LMFDB - Embedded newform 1200.3.bg.c.193.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.bg (of order $4$, degree $2$, not 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:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 600)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.2
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.c.1057.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +(-0.325765 + 0.325765i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{3} +(-0.325765 + 0.325765i) q^{7} +3.00000i q^{9} +11.7980 q^{11} +(3.67423 + 3.67423i) q^{13} +(11.3485 - 11.3485i) q^{17} -30.3939i q^{19} -0.797959 q^{21} +(-9.55051 - 9.55051i) q^{23} +(-3.67423 + 3.67423i) q^{27} +15.3939i q^{29} +21.2020 q^{31} +(14.4495 + 14.4495i) q^{33} +(3.10102 - 3.10102i) q^{37} +9.00000i q^{39} +18.2020 q^{41} +(-19.4268 - 19.4268i) q^{43} +(-27.7526 + 27.7526i) q^{47} +48.7878i q^{49} +27.7980 q^{51} +(56.8990 + 56.8990i) q^{53} +(37.2247 - 37.2247i) q^{57} -82.0000i q^{59} +94.5959 q^{61} +(-0.977296 - 0.977296i) q^{63} +(12.7298 - 12.7298i) q^{67} -23.3939i q^{69} +77.7980 q^{71} +(90.2929 + 90.2929i) q^{73} +(-3.84337 + 3.84337i) q^{77} -103.980i q^{79} -9.00000 q^{81} +(-22.8536 - 22.8536i) q^{83} +(-18.8536 + 18.8536i) q^{87} -159.596i q^{89} -2.39388 q^{91} +(25.9671 + 25.9671i) q^{93} +(-56.7298 + 56.7298i) q^{97} +35.3939i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{7}+O(q^{10}) $ 4 * q - 16 * q^7	$ 4 q - 16 q^{7} + 8 q^{11} + 16 q^{17} + 36 q^{21} - 48 q^{23} + 124 q^{31} + 48 q^{33} + 32 q^{37} + 112 q^{41} - 112 q^{43} - 160 q^{47} + 72 q^{51} + 208 q^{53} + 144 q^{57} + 300 q^{61} - 48 q^{63} + 144 q^{67} + 272 q^{71} + 224 q^{73} + 112 q^{77} - 36 q^{81} - 160 q^{83} - 144 q^{87} + 108 q^{91} - 48 q^{93} - 320 q^{97}+O(q^{100}) $ 4 * q - 16 * q^7 + 8 * q^11 + 16 * q^17 + 36 * q^21 - 48 * q^23 + 124 * q^31 + 48 * q^33 + 32 * q^37 + 112 * q^41 - 112 * q^43 - 160 * q^47 + 72 * q^51 + 208 * q^53 + 144 * q^57 + 300 * q^61 - 48 * q^63 + 144 * q^67 + 272 * q^71 + 224 * q^73 + 112 * q^77 - 36 * q^81 - 160 * q^83 - 144 * q^87 + 108 * q^91 - 48 * q^93 - 320 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$3$	1.22474	+	1.22474i	0.408248	+	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.325765	+	0.325765i	−0.0465379	+	0.0465379i	−0.729993	−	0.683455i	$-0.760477\pi$
$7$	−0.325765	+	0.325765i	−0.0465379	+	0.0465379i	0.683455	+	0.729993i	$0.260477\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	11.7980			1.07254			0.536271	−	0.844046i	$-0.319833\pi$
$11$	11.7980			1.07254			0.536271	+	0.844046i	$0.319833\pi$
$12$		0			0
$13$	3.67423	+	3.67423i	0.282633	+	0.282633i	0.834158	−	0.551525i	$-0.185954\pi$
$13$	3.67423	+	3.67423i	0.282633	+	0.282633i	−0.551525	+	0.834158i	$0.685954\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	11.3485	−	11.3485i	0.667557	−	0.667557i	−0.289593	−	0.957150i	$-0.593520\pi$
$17$	11.3485	−	11.3485i	0.667557	−	0.667557i	0.957150	+	0.289593i	$0.0935199\pi$
$18$		0			0
$19$		−	30.3939i		−	1.59968i	−0.600215	−	0.799839i	$-0.704918\pi$
$19$		−	30.3939i		−	1.59968i	0.600215	−	0.799839i	$-0.295082\pi$
$20$		0			0
$21$	−0.797959			−0.0379980
$22$		0			0
$23$	−9.55051	−	9.55051i	−0.415240	−	0.415240i	0.468320	−	0.883559i	$-0.344860\pi$
$23$	−9.55051	−	9.55051i	−0.415240	−	0.415240i	−0.883559	+	0.468320i	$0.844860\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$		0			0
$29$			15.3939i			0.530823i	0.964135	+	0.265412i	$0.0855079\pi$
$29$			15.3939i			0.530823i	−0.964135	+	0.265412i	$0.914492\pi$
$30$		0			0
$31$	21.2020			0.683937			0.341968	−	0.939711i	$-0.388906\pi$
$31$	21.2020			0.683937			0.341968	+	0.939711i	$0.388906\pi$
$32$		0			0
$33$	14.4495	+	14.4495i	0.437863	+	0.437863i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	3.10102	−	3.10102i	0.0838114	−	0.0838114i	−0.663958	−	0.747770i	$-0.731125\pi$
$37$	3.10102	−	3.10102i	0.0838114	−	0.0838114i	0.747770	+	0.663958i	$0.231125\pi$
$38$		0			0
$39$			9.00000i			0.230769i
$40$		0			0
$41$	18.2020			0.443952			0.221976	−	0.975052i	$-0.428749\pi$
$41$	18.2020			0.443952			0.221976	+	0.975052i	$0.428749\pi$
$42$		0			0
$43$	−19.4268	−	19.4268i	−0.451786	−	0.451786i	0.444161	−	0.895947i	$-0.353502\pi$
$43$	−19.4268	−	19.4268i	−0.451786	−	0.451786i	−0.895947	+	0.444161i	$0.853502\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−27.7526	+	27.7526i	−0.590480	+	0.590480i	−0.937761	−	0.347281i	$-0.887105\pi$
$47$	−27.7526	+	27.7526i	−0.590480	+	0.590480i	0.347281	+	0.937761i	$0.387105\pi$
$48$		0			0
$49$			48.7878i			0.995668i
$50$		0			0
$51$	27.7980			0.545058
$52$		0			0
$53$	56.8990	+	56.8990i	1.07357	+	1.07357i	0.997070	+	0.0764957i	$0.0243732\pi$
$53$	56.8990	+	56.8990i	1.07357	+	1.07357i	0.0764957	+	0.997070i	$0.475627\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	37.2247	−	37.2247i	0.653066	−	0.653066i
$58$		0			0
$59$		−	82.0000i		−	1.38983i	−0.719092	−	0.694915i	$-0.755442\pi$
$59$		−	82.0000i		−	1.38983i	0.719092	−	0.694915i	$-0.244558\pi$
$60$		0			0
$61$	94.5959			1.55075			0.775376	−	0.631499i	$-0.217560\pi$
$61$	94.5959			1.55075			0.775376	+	0.631499i	$0.217560\pi$
$62$		0			0
$63$	−0.977296	−	0.977296i	−0.0155126	−	0.0155126i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	12.7298	−	12.7298i	0.189998	−	0.189998i	−0.605697	−	0.795695i	$-0.707106\pi$
$67$	12.7298	−	12.7298i	0.189998	−	0.189998i	0.795695	+	0.605697i	$0.207106\pi$
$68$		0			0
$69$		−	23.3939i		−	0.339042i
$70$		0			0
$71$	77.7980			1.09575			0.547873	−	0.836562i	$-0.315438\pi$
$71$	77.7980			1.09575			0.547873	+	0.836562i	$0.315438\pi$
$72$		0			0
$73$	90.2929	+	90.2929i	1.23689	+	1.23689i	0.961266	+	0.275622i	$0.0888840\pi$
$73$	90.2929	+	90.2929i	1.23689	+	1.23689i	0.275622	+	0.961266i	$0.411116\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−3.84337	+	3.84337i	−0.0499139	+	0.0499139i
$78$		0			0
$79$		−	103.980i		−	1.31620i	−0.752932	−	0.658099i	$-0.771361\pi$
$79$		−	103.980i		−	1.31620i	0.752932	−	0.658099i	$-0.228639\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−22.8536	−	22.8536i	−0.275344	−	0.275344i	0.555903	−	0.831247i	$-0.312373\pi$
$83$	−22.8536	−	22.8536i	−0.275344	−	0.275344i	−0.831247	+	0.555903i	$0.812373\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−18.8536	+	18.8536i	−0.216708	+	0.216708i
$88$		0			0
$89$		−	159.596i		−	1.79321i	−0.442829	−	0.896606i	$-0.646025\pi$
$89$		−	159.596i		−	1.79321i	0.442829	−	0.896606i	$-0.353975\pi$
$90$		0			0
$91$	−2.39388			−0.0263063
$92$		0			0
$93$	25.9671	+	25.9671i	0.279216	+	0.279216i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−56.7298	+	56.7298i	−0.584844	+	0.584844i	−0.936230	−	0.351387i	$-0.885710\pi$
$97$	−56.7298	+	56.7298i	−0.584844	+	0.584844i	0.351387	+	0.936230i	$0.385710\pi$
$98$		0			0
$99$			35.3939i			0.357514i
$100$		0			0
$101$	46.2020			0.457446			0.228723	−	0.973492i	$-0.426545\pi$
$101$	46.2020			0.457446			0.228723	+	0.973492i	$0.426545\pi$
$102$		0			0
$103$	141.576	+	141.576i	1.37452	+	1.37452i	0.853615	+	0.520905i	$0.174405\pi$
$103$	141.576	+	141.576i	1.37452	+	1.37452i	0.520905	+	0.853615i	$0.325595\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−114.384	+	114.384i	−1.06901	+	1.06901i	−0.0715708	+	0.997436i	$0.522801\pi$
$107$	−114.384	+	114.384i	−1.06901	+	1.06901i	−0.997436	+	0.0715708i	$0.977199\pi$
$108$		0			0
$109$			160.171i			1.46946i	0.678358	+	0.734731i	$0.262692\pi$
$109$			160.171i			1.46946i	−0.678358	+	0.734731i	$0.737308\pi$
$110$		0			0
$111$	7.59592			0.0684317
$112$		0			0
$113$	−21.2122	−	21.2122i	−0.187719	−	0.187719i	0.606990	−	0.794709i	$-0.292377\pi$
$113$	−21.2122	−	21.2122i	−0.187719	−	0.187719i	−0.794709	+	0.606990i	$0.792377\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−11.0227	+	11.0227i	−0.0942111	+	0.0942111i
$118$		0			0
$119$			7.39388i			0.0621334i
$120$		0			0
$121$	18.1918			0.150346
$122$		0			0
$123$	22.2929	+	22.2929i	0.181243	+	0.181243i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	112.495	−	112.495i	0.885787	−	0.885787i	−0.108329	−	0.994115i	$-0.534550\pi$
$127$	112.495	−	112.495i	0.885787	−	0.885787i	0.994115	+	0.108329i	$0.0345498\pi$
$128$		0			0
$129$		−	47.5857i		−	0.368881i
$130$		0			0
$131$	144.586			1.10371			0.551854	−	0.833941i	$-0.313921\pi$
$131$	144.586			1.10371			0.551854	+	0.833941i	$0.313921\pi$
$132$		0			0
$133$	9.90127	+	9.90127i	0.0744457	+	0.0744457i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	15.6617	−	15.6617i	0.114319	−	0.114319i	−0.647633	−	0.761952i	$-0.724241\pi$
$137$	15.6617	−	15.6617i	0.114319	−	0.114319i	0.761952	+	0.647633i	$0.224241\pi$
$138$		0			0
$139$		−	97.1918i		−	0.699222i	−0.936895	−	0.349611i	$-0.886314\pi$
$139$		−	97.1918i		−	0.699222i	0.936895	−	0.349611i	$-0.113686\pi$
$140$		0			0
$141$	−67.9796			−0.482125
$142$		0			0
$143$	43.3485	+	43.3485i	0.303136	+	0.303136i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−59.7526	+	59.7526i	−0.406480	+	0.406480i
$148$		0			0
$149$		−	177.151i		−	1.18893i	−0.804120	−	0.594466i	$-0.797363\pi$
$149$		−	177.151i		−	1.18893i	0.804120	−	0.594466i	$-0.202637\pi$
$150$		0			0
$151$	−112.778			−0.746871			−0.373436	−	0.927656i	$-0.621820\pi$
$151$	−112.778			−0.746871			−0.373436	+	0.927656i	$0.621820\pi$
$152$		0			0
$153$	34.0454	+	34.0454i	0.222519	+	0.222519i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−152.911	+	152.911i	−0.973958	+	0.973958i	−0.999669	−	0.0257110i	$-0.991815\pi$
$157$	−152.911	+	152.911i	−0.973958	+	0.973958i	0.0257110	+	0.999669i	$0.491815\pi$
$158$		0			0
$159$			139.373i			0.876563i
$160$		0			0
$161$	6.22245			0.0386488
$162$		0			0
$163$	156.573	+	156.573i	0.960572	+	0.960572i	0.999252	−	0.0386798i	$-0.0123152\pi$
$163$	156.573	+	156.573i	0.960572	+	0.960572i	−0.0386798	+	0.999252i	$0.512315\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.81837	−	3.81837i	0.0228645	−	0.0228645i	−0.695582	−	0.718447i	$-0.744853\pi$
$167$	3.81837	−	3.81837i	0.0228645	−	0.0228645i	0.718447	+	0.695582i	$0.244853\pi$
$168$		0			0
$169$		−	142.000i		−	0.840237i
$170$		0			0
$171$	91.1816			0.533226
$172$		0			0
$173$	−157.015	−	157.015i	−0.907600	−	0.907600i	0.0884782	−	0.996078i	$-0.471800\pi$
$173$	−157.015	−	157.015i	−0.907600	−	0.907600i	−0.996078	+	0.0884782i	$0.971800\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	100.429	−	100.429i	0.567396	−	0.567396i
$178$		0			0
$179$		−	13.2327i		−	0.0739255i	−0.999317	−	0.0369627i	$-0.988232\pi$
$179$		−	13.2327i		−	0.0739255i	0.999317	−	0.0369627i	$-0.0117683\pi$
$180$		0			0
$181$	236.959			1.30917			0.654583	−	0.755990i	$-0.272844\pi$
$181$	236.959			1.30917			0.654583	+	0.755990i	$0.272844\pi$
$182$		0			0
$183$	115.856	+	115.856i	0.633092	+	0.633092i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	133.889	−	133.889i	0.715983	−	0.715983i
$188$		0			0
$189$		−	2.39388i		−	0.0126660i
$190$		0			0
$191$	24.2020			0.126712			0.0633561	−	0.997991i	$-0.479820\pi$
$191$	24.2020			0.126712			0.0633561	+	0.997991i	$0.479820\pi$
$192$		0			0
$193$	−63.7401	−	63.7401i	−0.330259	−	0.330259i	0.522426	−	0.852685i	$-0.325027\pi$
$193$	−63.7401	−	63.7401i	−0.330259	−	0.330259i	−0.852685	+	0.522426i	$0.825027\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−245.914	+	245.914i	−1.24829	+	1.24829i	−0.291820	+	0.956473i	$0.594261\pi$
$197$	−245.914	+	245.914i	−1.24829	+	1.24829i	−0.956473	+	0.291820i	$0.905739\pi$
$198$		0			0
$199$			102.798i			0.516573i	0.966068	+	0.258286i	$0.0831578\pi$
$199$			102.798i			0.516573i	−0.966068	+	0.258286i	$0.916842\pi$
$200$		0			0
$201$	31.1816			0.155132
$202$		0			0
$203$	−5.01479	−	5.01479i	−0.0247034	−	0.0247034i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	28.6515	−	28.6515i	0.138413	−	0.138413i
$208$		0			0
$209$		−	358.586i		−	1.71572i
$210$		0			0
$211$	6.43470			0.0304962			0.0152481	−	0.999884i	$-0.495146\pi$
$211$	6.43470			0.0304962			0.0152481	+	0.999884i	$0.495146\pi$
$212$		0			0
$213$	95.2827	+	95.2827i	0.447336	+	0.447336i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−6.90689	+	6.90689i	−0.0318290	+	0.0318290i
$218$		0			0
$219$			221.171i			1.00992i
$220$		0			0
$221$	83.3939			0.377348
$222$		0			0
$223$	33.3156	+	33.3156i	0.149397	+	0.149397i	0.777849	−	0.628452i	$-0.216311\pi$
$223$	33.3156	+	33.3156i	0.149397	+	0.149397i	−0.628452	+	0.777849i	$0.716311\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−72.5857	+	72.5857i	−0.319761	+	0.319761i	−0.848675	−	0.528914i	$-0.822599\pi$
$227$	−72.5857	+	72.5857i	−0.319761	+	0.319761i	0.528914	+	0.848675i	$0.322599\pi$
$228$		0			0
$229$			295.788i			1.29165i	0.763486	+	0.645825i	$0.223486\pi$
$229$			295.788i			1.29165i	−0.763486	+	0.645825i	$0.776514\pi$
$230$		0			0
$231$	−9.41429			−0.0407545
$232$		0			0
$233$	−39.2827	−	39.2827i	−0.168595	−	0.168595i	0.617767	−	0.786362i	$-0.288038\pi$
$233$	−39.2827	−	39.2827i	−0.168595	−	0.168595i	−0.786362	+	0.617767i	$0.788038\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	127.348	−	127.348i	0.537335	−	0.537335i
$238$		0			0
$239$			89.7571i			0.375553i	0.982212	+	0.187776i	$0.0601280\pi$
$239$			89.7571i			0.375553i	−0.982212	+	0.187776i	$0.939872\pi$
$240$		0			0
$241$	−4.23266			−0.0175629			−0.00878144	−	0.999961i	$-0.502795\pi$
$241$	−4.23266			−0.0175629			−0.00878144	+	0.999961i	$0.502795\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	111.674	−	111.674i	0.452122	−	0.452122i
$248$		0			0
$249$		−	55.9796i		−	0.224818i
$250$		0			0
$251$	−299.151			−1.19184			−0.595918	−	0.803045i	$-0.703212\pi$
$251$	−299.151			−1.19184			−0.595918	+	0.803045i	$0.703212\pi$
$252$		0			0
$253$	−112.677	−	112.677i	−0.445362	−	0.445362i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	229.262	−	229.262i	0.892071	−	0.892071i	−0.102647	−	0.994718i	$-0.532731\pi$
$257$	229.262	−	229.262i	0.892071	−	0.892071i	0.994718	+	0.102647i	$0.0327312\pi$
$258$		0			0
$259$			2.02041i			0.00780081i
$260$		0			0
$261$	−46.1816			−0.176941
$262$		0			0
$263$	−207.464	−	207.464i	−0.788838	−	0.788838i	0.192466	−	0.981304i	$-0.438351\pi$
$263$	−207.464	−	207.464i	−0.788838	−	0.788838i	−0.981304	+	0.192466i	$0.938351\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	195.464	−	195.464i	0.732076	−	0.732076i
$268$		0			0
$269$		−	100.161i		−	0.372347i	−0.982517	−	0.186173i	$-0.940391\pi$
$269$		−	100.161i		−	0.372347i	0.982517	−	0.186173i	$-0.0596086\pi$
$270$		0			0
$271$	−168.424			−0.621493			−0.310746	−	0.950493i	$-0.600579\pi$
$271$	−168.424			−0.621493			−0.310746	+	0.950493i	$0.600579\pi$
$272$		0			0
$273$	−2.93189	−	2.93189i	−0.0107395	−	0.0107395i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−137.563	+	137.563i	−0.496617	+	0.496617i	−0.910383	−	0.413766i	$-0.864213\pi$
$277$	−137.563	+	137.563i	−0.496617	+	0.496617i	0.413766	+	0.910383i	$0.364213\pi$
$278$		0			0
$279$			63.6061i			0.227979i
$280$		0			0
$281$	−254.586			−0.905999			−0.452999	−	0.891511i	$-0.649646\pi$
$281$	−254.586			−0.905999			−0.452999	+	0.891511i	$0.649646\pi$
$282$		0			0
$283$	−121.472	−	121.472i	−0.429230	−	0.429230i	0.459136	−	0.888366i	$-0.348159\pi$
$283$	−121.472	−	121.472i	−0.429230	−	0.429230i	−0.888366	+	0.459136i	$0.848159\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−5.92959	+	5.92959i	−0.0206606	+	0.0206606i
$288$		0			0
$289$			31.4245i			0.108735i
$290$		0			0
$291$	−138.959			−0.477523
$292$		0			0
$293$	−353.510	−	353.510i	−1.20652	−	1.20652i	−0.972147	−	0.234370i	$-0.924697\pi$
$293$	−353.510	−	353.510i	−1.20652	−	1.20652i	−0.234370	−	0.972147i	$-0.575303\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−43.3485	+	43.3485i	−0.145954	+	0.145954i
$298$		0			0
$299$		−	70.1816i		−	0.234721i
$300$		0			0
$301$	12.6571			0.0420503
$302$		0			0
$303$	56.5857	+	56.5857i	0.186752	+	0.186752i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	142.396	−	142.396i	0.463831	−	0.463831i	−0.436078	−	0.899909i	$-0.643633\pi$
$307$	142.396	−	142.396i	0.463831	−	0.463831i	0.899909	+	0.436078i	$0.143633\pi$
$308$		0			0
$309$			346.788i			1.12229i
$310$		0			0
$311$	479.535			1.54191			0.770956	−	0.636888i	$-0.219779\pi$
$311$	479.535			1.54191			0.770956	+	0.636888i	$0.219779\pi$
$312$		0			0
$313$	53.6288	+	53.6288i	0.171338	+	0.171338i	0.787567	−	0.616229i	$-0.211340\pi$
$313$	53.6288	+	53.6288i	0.171338	+	0.171338i	−0.616229	+	0.787567i	$0.711340\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−78.0704	+	78.0704i	−0.246279	+	0.246279i	−0.819442	−	0.573163i	$-0.805716\pi$
$317$	−78.0704	+	78.0704i	−0.246279	+	0.246279i	0.573163	+	0.819442i	$0.305716\pi$
$318$		0			0
$319$			181.616i			0.569330i
$320$		0			0
$321$	−280.182			−0.872840
$322$		0			0
$323$	−344.924	−	344.924i	−1.06788	−	1.06788i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−196.169	+	196.169i	−0.599906	+	0.599906i
$328$		0			0
$329$		−	18.0816i		−	0.0549594i
$330$		0			0
$331$	−120.424			−0.363820			−0.181910	−	0.983315i	$-0.558228\pi$
$331$	−120.424			−0.363820			−0.181910	+	0.983315i	$0.558228\pi$
$332$		0			0
$333$	9.30306	+	9.30306i	0.0279371	+	0.0279371i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	468.062	−	468.062i	1.38891	−	1.38891i	0.561291	−	0.827619i	$-0.310305\pi$
$337$	468.062	−	468.062i	1.38891	−	1.38891i	0.827619	−	0.561291i	$-0.189695\pi$
$338$		0			0
$339$		−	51.9592i		−	0.153272i
$340$		0			0
$341$	250.141			0.733551
$342$		0			0
$343$	−31.8559	−	31.8559i	−0.0928742	−	0.0928742i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−194.318	+	194.318i	−0.559994	+	0.559994i	−0.929306	−	0.369312i	$-0.879593\pi$
$347$	−194.318	+	194.318i	−0.559994	+	0.559994i	0.369312	+	0.929306i	$0.379593\pi$
$348$		0			0
$349$			599.576i			1.71798i	0.511991	+	0.858991i	$0.328908\pi$
$349$			599.576i			1.71798i	−0.511991	+	0.858991i	$0.671092\pi$
$350$		0			0
$351$	−27.0000			−0.0769231
$352$		0			0
$353$	−146.409	−	146.409i	−0.414755	−	0.414755i	0.468636	−	0.883391i	$-0.344746\pi$
$353$	−146.409	−	146.409i	−0.414755	−	0.414755i	−0.883391	+	0.468636i	$0.844746\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−9.05561	+	9.05561i	−0.0253659	+	0.0253659i
$358$		0			0
$359$		−	452.949i		−	1.26170i	−0.775906	−	0.630848i	$-0.782707\pi$
$359$		−	452.949i		−	1.26170i	0.775906	−	0.630848i	$-0.217293\pi$
$360$		0			0
$361$	−562.788			−1.55897
$362$		0			0
$363$	22.2804	+	22.2804i	0.0613784	+	0.0613784i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−221.901	+	221.901i	−0.604636	+	0.604636i	−0.941539	−	0.336904i	$-0.890620\pi$
$367$	−221.901	+	221.901i	−0.604636	+	0.604636i	0.336904	+	0.941539i	$0.390620\pi$
$368$		0			0
$369$			54.6061i			0.147984i
$370$		0			0
$371$	−37.0714			−0.0999230
$372$		0			0
$373$	243.518	+	243.518i	0.652862	+	0.652862i	0.953681	−	0.300819i	$-0.0972600\pi$
$373$	243.518	+	243.518i	0.652862	+	0.652862i	−0.300819	+	0.953681i	$0.597260\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−56.5607	+	56.5607i	−0.150028	+	0.150028i
$378$		0			0
$379$			164.839i			0.434931i	0.976068	+	0.217465i	$0.0697789\pi$
$379$			164.839i			0.434931i	−0.976068	+	0.217465i	$0.930221\pi$
$380$		0			0
$381$	275.555			0.723242
$382$		0			0
$383$	−455.110	−	455.110i	−1.18828	−	1.18828i	−0.977543	−	0.210734i	$-0.932415\pi$
$383$	−455.110	−	455.110i	−1.18828	−	1.18828i	−0.210734	−	0.977543i	$-0.567585\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	58.2804	−	58.2804i	0.150595	−	0.150595i
$388$		0			0
$389$			619.696i			1.59305i	0.604607	+	0.796524i	$0.293330\pi$
$389$			619.696i			1.59305i	−0.604607	+	0.796524i	$0.706670\pi$
$390$		0			0
$391$	−216.767			−0.554392
$392$		0			0
$393$	177.081	+	177.081i	0.450587	+	0.450587i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	108.952	−	108.952i	0.274439	−	0.274439i	−0.556445	−	0.830884i	$-0.687835\pi$
$397$	108.952	−	108.952i	0.274439	−	0.274439i	0.830884	+	0.556445i	$0.187835\pi$
$398$		0			0
$399$			24.2531i			0.0607846i
$400$		0			0
$401$	−412.182			−1.02788			−0.513942	−	0.857825i	$-0.671815\pi$
$401$	−412.182			−1.02788			−0.513942	+	0.857825i	$0.671815\pi$
$402$		0			0
$403$	77.9013	+	77.9013i	0.193303	+	0.193303i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	36.5857	−	36.5857i	0.0898912	−	0.0898912i
$408$		0			0
$409$		−	5.86939i		−	0.0143506i	−0.999974	−	0.00717530i	$-0.997716\pi$
$409$		−	5.86939i		−	0.0143506i	0.999974	−	0.00717530i	$-0.00228399\pi$
$410$		0			0
$411$	38.3633			0.0933413
$412$		0			0
$413$	26.7128	+	26.7128i	0.0646798	+	0.0646798i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	119.035	−	119.035i	0.285456	−	0.285456i
$418$		0			0
$419$		−	517.757i		−	1.23570i	−0.786297	−	0.617849i	$-0.788004\pi$
$419$		−	517.757i		−	1.23570i	0.786297	−	0.617849i	$-0.211996\pi$
$420$		0			0
$421$	−770.322			−1.82974			−0.914872	−	0.403743i	$-0.867709\pi$
$421$	−770.322			−1.82974			−0.914872	+	0.403743i	$0.867709\pi$
$422$		0			0
$423$	−83.2577	−	83.2577i	−0.196827	−	0.196827i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−30.8161	+	30.8161i	−0.0721688	+	0.0721688i
$428$		0			0
$429$			106.182i			0.247510i
$430$		0			0
$431$	−674.504			−1.56497			−0.782487	−	0.622667i	$-0.786049\pi$
$431$	−674.504			−1.56497			−0.782487	+	0.622667i	$0.786049\pi$
$432$		0			0
$433$	116.796	+	116.796i	0.269736	+	0.269736i	0.828994	−	0.559258i	$-0.188914\pi$
$433$	116.796	+	116.796i	0.269736	+	0.269736i	−0.559258	+	0.828994i	$0.688914\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−290.277	+	290.277i	−0.664250	+	0.664250i
$438$		0			0
$439$			133.141i			0.303282i	0.988436	+	0.151641i	$0.0484558\pi$
$439$			133.141i			0.303282i	−0.988436	+	0.151641i	$0.951544\pi$
$440$		0			0
$441$	−146.363			−0.331889
$442$		0			0
$443$	−41.3031	−	41.3031i	−0.0932349	−	0.0932349i	0.658951	−	0.752186i	$-0.271000\pi$
$443$	−41.3031	−	41.3031i	−0.0932349	−	0.0932349i	−0.752186	+	0.658951i	$0.771000\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	216.965	−	216.965i	0.485380	−	0.485380i
$448$		0			0
$449$		−	424.767i		−	0.946030i	−0.881054	−	0.473015i	$-0.843166\pi$
$449$		−	424.767i		−	0.946030i	0.881054	−	0.473015i	$-0.156834\pi$
$450$		0			0
$451$	214.747			0.476157
$452$		0			0
$453$	−138.124	−	138.124i	−0.304909	−	0.304909i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	553.798	−	553.798i	1.21181	−	1.21181i	0.241381	−	0.970430i	$-0.422400\pi$
$457$	553.798	−	553.798i	1.21181	−	1.21181i	0.970430	−	0.241381i	$-0.0776005\pi$
$458$		0			0
$459$			83.3939i			0.181686i
$460$		0			0
$461$	10.6469			0.0230953			0.0115477	−	0.999933i	$-0.496324\pi$
$461$	10.6469			0.0230953			0.0115477	+	0.999933i	$0.496324\pi$
$462$		0			0
$463$	−435.464	−	435.464i	−0.940528	−	0.940528i	0.0578005	−	0.998328i	$-0.481591\pi$
$463$	−435.464	−	435.464i	−0.940528	−	0.940528i	−0.998328	+	0.0578005i	$0.981591\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−344.974	+	344.974i	−0.738702	+	0.738702i	−0.972327	−	0.233625i	$-0.924941\pi$
$467$	−344.974	+	344.974i	−0.738702	+	0.738702i	0.233625	+	0.972327i	$0.424941\pi$
$468$		0			0
$469$			8.29389i			0.0176842i
$470$		0			0
$471$	−374.555			−0.795234
$472$		0			0
$473$	−229.196	−	229.196i	−0.484559	−	0.484559i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−170.697	+	170.697i	−0.357855	+	0.357855i
$478$		0			0
$479$		−	349.555i		−	0.729760i	−0.931055	−	0.364880i	$-0.881110\pi$
$479$		−	349.555i		−	0.729760i	0.931055	−	0.364880i	$-0.118890\pi$
$480$		0			0
$481$	22.7878			0.0473758
$482$		0			0
$483$	7.62092	+	7.62092i	0.0157783	+	0.0157783i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−345.785	+	345.785i	−0.710032	+	0.710032i	−0.966542	−	0.256510i	$-0.917427\pi$
$487$	−345.785	+	345.785i	−0.710032	+	0.710032i	0.256510	+	0.966542i	$0.417427\pi$
$488$		0			0
$489$			383.524i			0.784304i
$490$		0			0
$491$	−611.192			−1.24479			−0.622395	−	0.782703i	$-0.713840\pi$
$491$	−611.192			−1.24479			−0.622395	+	0.782703i	$0.713840\pi$
$492$		0			0
$493$	174.697	+	174.697i	0.354355	+	0.354355i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−25.3439	+	25.3439i	−0.0509937	+	0.0509937i
$498$		0			0
$499$			716.292i			1.43545i	0.696324	+	0.717727i	$0.254818\pi$
$499$			716.292i			1.43545i	−0.696324	+	0.717727i	$0.745182\pi$
$500$		0			0
$501$	9.35306			0.0186688
$502$		0			0
$503$	−503.333	−	503.333i	−1.00066	−	1.00066i	−1.00000		0.000661543i	$-0.999789\pi$
$503$	−503.333	−	503.333i	−1.00066	−	1.00066i	−0.000661543	−	1.00000i	$-0.500211\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	173.914	−	173.914i	0.343025	−	0.343025i
$508$		0			0
$509$			811.090i			1.59350i	0.604311	+	0.796748i	$0.293448\pi$
$509$			811.090i			1.59350i	−0.604311	+	0.796748i	$0.706552\pi$
$510$		0			0
$511$	−58.8286			−0.115124
$512$		0			0
$513$	111.674	+	111.674i	0.217689	+	0.217689i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−327.423	+	327.423i	−0.633314	+	0.633314i
$518$		0			0
$519$		−	384.606i		−	0.741052i
$520$		0			0
$521$	−247.880			−0.475777			−0.237888	−	0.971293i	$-0.576455\pi$
$521$	−247.880			−0.475777			−0.237888	+	0.971293i	$0.576455\pi$
$522$		0			0
$523$	−478.578	−	478.578i	−0.915063	−	0.915063i	0.0816023	−	0.996665i	$-0.473996\pi$
$523$	−478.578	−	478.578i	−0.915063	−	0.915063i	−0.996665	+	0.0816023i	$0.973996\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	240.611	−	240.611i	0.456567	−	0.456567i
$528$		0			0
$529$		−	346.576i		−	0.655152i
$530$		0			0
$531$	246.000			0.463277
$532$		0			0
$533$	66.8786	+	66.8786i	0.125476	+	0.125476i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	16.2066	−	16.2066i	0.0301799	−	0.0301799i
$538$		0			0
$539$			575.596i			1.06790i
$540$		0			0
$541$	522.110			0.965084			0.482542	−	0.875873i	$-0.339714\pi$
$541$	522.110			0.965084			0.482542	+	0.875873i	$0.339714\pi$
$542$		0			0
$543$	290.215	+	290.215i	0.534465	+	0.534465i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−150.384	+	150.384i	−0.274924	+	0.274924i	−0.831079	−	0.556154i	$-0.812276\pi$
$547$	−150.384	+	150.384i	−0.274924	+	0.274924i	0.556154	+	0.831079i	$0.312276\pi$
$548$		0			0
$549$			283.788i			0.516918i
$550$		0			0
$551$	467.880			0.849146
$552$		0			0
$553$	33.8730	+	33.8730i	0.0612531	+	0.0612531i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−41.1464	+	41.1464i	−0.0738715	+	0.0738715i	−0.743077	−	0.669206i	$-0.766635\pi$
$557$	−41.1464	+	41.1464i	−0.0738715	+	0.0738715i	0.669206	+	0.743077i	$0.266635\pi$
$558$		0			0
$559$		−	142.757i		−	0.255379i
$560$		0			0
$561$	327.959			0.584597
$562$		0			0
$563$	634.954	+	634.954i	1.12780	+	1.12780i	0.990534	+	0.137270i	$0.0438329\pi$
$563$	634.954	+	634.954i	1.12780	+	1.12780i	0.137270	+	0.990534i	$0.456167\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	2.93189	−	2.93189i	0.00517088	−	0.00517088i
$568$		0			0
$569$		−	775.271i		−	1.36252i	−0.732044	−	0.681258i	$-0.761433\pi$
$569$		−	775.271i		−	1.36252i	0.732044	−	0.681258i	$-0.238567\pi$
$570$		0			0
$571$	151.929			0.266075			0.133037	−	0.991111i	$-0.457527\pi$
$571$	151.929			0.266075			0.133037	+	0.991111i	$0.457527\pi$
$572$		0			0
$573$	29.6413	+	29.6413i	0.0517301	+	0.0517301i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	302.875	−	302.875i	0.524914	−	0.524914i	−0.394138	−	0.919051i	$-0.628957\pi$
$577$	302.875	−	302.875i	0.524914	−	0.524914i	0.919051	+	0.394138i	$0.128957\pi$
$578$		0			0
$579$		−	156.131i		−	0.269656i
$580$		0			0
$581$	14.8898			0.0256279
$582$		0			0
$583$	671.292	+	671.292i	1.15144	+	1.15144i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	102.374	−	102.374i	0.174403	−	0.174403i	−0.614508	−	0.788911i	$-0.710645\pi$
$587$	102.374	−	102.374i	0.174403	−	0.174403i	0.788911	+	0.614508i	$0.210645\pi$
$588$		0			0
$589$		−	644.412i		−	1.09408i
$590$		0			0
$591$	−602.363			−1.01923
$592$		0			0
$593$	177.576	+	177.576i	0.299453	+	0.299453i	0.840799	−	0.541347i	$-0.182085\pi$
$593$	177.576	+	177.576i	0.299453	+	0.299453i	−0.541347	+	0.840799i	$0.682085\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−125.901	+	125.901i	−0.210890	+	0.210890i
$598$		0			0
$599$		−	248.282i		−	0.414494i	−0.978289	−	0.207247i	$-0.933550\pi$
$599$		−	248.282i		−	0.414494i	0.978289	−	0.207247i	$-0.0664503\pi$
$600$		0			0
$601$	−469.706			−0.781541			−0.390770	−	0.920488i	$-0.627791\pi$
$601$	−469.706			−0.781541			−0.390770	+	0.920488i	$0.627791\pi$
$602$		0			0
$603$	38.1895	+	38.1895i	0.0633326	+	0.0633326i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−385.666	+	385.666i	−0.635365	+	0.635365i	−0.949408	−	0.314044i	$-0.898316\pi$
$607$	−385.666	+	385.666i	−0.635365	+	0.635365i	0.314044	+	0.949408i	$0.398316\pi$
$608$		0			0
$609$		−	12.2837i		−	0.0201702i
$610$		0			0
$611$	−203.939			−0.333779
$612$		0			0
$613$	−520.232	−	520.232i	−0.848665	−	0.848665i	0.141302	−	0.989967i	$-0.454871\pi$
$613$	−520.232	−	520.232i	−0.848665	−	0.848665i	−0.989967	+	0.141302i	$0.954871\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−120.767	+	120.767i	−0.195733	+	0.195733i	−0.798168	−	0.602435i	$-0.794197\pi$
$617$	−120.767	+	120.767i	−0.195733	+	0.195733i	0.602435	+	0.798168i	$0.294197\pi$
$618$		0			0
$619$			795.039i			1.28439i	0.766540	+	0.642196i	$0.221977\pi$
$619$			795.039i			1.28439i	−0.766540	+	0.642196i	$0.778023\pi$
$620$		0			0
$621$	70.1816			0.113014
$622$		0			0
$623$	51.9908	+	51.9908i	0.0834524	+	0.0834524i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	439.176	−	439.176i	0.700440	−	0.700440i
$628$		0			0
$629$		−	70.3837i		−	0.111898i
$630$		0			0
$631$	834.271			1.32214			0.661071	−	0.750323i	$-0.270102\pi$
$631$	834.271			1.32214			0.661071	+	0.750323i	$0.270102\pi$
$632$		0			0
$633$	7.88086	+	7.88086i	0.0124500	+	0.0124500i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−179.258	+	179.258i	−0.281409	+	0.281409i
$638$		0			0
$639$			233.394i			0.365249i
$640$		0			0
$641$	858.665			1.33957			0.669786	−	0.742554i	$-0.266386\pi$
$641$	858.665			1.33957			0.669786	+	0.742554i	$0.266386\pi$
$642$		0			0
$643$	425.040	+	425.040i	0.661026	+	0.661026i	0.955622	−	0.294596i	$-0.0951850\pi$
$643$	425.040	+	425.040i	0.661026	+	0.661026i	−0.294596	+	0.955622i	$0.595185\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	727.373	−	727.373i	1.12422	−	1.12422i	0.133126	−	0.991099i	$-0.457499\pi$
$647$	727.373	−	727.373i	1.12422	−	1.12422i	0.991099	−	0.133126i	$-0.0425014\pi$
$648$		0			0
$649$		−	967.433i		−	1.49065i
$650$		0			0
$651$	−16.9184			−0.0259883
$652$		0			0
$653$	−561.773	−	561.773i	−0.860296	−	0.860296i	0.131077	−	0.991372i	$-0.458157\pi$
$653$	−561.773	−	561.773i	−0.860296	−	0.860296i	−0.991372	+	0.131077i	$0.958157\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−270.879	+	270.879i	−0.412296	+	0.412296i
$658$		0			0
$659$		−	678.606i		−	1.02975i	−0.857265	−	0.514876i	$-0.827838\pi$
$659$		−	678.606i		−	1.02975i	0.857265	−	0.514876i	$-0.172162\pi$
$660$		0			0
$661$	−241.069			−0.364704			−0.182352	−	0.983233i	$-0.558371\pi$
$661$	−241.069			−0.364704			−0.182352	+	0.983233i	$0.558371\pi$
$662$		0			0
$663$	102.136	+	102.136i	0.154052	+	0.154052i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	147.019	−	147.019i	0.220419	−	0.220419i
$668$		0			0
$669$			81.6061i			0.121982i
$670$		0			0
$671$	1116.04			1.66325
$672$		0			0
$673$	39.6867	+	39.6867i	0.0589699	+	0.0589699i	0.735977	−	0.677007i	$-0.236723\pi$
$673$	39.6867	+	39.6867i	0.0589699	+	0.0589699i	−0.677007	+	0.735977i	$0.736723\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−285.873	+	285.873i	−0.422264	+	0.422264i	−0.885983	−	0.463718i	$-0.846515\pi$
$677$	−285.873	+	285.873i	−0.422264	+	0.422264i	0.463718	+	0.885983i	$0.346515\pi$
$678$		0			0
$679$		−	36.9612i		−	0.0544348i
$680$		0			0
$681$	−177.798			−0.261084
$682$		0			0
$683$	180.767	+	180.767i	0.264667	+	0.264667i	0.826947	−	0.562280i	$-0.190076\pi$
$683$	180.767	+	180.767i	0.264667	+	0.264667i	−0.562280	+	0.826947i	$0.690076\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−362.265	+	362.265i	−0.527314	+	0.527314i
$688$		0			0
$689$			418.120i			0.606851i
$690$		0			0
$691$	265.273			0.383898			0.191949	−	0.981405i	$-0.438519\pi$
$691$	265.273			0.383898			0.191949	+	0.981405i	$0.438519\pi$
$692$		0			0
$693$	−11.5301	−	11.5301i	−0.0166380	−	0.0166380i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	206.565	−	206.565i	0.296363	−	0.296363i
$698$		0			0
$699$		−	96.2225i		−	0.137657i
$700$		0			0
$701$	1371.27			1.95616			0.978082	−	0.208219i	$-0.0667665\pi$
$701$	1371.27			1.95616			0.978082	+	0.208219i	$0.0667665\pi$
$702$		0			0
$703$	−94.2520	−	94.2520i	−0.134071	−	0.134071i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15.0510	+	15.0510i	−0.0212886	+	0.0212886i
$708$		0			0
$709$		−	213.829i		−	0.301592i	−0.988565	−	0.150796i	$-0.951816\pi$
$709$		−	213.829i		−	0.301592i	0.988565	−	0.150796i	$-0.0481836\pi$
$710$		0			0
$711$	311.939			0.438732
$712$		0			0
$713$	−202.490	−	202.490i	−0.283998	−	0.283998i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−109.930	+	109.930i	−0.153319	+	0.153319i
$718$		0			0
$719$			1102.34i			1.53316i	0.642150	+	0.766579i	$0.278043\pi$
$719$			1102.34i			1.53316i	−0.642150	+	0.766579i	$0.721957\pi$
$720$		0			0
$721$	−92.2408			−0.127935
$722$		0			0
$723$	−5.18392	−	5.18392i	−0.00717002	−	0.00717002i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−119.593	+	119.593i	−0.164502	+	0.164502i	−0.784558	−	0.620056i	$-0.787110\pi$
$727$	−119.593	+	119.593i	−0.164502	+	0.164502i	0.620056	+	0.784558i	$0.287110\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−440.929			−0.603185
$732$		0			0
$733$	−206.697	−	206.697i	−0.281988	−	0.281988i	0.551914	−	0.833901i	$-0.313898\pi$
$733$	−206.697	−	206.697i	−0.281988	−	0.281988i	−0.833901	+	0.551914i	$0.813898\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	150.186	−	150.186i	0.203780	−	0.203780i
$738$		0			0
$739$			179.131i			0.242396i	0.992628	+	0.121198i	$0.0386736\pi$
$739$			179.131i			0.242396i	−0.992628	+	0.121198i	$0.961326\pi$
$740$		0			0
$741$	273.545			0.369156
$742$		0			0
$743$	984.777	+	984.777i	1.32541	+	1.32541i	0.909330	+	0.416076i	$0.136595\pi$
$743$	984.777	+	984.777i	1.32541	+	1.32541i	0.416076	+	0.909330i	$0.363405\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	68.5607	−	68.5607i	0.0917814	−	0.0917814i
$748$		0			0
$749$		−	74.5245i		−	0.0994986i
$750$		0			0
$751$	−343.576			−0.457491			−0.228745	−	0.973486i	$-0.573462\pi$
$751$	−343.576			−0.457491			−0.228745	+	0.973486i	$0.573462\pi$
$752$		0			0
$753$	−366.384	−	366.384i	−0.486565	−	0.486565i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	585.588	−	585.588i	0.773564	−	0.773564i	−0.205164	−	0.978728i	$-0.565773\pi$
$757$	585.588	−	585.588i	0.773564	−	0.773564i	0.978728	+	0.205164i	$0.0657726\pi$
$758$		0			0
$759$		−	276.000i		−	0.363636i
$760$		0			0
$761$	−477.857			−0.627933			−0.313967	−	0.949434i	$-0.601658\pi$
$761$	−477.857			−0.627933			−0.313967	+	0.949434i	$0.601658\pi$
$762$		0			0
$763$	−52.1783	−	52.1783i	−0.0683857	−	0.0683857i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	301.287	−	301.287i	0.392813	−	0.392813i
$768$		0			0
$769$		−	1116.56i		−	1.45196i	−0.687717	−	0.725979i	$-0.741387\pi$
$769$		−	1116.56i		−	1.45196i	0.687717	−	0.725979i	$-0.258613\pi$
$770$		0			0
$771$	561.576			0.728373
$772$		0			0
$773$	375.151	+	375.151i	0.485318	+	0.485318i	0.906825	−	0.421507i	$-0.138499\pi$
$773$	375.151	+	375.151i	0.485318	+	0.485318i	−0.421507	+	0.906825i	$0.638499\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−2.47449	+	2.47449i	−0.00318467	+	0.00318467i
$778$		0			0
$779$		−	553.231i		−	0.710180i
$780$		0			0
$781$	917.857			1.17523
$782$		0			0
$783$	−56.5607	−	56.5607i	−0.0722359	−	0.0722359i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	559.229	−	559.229i	0.710584	−	0.710584i	−0.256074	−	0.966657i	$-0.582429\pi$
$787$	559.229	−	559.229i	0.710584	−	0.710584i	0.966657	+	0.256074i	$0.0824290\pi$
$788$		0			0
$789$		−	508.182i		−	0.644083i
$790$		0			0
$791$	13.8204			0.0174721
$792$		0			0
$793$	347.568	+	347.568i	0.438295	+	0.438295i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−437.839	+	437.839i	−0.549359	+	0.549359i	−0.926255	−	0.376897i	$-0.876991\pi$
$797$	−437.839	+	437.839i	−0.549359	+	0.549359i	0.376897	+	0.926255i	$0.376991\pi$
$798$		0			0
$799$			629.898i			0.788358i
$800$		0			0
$801$	478.788			0.597738
$802$		0			0
$803$	1065.27	+	1065.27i	1.32661	+	1.32661i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	122.672	−	122.672i	0.152010	−	0.152010i
$808$		0			0
$809$		−	210.565i		−	0.260278i	−0.991496	−	0.130139i	$-0.958458\pi$
$809$		−	210.565i		−	0.260278i	0.991496	−	0.130139i	$-0.0415424\pi$
$810$		0			0
$811$	858.818			1.05896			0.529481	−	0.848322i	$-0.322387\pi$
$811$	858.818			1.05896			0.529481	+	0.848322i	$0.322387\pi$
$812$		0			0
$813$	−206.277	−	206.277i	−0.253723	−	0.253723i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−590.455	+	590.455i	−0.722712	+	0.722712i
$818$		0			0
$819$		−	7.18163i		−	0.00876878i
$820$		0			0
$821$	−637.878			−0.776952			−0.388476	−	0.921459i	$-0.626998\pi$
$821$	−637.878			−0.776952			−0.388476	+	0.921459i	$0.626998\pi$
$822$		0			0
$823$	636.855	+	636.855i	0.773821	+	0.773821i	0.978772	−	0.204951i	$-0.0657035\pi$
$823$	636.855	+	636.855i	0.773821	+	0.773821i	−0.204951	+	0.978772i	$0.565704\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	43.7684	−	43.7684i	0.0529243	−	0.0529243i	−0.680149	−	0.733074i	$-0.738085\pi$
$827$	43.7684	−	43.7684i	0.0529243	−	0.0529243i	0.733074	+	0.680149i	$0.238085\pi$
$828$		0			0
$829$			325.110i			0.392172i	0.980587	+	0.196086i	$0.0628231\pi$
$829$			325.110i			0.392172i	−0.980587	+	0.196086i	$0.937177\pi$
$830$		0			0
$831$	−336.959			−0.405486
$832$		0			0
$833$	553.666	+	553.666i	0.664665	+	0.664665i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−77.9013	+	77.9013i	−0.0930720	+	0.0930720i
$838$		0			0
$839$			279.394i			0.333008i	0.986041	+	0.166504i	$0.0532479\pi$
$839$			279.394i			0.333008i	−0.986041	+	0.166504i	$0.946752\pi$
$840$		0			0
$841$	604.029			0.718227
$842$		0			0
$843$	−311.803	−	311.803i	−0.369873	−	0.369873i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−5.92627	+	5.92627i	−0.00699678	+	0.00699678i
$848$		0			0
$849$		−	297.545i		−	0.350465i
$850$		0			0
$851$	−59.2327			−0.0696036
$852$		0			0
$853$	708.269	+	708.269i	0.830327	+	0.830327i	0.987561	−	0.157234i	$-0.0502577\pi$
$853$	708.269	+	708.269i	0.830327	+	0.830327i	−0.157234	+	0.987561i	$0.550258\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	77.9888	−	77.9888i	0.0910021	−	0.0910021i	−0.660140	−	0.751142i	$-0.729503\pi$
$857$	77.9888	−	77.9888i	0.0910021	−	0.0910021i	0.751142	+	0.660140i	$0.229503\pi$
$858$		0			0
$859$		−	937.837i		−	1.09178i	−0.837858	−	0.545889i	$-0.816192\pi$
$859$		−	937.837i		−	1.09178i	0.837858	−	0.545889i	$-0.183808\pi$
$860$		0			0
$861$	−14.5245			−0.0168693
$862$		0			0
$863$	−1144.00	−	1144.00i	−1.32561	−	1.32561i	−0.909159	−	0.416449i	$-0.863274\pi$
$863$	−1144.00	−	1144.00i	−1.32561	−	1.32561i	−0.416449	−	0.909159i	$-0.636726\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−38.4870	+	38.4870i	−0.0443910	+	0.0443910i
$868$		0			0
$869$		−	1226.75i		−	1.41168i
$870$		0			0
$871$	93.5449			0.107399
$872$		0			0
$873$	−170.190	−	170.190i	−0.194948	−	0.194948i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	191.740	−	191.740i	0.218632	−	0.218632i	−0.589290	−	0.807922i	$-0.700592\pi$
$877$	191.740	−	191.740i	0.218632	−	0.218632i	0.807922	+	0.589290i	$0.200592\pi$
$878$		0			0
$879$		−	865.918i		−	0.985118i
$880$		0			0
$881$	1300.97			1.47669			0.738347	−	0.674421i	$-0.235607\pi$
$881$	1300.97			1.47669			0.738347	+	0.674421i	$0.235607\pi$
$882$		0			0
$883$	316.623	+	316.623i	0.358577	+	0.358577i	0.863288	−	0.504711i	$-0.168401\pi$
$883$	316.623	+	316.623i	0.358577	+	0.358577i	−0.504711	+	0.863288i	$0.668401\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−426.894	+	426.894i	−0.481279	+	0.481279i	−0.905540	−	0.424261i	$-0.860534\pi$
$887$	−426.894	+	426.894i	−0.481279	+	0.481279i	0.424261	+	0.905540i	$0.360534\pi$
$888$		0			0
$889$			73.2939i			0.0824453i
$890$		0			0
$891$	−106.182			−0.119171
$892$		0			0
$893$	843.508	+	843.508i	0.944577	+	0.944577i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	85.9546	−	85.9546i	0.0958245	−	0.0958245i
$898$		0			0
$899$			326.382i			0.363050i
$900$		0			0
$901$	1291.43			1.43333
$902$		0			0
$903$	15.5018	+	15.5018i	0.0171670	+	0.0171670i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−591.687	+	591.687i	−0.652356	+	0.652356i	−0.953560	−	0.301204i	$-0.902612\pi$
$907$	−591.687	+	591.687i	−0.652356	+	0.652356i	0.301204	+	0.953560i	$0.402612\pi$
$908$		0			0
$909$			138.606i			0.152482i
$910$		0			0
$911$	972.947			1.06800			0.533999	−	0.845485i	$-0.320688\pi$
$911$	972.947			1.06800			0.533999	+	0.845485i	$0.320688\pi$
$912$		0			0
$913$	−269.626	−	269.626i	−0.295318	−	0.295318i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−47.1010	+	47.1010i	−0.0513643	+	0.0513643i
$918$		0			0
$919$		−	226.031i		−	0.245953i	−0.992410	−	0.122976i	$-0.960756\pi$
$919$		−	226.031i		−	0.245953i	0.992410	−	0.122976i	$-0.0392439\pi$
$920$		0			0
$921$	348.798			0.378717
$922$		0			0
$923$	285.848	+	285.848i	0.309694	+	0.309694i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−424.727	+	424.727i	−0.458173	+	0.458173i
$928$		0			0
$929$			1161.66i			1.25044i	0.780450	+	0.625218i	$0.214990\pi$
$929$			1161.66i			1.25044i	−0.780450	+	0.625218i	$0.785010\pi$
$930$		0			0
$931$	1482.85			1.59275
$932$		0			0
$933$	587.308	+	587.308i	0.629483	+	0.629483i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−927.997	+	927.997i	−0.990391	+	0.990391i	−0.999954	−	0.00956294i	$-0.996956\pi$
$937$	−927.997	+	927.997i	−0.990391	+	0.990391i	0.00956294	+	0.999954i	$0.496956\pi$
$938$		0			0
$939$			131.363i			0.139897i
$940$		0			0
$941$	−1039.86			−1.10506			−0.552528	−	0.833495i	$-0.686337\pi$
$941$	−1039.86			−1.10506			−0.552528	+	0.833495i	$0.686337\pi$
$942$		0			0
$943$	−173.839	−	173.839i	−0.184347	−	0.184347i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1203.27	+	1203.27i	−1.27061	+	1.27061i	−0.324840	+	0.945769i	$0.605311\pi$
$947$	−1203.27	+	1203.27i	−1.27061	+	1.27061i	−0.945769	+	0.324840i	$0.894689\pi$
$948$		0			0
$949$			663.514i			0.699172i
$950$		0			0
$951$	−191.233			−0.201086
$952$		0			0
$953$	1002.66	+	1002.66i	1.05211	+	1.05211i	0.998566	+	0.0535393i	$0.0170502\pi$
$953$	1002.66	+	1002.66i	1.05211	+	1.05211i	0.0535393	+	0.998566i	$0.482950\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−222.434	+	222.434i	−0.232428	+	0.232428i
$958$		0			0
$959$			10.2041i			0.0106404i
$960$		0			0
$961$	−511.473			−0.532230
$962$		0			0
$963$	−343.151	−	343.151i	−0.356335	−	0.356335i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−382.434	+	382.434i	−0.395485	+	0.395485i	−0.876637	−	0.481152i	$-0.840218\pi$
$967$	−382.434	+	382.434i	−0.395485	+	0.395485i	0.481152	+	0.876637i	$0.340218\pi$
$968$		0			0
$969$		−	844.888i		−	0.871917i
$970$		0			0
$971$	−1241.64			−1.27872			−0.639360	−	0.768908i	$-0.720801\pi$
$971$	−1241.64			−1.27872			−0.639360	+	0.768908i	$0.720801\pi$
$972$		0			0
$973$	31.6617	+	31.6617i	0.0325403	+	0.0325403i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1155.44	−	1155.44i	1.18264	−	1.18264i	0.203582	−	0.979058i	$-0.434742\pi$
$977$	1155.44	−	1155.44i	1.18264	−	1.18264i	0.979058	−	0.203582i	$-0.0652584\pi$
$978$		0			0
$979$		−	1882.91i		−	1.92330i
$980$		0			0
$981$	−480.514			−0.489821
$982$		0			0
$983$	582.352	+	582.352i	0.592423	+	0.592423i	0.938285	−	0.345862i	$-0.112414\pi$
$983$	582.352	+	582.352i	0.592423	+	0.592423i	−0.345862	+	0.938285i	$0.612414\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	22.1454	−	22.1454i	0.0224371	−	0.0224371i
$988$		0			0
$989$			371.071i			0.375199i
$990$		0			0
$991$	−1762.94			−1.77895			−0.889474	−	0.456987i	$-0.848929\pi$
$991$	−1762.94			−1.77895			−0.889474	+	0.456987i	$0.848929\pi$
$992$		0			0
$993$	−147.489	−	147.489i	−0.148529	−	0.148529i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−734.170	+	734.170i	−0.736380	+	0.736380i	−0.971875	−	0.235496i	$-0.924329\pi$
$997$	−734.170	+	734.170i	−0.736380	+	0.736380i	0.235496	+	0.971875i	$0.424329\pi$
$998$		0			0
$999$			22.7878i			0.0228106i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.c.193.2		4
4.3	odd	2		600.3.u.g.193.1	yes	4
5.2	odd	4	inner	1200.3.bg.c.1057.2		4
5.3	odd	4		1200.3.bg.n.1057.1		4
5.4	even	2		1200.3.bg.n.193.1		4
12.11	even	2		1800.3.v.p.793.1		4
20.3	even	4		600.3.u.b.457.2	yes	4
20.7	even	4		600.3.u.g.457.1	yes	4
20.19	odd	2		600.3.u.b.193.2	✓	4
60.23	odd	4		1800.3.v.i.1657.2		4
60.47	odd	4		1800.3.v.p.1657.1		4
60.59	even	2		1800.3.v.i.793.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.u.b.193.2	✓	4	20.19	odd	2
600.3.u.b.457.2	yes	4	20.3	even	4
600.3.u.g.193.1	yes	4	4.3	odd	2
600.3.u.g.457.1	yes	4	20.7	even	4
1200.3.bg.c.193.2		4	1.1	even	1	trivial
1200.3.bg.c.1057.2		4	5.2	odd	4	inner
1200.3.bg.n.193.1		4	5.4	even	2
1200.3.bg.n.1057.1		4	5.3	odd	4
1800.3.v.i.793.2		4	60.59	even	2
1800.3.v.i.1657.2		4	60.23	odd	4
1800.3.v.p.793.1		4	12.11	even	2
1800.3.v.p.1657.1		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1200.bg (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +(-0.325765 + 0.325765i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{3} +(-0.325765 + 0.325765i) q^{7} +3.00000i q^{9} +11.7980 q^{11} +(3.67423 + 3.67423i) q^{13} +(11.3485 - 11.3485i) q^{17} -30.3939i q^{19} -0.797959 q^{21} +(-9.55051 - 9.55051i) q^{23} +(-3.67423 + 3.67423i) q^{27} +15.3939i q^{29} +21.2020 q^{31} +(14.4495 + 14.4495i) q^{33} +(3.10102 - 3.10102i) q^{37} +9.00000i q^{39} +18.2020 q^{41} +(-19.4268 - 19.4268i) q^{43} +(-27.7526 + 27.7526i) q^{47} +48.7878i q^{49} +27.7980 q^{51} +(56.8990 + 56.8990i) q^{53} +(37.2247 - 37.2247i) q^{57} -82.0000i q^{59} +94.5959 q^{61} +(-0.977296 - 0.977296i) q^{63} +(12.7298 - 12.7298i) q^{67} -23.3939i q^{69} +77.7980 q^{71} +(90.2929 + 90.2929i) q^{73} +(-3.84337 + 3.84337i) q^{77} -103.980i q^{79} -9.00000 q^{81} +(-22.8536 - 22.8536i) q^{83} +(-18.8536 + 18.8536i) q^{87} -159.596i q^{89} -2.39388 q^{91} +(25.9671 + 25.9671i) q^{93} +(-56.7298 + 56.7298i) q^{97} +35.3939i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{7}+O(q^{10}) \) 4 * q - 16 * q^7	\( 4 q - 16 q^{7} + 8 q^{11} + 16 q^{17} + 36 q^{21} - 48 q^{23} + 124 q^{31} + 48 q^{33} + 32 q^{37} + 112 q^{41} - 112 q^{43} - 160 q^{47} + 72 q^{51} + 208 q^{53} + 144 q^{57} + 300 q^{61} - 48 q^{63} + 144 q^{67} + 272 q^{71} + 224 q^{73} + 112 q^{77} - 36 q^{81} - 160 q^{83} - 144 q^{87} + 108 q^{91} - 48 q^{93} - 320 q^{97}+O(q^{100}) \) 4 * q - 16 * q^7 + 8 * q^11 + 16 * q^17 + 36 * q^21 - 48 * q^23 + 124 * q^31 + 48 * q^33 + 32 * q^37 + 112 * q^41 - 112 * q^43 - 160 * q^47 + 72 * q^51 + 208 * q^53 + 144 * q^57 + 300 * q^61 - 48 * q^63 + 144 * q^67 + 272 * q^71 + 224 * q^73 + 112 * q^77 - 36 * q^81 - 160 * q^83 - 144 * q^87 + 108 * q^91 - 48 * q^93 - 320 * q^97

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