LMFDB - Embedded newform 252.4.b.c.55.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,4,Mod(55,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("252.55");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	252.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:	$14.8684813214$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			55.4
Root			$-1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	252.55
Dual form			252.4.b.c.55.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.32288 + 2.50000i) q^{2} +(-4.50000 + 6.61438i) q^{4} -18.5203i q^{7} +(-22.4889 - 2.50000i) q^{8} +O(q^{10})$	$q+(1.32288 + 2.50000i) q^{2} +(-4.50000 + 6.61438i) q^{4} -18.5203i q^{7} +(-22.4889 - 2.50000i) q^{8} -68.0000i q^{11} +(46.3006 - 24.5000i) q^{14} +(-23.5000 - 59.5294i) q^{16} +(170.000 - 89.9555i) q^{22} -40.0000i q^{23} +125.000 q^{25} +(122.500 + 83.3412i) q^{28} -264.575 q^{29} +(117.736 - 137.500i) q^{32} +450.000 q^{37} -534.442i q^{43} +(449.778 + 306.000i) q^{44} +(100.000 - 52.9150i) q^{46} -343.000 q^{49} +(165.359 + 312.500i) q^{50} -497.401 q^{53} +(-46.3006 + 416.500i) q^{56} +(-350.000 - 661.438i) q^{58} +(499.500 + 112.444i) q^{64} +809.600i q^{67} -688.000i q^{71} +(595.294 + 1125.00i) q^{74} -1259.38 q^{77} -238.118i q^{79} +(1336.10 - 707.000i) q^{86} +(-170.000 + 1529.24i) q^{88} +(264.575 + 180.000i) q^{92} +(-453.746 - 857.500i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 18 q^{4}+O(q^{10}) $ 4 * q - 18 * q^4	$ 4 q - 18 q^{4} - 94 q^{16} + 680 q^{22} + 500 q^{25} + 490 q^{28} + 1800 q^{37} + 400 q^{46} - 1372 q^{49} - 1400 q^{58} + 1998 q^{64} - 680 q^{88}+O(q^{100}) $ 4 * q - 18 * q^4 - 94 * q^16 + 680 * q^22 + 500 * q^25 + 490 * q^28 + 1800 * q^37 + 400 * q^46 - 1372 * q^49 - 1400 * q^58 + 1998 * q^64 - 680 * q^88

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	1.32288	+	2.50000i	0.467707	+	0.883883i
$2$	1.32288	+	2.50000i	0.467707	+	0.883883i
$3$		0			0
$3$		0			0
$4$	−4.50000	+	6.61438i	−0.562500	+	0.826797i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		−	18.5203i		−	1.00000i
$7$		−	18.5203i		−	1.00000i
$8$	−22.4889	−	2.50000i	−0.993878	−	0.110485i
$9$		0			0
$10$		0			0
$11$		−	68.0000i		−	1.86389i	−0.362602	−	0.931944i	$-0.618111\pi$
$11$		−	68.0000i		−	1.86389i	0.362602	−	0.931944i	$-0.381889\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	46.3006	−	24.5000i	0.883883	−	0.467707i
$15$		0			0
$16$	−23.5000	−	59.5294i	−0.367188	−	0.930147i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$	170.000	−	89.9555i	1.64746	−	0.871754i
$23$		−	40.0000i		−	0.362634i	−0.983425	−	0.181317i	$-0.941964\pi$
$23$		−	40.0000i		−	0.362634i	0.983425	−	0.181317i	$-0.0580360\pi$
$24$		0			0
$25$	125.000			1.00000
$26$		0			0
$27$		0			0
$28$	122.500	+	83.3412i	0.826797	+	0.562500i
$29$	−264.575			−1.69415			−0.847075	−	0.531473i	$-0.821639\pi$
$29$	−264.575			−1.69415			−0.847075	+	0.531473i	$0.821639\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	117.736	−	137.500i	0.650405	−	0.759587i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	450.000			1.99945			0.999724	−	0.0235113i	$-0.00748457\pi$
$37$	450.000			1.99945			0.999724	+	0.0235113i	$0.00748457\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	534.442i		−	1.89539i	−0.319183	−	0.947693i	$-0.603408\pi$
$43$		−	534.442i		−	1.89539i	0.319183	−	0.947693i	$-0.396592\pi$
$44$	449.778	+	306.000i	1.54106	+	1.04844i
$45$		0			0
$46$	100.000	−	52.9150i	0.320526	−	0.169606i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−343.000			−1.00000
$50$	165.359	+	312.500i	0.467707	+	0.883883i
$51$		0			0
$52$		0			0
$53$	−497.401			−1.28912			−0.644560	−	0.764554i	$-0.722959\pi$
$53$	−497.401			−1.28912			−0.644560	+	0.764554i	$0.722959\pi$
$54$		0			0
$55$		0			0
$56$	−46.3006	+	416.500i	−0.110485	+	0.993878i
$57$		0			0
$58$	−350.000	−	661.438i	−0.792366	−	1.49743i
$59$		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
$63$		0			0
$64$	499.500	+	112.444i	0.975586	+	0.219618i
$65$		0			0
$66$		0			0
$67$			809.600i			1.47624i	0.674667	+	0.738122i	$0.264287\pi$
$67$			809.600i			1.47624i	−0.674667	+	0.738122i	$0.735713\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	688.000i		−	1.15001i	−0.818151	−	0.575004i	$-0.805000\pi$
$71$		−	688.000i		−	1.15001i	0.818151	−	0.575004i	$-0.195000\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	595.294	+	1125.00i	0.935156	+	1.76728i
$75$		0			0
$76$		0			0
$77$	−1259.38			−1.86389
$78$		0			0
$79$		−	238.118i		−	0.339118i	−0.985520	−	0.169559i	$-0.945766\pi$
$79$		−	238.118i		−	0.339118i	0.985520	−	0.169559i	$-0.0542343\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	1336.10	−	707.000i	1.67530	−	0.886486i
$87$		0			0
$88$	−170.000	+	1529.24i	−0.205933	+	1.85248i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	264.575	+	180.000i	0.299825	+	0.203981i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−453.746	−	857.500i	−0.467707	−	0.883883i
$99$		0			0
$100$	−562.500	+	826.797i	−0.562500	+	0.826797i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$	−658.000	−	1243.50i	−0.602930	−	1.13943i
$107$		−	1580.00i		−	1.42752i	−0.700392	−	0.713759i	$-0.746991\pi$
$107$		−	1580.00i		−	1.42752i	0.700392	−	0.713759i	$-0.253009\pi$
$108$		0			0
$109$	−54.0000			−0.0474519			−0.0237260	−	0.999718i	$-0.507553\pi$
$109$	−54.0000			−0.0474519			−0.0237260	+	0.999718i	$0.507553\pi$
$110$		0			0
$111$		0			0
$112$	−1102.50	+	435.226i	−0.930147	+	0.367188i
$113$	−2307.10			−1.92065			−0.960324	−	0.278886i	$-0.910035\pi$
$113$	−2307.10			−1.92065			−0.960324	+	0.278886i	$0.910035\pi$
$114$		0			0
$115$		0			0
$116$	1190.59	−	1750.00i	0.952960	−	1.40072i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3293.00			−2.47408
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$			2047.81i			1.43082i	0.698706	+	0.715409i	$0.253760\pi$
$127$			2047.81i			1.43082i	−0.698706	+	0.715409i	$0.746240\pi$
$128$	379.665	+	1397.50i	0.262172	+	0.965021i
$129$		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
$134$	−2024.00	+	1071.00i	−1.30483	+	0.690450i
$135$		0			0
$136$		0			0
$137$	783.142			0.488382			0.244191	−	0.969727i	$-0.421478\pi$
$137$	783.142			0.488382			0.244191	+	0.969727i	$0.421478\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	1720.00	−	910.138i	1.01647	−	0.537867i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−2025.00	+	2976.47i	−1.12469	+	1.65314i
$149$	3545.31			1.94928			0.974640	−	0.223777i	$-0.0718387\pi$
$149$	3545.31			1.94928			0.974640	+	0.223777i	$0.0718387\pi$
$150$		0			0
$151$			2248.89i			1.21200i	0.795465	+	0.606000i	$0.207227\pi$
$151$			2248.89i			1.21200i	−0.795465	+	0.606000i	$0.792773\pi$
$152$		0			0
$153$		0			0
$154$	−1666.00	−	3148.44i	−0.871754	−	1.64746i
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	595.294	−	315.000i	0.299741	−	0.158608i
$159$		0			0
$160$		0			0
$161$	−740.810			−0.362634
$162$		0			0
$163$		−	3762.26i		−	1.80787i	−0.427670	−	0.903935i	$-0.640665\pi$
$163$		−	3762.26i		−	1.80787i	0.427670	−	0.903935i	$-0.359335\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	2197.00			1.00000
$170$		0			0
$171$		0			0
$172$	3535.00	+	2404.99i	1.56710	+	1.06615i
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		−	2315.03i		−	1.00000i
$176$	−4048.00	+	1598.00i	−1.73369	+	0.684396i
$177$		0			0
$178$		0			0
$179$			2084.00i			0.870198i	0.900383	+	0.435099i	$0.143287\pi$
$179$			2084.00i			0.870198i	−0.900383	+	0.435099i	$0.856713\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−100.000	+	899.555i	−0.0400657	+	0.360414i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			4072.00i			1.54262i	0.636462	+	0.771308i	$0.280397\pi$
$191$			4072.00i			1.54262i	−0.636462	+	0.771308i	$0.719603\pi$
$192$		0			0
$193$	4590.00			1.71189			0.855947	−	0.517064i	$-0.172975\pi$
$193$	4590.00			1.71189			0.855947	+	0.517064i	$0.172975\pi$
$194$		0			0
$195$		0			0
$196$	1543.50	−	2268.73i	0.562500	−	0.826797i
$197$	5069.26			1.83335			0.916675	−	0.399634i	$-0.130863\pi$
$197$	5069.26			1.83335			0.916675	+	0.399634i	$0.130863\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−2811.11	−	312.500i	−0.993878	−	0.110485i
$201$		0			0
$202$		0			0
$203$			4900.00i			1.69415i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$			1772.65i			0.578363i	0.957274	+	0.289181i	$0.0933830\pi$
$211$			1772.65i			0.578363i	−0.957274	+	0.289181i	$0.906617\pi$
$212$	2238.31	−	3290.00i	0.725130	−	1.06584i
$213$		0			0
$214$	3950.00	−	2090.14i	1.26176	−	0.667660i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−71.4353	−	135.000i	−0.0221936	−	0.0419420i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−2546.54	−	2180.50i	−0.759587	−	0.650405i
$225$		0			0
$226$	−3052.00	−	5767.74i	−0.898301	−	1.69763i
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	5950.00	+	661.438i	1.68378	+	0.187179i
$233$	5312.67			1.49375			0.746877	−	0.664963i	$-0.231553\pi$
$233$	5312.67			1.49375			0.746877	+	0.664963i	$0.231553\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	7376.00i		−	1.99629i	−0.0608655	−	0.998146i	$-0.519386\pi$
$239$		−	7376.00i		−	1.99629i	0.0608655	−	0.998146i	$-0.480614\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−4356.23	−	8232.50i	−1.15714	−	2.18680i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−2720.00			−0.675909
$254$	−5119.53	+	2709.00i	−1.26468	+	0.669204i
$255$		0			0
$256$	−2991.50	+	2797.88i	−0.730347	+	0.683077i
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		−	8334.12i		−	1.99945i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			7520.00i			1.76313i	0.472063	+	0.881565i	$0.343509\pi$
$263$			7520.00i			1.76313i	−0.472063	+	0.881565i	$0.656491\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−5355.00	−	3643.20i	−1.22055	−	0.830387i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$	1036.00	+	1957.86i	0.228420	+	0.431673i
$275$		−	8500.00i		−	1.86389i
$276$		0			0
$277$	7310.00			1.58561			0.792807	−	0.609472i	$-0.208619\pi$
$277$	7310.00			1.58561			0.792807	+	0.609472i	$0.208619\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	8360.57			1.77491			0.887456	−	0.460893i	$-0.152471\pi$
$281$	8360.57			1.77491			0.887456	+	0.460893i	$0.152471\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	4550.69	+	3096.00i	0.950824	+	0.646880i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4913.00			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	−10120.0	−	1125.00i	−1.98721	−	0.220910i
$297$		0			0
$298$	4690.00	+	8863.27i	0.911693	+	1.72294i
$299$		0			0
$300$		0			0
$301$	−9898.00			−1.89539
$302$	−5622.22	+	2975.00i	−1.07127	+	0.566861i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	5667.20	−	8330.00i	1.04844	−	1.54106i
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$	1575.00	+	1071.53i	0.280382	+	0.190754i
$317$	−8879.14			−1.57319			−0.786597	−	0.617467i	$-0.788159\pi$
$317$	−8879.14			−1.57319			−0.786597	+	0.617467i	$0.788159\pi$
$318$		0			0
$319$			17991.1i			3.15771i
$320$		0			0
$321$		0			0
$322$	−980.000	−	1852.03i	−0.169606	−	0.320526i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	9405.65	−	4977.00i	1.59795	−	0.845554i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	5106.30i		−	0.847938i	−0.905677	−	0.423969i	$-0.860636\pi$
$331$		−	5106.30i		−	0.847938i	0.905677	−	0.423969i	$-0.139364\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3330.00			−0.538269			−0.269135	−	0.963103i	$-0.586738\pi$
$337$	−3330.00			−0.538269			−0.269135	+	0.963103i	$0.586738\pi$
$338$	2906.36	+	5492.50i	0.467707	+	0.883883i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			6352.45i			1.00000i
$344$	−1336.10	+	12019.0i	−0.209413	+	1.88378i
$345$		0			0
$346$		0			0
$347$			4100.00i			0.634293i	0.948377	+	0.317146i	$0.102725\pi$
$347$			4100.00i			0.634293i	−0.948377	+	0.317146i	$0.897275\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	5787.58	−	3062.50i	0.883883	−	0.467707i
$351$		0			0
$352$	−9350.00	−	8006.04i	−1.41579	−	1.21228i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−5210.00	+	2756.87i	−0.769154	+	0.406998i
$359$			8104.00i			1.19140i	0.803207	+	0.595700i	$0.203125\pi$
$359$			8104.00i			1.19140i	−0.803207	+	0.595700i	$0.796875\pi$
$360$		0			0
$361$	−6859.00			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−2381.18	+	940.000i	−0.337303	+	0.133155i
$369$		0			0
$370$		0			0
$371$			9212.00i			1.28912i
$372$		0			0
$373$	−13970.0			−1.93925			−0.969624	−	0.244602i	$-0.921343\pi$
$373$	−13970.0			−1.93925			−0.969624	+	0.244602i	$0.921343\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	8704.52i		−	1.17974i	−0.807498	−	0.589870i	$-0.799179\pi$
$379$		−	8704.52i		−	1.17974i	0.807498	−	0.589870i	$-0.200821\pi$
$380$		0			0
$381$		0			0
$382$	−10180.0	+	5386.75i	−1.36349	+	0.721492i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	6072.00	+	11475.0i	0.800665	+	1.51311i
$387$		0			0
$388$		0			0
$389$	−11165.1			−1.45525			−0.727624	−	0.685976i	$-0.759375\pi$
$389$	−11165.1			−1.45525			−0.727624	+	0.685976i	$0.759375\pi$
$390$		0			0
$391$		0			0
$392$	7713.69	+	857.500i	0.993878	+	0.110485i
$393$		0			0
$394$	6706.00	+	12673.1i	0.857471	+	1.62047i
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−2937.50	−	7441.18i	−0.367188	−	0.930147i
$401$	−15980.3			−1.99007			−0.995037	−	0.0995016i	$-0.968275\pi$
$401$	−15980.3			−1.99007			−0.995037	+	0.0995016i	$0.968275\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	−12250.0	+	6482.09i	−1.49743	+	0.792366i
$407$		−	30600.0i		−	3.72675i
$408$		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
$421$	15262.0			1.76680			0.883402	−	0.468616i	$-0.155247\pi$
$421$	15262.0			1.76680			0.883402	+	0.468616i	$0.155247\pi$
$422$	−4431.63	+	2345.00i	−0.511205	+	0.270504i
$423$		0			0
$424$	11186.0	+	1243.50i	1.28123	+	0.142429i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	10450.7	+	7110.00i	1.18027	+	0.802979i
$429$		0			0
$430$		0			0
$431$			8608.00i			0.962025i	0.876714	+	0.481012i	$0.159731\pi$
$431$			8608.00i			0.962025i	−0.876714	+	0.481012i	$0.840269\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	243.000	−	357.176i	0.0266917	−	0.0392331i
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	18580.0i		−	1.99269i	−0.0854102	−	0.996346i	$-0.527220\pi$
$443$		−	18580.0i		−	1.99269i	0.0854102	−	0.996346i	$-0.472780\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	2082.50	−	9250.87i	0.219618	−	0.975586i
$449$	18837.7			1.97997			0.989987	−	0.141158i	$-0.0450827\pi$
$449$	18837.7			1.97997			0.989987	+	0.141158i	$0.0450827\pi$
$450$		0			0
$451$		0			0
$452$	10381.9	−	15260.0i	1.08036	−	1.58799i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−8010.00			−0.819895			−0.409947	−	0.912109i	$-0.634453\pi$
$457$	−8010.00			−0.819895			−0.409947	+	0.912109i	$0.634453\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$			18049.3i			1.81171i	0.423585	+	0.905856i	$0.360771\pi$
$463$			18049.3i			1.81171i	−0.423585	+	0.905856i	$0.639229\pi$
$464$	6217.52	+	15750.0i	0.622071	+	1.57581i
$465$		0			0
$466$	7028.00	+	13281.7i	0.698639	+	1.32030i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$	14994.0			1.47624
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−36342.0			−3.53279
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	18440.0	−	9757.53i	1.76449	−	0.933680i
$479$		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
$484$	14818.5	−	21781.1i	1.39167	−	2.04556i
$485$		0			0
$486$		0			0
$487$		−	3296.61i		−	0.306742i	−0.988169	−	0.153371i	$-0.950987\pi$
$487$		−	3296.61i		−	0.306742i	0.988169	−	0.153371i	$-0.0490130\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	20372.0i		−	1.87246i	−0.351394	−	0.936228i	$-0.614292\pi$
$491$		−	20372.0i		−	1.87246i	0.351394	−	0.936228i	$-0.385708\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12741.9			−1.15001
$498$		0			0
$499$		−	21086.6i		−	1.89172i	−0.324577	−	0.945859i	$-0.605222\pi$
$499$		−	21086.6i		−	1.89172i	0.324577	−	0.945859i	$-0.394778\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$	−3598.22	−	6800.00i	−0.316127	−	0.597425i
$507$		0			0
$508$	−13545.0	−	9215.15i	−1.18300	−	0.804835i
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−10952.1	−	3777.50i	−0.945349	−	0.326062i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	20835.3	−	11025.0i	1.76728	−	0.935156i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$	−18800.0	+	9948.02i	−1.55840	+	0.824628i
$527$		0			0
$528$		0			0
$529$	10567.0			0.868497
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	2024.00	−	18207.0i	0.163103	−	1.46721i
$537$		0			0
$538$		0			0
$539$			23324.0i			1.86389i
$540$		0			0
$541$	15878.0			1.26183			0.630914	−	0.775853i	$-0.282680\pi$
$541$	15878.0			1.26183			0.630914	+	0.775853i	$0.282680\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	22049.7i		−	1.72354i	−0.507299	−	0.861770i	$-0.669356\pi$
$547$		−	22049.7i		−	1.72354i	0.507299	−	0.861770i	$-0.330644\pi$
$548$	−3524.14	+	5180.00i	−0.274715	+	0.403793i
$549$		0			0
$550$	21250.0	−	11244.4i	1.64746	−	0.871754i
$551$		0			0
$552$		0			0
$553$	−4410.00			−0.339118
$554$	9670.22	+	18275.0i	0.741603	+	1.40150i
$555$		0			0
$556$		0			0
$557$	16498.9			1.25508			0.627541	−	0.778583i	$-0.284061\pi$
$557$	16498.9			1.25508			0.627541	+	0.778583i	$0.284061\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	11060.0	+	20901.4i	0.830139	+	1.56881i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−1720.00	+	15472.4i	−0.127059	+	1.14297i
$569$	3598.22			0.265106			0.132553	−	0.991176i	$-0.457683\pi$
$569$	3598.22			0.265106			0.132553	+	0.991176i	$0.457683\pi$
$570$		0			0
$571$		−	26431.1i		−	1.93714i	−0.248747	−	0.968569i	$-0.580019\pi$
$571$		−	26431.1i		−	1.93714i	0.248747	−	0.968569i	$-0.419981\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	5000.00i		−	0.362634i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	6499.29	+	12282.5i	0.467707	+	0.883883i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$			33823.3i			2.40277i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−10575.0	−	26788.2i	−0.734172	−	1.85978i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	−15953.9	+	23450.0i	−1.09647	+	1.61166i
$597$		0			0
$598$		0			0
$599$			24736.0i			1.68729i	0.536903	+	0.843644i	$0.319594\pi$
$599$			24736.0i			1.68729i	−0.536903	+	0.843644i	$0.680406\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	−13093.8	−	24745.0i	−0.886486	−	1.67530i
$603$		0			0
$604$	−14875.0	−	10120.0i	−1.00208	−	0.681750i
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−15010.0			−0.988986			−0.494493	−	0.869182i	$-0.664646\pi$
$613$	−15010.0			−0.988986			−0.494493	+	0.869182i	$0.664646\pi$
$614$		0			0
$615$		0			0
$616$	28322.0	+	3148.44i	1.85248	+	0.205933i
$617$	−2497.59			−0.162965			−0.0814823	−	0.996675i	$-0.525965\pi$
$617$	−2497.59			−0.162965			−0.0814823	+	0.996675i	$0.525965\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15625.0			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$			17858.8i			1.12670i	0.826218	+	0.563351i	$0.190488\pi$
$631$			17858.8i			1.12670i	−0.826218	+	0.563351i	$0.809512\pi$
$632$	−595.294	+	5355.00i	−0.0374676	+	0.337042i
$633$		0			0
$634$	−11746.0	−	22197.9i	−0.735794	−	1.39052i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−44977.8	+	23800.0i	−2.79105	+	1.47688i
$639$		0			0
$640$		0			0
$641$	31219.9			1.92373			0.961865	−	0.273526i	$-0.0881899\pi$
$641$	31219.9			1.92373			0.961865	+	0.273526i	$0.0881899\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	3333.65	−	4900.00i	0.203981	−	0.299825i
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	24885.0	+	16930.2i	1.49474	+	1.01693i
$653$	19546.8			1.17140			0.585701	−	0.810527i	$-0.300819\pi$
$653$	19546.8			1.17140			0.585701	+	0.810527i	$0.300819\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	1804.00i		−	0.106637i	−0.998578	−	0.0533186i	$-0.983020\pi$
$659$		−	1804.00i		−	0.106637i	0.998578	−	0.0533186i	$-0.0169799\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	12765.8	−	6755.00i	0.749479	−	0.396587i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			10583.0i			0.614356i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−33570.0			−1.92278			−0.961388	−	0.275196i	$-0.911257\pi$
$673$	−33570.0			−1.92278			−0.961388	+	0.275196i	$0.911257\pi$
$674$	−4405.18	−	8325.00i	−0.251752	−	0.475767i
$675$		0			0
$676$	−9886.50	+	14531.8i	−0.562500	+	0.826797i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$			34060.0i			1.90815i	0.299560	+	0.954077i	$0.403160\pi$
$683$			34060.0i			1.90815i	−0.299560	+	0.954077i	$0.596840\pi$
$684$		0			0
$685$		0			0
$686$	−15881.1	+	8403.50i	−0.883883	+	0.467707i
$687$		0			0
$688$	−31815.0	+	12559.4i	−1.76299	+	0.695962i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$	−10250.0	+	5423.79i	−0.560641	+	0.296663i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	15312.5	+	10417.6i	0.826797	+	0.562500i
$701$	36881.8			1.98717			0.993584	−	0.113093i	$-0.0360758\pi$
$701$	36881.8			1.98717			0.993584	+	0.113093i	$0.0360758\pi$
$702$		0			0
$703$		0			0
$704$	7646.22	−	33966.0i	0.409343	−	1.81838i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	12546.0			0.664563			0.332281	−	0.943180i	$-0.392182\pi$
$709$	12546.0			0.664563			0.332281	+	0.943180i	$0.392182\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−13784.4	−	9378.00i	−0.719477	−	0.489486i
$717$		0			0
$718$	−20260.0	+	10720.6i	−1.05306	+	0.557227i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−9073.60	−	17147.5i	−0.467707	−	0.883883i
$723$		0			0
$724$		0			0
$725$	−33071.9			−1.69415
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	−5500.00	−	4709.44i	−0.275452	−	0.235859i
$737$	55052.8			2.75155
$738$		0			0
$739$		−	31193.4i		−	1.55273i	−0.630283	−	0.776365i	$-0.717061\pi$
$739$		−	31193.4i		−	1.55273i	0.630283	−	0.776365i	$-0.282939\pi$
$740$		0			0
$741$		0			0
$742$	−23030.0	+	12186.3i	−1.13943	+	0.602930i
$743$		−	25160.0i		−	1.24230i	−0.783691	−	0.621151i	$-0.786665\pi$
$743$		−	25160.0i		−	1.24230i	0.783691	−	0.621151i	$-0.213335\pi$
$744$		0			0
$745$		0			0
$746$	−18480.6	−	34925.0i	−0.907000	−	1.71407i
$747$		0			0
$748$		0			0
$749$	−29262.0			−1.42752
$750$		0			0
$751$			41088.5i			1.99646i	0.0594732	+	0.998230i	$0.481058\pi$
$751$			41088.5i			1.99646i	−0.0594732	+	0.998230i	$0.518942\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−34830.0			−1.67228			−0.836141	−	0.548514i	$-0.815194\pi$
$757$	−34830.0			−1.67228			−0.836141	+	0.548514i	$0.815194\pi$
$758$	21761.3	−	11515.0i	1.04275	−	0.551773i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$			1000.09i			0.0474519i
$764$	−26933.7	−	18324.0i	−1.27543	−	0.867721i
$765$		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
$771$		0			0
$772$	−20655.0	+	30360.0i	−0.962940	+	1.41539i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−14770.0	−	27912.7i	−0.680630	−	1.28627i
$779$		0			0
$780$		0			0
$781$	−46784.0			−2.14349
$782$		0			0
$783$		0			0
$784$	8060.50	+	20418.6i	0.367188	+	0.930147i
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−22811.7	+	33530.0i	−1.03126	+	1.51581i
$789$		0			0
$790$		0			0
$791$			42728.0i			1.92065i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	14717.0	−	17187.5i	0.650405	−	0.759587i
$801$		0			0
$802$	−21140.0	−	39950.8i	−0.930772	−	1.75899i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26880.8			1.16821			0.584104	−	0.811679i	$-0.301446\pi$
$809$	26880.8			1.16821			0.584104	+	0.811679i	$0.301446\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−32410.5	−	22050.0i	−1.40072	−	0.952960i
$813$		0			0
$814$	76500.0	−	40480.0i	3.29401	−	1.74303i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	17832.4			0.758044			0.379022	−	0.925388i	$-0.376261\pi$
$821$	17832.4			0.758044			0.379022	+	0.925388i	$0.376261\pi$
$822$		0			0
$823$		−	9572.33i		−	0.405432i	−0.979238	−	0.202716i	$-0.935023\pi$
$823$		−	9572.33i		−	0.405432i	0.979238	−	0.202716i	$-0.0649768\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	23980.0i		−	1.00830i	−0.863615	−	0.504151i	$-0.831805\pi$
$827$		−	23980.0i		−	1.00830i	0.863615	−	0.504151i	$-0.168195\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		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
$841$	45611.0			1.87015
$842$	20189.7	+	38155.0i	0.826347	+	1.56165i
$843$		0			0
$844$	−11725.0	−	7976.94i	−0.478189	−	0.325329i
$845$		0			0
$846$		0			0
$847$			60987.2i			2.47408i
$848$	11688.9	+	29610.0i	0.473348	+	1.19907i
$849$		0			0
$850$		0			0
$851$		−	18000.0i		−	0.725067i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	−3950.00	+	35532.4i	−0.157720	+	1.41878i
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−21520.0	+	11387.3i	−0.850318	+	0.449946i
$863$		−	20200.0i		−	0.796774i	−0.917217	−	0.398387i	$-0.869570\pi$
$863$		−	20200.0i		−	0.796774i	0.917217	−	0.398387i	$-0.130430\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−16192.0			−0.632078
$870$		0			0
$871$		0			0
$872$	1214.40	+	135.000i	0.0471614	+	0.00524275i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	6550.00			0.252198			0.126099	−	0.992018i	$-0.459754\pi$
$877$	6550.00			0.252198			0.126099	+	0.992018i	$0.459754\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		−	43014.6i		−	1.63936i	−0.572820	−	0.819681i	$-0.694150\pi$
$883$		−	43014.6i		−	1.63936i	0.572820	−	0.819681i	$-0.305850\pi$
$884$		0			0
$885$		0			0
$886$	46450.0	−	24579.0i	1.76131	−	0.931996i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	37926.0			1.43082
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	25882.1	−	7031.50i	0.965021	−	0.262172i
$897$		0			0
$898$	24920.0	+	47094.4i	0.926048	+	1.75007i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	51884.0	+	5767.74i	1.90889	+	0.212204i
$905$		0			0
$906$		0			0
$907$		−	14250.0i		−	0.521680i	−0.965382	−	0.260840i	$-0.916000\pi$
$907$		−	14250.0i		−	0.521680i	0.965382	−	0.260840i	$-0.0839996\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			39632.0i			1.44135i	0.693275	+	0.720673i	$0.256167\pi$
$911$			39632.0i			1.44135i	−0.693275	+	0.720673i	$0.743833\pi$
$912$		0			0
$913$		0			0
$914$	−10596.2	−	20025.0i	−0.383471	−	0.724692i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	51301.1i		−	1.84142i	−0.390244	−	0.920711i	$-0.627609\pi$
$919$		−	51301.1i		−	1.84142i	0.390244	−	0.920711i	$-0.372391\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	56250.0			1.99945
$926$	−45123.3	+	23877.0i	−1.60134	+	0.847351i
$927$		0			0
$928$	−31150.0	+	36379.1i	−1.10188	+	1.28686i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−23907.0	+	35140.0i	−0.840236	+	1.23503i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	19835.2	+	37485.0i	0.690450	+	1.30483i
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−48076.0	−	90855.1i	−1.65231	−	3.12257i
$947$			48820.0i			1.67522i	0.546266	+	0.837612i	$0.316049\pi$
$947$			48820.0i			1.67522i	−0.546266	+	0.837612i	$0.683951\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−51031.3			−1.73459			−0.867295	−	0.497794i	$-0.834143\pi$
$953$	−51031.3			−1.73459			−0.867295	+	0.497794i	$0.834143\pi$
$954$		0			0
$955$		0			0
$956$	48787.7	+	33192.0i	1.65053	+	1.12291i
$957$		0			0
$958$		0			0
$959$		−	14504.0i		−	0.488382i
$960$		0			0
$961$	−29791.0			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	30145.7i		−	1.00250i	−0.865302	−	0.501251i	$-0.832873\pi$
$967$		−	30145.7i		−	1.00250i	0.865302	−	0.501251i	$-0.167127\pi$
$968$	74055.9	+	8232.50i	2.45893	+	0.273350i
$969$		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
$974$	8241.52	−	4361.00i	0.271124	−	0.143466i
$975$		0			0
$976$		0			0
$977$	−48216.2			−1.57889			−0.789443	−	0.613824i	$-0.789631\pi$
$977$	−48216.2			−1.57889			−0.789443	+	0.613824i	$0.789631\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	50930.0	−	26949.6i	1.65503	−	0.875761i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−21377.7			−0.687331
$990$		0			0
$991$			24155.7i			0.774300i	0.922017	+	0.387150i	$0.126540\pi$
$991$			24155.7i			0.774300i	−0.922017	+	0.387150i	$0.873460\pi$
$992$		0			0
$993$		0			0
$994$	−16856.0	−	31854.8i	−0.537867	−	1.01647i
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	52716.6	−	27895.0i	1.67206	−	0.884770i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.4.b.c.55.4	yes	4
3.2	odd	2	inner	252.4.b.c.55.1	✓	4
4.3	odd	2	inner	252.4.b.c.55.3	yes	4
7.6	odd	2	CM	252.4.b.c.55.4	yes	4
12.11	even	2	inner	252.4.b.c.55.2	yes	4
21.20	even	2	inner	252.4.b.c.55.1	✓	4
28.27	even	2	inner	252.4.b.c.55.3	yes	4
84.83	odd	2	inner	252.4.b.c.55.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.b.c.55.1	✓	4	3.2	odd	2	inner
252.4.b.c.55.1	✓	4	21.20	even	2	inner
252.4.b.c.55.2	yes	4	12.11	even	2	inner
252.4.b.c.55.2	yes	4	84.83	odd	2	inner
252.4.b.c.55.3	yes	4	4.3	odd	2	inner
252.4.b.c.55.3	yes	4	28.27	even	2	inner
252.4.b.c.55.4	yes	4	1.1	even	1	trivial
252.4.b.c.55.4	yes	4	7.6	odd	2	CM

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.32288 + 2.50000i) q^{2} +(-4.50000 + 6.61438i) q^{4} -18.5203i q^{7} +(-22.4889 - 2.50000i) q^{8} +O(q^{10})\)	\(q+(1.32288 + 2.50000i) q^{2} +(-4.50000 + 6.61438i) q^{4} -18.5203i q^{7} +(-22.4889 - 2.50000i) q^{8} -68.0000i q^{11} +(46.3006 - 24.5000i) q^{14} +(-23.5000 - 59.5294i) q^{16} +(170.000 - 89.9555i) q^{22} -40.0000i q^{23} +125.000 q^{25} +(122.500 + 83.3412i) q^{28} -264.575 q^{29} +(117.736 - 137.500i) q^{32} +450.000 q^{37} -534.442i q^{43} +(449.778 + 306.000i) q^{44} +(100.000 - 52.9150i) q^{46} -343.000 q^{49} +(165.359 + 312.500i) q^{50} -497.401 q^{53} +(-46.3006 + 416.500i) q^{56} +(-350.000 - 661.438i) q^{58} +(499.500 + 112.444i) q^{64} +809.600i q^{67} -688.000i q^{71} +(595.294 + 1125.00i) q^{74} -1259.38 q^{77} -238.118i q^{79} +(1336.10 - 707.000i) q^{86} +(-170.000 + 1529.24i) q^{88} +(264.575 + 180.000i) q^{92} +(-453.746 - 857.500i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{4}+O(q^{10}) \) 4 * q - 18 * q^4	\( 4 q - 18 q^{4} - 94 q^{16} + 680 q^{22} + 500 q^{25} + 490 q^{28} + 1800 q^{37} + 400 q^{46} - 1372 q^{49} - 1400 q^{58} + 1998 q^{64} - 680 q^{88}+O(q^{100}) \) 4 * q - 18 * q^4 - 94 * q^16 + 680 * q^22 + 500 * q^25 + 490 * q^28 + 1800 * q^37 + 400 * q^46 - 1372 * q^49 - 1400 * q^58 + 1998 * q^64 - 680 * q^88

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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