LMFDB - Embedded newform 1920.2.b.h.191.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(191,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1920.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.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:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.619810816.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{5} + 14x^{4} - 8x^{3} + 2x^{2} + 2x + 1 $ x^8 - 2x^5 + 14x^4 - 8x^3 + 2x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.3
Root			$0.561103 - 0.561103i$ of defining polynomial
Character	$\chi$	$=$	1920.191
Dual form			1920.2.b.h.191.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.329998 - 1.70032i) q^{3} +1.00000 q^{5} -2.66000i q^{7} +(-2.78220 + 1.12221i) q^{9} -2.24441i q^{11} +2.74065i q^{13} +(-0.329998 - 1.70032i) q^{15} +0.496238i q^{17} -8.30506 q^{19} +(-4.52285 + 0.877793i) q^{21} -3.15623 q^{23} +1.00000 q^{25} +(2.82624 + 4.36032i) q^{27} -6.55688 q^{29} -1.50376i q^{31} +(-3.81623 + 0.740652i) q^{33} -2.66000i q^{35} -9.38064i q^{37} +(4.66000 - 0.904410i) q^{39} +10.3657i q^{41} +2.14130 q^{43} +(-2.78220 + 1.12221i) q^{45} +8.96505 q^{47} -0.0755782 q^{49} +(0.843766 - 0.163758i) q^{51} -10.1213 q^{53} -2.24441i q^{55} +(2.74065 + 14.1213i) q^{57} -0.435595i q^{59} -5.87687i q^{61} +(2.98507 + 7.40065i) q^{63} +2.74065i q^{65} -10.1413 q^{67} +(1.04155 + 5.36662i) q^{69} -4.32752 q^{71} -10.6101 q^{73} +(-0.329998 - 1.70032i) q^{75} -5.97013 q^{77} -13.2975i q^{79} +(6.48130 - 6.24441i) q^{81} +6.08066i q^{83} +0.496238i q^{85} +(2.16376 + 11.1488i) q^{87} +8.12129i q^{89} +7.29012 q^{91} +(-2.55688 + 0.496238i) q^{93} -8.30506 q^{95} +7.48130 q^{97} +(2.51870 + 6.24441i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{3} + 8 q^{5} - 4 q^{9} + 2 q^{15} - 8 q^{19} + 4 q^{21} - 12 q^{23} + 8 q^{25} + 14 q^{27} - 8 q^{29} - 8 q^{33} + 28 q^{39} - 36 q^{43} - 4 q^{45} + 4 q^{47} + 20 q^{51} - 16 q^{63} - 28 q^{67}+ \cdots + 64 q^{99}+O(q^{100}) $ 8 * q + 2 * q^3 + 8 * q^5 - 4 * q^9 + 2 * q^15 - 8 * q^19 + 4 * q^21 - 12 * q^23 + 8 * q^25 + 14 * q^27 - 8 * q^29 - 8 * q^33 + 28 * q^39 - 36 * q^43 - 4 * q^45 + 4 * q^47 + 20 * q^51 - 16 * q^63 - 28 * q^67 + 12 * q^69 - 24 * q^71 + 32 * q^73 + 2 * q^75 + 32 * q^77 + 8 * q^81 + 12 * q^87 - 40 * q^91 + 24 * q^93 - 8 * q^95 + 16 * q^97 + 64 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$.

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.329998	−	1.70032i	−0.190524	−	0.981682i
$3$	−0.329998	−	1.70032i	−0.190524	−	0.981682i
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$		−	2.66000i		−	1.00538i	−0.864466	−	0.502692i	$-0.832343\pi$
$7$		−	2.66000i		−	1.00538i	0.864466	−	0.502692i	$-0.167657\pi$
$8$		0			0
$9$	−2.78220	+	1.12221i	−0.927401	+	0.374069i
$10$		0			0
$11$		−	2.24441i		−	0.676716i	−0.941017	−	0.338358i	$-0.890128\pi$
$11$		−	2.24441i		−	0.676716i	0.941017	−	0.338358i	$-0.109872\pi$
$12$		0			0
$13$			2.74065i			0.760120i	0.924962	+	0.380060i	$0.124097\pi$
$13$			2.74065i			0.760120i	−0.924962	+	0.380060i	$0.875903\pi$
$14$		0			0
$15$	−0.329998	−	1.70032i	−0.0852051	−	0.439022i
$16$		0			0
$17$			0.496238i			0.120355i	0.998188	+	0.0601777i	$0.0191667\pi$
$17$			0.496238i			0.120355i	−0.998188	+	0.0601777i	$0.980833\pi$
$18$		0			0
$19$	−8.30506			−1.90531			−0.952655	−	0.304052i	$-0.901660\pi$
$19$	−8.30506			−1.90531			−0.952655	+	0.304052i	$0.901660\pi$
$20$		0			0
$21$	−4.52285	+	0.877793i	−0.986968	+	0.191550i
$22$		0			0
$23$	−3.15623			−0.658120			−0.329060	−	0.944309i	$-0.606732\pi$
$23$	−3.15623			−0.658120			−0.329060	+	0.944309i	$0.606732\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	2.82624	+	4.36032i	0.543909	+	0.839144i
$28$		0			0
$29$	−6.55688			−1.21758			−0.608791	−	0.793330i	$-0.708345\pi$
$29$	−6.55688			−1.21758			−0.608791	+	0.793330i	$0.708345\pi$
$30$		0			0
$31$		−	1.50376i		−	0.270084i	−0.990840	−	0.135042i	$-0.956883\pi$
$31$		−	1.50376i		−	0.270084i	0.990840	−	0.135042i	$-0.0431169\pi$
$32$		0			0
$33$	−3.81623	+	0.740652i	−0.664320	+	0.128931i
$34$		0			0
$35$		−	2.66000i		−	0.449621i
$36$		0			0
$37$		−	9.38064i		−	1.54217i	−0.636734	−	0.771083i	$-0.719715\pi$
$37$		−	9.38064i		−	1.54217i	0.636734	−	0.771083i	$-0.280285\pi$
$38$		0			0
$39$	4.66000	−	0.904410i	0.746197	−	0.144821i
$40$		0			0
$41$			10.3657i			1.61885i	0.587223	+	0.809425i	$0.300221\pi$
$41$			10.3657i			1.61885i	−0.587223	+	0.809425i	$0.699779\pi$
$42$		0			0
$43$	2.14130			0.326545			0.163273	−	0.986581i	$-0.447795\pi$
$43$	2.14130			0.326545			0.163273	+	0.986581i	$0.447795\pi$
$44$		0			0
$45$	−2.78220	+	1.12221i	−0.414746	+	0.167289i
$46$		0			0
$47$	8.96505			1.30769			0.653844	−	0.756630i	$-0.273155\pi$
$47$	8.96505			1.30769			0.653844	+	0.756630i	$0.273155\pi$
$48$		0			0
$49$	−0.0755782			−0.0107969
$50$		0			0
$51$	0.843766	−	0.163758i	0.118151	−	0.0229307i
$52$		0			0
$53$	−10.1213			−1.39027			−0.695133	−	0.718881i	$-0.744654\pi$
$53$	−10.1213			−1.39027			−0.695133	+	0.718881i	$0.744654\pi$
$54$		0			0
$55$		−	2.24441i		−	0.302637i
$56$		0			0
$57$	2.74065	+	14.1213i	0.363008	+	1.87041i
$58$		0			0
$59$		−	0.435595i		−	0.0567096i	−0.999598	−	0.0283548i	$-0.990973\pi$
$59$		−	0.435595i		−	0.0567096i	0.999598	−	0.0283548i	$-0.00902682\pi$
$60$		0			0
$61$		−	5.87687i		−	0.752457i	−0.926527	−	0.376228i	$-0.877221\pi$
$61$		−	5.87687i		−	0.752457i	0.926527	−	0.376228i	$-0.122779\pi$
$62$		0			0
$63$	2.98507	+	7.40065i	0.376083	+	0.932394i
$64$		0			0
$65$			2.74065i			0.339936i
$66$		0			0
$67$	−10.1413			−1.23896			−0.619478	−	0.785014i	$-0.712656\pi$
$67$	−10.1413			−1.23896			−0.619478	+	0.785014i	$0.712656\pi$
$68$		0			0
$69$	1.04155	+	5.36662i	0.125388	+	0.646065i
$70$		0			0
$71$	−4.32752			−0.513582			−0.256791	−	0.966467i	$-0.582665\pi$
$71$	−4.32752			−0.513582			−0.256791	+	0.966467i	$0.582665\pi$
$72$		0			0
$73$	−10.6101			−1.24182			−0.620910	−	0.783882i	$-0.713237\pi$
$73$	−10.6101			−1.24182			−0.620910	+	0.783882i	$0.713237\pi$
$74$		0			0
$75$	−0.329998	−	1.70032i	−0.0381049	−	0.196336i
$76$		0			0
$77$	−5.97013			−0.680360
$78$		0			0
$79$		−	13.2975i		−	1.49609i	−0.663648	−	0.748045i	$-0.730993\pi$
$79$		−	13.2975i		−	1.49609i	0.663648	−	0.748045i	$-0.269007\pi$
$80$		0			0
$81$	6.48130	−	6.24441i	0.720145	−	0.693824i
$82$		0			0
$83$			6.08066i			0.667439i	0.942672	+	0.333719i	$0.108304\pi$
$83$			6.08066i			0.667439i	−0.942672	+	0.333719i	$0.891696\pi$
$84$		0			0
$85$			0.496238i			0.0538246i
$86$		0			0
$87$	2.16376	+	11.1488i	0.231979	+	1.19528i
$88$		0			0
$89$			8.12129i			0.860855i	0.902625	+	0.430427i	$0.141637\pi$
$89$			8.12129i			0.860855i	−0.902625	+	0.430427i	$0.858363\pi$
$90$		0			0
$91$	7.29012			0.764213
$92$		0			0
$93$	−2.55688	+	0.496238i	−0.265136	+	0.0514575i
$94$		0			0
$95$	−8.30506			−0.852081
$96$		0			0
$97$	7.48130			0.759611			0.379806	−	0.925066i	$-0.375991\pi$
$97$	7.48130			0.759611			0.379806	+	0.925066i	$0.375991\pi$
$98$		0			0
$99$	2.51870	+	6.24441i	0.253138	+	0.627587i
$100$		0			0
$101$	0.0831016			0.00826892			0.00413446	−	0.999991i	$-0.498684\pi$
$101$	0.0831016			0.00826892			0.00413446	+	0.999991i	$0.498684\pi$
$102$		0			0
$103$		−	0.851177i		−	0.0838690i	−0.999120	−	0.0419345i	$-0.986648\pi$
$103$		−	0.851177i		−	0.0838690i	0.999120	−	0.0419345i	$-0.0133521\pi$
$104$		0			0
$105$	−4.52285	+	0.877793i	−0.441385	+	0.0856638i
$106$		0			0
$107$			9.04815i			0.874718i	0.899287	+	0.437359i	$0.144086\pi$
$107$			9.04815i			0.874718i	−0.899287	+	0.437359i	$0.855914\pi$
$108$		0			0
$109$		−	9.37322i		−	0.897792i	−0.893584	−	0.448896i	$-0.851817\pi$
$109$		−	9.37322i		−	0.897792i	0.893584	−	0.448896i	$-0.148183\pi$
$110$		0			0
$111$	−15.9501	+	3.09559i	−1.51392	+	0.293820i
$112$		0			0
$113$			14.4664i			1.36088i	0.732803	+	0.680441i	$0.238212\pi$
$113$			14.4664i			1.36088i	−0.732803	+	0.680441i	$0.761788\pi$
$114$		0			0
$115$	−3.15623			−0.294320
$116$		0			0
$117$	−3.07558	−	7.62505i	−0.284337	−	0.704936i
$118$		0			0
$119$	1.31999			0.121003
$120$		0			0
$121$	5.96261			0.542055
$122$		0			0
$123$	17.6250	−	3.42066i	1.58920	−	0.308431i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			11.2701i			1.00006i	0.866008	+	0.500030i	$0.166678\pi$
$127$			11.2701i			1.00006i	−0.866008	+	0.500030i	$0.833322\pi$
$128$		0			0
$129$	−0.706625	−	3.64090i	−0.0622148	−	0.320564i
$130$		0			0
$131$		−	17.0057i		−	1.48579i	−0.669406	−	0.742897i	$-0.733451\pi$
$131$		−	17.0057i		−	1.48579i	0.669406	−	0.742897i	$-0.266549\pi$
$132$		0			0
$133$			22.0914i			1.91557i
$134$		0			0
$135$	2.82624	+	4.36032i	0.243244	+	0.375277i
$136$		0			0
$137$		−	7.01493i		−	0.599326i	−0.954045	−	0.299663i	$-0.903126\pi$
$137$		−	7.01493i		−	0.599326i	0.954045	−	0.299663i	$-0.0968743\pi$
$138$		0			0
$139$	7.93752			0.673251			0.336626	−	0.941639i	$-0.390714\pi$
$139$	7.93752			0.673251			0.336626	+	0.941639i	$0.390714\pi$
$140$		0			0
$141$	−2.95845	−	15.2435i	−0.249146	−	1.28373i
$142$		0			0
$143$	6.15116			0.514386
$144$		0			0
$145$	−6.55688			−0.544519
$146$		0			0
$147$	0.0249407	+	0.128507i	0.00205707	+	0.0105991i
$148$		0			0
$149$	−17.6176			−1.44329			−0.721647	−	0.692262i	$-0.756614\pi$
$149$	−17.6176			−1.44329			−0.721647	+	0.692262i	$0.756614\pi$
$150$		0			0
$151$		−	8.34508i		−	0.679113i	−0.940586	−	0.339557i	$-0.889723\pi$
$151$		−	8.34508i		−	0.679113i	0.940586	−	0.339557i	$-0.110277\pi$
$152$		0			0
$153$	−0.556882	−	1.38064i	−0.0450212	−	0.111618i
$154$		0			0
$155$		−	1.50376i		−	0.120785i
$156$		0			0
$157$			16.3882i			1.30792i	0.756530	+	0.653959i	$0.226893\pi$
$157$			16.3882i			1.30792i	−0.756530	+	0.653959i	$0.773107\pi$
$158$		0			0
$159$	3.34000	+	17.2095i	0.264880	+	1.36480i
$160$		0			0
$161$			8.39557i			0.661664i
$162$		0			0
$163$	−11.6226			−0.910353			−0.455176	−	0.890401i	$-0.650424\pi$
$163$	−11.6226			−0.910353			−0.455176	+	0.890401i	$0.650424\pi$
$164$		0			0
$165$	−3.81623	+	0.740652i	−0.297093	+	0.0576597i
$166$		0			0
$167$	−4.63754			−0.358863			−0.179432	−	0.983770i	$-0.557426\pi$
$167$	−4.63754			−0.358863			−0.179432	+	0.983770i	$0.557426\pi$
$168$		0			0
$169$	5.48883			0.422217
$170$		0			0
$171$	23.1064	−	9.31999i	1.76699	−	0.712718i
$172$		0			0
$173$	−7.27997			−0.553486			−0.276743	−	0.960944i	$-0.589255\pi$
$173$	−7.27997			−0.553486			−0.276743	+	0.960944i	$0.589255\pi$
$174$		0			0
$175$		−	2.66000i		−	0.201077i
$176$		0			0
$177$	−0.740652	+	0.143745i	−0.0556708	+	0.0108046i
$178$		0			0
$179$		−	3.07558i		−	0.229880i	−0.993372	−	0.114940i	$-0.963332\pi$
$179$		−	3.07558i		−	0.229880i	0.993372	−	0.114940i	$-0.0366675\pi$
$180$		0			0
$181$		−	17.6026i		−	1.30839i	−0.756326	−	0.654195i	$-0.773007\pi$
$181$		−	17.6026i		−	1.30839i	0.756326	−	0.654195i	$-0.226993\pi$
$182$		0			0
$183$	−9.99259	+	1.93936i	−0.738673	+	0.143361i
$184$		0			0
$185$		−	9.38064i		−	0.689678i
$186$		0			0
$187$	1.11376			0.0814465
$188$		0			0
$189$	11.5984	−	7.51778i	0.843662	−	0.546838i
$190$		0			0
$191$	−17.5626			−1.27078			−0.635391	−	0.772190i	$-0.719161\pi$
$191$	−17.5626			−1.27078			−0.635391	+	0.772190i	$0.719161\pi$
$192$		0			0
$193$	−27.7239			−1.99561			−0.997804	−	0.0662324i	$-0.978902\pi$
$193$	−27.7239			−1.99561			−0.997804	+	0.0662324i	$0.978902\pi$
$194$		0			0
$195$	4.66000	−	0.904410i	0.333709	−	0.0647661i
$196$		0			0
$197$	12.1363			0.864678			0.432339	−	0.901711i	$-0.357688\pi$
$197$	12.1363			0.864678			0.432339	+	0.901711i	$0.357688\pi$
$198$		0			0
$199$			9.78636i			0.693737i	0.937914	+	0.346868i	$0.112755\pi$
$199$			9.78636i			0.693737i	−0.937914	+	0.346868i	$0.887245\pi$
$200$		0			0
$201$	3.34661	+	17.2435i	0.236052	+	1.21626i
$202$		0			0
$203$			17.4413i			1.22414i
$204$		0			0
$205$			10.3657i			0.723972i
$206$		0			0
$207$	8.78128	−	3.54195i	0.610341	−	0.246182i
$208$		0			0
$209$			18.6400i			1.28935i
$210$		0			0
$211$	−19.6100			−1.35001			−0.675004	−	0.737814i	$-0.735858\pi$
$211$	−19.6100			−1.35001			−0.675004	+	0.737814i	$0.735858\pi$
$212$		0			0
$213$	1.42807	+	7.35818i	0.0978498	+	0.504174i
$214$		0			0
$215$	2.14130			0.146035
$216$		0			0
$217$	−4.00000			−0.271538
$218$		0			0
$219$	3.50132	+	18.0406i	0.236597	+	1.21907i
$220$		0			0
$221$	−1.36002			−0.0914846
$222$		0			0
$223$		−	8.34753i		−	0.558992i	−0.960147	−	0.279496i	$-0.909833\pi$
$223$		−	8.34753i		−	0.558992i	0.960147	−	0.279496i	$-0.0901673\pi$
$224$		0			0
$225$	−2.78220	+	1.12221i	−0.185480	+	0.0748138i
$226$		0			0
$227$			15.8895i			1.05462i	0.849673	+	0.527311i	$0.176800\pi$
$227$			15.8895i			1.05462i	−0.849673	+	0.527311i	$0.823200\pi$
$228$		0			0
$229$			5.64751i			0.373198i	0.982436	+	0.186599i	$0.0597465\pi$
$229$			5.64751i			0.373198i	−0.982436	+	0.186599i	$0.940254\pi$
$230$		0			0
$231$	1.97013	+	10.1512i	0.129625	+	0.667897i
$232$		0			0
$233$		−	22.7539i		−	1.49065i	−0.666699	−	0.745327i	$-0.732293\pi$
$233$		−	22.7539i		−	1.49065i	0.666699	−	0.745327i	$-0.267707\pi$
$234$		0			0
$235$	8.96505			0.584816
$236$		0			0
$237$	−22.6101	+	4.38816i	−1.46868	+	0.285042i
$238$		0			0
$239$	0.610115			0.0394650			0.0197325	−	0.999805i	$-0.493719\pi$
$239$	0.610115			0.0394650			0.0197325	+	0.999805i	$0.493719\pi$
$240$		0			0
$241$	10.1893			0.656353			0.328177	−	0.944616i	$-0.393566\pi$
$241$	10.1893			0.656353			0.328177	+	0.944616i	$0.393566\pi$
$242$		0			0
$243$	−12.7563	−	8.95967i	−0.818320	−	0.574763i
$244$		0			0
$245$	−0.0755782			−0.00482851
$246$		0			0
$247$		−	22.7613i		−	1.44826i
$248$		0			0
$249$	10.3391	−	2.00660i	0.655213	−	0.127163i
$250$		0			0
$251$		−	2.24441i		−	0.141666i	−0.997488	−	0.0708331i	$-0.977434\pi$
$251$		−	2.24441i		−	0.141666i	0.997488	−	0.0708331i	$-0.0225658\pi$
$252$		0			0
$253$			7.08389i			0.445361i
$254$		0			0
$255$	0.843766	−	0.163758i	0.0528387	−	0.0102549i
$256$		0			0
$257$			2.51129i			0.156650i	0.996928	+	0.0783248i	$0.0249571\pi$
$257$			2.51129i			0.156650i	−0.996928	+	0.0783248i	$0.975043\pi$
$258$		0			0
$259$	−24.9525			−1.55047
$260$		0			0
$261$	18.2426	−	7.35818i	1.12919	−	0.455460i
$262$		0			0
$263$	23.6451			1.45802			0.729008	−	0.684505i	$-0.239981\pi$
$263$	23.6451			1.45802			0.729008	+	0.684505i	$0.239981\pi$
$264$		0			0
$265$	−10.1213			−0.621746
$266$		0			0
$267$	13.8088	−	2.68001i	0.845086	−	0.164014i
$268$		0			0
$269$	24.5802			1.49868			0.749342	−	0.662183i	$-0.230370\pi$
$269$	24.5802			1.49868			0.749342	+	0.662183i	$0.230370\pi$
$270$		0			0
$271$		−	29.8676i		−	1.81433i	−0.420776	−	0.907164i	$-0.638242\pi$
$271$		−	29.8676i		−	1.81433i	0.420776	−	0.907164i	$-0.361758\pi$
$272$		0			0
$273$	−2.40573	−	12.3956i	−0.145601	−	0.750214i
$274$		0			0
$275$		−	2.24441i		−	0.135343i
$276$		0			0
$277$		−	22.8471i		−	1.37275i	−0.727248	−	0.686375i	$-0.759201\pi$
$277$		−	22.8471i		−	1.37275i	0.727248	−	0.686375i	$-0.240799\pi$
$278$		0			0
$279$	1.68753	+	4.18377i	0.101030	+	0.250476i
$280$		0			0
$281$		−	28.2145i		−	1.68314i	−0.540149	−	0.841569i	$-0.681632\pi$
$281$		−	28.2145i		−	1.68314i	0.540149	−	0.841569i	$-0.318368\pi$
$282$		0			0
$283$	−22.4688			−1.33563			−0.667816	−	0.744326i	$-0.732771\pi$
$283$	−22.4688			−1.33563			−0.667816	+	0.744326i	$0.732771\pi$
$284$		0			0
$285$	2.74065	+	14.1213i	0.162342	+	0.836473i
$286$		0			0
$287$	27.5727			1.62757
$288$		0			0
$289$	16.7537			0.985515
$290$		0			0
$291$	−2.46881	−	12.7206i	−0.144724	−	0.745697i
$292$		0			0
$293$	16.9626			0.990966			0.495483	−	0.868618i	$-0.334991\pi$
$293$	16.9626			0.990966			0.495483	+	0.868618i	$0.334991\pi$
$294$		0			0
$295$		−	0.435595i		−	0.0253613i
$296$		0			0
$297$	9.78636	−	6.34324i	0.567862	−	0.368072i
$298$		0			0
$299$		−	8.65014i		−	0.500250i
$300$		0			0
$301$		−	5.69585i		−	0.328303i
$302$		0			0
$303$	−0.0274234	−	0.141300i	−0.00157543	−	0.00811745i
$304$		0			0
$305$		−	5.87687i		−	0.336509i
$306$		0			0
$307$	−10.9576			−0.625386			−0.312693	−	0.949854i	$-0.601231\pi$
$307$	−10.9576			−0.625386			−0.312693	+	0.949854i	$0.601231\pi$
$308$		0			0
$309$	−1.44728	+	0.280887i	−0.0823327	+	0.0159791i
$310$		0			0
$311$	30.5550			1.73262			0.866309	−	0.499509i	$-0.166486\pi$
$311$	30.5550			1.73262			0.866309	+	0.499509i	$0.166486\pi$
$312$		0			0
$313$	−10.9476			−0.618793			−0.309396	−	0.950933i	$-0.600127\pi$
$313$	−10.9476			−0.618793			−0.309396	+	0.950933i	$0.600127\pi$
$314$		0			0
$315$	2.98507	+	7.40065i	0.168189	+	0.416979i
$316$		0			0
$317$	22.3639			1.25608			0.628040	−	0.778181i	$-0.283857\pi$
$317$	22.3639			1.25608			0.628040	+	0.778181i	$0.283857\pi$
$318$		0			0
$319$			14.7164i			0.823958i
$320$		0			0
$321$	15.3848	−	2.98587i	0.858695	−	0.166655i
$322$		0			0
$323$		−	4.12129i		−	0.229315i
$324$		0			0
$325$			2.74065i			0.152024i
$326$		0			0
$327$	−15.9375	+	3.09314i	−0.881347	+	0.171051i
$328$		0			0
$329$		−	23.8470i		−	1.31473i
$330$		0			0
$331$	−20.6326			−1.13407			−0.567034	−	0.823694i	$-0.691909\pi$
$331$	−20.6326			−1.13407			−0.567034	+	0.823694i	$0.691909\pi$
$332$		0			0
$333$	10.5270	+	26.0988i	0.576877	+	1.43021i
$334$		0			0
$335$	−10.1413			−0.554078
$336$		0			0
$337$	3.11376			0.169618			0.0848088	−	0.996397i	$-0.472972\pi$
$337$	3.11376			0.169618			0.0848088	+	0.996397i	$0.472972\pi$
$338$		0			0
$339$	24.5975	−	4.77387i	1.33595	−	0.259281i
$340$		0			0
$341$	−3.37506			−0.182770
$342$		0			0
$343$		−	18.4189i		−	0.994529i
$344$		0			0
$345$	1.04155	+	5.36662i	0.0560752	+	0.288929i
$346$		0			0
$347$		−	11.7282i		−	0.629601i	−0.949158	−	0.314800i	$-0.898062\pi$
$347$		−	11.7282i		−	0.629601i	0.949158	−	0.314800i	$-0.101938\pi$
$348$		0			0
$349$		−	20.1213i		−	1.07707i	−0.842604	−	0.538534i	$-0.818978\pi$
$349$		−	20.1213i		−	1.07707i	0.842604	−	0.538534i	$-0.181022\pi$
$350$		0			0
$351$	−11.9501	+	7.74573i	−0.637850	+	0.413436i
$352$		0			0
$353$		−	1.22764i		−	0.0653407i	−0.999466	−	0.0326703i	$-0.989599\pi$
$353$		−	1.22764i		−	0.0653407i	0.999466	−	0.0326703i	$-0.0104011\pi$
$354$		0			0
$355$	−4.32752			−0.229681
$356$		0			0
$357$	−0.435595	−	2.24441i	−0.0230541	−	0.118787i
$358$		0			0
$359$	−37.2351			−1.96519			−0.982595	−	0.185759i	$-0.940526\pi$
$359$	−37.2351			−1.96519			−0.982595	+	0.185759i	$0.940526\pi$
$360$		0			0
$361$	49.9740			2.63021
$362$		0			0
$363$	−1.96765	−	10.1384i	−0.103275	−	0.532126i
$364$		0			0
$365$	−10.6101			−0.555359
$366$		0			0
$367$		−	18.9826i		−	0.990885i	−0.868641	−	0.495442i	$-0.835006\pi$
$367$		−	18.9826i		−	0.990885i	0.868641	−	0.495442i	$-0.164994\pi$
$368$		0			0
$369$	−11.6325	−	28.8395i	−0.605562	−	1.50132i
$370$		0			0
$371$			26.9226i			1.39775i
$372$		0			0
$373$			33.6232i			1.74094i	0.492218	+	0.870472i	$0.336186\pi$
$373$			33.6232i			1.74094i	−0.492218	+	0.870472i	$0.663814\pi$
$374$		0			0
$375$	−0.329998	−	1.70032i	−0.0170410	−	0.0878043i
$376$		0			0
$377$		−	17.9701i		−	0.925509i
$378$		0			0
$379$	16.3051			0.837535			0.418767	−	0.908094i	$-0.362462\pi$
$379$	16.3051			0.837535			0.418767	+	0.908094i	$0.362462\pi$
$380$		0			0
$381$	19.1628	−	3.71911i	0.981742	−	0.190536i
$382$		0			0
$383$	−3.44372			−0.175966			−0.0879831	−	0.996122i	$-0.528042\pi$
$383$	−3.44372			−0.175966			−0.0879831	+	0.996122i	$0.528042\pi$
$384$		0			0
$385$	−5.97013			−0.304266
$386$		0			0
$387$	−5.95753	+	2.40298i	−0.302838	+	0.122150i
$388$		0			0
$389$	22.2426			1.12774			0.563872	−	0.825862i	$-0.309311\pi$
$389$	22.2426			1.12774			0.563872	+	0.825862i	$0.309311\pi$
$390$		0			0
$391$		−	1.56624i		−	0.0792084i
$392$		0			0
$393$	−28.9152	+	5.61184i	−1.45858	+	0.283080i
$394$		0			0
$395$		−	13.2975i		−	0.669072i
$396$		0			0
$397$		−	8.23678i		−	0.413392i	−0.978405	−	0.206696i	$-0.933729\pi$
$397$		−	8.23678i		−	0.413392i	0.978405	−	0.206696i	$-0.0662711\pi$
$398$		0			0
$399$	37.5626	−	7.29012i	1.88048	−	0.364963i
$400$		0			0
$401$			4.16620i			0.208050i	0.994575	+	0.104025i	$0.0331722\pi$
$401$			4.16620i			0.208050i	−0.994575	+	0.104025i	$0.966828\pi$
$402$		0			0
$403$	4.12129			0.205296
$404$		0			0
$405$	6.48130	−	6.24441i	0.322059	−	0.310287i
$406$		0			0
$407$	−21.0540			−1.04361
$408$		0			0
$409$	−11.4132			−0.564349			−0.282174	−	0.959363i	$-0.591056\pi$
$409$	−11.4132			−0.564349			−0.282174	+	0.959363i	$0.591056\pi$
$410$		0			0
$411$	−11.9277	+	2.31491i	−0.588348	+	0.114186i
$412$		0			0
$413$	−1.15868			−0.0570149
$414$		0			0
$415$			6.08066i			0.298488i
$416$		0			0
$417$	−2.61936	−	13.4964i	−0.128271	−	0.660919i
$418$		0			0
$419$		−	15.9319i		−	0.778326i	−0.921169	−	0.389163i	$-0.872764\pi$
$419$		−	15.9319i		−	0.778326i	0.921169	−	0.389163i	$-0.127236\pi$
$420$		0			0
$421$		−	20.3507i		−	0.991830i	−0.868371	−	0.495915i	$-0.834833\pi$
$421$		−	20.3507i		−	0.991830i	0.868371	−	0.495915i	$-0.165167\pi$
$422$		0			0
$423$	−24.9426	+	10.0606i	−1.21275	+	0.489165i
$424$		0			0
$425$			0.496238i			0.0240711i
$426$		0			0
$427$	−15.6325			−0.756508
$428$		0			0
$429$	−2.02987	−	10.4590i	−0.0980030	−	0.504963i
$430$		0			0
$431$	−20.2377			−0.974815			−0.487407	−	0.873175i	$-0.662057\pi$
$431$	−20.2377			−0.974815			−0.487407	+	0.873175i	$0.662057\pi$
$432$		0			0
$433$	−6.44391			−0.309675			−0.154837	−	0.987940i	$-0.549485\pi$
$433$	−6.44391			−0.309675			−0.154837	+	0.987940i	$0.549485\pi$
$434$		0			0
$435$	2.16376	+	11.1488i	0.103744	+	0.534545i
$436$		0			0
$437$	26.2127			1.25392
$438$		0			0
$439$		−	36.0588i		−	1.72099i	−0.509456	−	0.860496i	$-0.670154\pi$
$439$		−	36.0588i		−	1.72099i	0.509456	−	0.860496i	$-0.329846\pi$
$440$		0			0
$441$	0.210274	−	0.0848144i	0.0100130	−	0.00403878i
$442$		0			0
$443$		−	11.3157i		−	0.537625i	−0.963192	−	0.268813i	$-0.913369\pi$
$443$		−	11.3157i		−	0.537625i	0.963192	−	0.268813i	$-0.0866313\pi$
$444$		0			0
$445$			8.12129i			0.384986i
$446$		0			0
$447$	5.81378	+	29.9557i	0.274983	+	1.41686i
$448$		0			0
$449$			10.2444i			0.483464i	0.970343	+	0.241732i	$0.0777154\pi$
$449$			10.2444i			0.483464i	−0.970343	+	0.241732i	$0.922285\pi$
$450$		0			0
$451$	23.2649			1.09550
$452$		0			0
$453$	−14.1893	+	2.75386i	−0.666673	+	0.129388i
$454$		0			0
$455$	7.29012			0.341766
$456$		0			0
$457$	33.3714			1.56105			0.780524	−	0.625126i	$-0.214952\pi$
$457$	33.3714			1.56105			0.780524	+	0.625126i	$0.214952\pi$
$458$		0			0
$459$	−2.16376	+	1.40249i	−0.100996	+	0.0654625i
$460$		0			0
$461$	23.8368			1.11019			0.555096	−	0.831786i	$-0.312681\pi$
$461$	23.8368			1.11019			0.555096	+	0.831786i	$0.312681\pi$
$462$		0			0
$463$		−	27.7288i		−	1.28867i	−0.764744	−	0.644334i	$-0.777135\pi$
$463$		−	27.7288i		−	1.28867i	0.764744	−	0.644334i	$-0.222865\pi$
$464$		0			0
$465$	−2.55688	+	0.496238i	−0.118573	+	0.0230125i
$466$		0			0
$467$		−	24.4996i		−	1.13371i	−0.823819	−	0.566853i	$-0.808161\pi$
$467$		−	24.4996i		−	1.13371i	0.823819	−	0.566853i	$-0.191839\pi$
$468$		0			0
$469$			26.9758i			1.24563i
$470$		0			0
$471$	27.8652	−	5.40806i	1.28396	−	0.249190i
$472$		0			0
$473$		−	4.80596i		−	0.220978i
$474$		0			0
$475$	−8.30506			−0.381062
$476$		0			0
$477$	28.1595	−	11.3582i	1.28933	−	0.520055i
$478$		0			0
$479$	−0.0449159			−0.00205226			−0.00102613	−	0.999999i	$-0.500327\pi$
$479$	−0.0449159			−0.00205226			−0.00102613	+	0.999999i	$0.500327\pi$
$480$		0			0
$481$	25.7091			1.17223
$482$		0			0
$483$	14.2752	−	2.77052i	0.649544	−	0.126063i
$484$		0			0
$485$	7.48130			0.339709
$486$		0			0
$487$			24.6301i			1.11610i	0.829808	+	0.558049i	$0.188450\pi$
$487$			24.6301i			1.11610i	−0.829808	+	0.558049i	$0.811550\pi$
$488$		0			0
$489$	3.83544	+	19.7622i	0.173444	+	0.893677i
$490$		0			0
$491$			26.8545i			1.21193i	0.795492	+	0.605964i	$0.207212\pi$
$491$			26.8545i			1.21193i	−0.795492	+	0.605964i	$0.792788\pi$
$492$		0			0
$493$		−	3.25378i		−	0.146543i
$494$		0			0
$495$	2.51870	+	6.24441i	0.113207	+	0.280666i
$496$		0			0
$497$			11.5112i			0.516347i
$498$		0			0
$499$	0.305057			0.0136562			0.00682812	−	0.999977i	$-0.497827\pi$
$499$	0.305057			0.0136562			0.00682812	+	0.999977i	$0.497827\pi$
$500$		0			0
$501$	1.53038	+	7.88532i	0.0683722	+	0.352290i
$502$		0			0
$503$	10.6526			0.474975			0.237488	−	0.971391i	$-0.423676\pi$
$503$	10.6526			0.474975			0.237488	+	0.971391i	$0.423676\pi$
$504$		0			0
$505$	0.0831016			0.00369797
$506$		0			0
$507$	−1.81130	−	9.33278i	−0.0804427	−	0.414483i
$508$		0			0
$509$	−23.0457			−1.02148			−0.510742	−	0.859734i	$-0.670629\pi$
$509$	−23.0457			−1.02148			−0.510742	+	0.859734i	$0.670629\pi$
$510$		0			0
$511$			28.2229i			1.24851i
$512$		0			0
$513$	−23.4721	−	36.2127i	−1.03632	−	1.59883i
$514$		0			0
$515$		−	0.851177i		−	0.0375073i
$516$		0			0
$517$		−	20.1213i		−	0.884933i
$518$		0			0
$519$	2.40237	+	12.3783i	0.105453	+	0.543347i
$520$		0			0
$521$		−	20.9476i		−	0.917729i	−0.888506	−	0.458865i	$-0.848256\pi$
$521$		−	20.9476i		−	0.917729i	0.888506	−	0.458865i	$-0.151744\pi$
$522$		0			0
$523$	8.68520			0.379777			0.189889	−	0.981806i	$-0.439187\pi$
$523$	8.68520			0.379777			0.189889	+	0.981806i	$0.439187\pi$
$524$		0			0
$525$	−4.52285	+	0.877793i	−0.197394	+	0.0383100i
$526$		0			0
$527$	0.746224			0.0325060
$528$		0			0
$529$	−13.0382			−0.566878
$530$		0			0
$531$	0.488827	+	1.21191i	0.0212133	+	0.0525925i
$532$		0			0
$533$	−28.4088			−1.23052
$534$		0			0
$535$			9.04815i			0.391186i
$536$		0			0
$537$	−5.22948	+	1.01493i	−0.225669	+	0.0437977i
$538$		0			0
$539$			0.169629i			0.00730643i
$540$		0			0
$541$		−	14.0914i		−	0.605837i	−0.953016	−	0.302919i	$-0.902039\pi$
$541$		−	14.0914i		−	0.605837i	0.953016	−	0.302919i	$-0.0979611\pi$
$542$		0			0
$543$	−29.9301	+	5.80882i	−1.28442	+	0.249280i
$544$		0			0
$545$		−	9.37322i		−	0.401505i
$546$		0			0
$547$	7.44647			0.318388			0.159194	−	0.987247i	$-0.449110\pi$
$547$	7.44647			0.318388			0.159194	+	0.987247i	$0.449110\pi$
$548$		0			0
$549$	6.59507	+	16.3507i	0.281471	+	0.697829i
$550$		0			0
$551$	54.4553			2.31987
$552$		0			0
$553$	−35.3714			−1.50414
$554$		0			0
$555$	−15.9501	+	3.09559i	−0.677045	+	0.131400i
$556$		0			0
$557$	−10.1065			−0.428225			−0.214112	−	0.976809i	$-0.568686\pi$
$557$	−10.1065			−0.428225			−0.214112	+	0.976809i	$0.568686\pi$
$558$		0			0
$559$			5.86856i			0.248214i
$560$		0			0
$561$	−0.367540	−	1.89376i	−0.0155175	−	0.0799546i
$562$		0			0
$563$			17.0482i			0.718494i	0.933243	+	0.359247i	$0.116966\pi$
$563$			17.0482i			0.718494i	−0.933243	+	0.359247i	$0.883034\pi$
$564$		0			0
$565$			14.4664i			0.608605i
$566$		0			0
$567$	−16.6101	−	17.2402i	−0.697559	−	0.724022i
$568$		0			0
$569$		−	30.8694i		−	1.29411i	−0.762443	−	0.647055i	$-0.776000\pi$
$569$		−	30.8694i		−	1.29411i	0.762443	−	0.647055i	$-0.224000\pi$
$570$		0			0
$571$	−1.33756			−0.0559751			−0.0279875	−	0.999608i	$-0.508910\pi$
$571$	−1.33756			−0.0559751			−0.0279875	+	0.999608i	$0.508910\pi$
$572$		0			0
$573$	5.79561	+	29.8621i	0.242115	+	1.24751i
$574$		0			0
$575$	−3.15623			−0.131624
$576$		0			0
$577$	24.2576			1.00986			0.504929	−	0.863161i	$-0.331519\pi$
$577$	24.2576			1.00986			0.504929	+	0.863161i	$0.331519\pi$
$578$		0			0
$579$	9.14882	+	47.1396i	0.380212	+	1.95905i
$580$		0			0
$581$	16.1745			0.671032
$582$		0			0
$583$			22.7164i			0.940815i
$584$		0			0
$585$	−3.07558	−	7.62505i	−0.127160	−	0.315257i
$586$		0			0
$587$			15.2344i			0.628793i	0.949292	+	0.314396i	$0.101802\pi$
$587$			15.2344i			0.628793i	−0.949292	+	0.314396i	$0.898198\pi$
$588$		0			0
$589$			12.4888i			0.514593i
$590$		0			0
$591$	−4.00497	−	20.6357i	−0.164742	−	0.848839i
$592$		0			0
$593$		−	30.5076i		−	1.25280i	−0.779503	−	0.626399i	$-0.784528\pi$
$593$		−	30.5076i		−	1.25280i	0.779503	−	0.626399i	$-0.215472\pi$
$594$		0			0
$595$	1.31999			0.0541144
$596$		0			0
$597$	16.6400	−	3.22948i	0.681029	−	0.132174i
$598$		0			0
$599$	−41.5878			−1.69923			−0.849615	−	0.527403i	$-0.823166\pi$
$599$	−41.5878			−1.69923			−0.849615	+	0.527403i	$0.823166\pi$
$600$		0			0
$601$	−29.6709			−1.21030			−0.605150	−	0.796111i	$-0.706887\pi$
$601$	−29.6709			−1.21030			−0.605150	+	0.796111i	$0.706887\pi$
$602$		0			0
$603$	28.2152	−	11.3806i	1.14901	−	0.463455i
$604$		0			0
$605$	5.96261			0.242414
$606$		0			0
$607$		−	4.87638i		−	0.197926i	−0.995091	−	0.0989631i	$-0.968447\pi$
$607$		−	4.87638i		−	0.197926i	0.995091	−	0.0989631i	$-0.0315526\pi$
$608$		0			0
$609$	29.6558	−	5.75559i	1.20171	−	0.233228i
$610$		0			0
$611$			24.5701i			0.993999i
$612$		0			0
$613$			44.9682i			1.81625i	0.418702	+	0.908124i	$0.362485\pi$
$613$			44.9682i			1.81625i	−0.418702	+	0.908124i	$0.637515\pi$
$614$		0			0
$615$	17.6250	−	3.42066i	0.710711	−	0.137934i
$616$		0			0
$617$			13.7762i			0.554609i	0.960782	+	0.277305i	$0.0894411\pi$
$617$			13.7762i			0.554609i	−0.960782	+	0.277305i	$0.910559\pi$
$618$		0			0
$619$	−18.6126			−0.748105			−0.374052	−	0.927408i	$-0.622032\pi$
$619$	−18.6126			−0.748105			−0.374052	+	0.927408i	$0.622032\pi$
$620$		0			0
$621$	−8.92026	−	13.7622i	−0.357958	−	0.552258i
$622$		0			0
$623$	21.6026			0.865490
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	31.6940	−	6.15116i	1.26574	−	0.245654i
$628$		0			0
$629$	4.65503			0.185608
$630$		0			0
$631$			48.0489i			1.91280i	0.292067	+	0.956398i	$0.405657\pi$
$631$			48.0489i			1.91280i	−0.292067	+	0.956398i	$0.594343\pi$
$632$		0			0
$633$	6.47126	+	33.3434i	0.257210	+	1.32528i
$634$		0			0
$635$			11.2701i			0.447241i
$636$		0			0
$637$		−	0.207134i		−	0.00820693i
$638$		0			0
$639$	12.0400	−	4.85637i	0.476296	−	0.192115i
$640$		0			0
$641$			22.1995i			0.876827i	0.898773	+	0.438414i	$0.144460\pi$
$641$			22.1995i			0.876827i	−0.898773	+	0.438414i	$0.855540\pi$
$642$		0			0
$643$	20.5751			0.811401			0.405700	−	0.914006i	$-0.367028\pi$
$643$	20.5751			0.811401			0.405700	+	0.914006i	$0.367028\pi$
$644$		0			0
$645$	−0.706625	−	3.64090i	−0.0278233	−	0.143360i
$646$		0			0
$647$	−48.7852			−1.91795			−0.958973	−	0.283498i	$-0.908505\pi$
$647$	−48.7852			−1.91795			−0.958973	+	0.283498i	$0.908505\pi$
$648$		0			0
$649$	−0.977654			−0.0383763
$650$		0			0
$651$	1.31999	+	6.80130i	0.0517346	+	0.266564i
$652$		0			0
$653$	46.2278			1.80903			0.904516	−	0.426440i	$-0.140232\pi$
$653$	46.2278			1.80903			0.904516	+	0.426440i	$0.140232\pi$
$654$		0			0
$655$		−	17.0057i		−	0.664467i
$656$		0			0
$657$	29.5195	−	11.9067i	1.15166	−	0.464526i
$658$		0			0
$659$		−	42.6631i		−	1.66192i	−0.556333	−	0.830960i	$-0.687792\pi$
$659$		−	42.6631i		−	1.66192i	0.556333	−	0.830960i	$-0.312208\pi$
$660$		0			0
$661$		−	41.8171i		−	1.62650i	−0.581916	−	0.813249i	$-0.697697\pi$
$661$		−	41.8171i		−	1.62650i	0.581916	−	0.813249i	$-0.302303\pi$
$662$		0			0
$663$	0.448803	+	2.31247i	0.0174300	+	0.0898088i
$664$		0			0
$665$			22.0914i			0.856668i
$666$		0			0
$667$	20.6951			0.801316
$668$		0			0
$669$	−14.1935	+	2.75467i	−0.548752	+	0.106502i
$670$		0			0
$671$	−13.1901			−0.509200
$672$		0			0
$673$	−8.76127			−0.337722			−0.168861	−	0.985640i	$-0.554009\pi$
$673$	−8.76127			−0.337722			−0.168861	+	0.985640i	$0.554009\pi$
$674$		0			0
$675$	2.82624	+	4.36032i	0.108782	+	0.167829i
$676$		0			0
$677$	−26.2278			−1.00801			−0.504007	−	0.863699i	$-0.668142\pi$
$677$	−26.2278			−1.00801			−0.504007	+	0.863699i	$0.668142\pi$
$678$		0			0
$679$		−	19.9002i		−	0.763701i
$680$		0			0
$681$	27.0173	−	5.24349i	1.03530	−	0.200931i
$682$		0			0
$683$			25.5469i			0.977525i	0.872417	+	0.488763i	$0.162552\pi$
$683$			25.5469i			0.977525i	−0.872417	+	0.488763i	$0.837448\pi$
$684$		0			0
$685$		−	7.01493i		−	0.268027i
$686$		0			0
$687$	9.60259	−	1.86367i	0.366362	−	0.0711033i
$688$		0			0
$689$		−	27.7389i		−	1.05677i
$690$		0			0
$691$	3.61000			0.137331			0.0686655	−	0.997640i	$-0.478126\pi$
$691$	3.61000			0.137331			0.0686655	+	0.997640i	$0.478126\pi$
$692$		0			0
$693$	16.6101	−	6.69972i	0.630966	−	0.254501i
$694$		0			0
$695$	7.93752			0.301087
$696$		0			0
$697$	−5.14386			−0.194838
$698$		0			0
$699$	−38.6889	+	7.50873i	−1.46335	+	0.284006i
$700$		0			0
$701$	−19.9850			−0.754821			−0.377411	−	0.926046i	$-0.623185\pi$
$701$	−19.9850			−0.754821			−0.377411	+	0.926046i	$0.623185\pi$
$702$		0			0
$703$			77.9067i			2.93831i
$704$		0			0
$705$	−2.95845	−	15.2435i	−0.111422	−	0.574103i
$706$		0			0
$707$		−	0.221050i		−	0.00831344i
$708$		0			0
$709$		−	15.0839i		−	0.566488i	−0.959048	−	0.283244i	$-0.908589\pi$
$709$		−	15.0839i		−	0.566488i	0.959048	−	0.283244i	$-0.0914106\pi$
$710$		0			0
$711$	14.9226	+	36.9964i	0.559641	+	1.38747i
$712$		0			0
$713$			4.74622i			0.177747i
$714$		0			0
$715$	6.15116			0.230040
$716$		0			0
$717$	−0.201337	−	1.03739i	−0.00751905	−	0.0387421i
$718$		0			0
$719$	−9.05770			−0.337795			−0.168898	−	0.985634i	$-0.554021\pi$
$719$	−9.05770			−0.337795			−0.168898	+	0.985634i	$0.554021\pi$
$720$		0			0
$721$	−2.26413			−0.0843205
$722$		0			0
$723$	−3.36246	−	17.3252i	−0.125051	−	0.644330i
$724$		0			0
$725$	−6.55688			−0.243516
$726$		0			0
$727$		−	16.2675i		−	0.603327i	−0.953414	−	0.301664i	$-0.902458\pi$
$727$		−	16.2675i		−	0.603327i	0.953414	−	0.301664i	$-0.0975420\pi$
$728$		0			0
$729$	−11.0248	+	24.6466i	−0.408325	+	0.912837i
$730$		0			0
$731$			1.06259i			0.0393015i
$732$		0			0
$733$			4.34324i			0.160421i	0.996778	+	0.0802107i	$0.0255593\pi$
$733$			4.34324i			0.160421i	−0.996778	+	0.0802107i	$0.974441\pi$
$734$		0			0
$735$	0.0249407	+	0.128507i	0.000919950	+	0.00474007i
$736$		0			0
$737$			22.7613i			0.838422i
$738$		0			0
$739$	23.6500			0.869980			0.434990	−	0.900435i	$-0.356752\pi$
$739$	23.6500			0.869980			0.434990	+	0.900435i	$0.356752\pi$
$740$		0			0
$741$	−38.7015	+	7.51117i	−1.42174	+	0.275930i
$742$		0			0
$743$	0.858588			0.0314985			0.0157493	−	0.999876i	$-0.494987\pi$
$743$	0.858588			0.0314985			0.0157493	+	0.999876i	$0.494987\pi$
$744$		0			0
$745$	−17.6176			−0.645460
$746$		0			0
$747$	−6.82375	−	16.9176i	−0.249668	−	0.618983i
$748$		0			0
$749$	24.0681			0.879428
$750$		0			0
$751$		−	14.0486i		−	0.512642i	−0.966592	−	0.256321i	$-0.917490\pi$
$751$		−	14.0486i		−	0.512642i	0.966592	−	0.256321i	$-0.0825105\pi$
$752$		0			0
$753$	−3.81623	+	0.740652i	−0.139071	+	0.0269909i
$754$		0			0
$755$		−	8.34508i		−	0.303709i
$756$		0			0
$757$			1.12324i			0.0408248i	0.999792	+	0.0204124i	$0.00649792\pi$
$757$			1.12324i			0.0408248i	−0.999792	+	0.0204124i	$0.993502\pi$
$758$		0			0
$759$	12.0449	−	2.33767i	0.437203	−	0.0848521i
$760$		0			0
$761$			35.0187i			1.26943i	0.772748	+	0.634713i	$0.218882\pi$
$761$			35.0187i			1.26943i	−0.772748	+	0.634713i	$0.781118\pi$
$762$		0			0
$763$	−24.9327			−0.902626
$764$		0			0
$765$	−0.556882	−	1.38064i	−0.0201341	−	0.0499170i
$766$		0			0
$767$	1.19381			0.0431061
$768$		0			0
$769$	−23.7688			−0.857125			−0.428562	−	0.903512i	$-0.640980\pi$
$769$	−23.7688			−0.857125			−0.428562	+	0.903512i	$0.640980\pi$
$770$		0			0
$771$	4.27000	−	0.828719i	0.153780	−	0.0298456i
$772$		0			0
$773$	−3.09894			−0.111461			−0.0557306	−	0.998446i	$-0.517749\pi$
$773$	−3.09894			−0.111461			−0.0557306	+	0.998446i	$0.517749\pi$
$774$		0			0
$775$		−	1.50376i		−	0.0540167i
$776$		0			0
$777$	8.23426	+	42.4273i	0.295402	+	1.52207i
$778$		0			0
$779$		−	86.0877i		−	3.08441i
$780$		0			0
$781$			9.71273i			0.347549i
$782$		0			0
$783$	−18.5313	−	28.5901i	−0.662255	−	1.02173i
$784$		0			0
$785$			16.3882i			0.584918i
$786$		0			0
$787$	47.4828			1.69258			0.846290	−	0.532723i	$-0.178831\pi$
$787$	47.4828			1.69258			0.846290	+	0.532723i	$0.178831\pi$
$788$		0			0
$789$	−7.80282	−	40.2043i	−0.277788	−	1.43131i
$790$		0			0
$791$	38.4805			1.36821
$792$		0			0
$793$	16.1065			0.571957
$794$		0			0
$795$	3.34000	+	17.2095i	0.118458	+	0.610357i
$796$		0			0
$797$	−41.3862			−1.46597			−0.732987	−	0.680242i	$-0.761875\pi$
$797$	−41.3862			−1.46597			−0.732987	+	0.680242i	$0.761875\pi$
$798$		0			0
$799$			4.44880i			0.157387i
$800$		0			0
$801$	−9.11376	−	22.5951i	−0.322019	−	0.798357i
$802$		0			0
$803$			23.8135i			0.840360i
$804$		0			0
$805$			8.39557i			0.295905i
$806$		0			0
$807$	−8.11143	−	41.7944i	−0.285536	−	1.47123i
$808$		0			0
$809$			26.2127i			0.921590i	0.887507	+	0.460795i	$0.152436\pi$
$809$			26.2127i			0.921590i	−0.887507	+	0.460795i	$0.847564\pi$
$810$		0			0
$811$	−21.4853			−0.754450			−0.377225	−	0.926122i	$-0.623122\pi$
$811$	−21.4853			−0.754450			−0.377225	+	0.926122i	$0.623122\pi$
$812$		0			0
$813$	−50.7846	+	9.85625i	−1.78109	+	0.345674i
$814$		0			0
$815$	−11.6226			−0.407122
$816$		0			0
$817$	−17.7836			−0.622170
$818$		0			0
$819$	−20.2826	+	8.18103i	−0.708731	+	0.285868i
$820$		0			0
$821$	23.9350			0.835337			0.417669	−	0.908599i	$-0.362847\pi$
$821$	23.9350			0.835337			0.417669	+	0.908599i	$0.362847\pi$
$822$		0			0
$823$		−	5.47611i		−	0.190885i	−0.995435	−	0.0954427i	$-0.969573\pi$
$823$		−	5.47611i		−	0.190885i	0.995435	−	0.0954427i	$-0.0304267\pi$
$824$		0			0
$825$	−3.81623	+	0.740652i	−0.132864	+	0.0257862i
$826$		0			0
$827$		−	22.1022i		−	0.768568i	−0.923215	−	0.384284i	$-0.874448\pi$
$827$		−	22.1022i		−	0.768568i	0.923215	−	0.384284i	$-0.125552\pi$
$828$		0			0
$829$			17.3732i			0.603397i	0.953403	+	0.301699i	$0.0975536\pi$
$829$			17.3732i			0.603397i	−0.953403	+	0.301699i	$0.902446\pi$
$830$		0			0
$831$	−38.8475	+	7.53950i	−1.34760	+	0.261542i
$832$		0			0
$833$		−	0.0375048i		−	0.00129946i
$834$		0			0
$835$	−4.63754			−0.160489
$836$		0			0
$837$	6.55688	−	4.24999i	0.226639	−	0.146901i
$838$		0			0
$839$	−18.2724			−0.630835			−0.315417	−	0.948953i	$-0.602145\pi$
$839$	−18.2724			−0.630835			−0.315417	+	0.948953i	$0.602145\pi$
$840$		0			0
$841$	13.9927			0.482507
$842$		0			0
$843$	−47.9739	+	9.31074i	−1.65231	+	0.320679i
$844$		0			0
$845$	5.48883			0.188821
$846$		0			0
$847$		−	15.8605i		−	0.544974i
$848$		0			0
$849$	7.41466	+	38.2043i	0.254471	+	1.31117i
$850$		0			0
$851$			29.6075i			1.01493i
$852$		0			0
$853$			2.49808i			0.0855325i	0.999085	+	0.0427663i	$0.0136171\pi$
$853$			2.49808i			0.0855325i	−0.999085	+	0.0427663i	$0.986383\pi$
$854$		0			0
$855$	23.1064	−	9.31999i	0.790221	−	0.318737i
$856$		0			0
$857$		−	25.9775i		−	0.887376i	−0.896181	−	0.443688i	$-0.853670\pi$
$857$		−	25.9775i		−	0.887376i	0.896181	−	0.443688i	$-0.146330\pi$
$858$		0			0
$859$	−32.8351			−1.12032			−0.560160	−	0.828384i	$-0.689260\pi$
$859$	−32.8351			−1.12032			−0.560160	+	0.828384i	$0.689260\pi$
$860$		0			0
$861$	−9.09894	−	46.8826i	−0.310091	−	1.59775i
$862$		0			0
$863$	37.2926			1.26945			0.634727	−	0.772737i	$-0.281113\pi$
$863$	37.2926			1.26945			0.634727	+	0.772737i	$0.281113\pi$
$864$		0			0
$865$	−7.27997			−0.247526
$866$		0			0
$867$	−5.52870	−	28.4868i	−0.187765	−	0.967462i
$868$		0			0
$869$	−29.8452			−1.01243
$870$		0			0
$871$		−	27.7938i		−	0.941756i
$872$		0			0
$873$	−20.8145	+	8.39557i	−0.704464	+	0.284147i
$874$		0			0
$875$		−	2.66000i		−	0.0899243i
$876$		0			0
$877$			13.0931i			0.442124i	0.975260	+	0.221062i	$0.0709523\pi$
$877$			13.0931i			0.442124i	−0.975260	+	0.221062i	$0.929048\pi$
$878$		0			0
$879$	−5.59763	−	28.8419i	−0.188803	−	0.972814i
$880$		0			0
$881$			34.6083i			1.16598i	0.812478	+	0.582991i	$0.198118\pi$
$881$			34.6083i			1.16598i	−0.812478	+	0.582991i	$0.801882\pi$
$882$		0			0
$883$	53.9202			1.81456			0.907280	−	0.420526i	$-0.138154\pi$
$883$	53.9202			1.81456			0.907280	+	0.420526i	$0.138154\pi$
$884$		0			0
$885$	−0.740652	+	0.143745i	−0.0248967	+	0.00483195i
$886$		0			0
$887$	−32.8949			−1.10450			−0.552252	−	0.833677i	$-0.686231\pi$
$887$	−32.8949			−1.10450			−0.552252	+	0.833677i	$0.686231\pi$
$888$		0			0
$889$	29.9784			1.00544
$890$		0			0
$891$	−14.0150	−	14.5467i	−0.469522	−	0.487334i
$892$		0			0
$893$	−74.4553			−2.49155
$894$		0			0
$895$		−	3.07558i		−	0.102805i
$896$		0			0
$897$	−14.7080	+	2.85453i	−0.491087	+	0.0953099i
$898$		0			0
$899$			9.85999i			0.328849i
$900$		0			0
$901$		−	5.02257i		−	0.167326i
$902$		0			0
$903$	−9.68479	+	1.87962i	−0.322290	+	0.0625498i
$904$		0			0
$905$		−	17.6026i		−	0.585130i
$906$		0			0
$907$	−35.3615			−1.17416			−0.587080	−	0.809529i	$-0.699723\pi$
$907$	−35.3615			−1.17416			−0.587080	+	0.809529i	$0.699723\pi$
$908$		0			0
$909$	−0.231206	+	0.0932572i	−0.00766860	+	0.00309315i
$910$		0			0
$911$	−7.61275			−0.252222			−0.126111	−	0.992016i	$-0.540249\pi$
$911$	−7.61275			−0.252222			−0.126111	+	0.992016i	$0.540249\pi$
$912$		0			0
$913$	13.6475			0.451667
$914$		0			0
$915$	−9.99259	+	1.93936i	−0.330345	+	0.0641131i
$916$		0			0
$917$	−45.2351			−1.49379
$918$		0			0
$919$			18.0474i			0.595330i	0.954670	+	0.297665i	$0.0962078\pi$
$919$			18.0474i			0.595330i	−0.954670	+	0.297665i	$0.903792\pi$
$920$		0			0
$921$	3.61600	+	18.6315i	0.119151	+	0.613930i
$922$		0			0
$923$		−	11.8602i		−	0.390384i
$924$		0			0
$925$		−	9.38064i		−	0.308433i
$926$		0			0
$927$	0.955196	+	2.36815i	0.0313728	+	0.0777801i
$928$		0			0
$929$			25.6958i			0.843053i	0.906816	+	0.421527i	$0.138506\pi$
$929$			25.6958i			0.843053i	−0.906816	+	0.421527i	$0.861494\pi$
$930$		0			0
$931$	0.627681			0.0205714
$932$		0			0
$933$	−10.0831	−	51.9535i	−0.330106	−	1.70088i
$934$		0			0
$935$	1.11376			0.0364240
$936$		0			0
$937$	−22.9124			−0.748516			−0.374258	−	0.927325i	$-0.622103\pi$
$937$	−22.9124			−0.748516			−0.374258	+	0.927325i	$0.622103\pi$
$938$		0			0
$939$	3.61267	+	18.6144i	0.117895	+	0.607458i
$940$		0			0
$941$	13.7657			0.448750			0.224375	−	0.974503i	$-0.427966\pi$
$941$	13.7657			0.448750			0.224375	+	0.974503i	$0.427966\pi$
$942$		0			0
$943$		−	32.7166i		−	1.06540i
$944$		0			0
$945$	11.5984	−	7.51778i	0.377297	−	0.244553i
$946$		0			0
$947$		−	10.3933i		−	0.337738i	−0.985638	−	0.168869i	$-0.945988\pi$
$947$		−	10.3933i		−	0.337738i	0.985638	−	0.168869i	$-0.0540116\pi$
$948$		0			0
$949$		−	29.0786i		−	0.943932i
$950$		0			0
$951$	−7.38003	−	38.0258i	−0.239314	−	1.23307i
$952$		0			0
$953$			17.4290i			0.564580i	0.959329	+	0.282290i	$0.0910940\pi$
$953$			17.4290i			0.564580i	−0.959329	+	0.282290i	$0.908906\pi$
$954$		0			0
$955$	−17.5626			−0.568311
$956$		0			0
$957$	25.0226	−	4.85637i	0.808865	−	0.156984i
$958$		0			0
$959$	−18.6597			−0.602553
$960$		0			0
$961$	28.7387			0.927055
$962$		0			0
$963$	−10.1539	−	25.1738i	−0.327205	−	0.811214i
$964$		0			0
$965$	−27.7239			−0.892463
$966$		0			0
$967$			29.7587i			0.956976i	0.878094	+	0.478488i	$0.158815\pi$
$967$			29.7587i			0.956976i	−0.878094	+	0.478488i	$0.841185\pi$
$968$		0			0
$969$	−7.00752	+	1.36002i	−0.225114	+	0.0436900i
$970$		0			0
$971$			8.71819i			0.279780i	0.990167	+	0.139890i	$0.0446749\pi$
$971$			8.71819i			0.279780i	−0.990167	+	0.139890i	$0.955325\pi$
$972$		0			0
$973$		−	21.1138i		−	0.676876i
$974$		0			0
$975$	4.66000	−	0.904410i	0.149239	−	0.0289643i
$976$		0			0
$977$			12.2500i			0.391912i	0.980613	+	0.195956i	$0.0627809\pi$
$977$			12.2500i			0.391912i	−0.980613	+	0.195956i	$0.937219\pi$
$978$		0			0
$979$	18.2275			0.582554
$980$		0			0
$981$	10.5187	+	26.0782i	0.335836	+	0.832613i
$982$		0			0
$983$	−4.74400			−0.151310			−0.0756551	−	0.997134i	$-0.524105\pi$
$983$	−4.74400			−0.151310			−0.0756551	+	0.997134i	$0.524105\pi$
$984$		0			0
$985$	12.1363			0.386696
$986$		0			0
$987$	−40.5476	+	7.86946i	−1.29065	+	0.250488i
$988$		0			0
$989$	−6.75844			−0.214906
$990$		0			0
$991$			23.7350i			0.753966i	0.926220	+	0.376983i	$0.123039\pi$
$991$			23.7350i			0.753966i	−0.926220	+	0.376983i	$0.876961\pi$
$992$		0			0
$993$	6.80871	+	35.0821i	0.216068	+	1.11330i
$994$		0			0
$995$			9.78636i			0.310249i
$996$		0			0
$997$			5.50192i			0.174248i	0.996197	+	0.0871238i	$0.0277676\pi$
$997$			5.50192i			0.174248i	−0.996197	+	0.0871238i	$0.972232\pi$
$998$		0			0
$999$	40.9026	−	26.5119i	1.29410	−	0.838799i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.b.h.191.3	yes	8
3.2	odd	2		1920.2.b.g.191.4	yes	8
4.3	odd	2		1920.2.b.b.191.6	yes	8
8.3	odd	2		1920.2.b.g.191.3	yes	8
8.5	even	2		1920.2.b.a.191.6	yes	8
12.11	even	2		1920.2.b.a.191.5	✓	8
24.5	odd	2		1920.2.b.b.191.5	yes	8
24.11	even	2	inner	1920.2.b.h.191.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.b.a.191.5	✓	8	12.11	even	2
1920.2.b.a.191.6	yes	8	8.5	even	2
1920.2.b.b.191.5	yes	8	24.5	odd	2
1920.2.b.b.191.6	yes	8	4.3	odd	2
1920.2.b.g.191.3	yes	8	8.3	odd	2
1920.2.b.g.191.4	yes	8	3.2	odd	2
1920.2.b.h.191.3	yes	8	1.1	even	1	trivial
1920.2.b.h.191.4	yes	8	24.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.329998 - 1.70032i) q^{3} +1.00000 q^{5} -2.66000i q^{7} +(-2.78220 + 1.12221i) q^{9} -2.24441i q^{11} +2.74065i q^{13} +(-0.329998 - 1.70032i) q^{15} +0.496238i q^{17} -8.30506 q^{19} +(-4.52285 + 0.877793i) q^{21} -3.15623 q^{23} +1.00000 q^{25} +(2.82624 + 4.36032i) q^{27} -6.55688 q^{29} -1.50376i q^{31} +(-3.81623 + 0.740652i) q^{33} -2.66000i q^{35} -9.38064i q^{37} +(4.66000 - 0.904410i) q^{39} +10.3657i q^{41} +2.14130 q^{43} +(-2.78220 + 1.12221i) q^{45} +8.96505 q^{47} -0.0755782 q^{49} +(0.843766 - 0.163758i) q^{51} -10.1213 q^{53} -2.24441i q^{55} +(2.74065 + 14.1213i) q^{57} -0.435595i q^{59} -5.87687i q^{61} +(2.98507 + 7.40065i) q^{63} +2.74065i q^{65} -10.1413 q^{67} +(1.04155 + 5.36662i) q^{69} -4.32752 q^{71} -10.6101 q^{73} +(-0.329998 - 1.70032i) q^{75} -5.97013 q^{77} -13.2975i q^{79} +(6.48130 - 6.24441i) q^{81} +6.08066i q^{83} +0.496238i q^{85} +(2.16376 + 11.1488i) q^{87} +8.12129i q^{89} +7.29012 q^{91} +(-2.55688 + 0.496238i) q^{93} -8.30506 q^{95} +7.48130 q^{97} +(2.51870 + 6.24441i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} + 8 q^{5} - 4 q^{9} + 2 q^{15} - 8 q^{19} + 4 q^{21} - 12 q^{23} + 8 q^{25} + 14 q^{27} - 8 q^{29} - 8 q^{33} + 28 q^{39} - 36 q^{43} - 4 q^{45} + 4 q^{47} + 20 q^{51} - 16 q^{63} - 28 q^{67}+ \cdots + 64 q^{99}+O(q^{100}) \) 8 * q + 2 * q^3 + 8 * q^5 - 4 * q^9 + 2 * q^15 - 8 * q^19 + 4 * q^21 - 12 * q^23 + 8 * q^25 + 14 * q^27 - 8 * q^29 - 8 * q^33 + 28 * q^39 - 36 * q^43 - 4 * q^45 + 4 * q^47 + 20 * q^51 - 16 * q^63 - 28 * q^67 + 12 * q^69 - 24 * q^71 + 32 * q^73 + 2 * q^75 + 32 * q^77 + 8 * q^81 + 12 * q^87 - 40 * q^91 + 24 * q^93 - 8 * q^95 + 16 * q^97 + 64 * q^99

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