	5.88220i	−1.46013	+	0.262797i
$502$		0			0
$503$	−39.6282			−1.76693			−0.883466	−	0.468495i	$-0.844797\pi$
$503$	−39.6282			−1.76693			−0.883466	+	0.468495i	$0.844797\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	64.5562	−	11.6190i	2.86704	−	0.516016i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−28.0000			−1.23865
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			7.13235i			0.313680i
$518$		0			0
$519$	44.6822	−	8.04198i	1.96133	−	0.353004i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$			37.0405i			1.61967i	0.586659	+	0.809834i	$0.300443\pi$
$523$			37.0405i			1.61967i	−0.586659	+	0.809834i	$0.699557\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−3.63020	−	20.1698i	−0.156655	−	0.870392i
$538$		0			0
$539$			38.8928i			1.67523i
$540$		0			0
$541$	36.8704			1.58518			0.792592	−	0.609753i	$-0.208731\pi$
$541$	36.8704			1.58518			0.792592	+	0.609753i	$0.208731\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	39.3434i		−	1.67305i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	9.81174	+	54.5152i	0.414252	+	2.30163i
$562$		0			0
$563$	44.7214			1.88478			0.942390	−	0.334515i	$-0.108573\pi$
$563$	44.7214			1.88478			0.942390	+	0.334515i	$0.108573\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−15.5628	−	18.0222i	−0.653577	−	0.756860i
$568$		0			0
$569$			47.3286i			1.98412i	0.125767	+	0.992060i	$0.459861\pi$
$569$			47.3286i			1.98412i	−0.125767	+	0.992060i	$0.540139\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$		0			0
$573$	8.08155	+	44.9020i	0.337611	+	1.87581i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	46.2448i		−	1.92519i	−0.270936	−	0.962597i	$-0.587333\pi$
$577$		−	46.2448i		−	1.92519i	0.270936	−	0.962597i	$-0.412667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			23.6643i			0.981761i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.94427			0.369170			0.184585	−	0.982817i	$-0.440906\pi$
$587$	8.94427			0.369170			0.184585	+	0.982817i	$0.440906\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.621055			0.0255037			0.0127518	−	0.999919i	$-0.495941\pi$
$593$	0.621055			0.0255037			0.0127518	+	0.999919i	$0.495941\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	48.5652i		−	1.98432i	−0.124975	−	0.992160i	$-0.539885\pi$
$599$		−	48.5652i		−	1.98432i	0.124975	−	0.992160i	$-0.460115\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$			5.06046i			0.205398i	0.994712	+	0.102699i	$0.0327478\pi$
$607$			5.06046i			0.205398i	−0.994712	+	0.102699i	$0.967252\pi$
$608$		0			0
$609$	21.8117	−	3.92572i	0.883856	−	0.159078i
$610$		0			0
$611$		−	9.15575i		−	0.370402i
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−38.8704			−1.54741			−0.773704	−	0.633548i	$-0.781598\pi$
$631$	−38.8704			−1.54741			−0.773704	+	0.633548i	$0.781598\pi$
$632$		0			0
$633$	45.8050	−	8.24406i	1.82058	−	0.327672i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	49.9265i		−	1.97816i
$638$		0			0
$639$	−12.3765	−	33.2689i	−0.489608	−	1.31610i
$640$		0			0
$641$		−	47.3286i		−	1.86937i	−0.355479	−	0.934684i	$-0.615682\pi$
$641$		−	47.3286i		−	1.86937i	0.355479	−	0.934684i	$-0.384318\pi$
$642$		0			0
$643$			19.3252i			0.762110i	0.924552	+	0.381055i	$0.124439\pi$
$643$			19.3252i			0.762110i	−0.924552	+	0.381055i	$0.875561\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17.8885			−0.703271			−0.351636	−	0.936137i	$-0.614374\pi$
$647$	−17.8885			−0.703271			−0.351636	+	0.936137i	$0.614374\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−11.0699	−	29.7566i	−0.431878	−	1.16092i
$658$		0			0
$659$			47.1253i			1.83574i	0.396878	+	0.917871i	$0.370093\pi$
$659$			47.1253i			1.83574i	−0.396878	+	0.917871i	$0.629907\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$	−12.5953	−	69.9808i	−0.489160	−	2.71783i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	2.68216	+	14.9024i	0.103698	+	0.576161i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	32.5886			1.25248			0.626242	−	0.779629i	$-0.284592\pi$
$677$	32.5886			1.25248			0.626242	+	0.779629i	$0.284592\pi$
$678$		0			0
$679$	−9.12957			−0.350361
$680$		0			0
$681$	13.0587	−	2.35033i	0.500410	−	0.0900647i
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$	−41.3328	+	15.3764i	−1.57010	+	0.584102i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	49.2851i		−	1.86147i	−0.365690	−	0.930737i	$-0.619167\pi$
$701$		−	49.2851i		−	1.86147i	0.365690	−	0.930737i	$-0.380833\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−46.6113			−1.75052			−0.875262	−	0.483650i	$-0.839311\pi$
$709$	−46.6113			−1.75052			−0.875262	+	0.483650i	$0.839311\pi$
$710$		0			0
$711$	41.8117	−	15.5546i	1.56806	−	0.583342i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	2.14642	+	11.9258i	0.0801595	+	0.445376i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−42.6113			−1.58693
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	5.29150i		−	0.196251i	−0.995174	−	0.0981255i	$-0.968715\pi$
$727$		−	5.29150i		−	0.196251i	0.995174	−	0.0981255i	$-0.0312847\pi$
$728$		0			0
$729$	13.0000	−	23.6643i	0.481481	−	0.876456i
$730$		0			0
$731$		0			0
$732$		0			0
$733$			42.5631i			1.57210i	0.618161	+	0.786051i	$0.287878\pi$
$733$			42.5631i			1.57210i	−0.618161	+	0.786051i	$0.712122\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−40.6113			−1.49391			−0.746955	−	0.664875i	$-0.768485\pi$
$739$	−40.6113			−1.49391			−0.746955	+	0.664875i	$0.768485\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−25.1489	+	9.35577i	−0.920152	+	0.342310i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	52.6113			1.91981			0.959906	−	0.280321i	$-0.0904408\pi$
$751$	52.6113			1.91981			0.959906	+	0.280321i	$0.0904408\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$		−	55.2180i		−	1.99902i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	7.62348	−	1.37209i	0.274553	−	0.0494145i
$772$		0			0
$773$	49.2351			1.77086			0.885431	−	0.464770i	$-0.153863\pi$
$773$	49.2351			1.77086			0.885431	+	0.464770i	$0.153863\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−65.7409			−2.35239
$782$		0			0
$783$	12.7954	+	21.6281i	0.457269	+	0.772925i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			51.5363i			1.83707i	0.395340	+	0.918535i	$0.370627\pi$
$787$			51.5363i			1.83707i	−0.395340	+	0.918535i	$0.629373\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21.0770			−0.746585			−0.373293	−	0.927714i	$-0.621771\pi$
$797$	−21.0770			−0.746585			−0.373293	+	0.927714i	$0.621771\pi$
$798$		0			0
$799$	−7.38872			−0.261394
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−58.8004			−2.07502
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	13.0687i		−	0.459471i	−0.973253	−	0.229736i	$-0.926214\pi$
$809$		−	13.0687i		−	0.459471i	0.973253	−	0.229736i	$-0.0737862\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	53.0587	−	19.7386i	1.85402	−	0.689723i
$820$		0			0
$821$			46.4054i			1.61956i	0.586734	+	0.809780i	$0.300414\pi$
$821$			46.4054i			1.61956i	−0.586734	+	0.809780i	$0.699586\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−40.2908			−1.39599
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	5.61128			0.193492
$842$		0			0
$843$	−7.86067	−	43.6748i	−0.270736	−	1.50424i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			52.5722i			1.80640i
$848$		0			0
$849$	−9.31784	−	51.7710i	−0.319787	−	1.77678i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	42.3320i		−	1.44942i	−0.689054	−	0.724710i	$-0.741974\pi$
$853$		−	42.3320i		−	1.44942i	0.689054	−	0.724710i	$-0.258026\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	49.1935			1.68042			0.840209	−	0.542263i	$-0.182432\pi$
$857$	49.1935			1.68042			0.840209	+	0.542263i	$0.182432\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−27.4955	+	4.94868i	−0.933795	+	0.168066i
$868$		0			0
$869$		−	82.6218i		−	2.80275i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−3.60941	−	9.70234i	−0.122160	−	0.328374i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$	−14.1883	+	2.55363i	−0.478558	+	0.0861318i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	35.7771			1.20128			0.600639	−	0.799521i	$-0.294913\pi$
$887$	35.7771			1.20128			0.600639	+	0.799521i	$0.294913\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−32.6822	−	37.8468i	−1.09489	−	1.26792i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			59.1608i			1.96008i	0.198789	+	0.980042i	$0.436299\pi$
$911$			59.1608i			1.96008i	−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$			49.6954i			1.64468i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	60.6113			1.99938			0.999691	−	0.0248659i	$-0.00791589\pi$
$919$	60.6113			1.99938			0.999691	+	0.0248659i	$0.00791589\pi$
$920$		0			0
$921$	7.05869	+	39.2189i	0.232592	+	1.29231i
$922$		0			0
$923$	84.3911			2.77777
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−16.8465	−	45.2846i	−0.553312	−	1.48734i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			24.6167i			0.804191i	0.915598	+	0.402096i	$0.131718\pi$
$937$			24.6167i			0.804191i	−0.915598	+	0.402096i	$0.868282\pi$
$938$		0			0
$939$	9.81174	+	54.5152i	0.320194	+	1.77903i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	75.4817			2.45024
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	45.8050	−	8.24406i	1.48066	−	0.266493i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	58.6822	−	21.8306i	1.87358	−	0.696998i
$982$		0			0
$983$	62.6515			1.99827			0.999136	−	0.0415592i	$-0.0132325\pi$
$983$	62.6515			1.99827			0.999136	+	0.0415592i	$0.0132325\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.04202	−	5.78960i	−0.0331679	−	0.184285i
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−52.0000			−1.65183			−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000			−1.65183			−0.825917	+	0.563791i	$0.809342\pi$
$992$		0			0
$993$	−13.6373	+	2.45446i	−0.432766	+	0.0778901i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			43.0251i			1.36262i	0.731995	+	0.681310i	$0.238589\pi$
$997$			43.0251i			1.36262i	−0.731995	+	0.681310i	$0.761411\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.2.d.j.1301.2		8
3.2	odd	2	inner	2100.2.d.j.1301.8		8
5.2	odd	4		420.2.f.a.209.6	yes	8
5.3	odd	4		420.2.f.a.209.3	✓	8
5.4	even	2	inner	2100.2.d.j.1301.7		8
7.6	odd	2	inner	2100.2.d.j.1301.7		8
15.2	even	4		420.2.f.a.209.5	yes	8
15.8	even	4		420.2.f.a.209.4	yes	8
15.14	odd	2	inner	2100.2.d.j.1301.1		8
20.3	even	4		1680.2.k.d.209.6		8
20.7	even	4		1680.2.k.d.209.3		8
21.20	even	2	inner	2100.2.d.j.1301.1		8
35.13	even	4		420.2.f.a.209.6	yes	8
35.27	even	4		420.2.f.a.209.3	✓	8
35.34	odd	2	CM	2100.2.d.j.1301.2		8
60.23	odd	4		1680.2.k.d.209.5		8
60.47	odd	4		1680.2.k.d.209.4		8
105.62	odd	4		420.2.f.a.209.4	yes	8
105.83	odd	4		420.2.f.a.209.5	yes	8
105.104	even	2	inner	2100.2.d.j.1301.8		8
140.27	odd	4		1680.2.k.d.209.6		8
140.83	odd	4		1680.2.k.d.209.3		8
420.83	even	4		1680.2.k.d.209.4		8
420.167	even	4		1680.2.k.d.209.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.f.a.209.3	✓	8	5.3	odd	4
420.2.f.a.209.3	✓	8	35.27	even	4
420.2.f.a.209.4	yes	8	15.8	even	4
420.2.f.a.209.4	yes	8	105.62	odd	4
420.2.f.a.209.5	yes	8	15.2	even	4
420.2.f.a.209.5	yes	8	105.83	odd	4
420.2.f.a.209.6	yes	8	5.2	odd	4
420.2.f.a.209.6	yes	8	35.13	even	4
1680.2.k.d.209.3		8	20.7	even	4
1680.2.k.d.209.3		8	140.83	odd	4
1680.2.k.d.209.4		8	60.47	odd	4
1680.2.k.d.209.4		8	420.83	even	4
1680.2.k.d.209.5		8	60.23	odd	4
1680.2.k.d.209.5		8	420.167	even	4
1680.2.k.d.209.6		8	20.3	even	4
1680.2.k.d.209.6		8	140.27	odd	4
2100.2.d.j.1301.1		8	15.14	odd	2	inner
2100.2.d.j.1301.1		8	21.20	even	2	inner
2100.2.d.j.1301.2		8	1.1	even	1	trivial
2100.2.d.j.1301.2		8	35.34	odd	2	CM
2100.2.d.j.1301.7		8	5.4	even	2	inner
2100.2.d.j.1301.7		8	7.6	odd	2	inner
2100.2.d.j.1301.8		8	3.2	odd	2	inner
2100.2.d.j.1301.8		8	105.104	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2100.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.70466 + 0.306808i) q^{3} -2.64575i q^{7} +(2.81174 - 1.04601i) q^{9} +O(q^{10})\)	\(q+(-1.70466 + 0.306808i) q^{3} -2.64575i q^{7} +(2.81174 - 1.04601i) q^{9} -5.55612i q^{11} +7.13235i q^{13} +5.75583 q^{17} +(0.811738 + 4.51011i) q^{21} +(-4.47214 + 2.64575i) q^{27} -4.83619i q^{29} +(1.70466 + 9.47129i) q^{33} +(-2.18826 - 12.1582i) q^{39} -1.28369 q^{47} -7.00000 q^{49} +(-9.81174 + 1.76593i) q^{51} +(-2.76748 - 7.43916i) q^{63} -11.8322i q^{71} -10.5830i q^{73} -14.7001 q^{77} +14.8704 q^{79} +(6.81174 - 5.88220i) q^{81} -8.94427 q^{83} +(1.48378 + 8.24406i) q^{87} +18.8704 q^{91} -3.45065i q^{97} +(-5.81174 - 15.6223i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{9}+O(q^{10}) \) 8 * q + 2 * q^9	\( 8 q + 2 q^{9} - 14 q^{21} - 38 q^{39} - 56 q^{49} - 58 q^{51} - 4 q^{79} + 34 q^{81} + 28 q^{91} - 26 q^{99}+O(q^{100}) \) 8 * q + 2 * q^9 - 14 * q^21 - 38 * q^39 - 56 * q^49 - 58 * q^51 - 4 * q^79 + 34 * q^81 + 28 * q^91 - 26 * q^99

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