LMFDB - Embedded newform 252.4.b.a.55.1

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:	$2$
Coefficient field:	$\Q(\sqrt{-7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 2 $ x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 - 1.32288i$ of defining polynomial
Character	$\chi$	$=$	252.55
Dual form			252.4.b.a.55.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 - 1.32288i) q^{2} +(4.50000 + 6.61438i) q^{4} -18.5203i q^{7} +(-2.50000 - 22.4889i) q^{8} +O(q^{10})$	$q+(-2.50000 - 1.32288i) q^{2} +(4.50000 + 6.61438i) q^{4} -18.5203i q^{7} +(-2.50000 - 22.4889i) q^{8} +26.4575i q^{11} +(-24.5000 + 46.3006i) q^{14} +(-23.5000 + 59.5294i) q^{16} +(35.0000 - 66.1438i) q^{22} -216.952i q^{23} +125.000 q^{25} +(122.500 - 83.3412i) q^{28} -166.000 q^{29} +(137.500 - 117.736i) q^{32} -450.000 q^{37} -534.442i q^{43} +(-175.000 + 119.059i) q^{44} +(-287.000 + 542.379i) q^{46} -343.000 q^{49} +(-312.500 - 165.359i) q^{50} -590.000 q^{53} +(-416.500 + 46.3006i) q^{56} +(415.000 + 219.597i) q^{58} +(-499.500 + 112.444i) q^{64} -809.600i q^{67} -978.928i q^{71} +(1125.00 + 595.294i) q^{74} +490.000 q^{77} +238.118i q^{79} +(-707.000 + 1336.10i) q^{86} +(595.000 - 66.1438i) q^{88} +(1435.00 - 976.282i) q^{92} +(857.500 + 453.746i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{2} + 9 q^{4} - 5 q^{8}+O(q^{10}) $ 2 * q - 5 * q^2 + 9 * q^4 - 5 * q^8	$ 2 q - 5 q^{2} + 9 q^{4} - 5 q^{8} - 49 q^{14} - 47 q^{16} + 70 q^{22} + 250 q^{25} + 245 q^{28} - 332 q^{29} + 275 q^{32} - 900 q^{37} - 350 q^{44} - 574 q^{46} - 686 q^{49} - 625 q^{50} - 1180 q^{53} - 833 q^{56} + 830 q^{58} - 999 q^{64} + 2250 q^{74} + 980 q^{77} - 1414 q^{86} + 1190 q^{88} + 2870 q^{92} + 1715 q^{98}+O(q^{100}) $ 2 * q - 5 * q^2 + 9 * q^4 - 5 * q^8 - 49 * q^14 - 47 * q^16 + 70 * q^22 + 250 * q^25 + 245 * q^28 - 332 * q^29 + 275 * q^32 - 900 * q^37 - 350 * q^44 - 574 * q^46 - 686 * q^49 - 625 * q^50 - 1180 * q^53 - 833 * q^56 + 830 * q^58 - 999 * q^64 + 2250 * q^74 + 980 * q^77 - 1414 * q^86 + 1190 * q^88 + 2870 * q^92 + 1715 * q^98

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 $

$2$	−2.50000	−	1.32288i	−0.883883	−	0.467707i
$2$	−2.50000	−	1.32288i	−0.883883	−	0.467707i
$3$		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
$8$	−2.50000	−	22.4889i	−0.110485	−	0.993878i
$9$		0			0
$10$		0			0
$11$			26.4575i			0.725204i	0.931944	+	0.362602i	$0.118111\pi$
$11$			26.4575i			0.725204i	−0.931944	+	0.362602i	$0.881889\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	−24.5000	+	46.3006i	−0.467707	+	0.883883i
$15$		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
$22$	35.0000	−	66.1438i	0.339183	−	0.640996i
$23$		−	216.952i		−	1.96685i	−0.181317	−	0.983425i	$-0.558036\pi$
$23$		−	216.952i		−	1.96685i	0.181317	−	0.983425i	$-0.441964\pi$
$24$		0			0
$25$	125.000			1.00000
$26$		0			0
$27$		0			0
$28$	122.500	−	83.3412i	0.826797	−	0.562500i
$29$	−166.000			−1.06295			−0.531473	−	0.847075i	$-0.678361\pi$
$29$	−166.000			−1.06295			−0.531473	+	0.847075i	$0.678361\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	137.500	−	117.736i	0.759587	−	0.650405i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−450.000			−1.99945			−0.999724	−	0.0235113i	$-0.992515\pi$
$37$	−450.000			−1.99945			−0.999724	+	0.0235113i	$0.992515\pi$
$38$		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	$-0.396592\pi$
$44$	−175.000	+	119.059i	−0.599596	+	0.407927i
$45$		0			0
$46$	−287.000	+	542.379i	−0.919910	+	1.73847i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−343.000			−1.00000
$50$	−312.500	−	165.359i	−0.883883	−	0.467707i
$51$		0			0
$52$		0			0
$53$	−590.000			−1.52911			−0.764554	−	0.644560i	$-0.777041\pi$
$53$	−590.000			−1.52911			−0.764554	+	0.644560i	$0.777041\pi$
$54$		0			0
$55$		0			0
$56$	−416.500	+	46.3006i	−0.993878	+	0.110485i
$57$		0			0
$58$	415.000	+	219.597i	0.939520	+	0.497147i
$59$		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	$-0.735713\pi$
$67$		−	809.600i		−	1.47624i	0.674667	−	0.738122i	$-0.264287\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	978.928i		−	1.63630i	−0.575004	−	0.818151i	$-0.695000\pi$
$71$		−	978.928i		−	1.63630i	0.575004	−	0.818151i	$-0.305000\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	1125.00	+	595.294i	1.76728	+	0.935156i
$75$		0			0
$76$		0			0
$77$	490.000			0.725204
$78$		0			0
$79$			238.118i			0.339118i	0.985520	+	0.169559i	$0.0542343\pi$
$79$			238.118i			0.339118i	−0.985520	+	0.169559i	$0.945766\pi$
$80$		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
$86$	−707.000	+	1336.10i	−0.886486	+	1.67530i
$87$		0			0
$88$	595.000	−	66.1438i	0.720764	−	0.0801244i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	1435.00	−	976.282i	1.62619	−	1.10635i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	857.500	+	453.746i	0.883883	+	0.467707i
$99$		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
$106$	1475.00	+	780.497i	1.35155	+	0.715175i
$107$			1550.41i			1.40078i	0.713759	+	0.700392i	$0.246991\pi$
$107$			1550.41i			1.40078i	−0.713759	+	0.700392i	$0.753009\pi$
$108$		0			0
$109$	54.0000			0.0474519			0.0237260	−	0.999718i	$-0.492447\pi$
$109$	54.0000			0.0474519			0.0237260	+	0.999718i	$0.492447\pi$
$110$		0			0
$111$		0			0
$112$	1102.50	+	435.226i	0.930147	+	0.367188i
$113$	670.000			0.557773			0.278886	−	0.960324i	$-0.410035\pi$
$113$	670.000			0.557773			0.278886	+	0.960324i	$0.410035\pi$
$114$		0			0
$115$		0			0
$116$	−747.000	−	1097.99i	−0.597907	−	0.878841i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	631.000			0.474080
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	2047.81i		−	1.43082i	−0.698706	−	0.715409i	$-0.746240\pi$
$127$		−	2047.81i		−	1.43082i	0.698706	−	0.715409i	$-0.253760\pi$
$128$	1397.50	+	379.665i	0.965021	+	0.262172i
$129$		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
$134$	−1071.00	+	2024.00i	−0.690450	+	1.30483i
$135$		0			0
$136$		0			0
$137$	3110.00			1.93945			0.969727	−	0.244191i	$-0.0785224\pi$
$137$	3110.00			1.93945			0.969727	+	0.244191i	$0.0785224\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−1295.00	+	2447.32i	−0.765310	+	1.44630i
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−2025.00	−	2976.47i	−1.12469	−	1.65314i
$149$	−814.000			−0.447554			−0.223777	−	0.974640i	$-0.571839\pi$
$149$	−814.000			−0.447554			−0.223777	+	0.974640i	$0.571839\pi$
$150$		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
$154$	−1225.00	−	648.209i	−0.640996	−	0.339183i
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$	315.000	−	595.294i	0.158608	−	0.299741i
$159$		0			0
$160$		0			0
$161$	−4018.00			−1.96685
$162$		0			0
$163$			3762.26i			1.80787i	0.427670	+	0.903935i	$0.359335\pi$
$163$			3762.26i			1.80787i	−0.427670	+	0.903935i	$0.640665\pi$
$164$		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
$176$	−1575.00	−	621.752i	−0.674546	−	0.266286i
$177$		0			0
$178$		0			0
$179$		−	4312.57i		−	1.80077i	−0.435099	−	0.900383i	$-0.643287\pi$
$179$		−	4312.57i		−	1.80077i	0.435099	−	0.900383i	$-0.356713\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−4879.00	+	542.379i	−1.95481	+	0.217308i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$			3360.10i			1.27292i	0.771308	+	0.636462i	$0.219603\pi$
$191$			3360.10i			1.27292i	−0.771308	+	0.636462i	$0.780397\pi$
$192$		0			0
$193$	−4590.00			−1.71189			−0.855947	−	0.517064i	$-0.827025\pi$
$193$	−4590.00			−1.71189			−0.855947	+	0.517064i	$0.827025\pi$
$194$		0			0
$195$		0			0
$196$	−1543.50	−	2268.73i	−0.562500	−	0.826797i
$197$	2210.00			0.799269			0.399634	−	0.916675i	$-0.369137\pi$
$197$	2210.00			0.799269			0.399634	+	0.916675i	$0.369137\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	−312.500	−	2811.11i	−0.110485	−	0.993878i
$201$		0			0
$202$		0			0
$203$			3074.36i			1.06295i
$204$		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	$0.906617\pi$
$212$	−2655.00	−	3902.48i	−0.860123	−	1.26426i
$213$		0			0
$214$	2051.00	−	3876.03i	0.655156	−	1.23813i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−135.000	−	71.4353i	−0.0419420	−	0.0221936i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−2180.50	−	2546.54i	−0.650405	−	0.759587i
$225$		0			0
$226$	−1675.00	−	886.327i	−0.493006	−	0.260874i
$227$		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
$232$	415.000	+	3733.16i	0.117440	+	1.05644i
$233$	−4730.00			−1.32993			−0.664963	−	0.746877i	$-0.731553\pi$
$233$	−4730.00			−1.32993			−0.664963	+	0.746877i	$0.731553\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	449.778i		−	0.121731i	−0.998146	−	0.0608655i	$-0.980614\pi$
$239$		−	449.778i		−	0.121731i	0.998146	−	0.0608655i	$-0.0193861\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−1577.50	−	834.735i	−0.419031	−	0.221730i
$243$		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
$253$	5740.00			1.42637
$254$	−2709.00	+	5119.53i	−0.669204	+	1.26468i
$255$		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
$263$		−	4026.83i		−	0.944126i	−0.881565	−	0.472063i	$-0.843509\pi$
$263$		−	4026.83i		−	0.944126i	0.881565	−	0.472063i	$-0.156491\pi$
$264$		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
$274$	−7775.00	−	4114.14i	−1.71425	−	0.907097i
$275$			3307.19i			0.725204i
$276$		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
$281$	4342.00			0.921786			0.460893	−	0.887456i	$-0.347529\pi$
$281$	4342.00			0.921786			0.460893	+	0.887456i	$0.347529\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$	6475.00	−	4405.18i	1.35289	−	0.920419i
$285$		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
$296$	1125.00	+	10120.0i	0.220910	+	1.98721i
$297$		0			0
$298$	2035.00	+	1076.82i	0.395585	+	0.209324i
$299$		0			0
$300$		0			0
$301$	−9898.00			−1.89539
$302$	2975.00	−	5622.22i	0.566861	−	1.07127i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	2205.00	+	3241.05i	0.407927	+	0.599596i
$309$		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
$317$	6970.00			1.23493			0.617467	−	0.786597i	$-0.288159\pi$
$317$	6970.00			1.23493			0.617467	+	0.786597i	$0.288159\pi$
$318$		0			0
$319$		−	4391.95i		−	0.770852i
$320$		0			0
$321$		0			0
$322$	10045.0	+	5315.31i	1.73847	+	0.919910i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	4977.00	−	9405.65i	0.845554	−	1.59795i
$327$		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	$-0.413262\pi$
$337$	3330.00			0.538269			0.269135	+	0.963103i	$0.413262\pi$
$338$	−5492.50	−	2906.36i	−0.883883	−	0.467707i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			6352.45i			1.00000i
$344$	−12019.0	+	1336.10i	−1.88378	+	0.209413i
$345$		0			0
$346$		0			0
$347$			12260.4i			1.89675i	0.317146	+	0.948377i	$0.397275\pi$
$347$			12260.4i			1.89675i	−0.317146	+	0.948377i	$0.602725\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−3062.50	+	5787.58i	−0.467707	+	0.883883i
$351$		0			0
$352$	3115.00	+	3637.91i	0.471676	+	0.550856i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−5705.00	+	10781.4i	−0.842231	+	1.59167i
$359$		−	10927.0i		−	1.60641i	−0.595700	−	0.803207i	$-0.703125\pi$
$359$		−	10927.0i		−	1.60641i	0.595700	−	0.803207i	$-0.296875\pi$
$360$		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	$0.5\pi$
$368$	12915.0	+	5098.36i	1.82946	+	0.722203i
$369$		0			0
$370$		0			0
$371$			10927.0i			1.52911i
$372$		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
$382$	4445.00	−	8400.26i	0.595356	−	1.12512i
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$	11475.0	+	6072.00i	1.51311	+	0.800665i
$387$		0			0
$388$		0			0
$389$	−10526.0			−1.37195			−0.685976	−	0.727624i	$-0.740625\pi$
$389$	−10526.0			−1.37195			−0.685976	+	0.727624i	$0.740625\pi$
$390$		0			0
$391$		0			0
$392$	857.500	+	7713.69i	0.110485	+	0.993878i
$393$		0			0
$394$	−5525.00	−	2923.56i	−0.706461	−	0.373824i
$395$		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
$401$	1598.00			0.199003			0.0995016	−	0.995037i	$-0.468275\pi$
$401$	1598.00			0.199003			0.0995016	+	0.995037i	$0.468275\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	4067.00	−	7685.91i	0.497147	−	0.939520i
$407$		−	11905.9i		−	1.45001i
$408$		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	$0.155247\pi$
$422$	2345.00	−	4431.63i	0.270504	−	0.511205i
$423$		0			0
$424$	1475.00	+	13268.4i	0.168944	+	1.51975i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−10255.0	+	6976.85i	−1.15816	+	0.787941i
$429$		0			0
$430$		0			0
$431$		−	15689.3i		−	1.75343i	−0.481012	−	0.876714i	$-0.659731\pi$
$431$		−	15689.3i		−	1.75343i	0.481012	−	0.876714i	$-0.340269\pi$
$432$		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
$443$			1592.74i			0.170820i	0.996346	+	0.0854102i	$0.0272201\pi$
$443$			1592.74i			0.170820i	−0.996346	+	0.0854102i	$0.972780\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	2082.50	+	9250.87i	0.219618	+	0.975586i
$449$	2686.00			0.282317			0.141158	−	0.989987i	$-0.454917\pi$
$449$	2686.00			0.282317			0.141158	+	0.989987i	$0.454917\pi$
$450$		0			0
$451$		0			0
$452$	3015.00	+	4431.63i	0.313747	+	0.461165i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	8010.00			0.819895			0.409947	−	0.912109i	$-0.365547\pi$
$457$	8010.00			0.819895			0.409947	+	0.912109i	$0.365547\pi$
$458$		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	$-0.639229\pi$
$463$		−	18049.3i		−	1.81171i	0.423585	−	0.905856i	$-0.360771\pi$
$464$	3901.00	−	9881.88i	0.390300	−	0.988696i
$465$		0			0
$466$	11825.0	+	6257.20i	1.17550	+	0.622016i
$467$		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
$473$	14140.0			1.37454
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−595.000	+	1124.44i	−0.0569344	+	0.107596i
$479$		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
$484$	2839.50	+	4173.67i	0.266670	+	0.391968i
$485$		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
$491$			7646.22i			0.702788i	0.936228	+	0.351394i	$0.114292\pi$
$491$			7646.22i			0.702788i	−0.936228	+	0.351394i	$0.885708\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18130.0			−1.63630
$498$		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
$506$	−14350.0	−	7593.31i	−1.26074	−	0.667122i
$507$		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
$512$	3777.50	+	10952.1i	0.326062	+	0.945349i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	11025.0	−	20835.3i	0.935156	−	1.76728i
$519$		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
$526$	−5327.00	+	10067.1i	−0.441575	+	0.834498i
$527$		0			0
$528$		0			0
$529$	−34901.0			−2.86850
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−18207.0	+	2024.00i	−1.46721	+	0.163103i
$537$		0			0
$538$		0			0
$539$		−	9074.93i		−	0.725204i
$540$		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	$0.330644\pi$
$547$			22049.7i			1.72354i	−0.507299	+	0.861770i	$0.669356\pi$
$548$	13995.0	+	20570.7i	1.09094	+	1.60354i
$549$		0			0
$550$	4375.00	−	8267.97i	0.339183	−	0.640996i
$551$		0			0
$552$		0			0
$553$	4410.00			0.339118
$554$	−18275.0	−	9670.22i	−1.40150	−	0.741603i
$555$		0			0
$556$		0			0
$557$	−20470.0			−1.55717			−0.778583	−	0.627541i	$-0.784061\pi$
$557$	−20470.0			−1.55717			−0.778583	+	0.627541i	$0.784061\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−10855.0	−	5743.93i	−0.814752	−	0.431126i
$563$		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
$568$	−22015.0	+	2447.32i	−1.62628	+	0.180787i
$569$	−26906.0			−1.98235			−0.991176	−	0.132553i	$-0.957683\pi$
$569$	−26906.0			−1.98235			−0.991176	+	0.132553i	$0.957683\pi$
$570$		0			0
$571$			26431.1i			1.93714i	0.248747	+	0.968569i	$0.419981\pi$
$571$			26431.1i			1.93714i	−0.248747	+	0.968569i	$0.580019\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	27119.0i		−	1.96685i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−12282.5	−	6499.29i	−0.883883	−	0.467707i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	15609.9i		−	1.10891i
$584$		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
$596$	−3663.00	−	5384.10i	−0.251749	−	0.370036i
$597$		0			0
$598$		0			0
$599$			15742.2i			1.07381i	0.843644	+	0.536903i	$0.180406\pi$
$599$			15742.2i			1.07381i	−0.843644	+	0.536903i	$0.819594\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	24745.0	+	13093.8i	1.67530	+	0.886486i
$603$		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
$616$	−1225.00	−	11019.6i	−0.0801244	−	0.720764i
$617$	30550.0			1.99335			0.996675	−	0.0814823i	$-0.0259654\pi$
$617$	30550.0			1.99335			0.996675	+	0.0814823i	$0.0259654\pi$
$618$		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	$-0.809512\pi$
$631$		−	17858.8i		−	1.12670i	0.826218	−	0.563351i	$-0.190488\pi$
$632$	5355.00	−	595.294i	0.337042	−	0.0374676i
$633$		0			0
$634$	−17425.0	−	9220.44i	−1.09154	−	0.577588i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−5810.00	+	10979.9i	−0.360533	+	0.681343i
$639$		0			0
$640$		0			0
$641$	8878.00			0.547051			0.273526	−	0.961865i	$-0.411810\pi$
$641$	8878.00			0.547051			0.273526	+	0.961865i	$0.411810\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	−18081.0	−	26576.6i	−1.10635	−	1.62619i
$645$		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
$653$	27050.0			1.62105			0.810527	−	0.585701i	$-0.199181\pi$
$653$	27050.0			1.62105			0.810527	+	0.585701i	$0.199181\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			33786.2i			1.99716i	0.0533186	+	0.998578i	$0.483020\pi$
$659$			33786.2i			1.99716i	−0.0533186	+	0.998578i	$0.516980\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$	−6755.00	+	12765.8i	−0.396587	+	0.749479i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$			36014.0i			2.09065i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	33570.0			1.92278			0.961388	−	0.275196i	$-0.0887428\pi$
$673$	33570.0			1.92278			0.961388	+	0.275196i	$0.0887428\pi$
$674$	−8325.00	−	4405.18i	−0.475767	−	0.251752i
$675$		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
$683$			10694.1i			0.599121i	0.954077	+	0.299560i	$0.0968400\pi$
$683$			10694.1i			0.599121i	−0.954077	+	0.299560i	$0.903160\pi$
$684$		0			0
$685$		0			0
$686$	8403.50	−	15881.1i	0.467707	−	0.883883i
$687$		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
$694$	16219.0	−	30651.0i	0.887125	−	1.67651i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	15312.5	−	10417.6i	0.826797	−	0.562500i
$701$	−4198.00			−0.226186			−0.113093	−	0.993584i	$-0.536076\pi$
$701$	−4198.00			−0.226186			−0.113093	+	0.993584i	$0.536076\pi$
$702$		0			0
$703$		0			0
$704$	−2975.00	−	13215.5i	−0.159268	−	0.707499i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−12546.0			−0.664563			−0.332281	−	0.943180i	$-0.607818\pi$
$709$	−12546.0			−0.664563			−0.332281	+	0.943180i	$0.607818\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	28525.0	−	19406.6i	1.48887	−	1.01293i
$717$		0			0
$718$	−14455.0	+	27317.4i	−0.751331	+	1.41988i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	17147.5	+	9073.60i	0.883883	+	0.467707i
$723$		0			0
$724$		0			0
$725$	−20750.0			−1.06295
$726$		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
$736$	−25543.0	−	29830.8i	−1.27925	−	1.49399i
$737$	21420.0			1.07058
$738$		0			0
$739$			31193.4i			1.55273i	0.630283	+	0.776365i	$0.282939\pi$
$739$			31193.4i			1.55273i	−0.630283	+	0.776365i	$0.717061\pi$
$740$		0			0
$741$		0			0
$742$	14455.0	−	27317.4i	0.715175	−	1.35155i
$743$			31743.7i			1.56738i	0.621151	+	0.783691i	$0.286665\pi$
$743$			31743.7i			1.56738i	−0.621151	+	0.783691i	$0.713335\pi$
$744$		0			0
$745$		0			0
$746$	34925.0	+	18480.6i	1.71407	+	0.907000i
$747$		0			0
$748$		0			0
$749$	28714.0			1.40078
$750$		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	$-0.184806\pi$
$757$	34830.0			1.67228			0.836141	+	0.548514i	$0.184806\pi$
$758$	−11515.0	+	21761.3i	−0.551773	+	1.04275i
$759$		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
$764$	−22225.0	+	15120.5i	−1.05245	+	0.716020i
$765$		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
$778$	26315.0	+	13924.6i	1.21265	+	0.641672i
$779$		0			0
$780$		0			0
$781$	25900.0			1.18665
$782$		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	$0.5\pi$
$788$	9945.00	+	14617.8i	0.449589	+	0.660833i
$789$		0			0
$790$		0			0
$791$		−	12408.6i		−	0.557773i
$792$		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
$800$	17187.5	−	14717.0i	0.759587	−	0.650405i
$801$		0			0
$802$	−3995.00	−	2113.96i	−0.175896	−	0.0930753i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−37354.0			−1.62336			−0.811679	−	0.584104i	$-0.801446\pi$
$809$	−37354.0			−1.62336			−0.811679	+	0.584104i	$0.801446\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−20335.0	+	13834.6i	−0.878841	+	0.597907i
$813$		0			0
$814$	−15750.0	+	29764.7i	−0.678178	+	1.28164i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	43538.0			1.85078			0.925388	−	0.379022i	$-0.123739\pi$
$821$	43538.0			1.85078			0.925388	+	0.379022i	$0.123739\pi$
$822$		0			0
$823$			9572.33i			0.405432i	0.979238	+	0.202716i	$0.0649768\pi$
$823$			9572.33i			0.405432i	−0.979238	+	0.202716i	$0.935023\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	41077.9i		−	1.72723i	−0.504151	−	0.863615i	$-0.668195\pi$
$827$		−	41077.9i		−	1.72723i	0.504151	−	0.863615i	$-0.331805\pi$
$828$		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
$841$	3167.00			0.129854
$842$	−38155.0	−	20189.7i	−1.56165	−	0.826347i
$843$		0			0
$844$	−11725.0	+	7976.94i	−0.478189	+	0.325329i
$845$		0			0
$846$		0			0
$847$		−	11686.3i		−	0.474080i
$848$	13865.0	−	35122.3i	0.561469	−	1.42230i
$849$		0			0
$850$		0			0
$851$			97628.2i			3.93261i
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	34867.0	−	3876.03i	1.39221	−	0.154766i
$857$		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
$862$	−20755.0	+	39223.3i	−0.820091	+	1.54983i
$863$		−	46507.0i		−	1.83443i	−0.398387	−	0.917217i	$-0.630430\pi$
$863$		−	46507.0i		−	1.83443i	0.398387	−	0.917217i	$-0.369570\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6300.00			−0.245930
$870$		0			0
$871$		0			0
$872$	−135.000	−	1214.40i	−0.00524275	−	0.0471614i
$873$		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
$886$	2107.00	−	3981.86i	0.0798940	−	0.150985i
$887$		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
$896$	7031.50	−	25882.1i	0.262172	−	0.965021i
$897$		0			0
$898$	−6715.00	−	3553.24i	−0.249535	−	0.132042i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1675.00	−	15067.6i	−0.0616257	−	0.554358i
$905$		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
$911$		−	38125.3i		−	1.38655i	−0.720673	−	0.693275i	$-0.756167\pi$
$911$		−	38125.3i		−	1.38655i	0.720673	−	0.693275i	$-0.243833\pi$
$912$		0			0
$913$		0			0
$914$	−20025.0	−	10596.2i	−0.724692	−	0.383471i
$915$		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
$926$	−23877.0	+	45123.3i	−0.847351	+	1.60134i
$927$		0			0
$928$	−22825.0	+	19544.2i	−0.807400	+	0.691346i
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	−21285.0	−	31286.0i	−0.748083	−	1.09958i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	37485.0	+	19835.2i	1.30483	+	0.690450i
$939$		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
$946$	−35350.0	−	18705.5i	−1.21493	−	0.642883i
$947$			31839.0i			1.09253i	0.837612	+	0.546266i	$0.183951\pi$
$947$			31839.0i			1.09253i	−0.837612	+	0.546266i	$0.816049\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−29290.0			−0.995589			−0.497794	−	0.867295i	$-0.665857\pi$
$953$	−29290.0			−0.995589			−0.497794	+	0.867295i	$0.665857\pi$
$954$		0			0
$955$		0			0
$956$	2975.00	−	2024.00i	0.100647	−	0.0684737i
$957$		0			0
$958$		0			0
$959$		−	57598.0i		−	1.93945i
$960$		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	$0.167127\pi$
$967$			30145.7i			1.00250i	−0.865302	+	0.501251i	$0.832873\pi$
$968$	−1577.50	−	14190.5i	−0.0523789	−	0.471177i
$969$		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
$974$	−4361.00	+	8241.52i	−0.143466	+	0.271124i
$975$		0			0
$976$		0			0
$977$	−37490.0			−1.22765			−0.613824	−	0.789443i	$-0.710369\pi$
$977$	−37490.0			−1.22765			−0.613824	+	0.789443i	$0.710369\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	10115.0	−	19115.6i	0.328699	−	0.621183i
$983$		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
$989$	−115948.			−3.72794
$990$		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
$994$	45325.0	+	23983.7i	1.44630	+	0.765310i
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$	−27895.0	+	52716.6i	−0.884770	+	1.67206i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.4.b.a.55.1		2
3.2	odd	2		28.4.d.a.27.2	yes	2
4.3	odd	2	inner	252.4.b.a.55.2		2
7.6	odd	2	CM	252.4.b.a.55.1		2
12.11	even	2		28.4.d.a.27.1	✓	2
21.2	odd	6		196.4.f.a.31.2		4
21.5	even	6		196.4.f.a.31.2		4
21.11	odd	6		196.4.f.a.19.1		4
21.17	even	6		196.4.f.a.19.1		4
21.20	even	2		28.4.d.a.27.2	yes	2
24.5	odd	2		448.4.f.a.447.1		2
24.11	even	2		448.4.f.a.447.2		2
28.27	even	2	inner	252.4.b.a.55.2		2
84.11	even	6		196.4.f.a.19.2		4
84.23	even	6		196.4.f.a.31.1		4
84.47	odd	6		196.4.f.a.31.1		4
84.59	odd	6		196.4.f.a.19.2		4
84.83	odd	2		28.4.d.a.27.1	✓	2
168.83	odd	2		448.4.f.a.447.2		2
168.125	even	2		448.4.f.a.447.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.4.d.a.27.1	✓	2	12.11	even	2
28.4.d.a.27.1	✓	2	84.83	odd	2
28.4.d.a.27.2	yes	2	3.2	odd	2
28.4.d.a.27.2	yes	2	21.20	even	2
196.4.f.a.19.1		4	21.11	odd	6
196.4.f.a.19.1		4	21.17	even	6
196.4.f.a.19.2		4	84.11	even	6
196.4.f.a.19.2		4	84.59	odd	6
196.4.f.a.31.1		4	84.23	even	6
196.4.f.a.31.1		4	84.47	odd	6
196.4.f.a.31.2		4	21.2	odd	6
196.4.f.a.31.2		4	21.5	even	6
252.4.b.a.55.1		2	1.1	even	1	trivial
252.4.b.a.55.1		2	7.6	odd	2	CM
252.4.b.a.55.2		2	4.3	odd	2	inner
252.4.b.a.55.2		2	28.27	even	2	inner
448.4.f.a.447.1		2	24.5	odd	2
448.4.f.a.447.1		2	168.125	even	2
448.4.f.a.447.2		2	24.11	even	2
448.4.f.a.447.2		2	168.83	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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.50000 - 1.32288i) q^{2} +(4.50000 + 6.61438i) q^{4} -18.5203i q^{7} +(-2.50000 - 22.4889i) q^{8} +O(q^{10})\)	\(q+(-2.50000 - 1.32288i) q^{2} +(4.50000 + 6.61438i) q^{4} -18.5203i q^{7} +(-2.50000 - 22.4889i) q^{8} +26.4575i q^{11} +(-24.5000 + 46.3006i) q^{14} +(-23.5000 + 59.5294i) q^{16} +(35.0000 - 66.1438i) q^{22} -216.952i q^{23} +125.000 q^{25} +(122.500 - 83.3412i) q^{28} -166.000 q^{29} +(137.500 - 117.736i) q^{32} -450.000 q^{37} -534.442i q^{43} +(-175.000 + 119.059i) q^{44} +(-287.000 + 542.379i) q^{46} -343.000 q^{49} +(-312.500 - 165.359i) q^{50} -590.000 q^{53} +(-416.500 + 46.3006i) q^{56} +(415.000 + 219.597i) q^{58} +(-499.500 + 112.444i) q^{64} -809.600i q^{67} -978.928i q^{71} +(1125.00 + 595.294i) q^{74} +490.000 q^{77} +238.118i q^{79} +(-707.000 + 1336.10i) q^{86} +(595.000 - 66.1438i) q^{88} +(1435.00 - 976.282i) q^{92} +(857.500 + 453.746i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{2} + 9 q^{4} - 5 q^{8}+O(q^{10}) \) 2 * q - 5 * q^2 + 9 * q^4 - 5 * q^8	\( 2 q - 5 q^{2} + 9 q^{4} - 5 q^{8} - 49 q^{14} - 47 q^{16} + 70 q^{22} + 250 q^{25} + 245 q^{28} - 332 q^{29} + 275 q^{32} - 900 q^{37} - 350 q^{44} - 574 q^{46} - 686 q^{49} - 625 q^{50} - 1180 q^{53} - 833 q^{56} + 830 q^{58} - 999 q^{64} + 2250 q^{74} + 980 q^{77} - 1414 q^{86} + 1190 q^{88} + 2870 q^{92} + 1715 q^{98}+O(q^{100}) \) 2 * q - 5 * q^2 + 9 * q^4 - 5 * q^8 - 49 * q^14 - 47 * q^16 + 70 * q^22 + 250 * q^25 + 245 * q^28 - 332 * q^29 + 275 * q^32 - 900 * q^37 - 350 * q^44 - 574 * q^46 - 686 * q^49 - 625 * q^50 - 1180 * q^53 - 833 * q^56 + 830 * q^58 - 999 * q^64 + 2250 * q^74 + 980 * q^77 - 1414 * q^86 + 1190 * q^88 + 2870 * q^92 + 1715 * q^98

Introduction

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