LMFDB - Embedded newform 448.3.h.a.97.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(97,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("448.97");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.h (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:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.98344960000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 17x^{4} + 81 $ x^8 - 17*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			97.6
Root			$0.144845 + 1.72598i$ of defining polynomial
Character	$\chi$	$=$	448.97
Dual form			448.3.h.a.97.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16228 q^{3} -3.74166 q^{5} +(-5.91608 + 3.74166i) q^{7} +1.00000 q^{9} +O(q^{10})$	$q+3.16228 q^{3} -3.74166 q^{5} +(-5.91608 + 3.74166i) q^{7} +1.00000 q^{9} +2.00000i q^{11} -11.2250 q^{13} -11.8322 q^{15} -6.32456i q^{17} -3.16228 q^{19} +(-18.7083 + 11.8322i) q^{21} -23.6643 q^{23} -11.0000 q^{25} -25.2982 q^{27} +23.6643i q^{29} -37.4166i q^{31} +6.32456i q^{33} +(22.1359 - 14.0000i) q^{35} +47.3286i q^{37} -35.4965 q^{39} +50.5964i q^{41} +10.0000i q^{43} -3.74166 q^{45} -82.3165i q^{47} +(21.0000 - 44.2719i) q^{49} -20.0000i q^{51} +70.9930i q^{53} -7.48331i q^{55} -10.0000 q^{57} +60.0833 q^{59} +56.1249 q^{61} +(-5.91608 + 3.74166i) q^{63} +42.0000 q^{65} +70.0000i q^{67} -74.8331 q^{69} -35.4965 q^{71} -107.517i q^{73} -34.7851 q^{75} +(-7.48331 - 11.8322i) q^{77} +11.8322 q^{79} -89.0000 q^{81} -123.329 q^{83} +23.6643i q^{85} +74.8331i q^{87} +56.9210i q^{89} +(66.4078 - 42.0000i) q^{91} -118.322i q^{93} +11.8322 q^{95} +145.465i q^{97} +2.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} - 88 q^{25} + 168 q^{49} - 80 q^{57} + 336 q^{65} - 712 q^{81}+O(q^{100}) $ 8 * q + 8 * q^9 - 88 * q^25 + 168 * q^49 - 80 * q^57 + 336 * q^65 - 712 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	3.16228			1.05409			0.527046	−	0.849837i	$-0.323299\pi$
$3$	3.16228			1.05409			0.527046	+	0.849837i	$0.323299\pi$
$4$		0			0
$5$	−3.74166			−0.748331			−0.374166	−	0.927362i	$-0.622071\pi$
$5$	−3.74166			−0.748331			−0.374166	+	0.927362i	$0.622071\pi$
$6$		0			0
$7$	−5.91608	+	3.74166i	−0.845154	+	0.534522i
$7$	−5.91608	+	3.74166i	−0.845154	+	0.534522i
$8$		0			0
$9$	1.00000			0.111111
$10$		0			0
$11$			2.00000i			0.181818i	0.995859	+	0.0909091i	$0.0289773\pi$
$11$			2.00000i			0.181818i	−0.995859	+	0.0909091i	$0.971023\pi$
$12$		0			0
$13$	−11.2250			−0.863459			−0.431730	−	0.902003i	$-0.642097\pi$
$13$	−11.2250			−0.863459			−0.431730	+	0.902003i	$0.642097\pi$
$14$		0			0
$15$	−11.8322			−0.788811
$16$		0			0
$17$		−	6.32456i		−	0.372033i	−0.982547	−	0.186016i	$-0.940442\pi$
$17$		−	6.32456i		−	0.372033i	0.982547	−	0.186016i	$-0.0595577\pi$
$18$		0			0
$19$	−3.16228			−0.166436			−0.0832178	−	0.996531i	$-0.526520\pi$
$19$	−3.16228			−0.166436			−0.0832178	+	0.996531i	$0.526520\pi$
$20$		0			0
$21$	−18.7083	+	11.8322i	−0.890871	+	0.563436i
$22$		0			0
$23$	−23.6643			−1.02888			−0.514442	−	0.857525i	$-0.672001\pi$
$23$	−23.6643			−1.02888			−0.514442	+	0.857525i	$0.672001\pi$
$24$		0			0
$25$	−11.0000			−0.440000
$26$		0			0
$27$	−25.2982			−0.936971
$28$		0			0
$29$			23.6643i			0.816011i	0.912979	+	0.408006i	$0.133776\pi$
$29$			23.6643i			0.816011i	−0.912979	+	0.408006i	$0.866224\pi$
$30$		0			0
$31$		−	37.4166i		−	1.20699i	−0.797368	−	0.603493i	$-0.793775\pi$
$31$		−	37.4166i		−	1.20699i	0.797368	−	0.603493i	$-0.206225\pi$
$32$		0			0
$33$			6.32456i			0.191653i
$34$		0			0
$35$	22.1359	−	14.0000i	0.632456	−	0.400000i
$36$		0			0
$37$			47.3286i			1.27915i	0.768728	+	0.639576i	$0.220890\pi$
$37$			47.3286i			1.27915i	−0.768728	+	0.639576i	$0.779110\pi$
$38$		0			0
$39$	−35.4965			−0.910166
$40$		0			0
$41$			50.5964i			1.23406i	0.786940	+	0.617030i	$0.211664\pi$
$41$			50.5964i			1.23406i	−0.786940	+	0.617030i	$0.788336\pi$
$42$		0			0
$43$			10.0000i			0.232558i	0.993217	+	0.116279i	$0.0370967\pi$
$43$			10.0000i			0.232558i	−0.993217	+	0.116279i	$0.962903\pi$
$44$		0			0
$45$	−3.74166			−0.0831479
$46$		0			0
$47$		−	82.3165i		−	1.75141i	−0.482843	−	0.875707i	$-0.660396\pi$
$47$		−	82.3165i		−	1.75141i	0.482843	−	0.875707i	$-0.339604\pi$
$48$		0			0
$49$	21.0000	−	44.2719i	0.428571	−	0.903508i
$50$		0			0
$51$		−	20.0000i		−	0.392157i
$52$		0			0
$53$			70.9930i			1.33949i	0.742591	+	0.669745i	$0.233597\pi$
$53$			70.9930i			1.33949i	−0.742591	+	0.669745i	$0.766403\pi$
$54$		0			0
$55$		−	7.48331i		−	0.136060i
$56$		0			0
$57$	−10.0000			−0.175439
$58$		0			0
$59$	60.0833			1.01836			0.509180	−	0.860660i	$-0.329949\pi$
$59$	60.0833			1.01836			0.509180	+	0.860660i	$0.329949\pi$
$60$		0			0
$61$	56.1249			0.920080			0.460040	−	0.887898i	$-0.347835\pi$
$61$	56.1249			0.920080			0.460040	+	0.887898i	$0.347835\pi$
$62$		0			0
$63$	−5.91608	+	3.74166i	−0.0939060	+	0.0593914i
$64$		0			0
$65$	42.0000			0.646154
$66$		0			0
$67$			70.0000i			1.04478i	0.852708	+	0.522388i	$0.174959\pi$
$67$			70.0000i			1.04478i	−0.852708	+	0.522388i	$0.825041\pi$
$68$		0			0
$69$	−74.8331			−1.08454
$70$		0			0
$71$	−35.4965			−0.499950			−0.249975	−	0.968252i	$-0.580422\pi$
$71$	−35.4965			−0.499950			−0.249975	+	0.968252i	$0.580422\pi$
$72$		0			0
$73$		−	107.517i		−	1.47284i	−0.676524	−	0.736421i	$-0.736514\pi$
$73$		−	107.517i		−	1.47284i	0.676524	−	0.736421i	$-0.263486\pi$
$74$		0			0
$75$	−34.7851			−0.463801
$76$		0			0
$77$	−7.48331	−	11.8322i	−0.0971859	−	0.153664i
$78$		0			0
$79$	11.8322			0.149774			0.0748871	−	0.997192i	$-0.476140\pi$
$79$	11.8322			0.149774			0.0748871	+	0.997192i	$0.476140\pi$
$80$		0			0
$81$	−89.0000			−1.09877
$82$		0			0
$83$	−123.329			−1.48589			−0.742945	−	0.669353i	$-0.766572\pi$
$83$	−123.329			−1.48589			−0.742945	+	0.669353i	$0.766572\pi$
$84$		0			0
$85$			23.6643i			0.278404i
$86$		0			0
$87$			74.8331i			0.860151i
$88$		0			0
$89$			56.9210i			0.639562i	0.947492	+	0.319781i	$0.103609\pi$
$89$			56.9210i			0.639562i	−0.947492	+	0.319781i	$0.896391\pi$
$90$		0			0
$91$	66.4078	−	42.0000i	0.729756	−	0.461538i
$92$		0			0
$93$		−	118.322i		−	1.27228i
$94$		0			0
$95$	11.8322			0.124549
$96$		0			0
$97$			145.465i			1.49964i	0.661644	+	0.749818i	$0.269859\pi$
$97$			145.465i			1.49964i	−0.661644	+	0.749818i	$0.730141\pi$
$98$		0			0
$99$			2.00000i			0.0202020i
$100$		0			0
$101$	−56.1249			−0.555692			−0.277846	−	0.960626i	$-0.589620\pi$
$101$	−56.1249			−0.555692			−0.277846	+	0.960626i	$0.589620\pi$
$102$		0			0
$103$			44.8999i			0.435921i	0.975958	+	0.217961i	$0.0699404\pi$
$103$			44.8999i			0.435921i	−0.975958	+	0.217961i	$0.930060\pi$
$104$		0			0
$105$	70.0000	−	44.2719i	0.666667	−	0.421637i
$106$		0			0
$107$		−	130.000i		−	1.21495i	−0.794338	−	0.607477i	$-0.792182\pi$
$107$		−	130.000i		−	1.21495i	0.794338	−	0.607477i	$-0.207818\pi$
$108$		0			0
$109$		−	165.650i		−	1.51973i	−0.650083	−	0.759863i	$-0.725266\pi$
$109$		−	165.650i		−	1.51973i	0.650083	−	0.759863i	$-0.274734\pi$
$110$		0			0
$111$			149.666i			1.34835i
$112$		0			0
$113$	−60.0000			−0.530973			−0.265487	−	0.964115i	$-0.585533\pi$
$113$	−60.0000			−0.530973			−0.265487	+	0.964115i	$0.585533\pi$
$114$		0			0
$115$	88.5438			0.769946
$116$		0			0
$117$	−11.2250			−0.0959399
$118$		0			0
$119$	23.6643	+	37.4166i	0.198860	+	0.314425i
$120$		0			0
$121$	117.000			0.966942
$122$		0			0
$123$			160.000i			1.30081i
$124$		0			0
$125$	134.700			1.07760
$126$		0			0
$127$	118.322			0.931666			0.465833	−	0.884873i	$-0.345755\pi$
$127$	118.322			0.931666			0.465833	+	0.884873i	$0.345755\pi$
$128$		0			0
$129$			31.6228i			0.245138i
$130$		0			0
$131$	−9.48683			−0.0724186			−0.0362093	−	0.999344i	$-0.511528\pi$
$131$	−9.48683			−0.0724186			−0.0362093	+	0.999344i	$0.511528\pi$
$132$		0			0
$133$	18.7083	−	11.8322i	0.140664	−	0.0889636i
$134$		0			0
$135$	94.6573			0.701165
$136$		0			0
$137$	−170.000			−1.24088			−0.620438	−	0.784256i	$-0.713045\pi$
$137$	−170.000			−1.24088			−0.620438	+	0.784256i	$0.713045\pi$
$138$		0			0
$139$	−117.004			−0.841757			−0.420879	−	0.907117i	$-0.638278\pi$
$139$	−117.004			−0.841757			−0.420879	+	0.907117i	$0.638278\pi$
$140$		0			0
$141$		−	260.308i		−	1.84615i
$142$		0			0
$143$		−	22.4499i		−	0.156993i
$144$		0			0
$145$		−	88.5438i		−	0.610647i
$146$		0			0
$147$	66.4078	−	140.000i	0.451754	−	0.952381i
$148$		0			0
$149$			94.6573i			0.635284i	0.948211	+	0.317642i	$0.102891\pi$
$149$			94.6573i			0.635284i	−0.948211	+	0.317642i	$0.897109\pi$
$150$		0			0
$151$	212.979			1.41046			0.705228	−	0.708981i	$-0.250845\pi$
$151$	212.979			1.41046			0.705228	+	0.708981i	$0.250845\pi$
$152$		0			0
$153$		−	6.32456i		−	0.0413370i
$154$		0			0
$155$			140.000i			0.903226i
$156$		0			0
$157$	273.141			1.73975			0.869876	−	0.493271i	$-0.164199\pi$
$157$	273.141			1.73975			0.869876	+	0.493271i	$0.164199\pi$
$158$		0			0
$159$			224.499i			1.41195i
$160$		0			0
$161$	140.000	−	88.5438i	0.869565	−	0.549961i
$162$		0			0
$163$		−	90.0000i		−	0.552147i	−0.961136	−	0.276074i	$-0.910967\pi$
$163$		−	90.0000i		−	0.552147i	0.961136	−	0.276074i	$-0.0890334\pi$
$164$		0			0
$165$		−	23.6643i		−	0.143420i
$166$		0			0
$167$		−	269.399i		−	1.61317i	−0.591118	−	0.806585i	$-0.701313\pi$
$167$		−	269.399i		−	1.61317i	0.591118	−	0.806585i	$-0.298687\pi$
$168$		0			0
$169$	−43.0000			−0.254438
$170$		0			0
$171$	−3.16228			−0.0184929
$172$		0			0
$173$	−11.2250			−0.0648842			−0.0324421	−	0.999474i	$-0.510328\pi$
$173$	−11.2250			−0.0648842			−0.0324421	+	0.999474i	$0.510328\pi$
$174$		0			0
$175$	65.0769	−	41.1582i	0.371868	−	0.235190i
$176$		0			0
$177$	190.000			1.07345
$178$		0			0
$179$			258.000i			1.44134i	0.693278	+	0.720670i	$0.256166\pi$
$179$			258.000i			1.44134i	−0.693278	+	0.720670i	$0.743834\pi$
$180$		0			0
$181$	130.958			0.723525			0.361762	−	0.932270i	$-0.382175\pi$
$181$	130.958			0.723525			0.361762	+	0.932270i	$0.382175\pi$
$182$		0			0
$183$	177.482			0.969849
$184$		0			0
$185$		−	177.088i		−	0.957230i
$186$		0			0
$187$	12.6491			0.0676423
$188$		0			0
$189$	149.666	−	94.6573i	0.791885	−	0.500832i
$190$		0			0
$191$	−11.8322			−0.0619485			−0.0309742	−	0.999520i	$-0.509861\pi$
$191$	−11.8322			−0.0619485			−0.0309742	+	0.999520i	$0.509861\pi$
$192$		0			0
$193$	−320.000			−1.65803			−0.829016	−	0.559226i	$-0.811099\pi$
$193$	−320.000			−1.65803			−0.829016	+	0.559226i	$0.811099\pi$
$194$		0			0
$195$	132.816			0.681106
$196$		0			0
$197$			236.643i			1.20123i	0.799537	+	0.600617i	$0.205078\pi$
$197$			236.643i			1.20123i	−0.799537	+	0.600617i	$0.794922\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$			221.359i			1.10129i
$202$		0			0
$203$	−88.5438	−	140.000i	−0.436176	−	0.689655i
$204$		0			0
$205$		−	189.315i		−	0.923486i
$206$		0			0
$207$	−23.6643			−0.114320
$208$		0			0
$209$		−	6.32456i		−	0.0302610i
$210$		0			0
$211$		−	42.0000i		−	0.199052i	−0.995035	−	0.0995261i	$-0.968267\pi$
$211$		−	42.0000i		−	0.199052i	0.995035	−	0.0995261i	$-0.0317327\pi$
$212$		0			0
$213$	−112.250			−0.526994
$214$		0			0
$215$		−	37.4166i		−	0.174031i
$216$		0			0
$217$	140.000	+	221.359i	0.645161	+	1.02009i
$218$		0			0
$219$		−	340.000i		−	1.55251i
$220$		0			0
$221$			70.9930i			0.321235i
$222$		0			0
$223$			104.766i			0.469805i	0.972019	+	0.234902i	$0.0754770\pi$
$223$			104.766i			0.469805i	−0.972019	+	0.234902i	$0.924523\pi$
$224$		0			0
$225$	−11.0000			−0.0488889
$226$		0			0
$227$	−91.7061			−0.403991			−0.201996	−	0.979386i	$-0.564743\pi$
$227$	−91.7061			−0.403991			−0.201996	+	0.979386i	$0.564743\pi$
$228$		0			0
$229$	−280.624			−1.22543			−0.612717	−	0.790303i	$-0.709923\pi$
$229$	−280.624			−1.22543			−0.612717	+	0.790303i	$0.709923\pi$
$230$		0			0
$231$	−23.6643	−	37.4166i	−0.102443	−	0.161977i
$232$		0			0
$233$	190.000			0.815451			0.407725	−	0.913105i	$-0.366322\pi$
$233$	190.000			0.815451			0.407725	+	0.913105i	$0.366322\pi$
$234$		0			0
$235$			308.000i			1.31064i
$236$		0			0
$237$	37.4166			0.157876
$238$		0			0
$239$	−354.965			−1.48521			−0.742604	−	0.669731i	$-0.766410\pi$
$239$	−354.965			−1.48521			−0.742604	+	0.669731i	$0.766410\pi$
$240$		0			0
$241$			373.149i			1.54834i	0.632981	+	0.774168i	$0.281831\pi$
$241$			373.149i			1.54834i	−0.632981	+	0.774168i	$0.718169\pi$
$242$		0			0
$243$	−53.7587			−0.221229
$244$		0			0
$245$	−78.5748	+	165.650i	−0.320713	+	0.676123i
$246$		0			0
$247$	35.4965			0.143710
$248$		0			0
$249$	−390.000			−1.56627
$250$		0			0
$251$	205.548			0.818917			0.409458	−	0.912329i	$-0.365718\pi$
$251$	205.548			0.818917			0.409458	+	0.912329i	$0.365718\pi$
$252$		0			0
$253$		−	47.3286i		−	0.187070i
$254$		0			0
$255$			74.8331i			0.293463i
$256$		0			0
$257$		−	328.877i		−	1.27968i	−0.768510	−	0.639838i	$-0.779001\pi$
$257$		−	328.877i		−	1.27968i	0.768510	−	0.639838i	$-0.220999\pi$
$258$		0			0
$259$	−177.088	−	280.000i	−0.683736	−	1.08108i
$260$		0			0
$261$			23.6643i			0.0906679i
$262$		0			0
$263$	−461.454			−1.75458			−0.877289	−	0.479962i	$-0.840651\pi$
$263$	−461.454			−1.75458			−0.877289	+	0.479962i	$0.840651\pi$
$264$		0			0
$265$		−	265.631i		−	1.00238i
$266$		0			0
$267$			180.000i			0.674157i
$268$		0			0
$269$	−505.124			−1.87778			−0.938892	−	0.344213i	$-0.888146\pi$
$269$	−505.124			−1.87778			−0.938892	+	0.344213i	$0.888146\pi$
$270$		0			0
$271$			149.666i			0.552274i	0.961118	+	0.276137i	$0.0890544\pi$
$271$			149.666i			0.552274i	−0.961118	+	0.276137i	$0.910946\pi$
$272$		0			0
$273$	210.000	−	132.816i	0.769231	−	0.486504i
$274$		0			0
$275$		−	22.0000i		−	0.0800000i
$276$		0			0
$277$			212.979i			0.768877i	0.923151	+	0.384438i	$0.125605\pi$
$277$			212.979i			0.768877i	−0.923151	+	0.384438i	$0.874395\pi$
$278$		0			0
$279$		−	37.4166i		−	0.134110i
$280$		0			0
$281$	−158.000			−0.562278			−0.281139	−	0.959667i	$-0.590712\pi$
$281$	−158.000			−0.562278			−0.281139	+	0.959667i	$0.590712\pi$
$282$		0			0
$283$	−237.171			−0.838059			−0.419030	−	0.907973i	$-0.637630\pi$
$283$	−237.171			−0.838059			−0.419030	+	0.907973i	$0.637630\pi$
$284$		0			0
$285$	37.4166			0.131286
$286$		0			0
$287$	−189.315	−	299.333i	−0.659633	−	1.04297i
$288$		0			0
$289$	249.000			0.861592
$290$		0			0
$291$			460.000i			1.58076i
$292$		0			0
$293$	−385.391			−1.31533			−0.657663	−	0.753312i	$-0.728455\pi$
$293$	−385.391			−1.31533			−0.657663	+	0.753312i	$0.728455\pi$
$294$		0			0
$295$	−224.811			−0.762071
$296$		0			0
$297$		−	50.5964i		−	0.170358i
$298$		0			0
$299$	265.631			0.888399
$300$		0			0
$301$	−37.4166	−	59.1608i	−0.124308	−	0.196548i
$302$		0			0
$303$	−177.482			−0.585750
$304$		0			0
$305$	−210.000			−0.688525
$306$		0			0
$307$	192.899			0.628335			0.314168	−	0.949368i	$-0.398275\pi$
$307$	192.899			0.628335			0.314168	+	0.949368i	$0.398275\pi$
$308$		0			0
$309$			141.986i			0.459501i
$310$		0			0
$311$			112.250i			0.360932i	0.983581	+	0.180466i	$0.0577605\pi$
$311$			112.250i			0.360932i	−0.983581	+	0.180466i	$0.942239\pi$
$312$		0			0
$313$		−	12.6491i		−	0.0404125i	−0.999796	−	0.0202062i	$-0.993568\pi$
$313$		−	12.6491i		−	0.0404125i	0.999796	−	0.0202062i	$-0.00643229\pi$
$314$		0			0
$315$	22.1359	−	14.0000i	0.0702728	−	0.0444444i
$316$		0			0
$317$			567.944i			1.79162i	0.444437	+	0.895810i	$0.353404\pi$
$317$			567.944i			1.79162i	−0.444437	+	0.895810i	$0.646596\pi$
$318$		0			0
$319$	−47.3286			−0.148366
$320$		0			0
$321$		−	411.096i		−	1.28067i
$322$		0			0
$323$			20.0000i			0.0619195i
$324$		0			0
$325$	123.475			0.379922
$326$		0			0
$327$		−	523.832i		−	1.60193i
$328$		0			0
$329$	308.000	+	486.991i	0.936170	+	1.48022i
$330$		0			0
$331$		−	498.000i		−	1.50453i	−0.658860	−	0.752266i	$-0.728961\pi$
$331$		−	498.000i		−	1.50453i	0.658860	−	0.752266i	$-0.271039\pi$
$332$		0			0
$333$			47.3286i			0.142128i
$334$		0			0
$335$		−	261.916i		−	0.781839i
$336$		0			0
$337$	−140.000			−0.415430			−0.207715	−	0.978189i	$-0.566603\pi$
$337$	−140.000			−0.415430			−0.207715	+	0.978189i	$0.566603\pi$
$338$		0			0
$339$	−189.737			−0.559695
$340$		0			0
$341$	74.8331			0.219452
$342$		0			0
$343$	41.4126	+	340.491i	0.120736	+	0.992685i
$344$		0			0
$345$	280.000			0.811594
$346$		0			0
$347$			170.000i			0.489914i	0.969534	+	0.244957i	$0.0787738\pi$
$347$			170.000i			0.489914i	−0.969534	+	0.244957i	$0.921226\pi$
$348$		0			0
$349$	−93.5414			−0.268027			−0.134014	−	0.990980i	$-0.542787\pi$
$349$	−93.5414			−0.268027			−0.134014	+	0.990980i	$0.542787\pi$
$350$		0			0
$351$	283.972			0.809037
$352$		0			0
$353$			594.508i			1.68416i	0.539353	+	0.842080i	$0.318669\pi$
$353$			594.508i			1.68416i	−0.539353	+	0.842080i	$0.681331\pi$
$354$		0			0
$355$	132.816			0.374129
$356$		0			0
$357$	74.8331	+	118.322i	0.209617	+	0.331433i
$358$		0			0
$359$	−638.937			−1.77977			−0.889884	−	0.456187i	$-0.849215\pi$
$359$	−638.937			−1.77977			−0.889884	+	0.456187i	$0.849215\pi$
$360$		0			0
$361$	−351.000			−0.972299
$362$		0			0
$363$	369.986			1.01925
$364$		0			0
$365$			402.293i			1.10217i
$366$		0			0
$367$		−	553.765i		−	1.50890i	−0.656359	−	0.754449i	$-0.727904\pi$
$367$		−	553.765i		−	1.50890i	0.656359	−	0.754449i	$-0.272096\pi$
$368$		0			0
$369$			50.5964i			0.137118i
$370$		0			0
$371$	−265.631	−	420.000i	−0.715987	−	1.13208i
$372$		0			0
$373$		−	70.9930i		−	0.190330i	−0.995462	−	0.0951648i	$-0.969662\pi$
$373$		−	70.9930i		−	0.190330i	0.995462	−	0.0951648i	$-0.0303378\pi$
$374$		0			0
$375$	425.958			1.13589
$376$		0			0
$377$		−	265.631i		−	0.704592i
$378$		0			0
$379$			158.000i			0.416887i	0.978034	+	0.208443i	$0.0668397\pi$
$379$			158.000i			0.416887i	−0.978034	+	0.208443i	$0.933160\pi$
$380$		0			0
$381$	374.166			0.982062
$382$		0			0
$383$		−	381.649i		−	0.996473i	−0.867041	−	0.498236i	$-0.833981\pi$
$383$		−	381.649i		−	0.996473i	0.867041	−	0.498236i	$-0.166019\pi$
$384$		0			0
$385$	28.0000	+	44.2719i	0.0727273	+	0.114992i
$386$		0			0
$387$			10.0000i			0.0258398i
$388$		0			0
$389$			94.6573i			0.243335i	0.992571	+	0.121667i	$0.0388241\pi$
$389$			94.6573i			0.243335i	−0.992571	+	0.121667i	$0.961176\pi$
$390$		0			0
$391$			149.666i			0.382778i
$392$		0			0
$393$	−30.0000			−0.0763359
$394$		0			0
$395$	−44.2719			−0.112081
$396$		0			0
$397$	86.0581			0.216771			0.108386	−	0.994109i	$-0.465432\pi$
$397$	86.0581			0.216771			0.108386	+	0.994109i	$0.465432\pi$
$398$		0			0
$399$	59.1608	−	37.4166i	0.148273	−	0.0937759i
$400$		0			0
$401$	88.0000			0.219451			0.109726	−	0.993962i	$-0.465003\pi$
$401$	88.0000			0.219451			0.109726	+	0.993962i	$0.465003\pi$
$402$		0			0
$403$			420.000i			1.04218i
$404$		0			0
$405$	333.008			0.822241
$406$		0			0
$407$	−94.6573			−0.232573
$408$		0			0
$409$		−	392.122i		−	0.958735i	−0.877614	−	0.479367i	$-0.840866\pi$
$409$		−	392.122i		−	0.958735i	0.877614	−	0.479367i	$-0.159134\pi$
$410$		0			0
$411$	−537.587			−1.30800
$412$		0			0
$413$	−355.457	+	224.811i	−0.860672	+	0.544337i
$414$		0			0
$415$	461.454			1.11194
$416$		0			0
$417$	−370.000			−0.887290
$418$		0			0
$419$	142.302			0.339624			0.169812	−	0.985476i	$-0.445684\pi$
$419$	142.302			0.339624			0.169812	+	0.985476i	$0.445684\pi$
$420$		0			0
$421$		−	567.944i		−	1.34903i	−0.738259	−	0.674517i	$-0.764352\pi$
$421$		−	567.944i		−	1.34903i	0.738259	−	0.674517i	$-0.235648\pi$
$422$		0			0
$423$		−	82.3165i		−	0.194602i
$424$		0			0
$425$			69.5701i			0.163694i
$426$		0			0
$427$	−332.039	+	210.000i	−0.777609	+	0.491803i
$428$		0			0
$429$		−	70.9930i		−	0.165485i
$430$		0			0
$431$	−23.6643			−0.0549056			−0.0274528	−	0.999623i	$-0.508740\pi$
$431$	−23.6643			−0.0549056			−0.0274528	+	0.999623i	$0.508740\pi$
$432$		0			0
$433$			120.167i			0.277521i	0.990326	+	0.138760i	$0.0443118\pi$
$433$			120.167i			0.277521i	−0.990326	+	0.138760i	$0.955688\pi$
$434$		0			0
$435$		−	280.000i		−	0.643678i
$436$		0			0
$437$	74.8331			0.171243
$438$		0			0
$439$			561.249i			1.27847i	0.769011	+	0.639235i	$0.220749\pi$
$439$			561.249i			1.27847i	−0.769011	+	0.639235i	$0.779251\pi$
$440$		0			0
$441$	21.0000	−	44.2719i	0.0476190	−	0.100390i
$442$		0			0
$443$		−	490.000i		−	1.10609i	−0.833150	−	0.553047i	$-0.813465\pi$
$443$		−	490.000i		−	1.10609i	0.833150	−	0.553047i	$-0.186535\pi$
$444$		0			0
$445$		−	212.979i		−	0.478604i
$446$		0			0
$447$			299.333i			0.669648i
$448$		0			0
$449$	−98.0000			−0.218263			−0.109131	−	0.994027i	$-0.534807\pi$
$449$	−98.0000			−0.218263			−0.109131	+	0.994027i	$0.534807\pi$
$450$		0			0
$451$	−101.193			−0.224374
$452$		0			0
$453$	673.498			1.48675
$454$		0			0
$455$	−248.475	+	157.150i	−0.546100	+	0.345384i
$456$		0			0
$457$	260.000			0.568928			0.284464	−	0.958687i	$-0.408184\pi$
$457$	260.000			0.568928			0.284464	+	0.958687i	$0.408184\pi$
$458$		0			0
$459$			160.000i			0.348584i
$460$		0			0
$461$	−392.874			−0.852221			−0.426111	−	0.904671i	$-0.640117\pi$
$461$	−392.874			−0.852221			−0.426111	+	0.904671i	$0.640117\pi$
$462$		0			0
$463$	177.482			0.383331			0.191666	−	0.981460i	$-0.438611\pi$
$463$	177.482			0.383331			0.191666	+	0.981460i	$0.438611\pi$
$464$		0			0
$465$			442.719i			0.952084i
$466$		0			0
$467$	401.609			0.859977			0.429989	−	0.902834i	$-0.358518\pi$
$467$	401.609			0.859977			0.429989	+	0.902834i	$0.358518\pi$
$468$		0			0
$469$	−261.916	−	414.126i	−0.558456	−	0.882997i
$470$		0			0
$471$	863.748			1.83386
$472$		0			0
$473$	−20.0000			−0.0422833
$474$		0			0
$475$	34.7851			0.0732317
$476$		0			0
$477$			70.9930i			0.148832i
$478$		0			0
$479$			187.083i			0.390570i	0.980747	+	0.195285i	$0.0625631\pi$
$479$			187.083i			0.390570i	−0.980747	+	0.195285i	$0.937437\pi$
$480$		0			0
$481$		−	531.263i		−	1.10450i
$482$		0			0
$483$	442.719	−	280.000i	0.916602	−	0.579710i
$484$		0			0
$485$		−	544.279i		−	1.12223i
$486$		0			0
$487$	−212.979			−0.437328			−0.218664	−	0.975800i	$-0.570170\pi$
$487$	−212.979			−0.437328			−0.218664	+	0.975800i	$0.570170\pi$
$488$		0			0
$489$		−	284.605i		−	0.582014i
$490$		0			0
$491$			658.000i			1.34012i	0.742306	+	0.670061i	$0.233732\pi$
$491$			658.000i			1.34012i	−0.742306	+	0.670061i	$0.766268\pi$
$492$		0			0
$493$	149.666			0.303583
$494$		0			0
$495$		−	7.48331i		−	0.0151178i
$496$		0			0
$497$	210.000	−	132.816i	0.422535	−	0.267235i
$498$		0			0
$499$			518.000i			1.03808i	0.854751	+	0.519038i	$0.173710\pi$
$499$			518.000i			1.03808i	−0.854751	+	0.519038i	$0.826290\pi$
$500$		0			0
$501$		−	851.915i		−	1.70043i
$502$		0			0
$503$			680.982i			1.35384i	0.736056	+	0.676920i	$0.236686\pi$
$503$			680.982i			1.35384i	−0.736056	+	0.676920i	$0.763314\pi$
$504$		0			0
$505$	210.000			0.415842
$506$		0			0
$507$	−135.978			−0.268201
$508$		0			0
$509$	−467.707			−0.918875			−0.459437	−	0.888210i	$-0.651949\pi$
$509$	−467.707			−0.918875			−0.459437	+	0.888210i	$0.651949\pi$
$510$		0			0
$511$	402.293	+	636.082i	0.787267	+	1.24478i
$512$		0			0
$513$	80.0000			0.155945
$514$		0			0
$515$		−	168.000i		−	0.326214i
$516$		0			0
$517$	164.633			0.318439
$518$		0			0
$519$	−35.4965			−0.0683940
$520$		0			0
$521$		−	404.772i		−	0.776913i	−0.921467	−	0.388456i	$-0.873008\pi$
$521$		−	404.772i		−	0.776913i	0.921467	−	0.388456i	$-0.126992\pi$
$522$		0			0
$523$	654.591			1.25161			0.625804	−	0.779980i	$-0.284771\pi$
$523$	654.591			1.25161			0.625804	+	0.779980i	$0.284771\pi$
$524$		0			0
$525$	205.791	−	130.154i	0.391983	−	0.247912i
$526$		0			0
$527$	−236.643			−0.449038
$528$		0			0
$529$	31.0000			0.0586011
$530$		0			0
$531$	60.0833			0.113151
$532$		0			0
$533$		−	567.944i		−	1.06556i
$534$		0			0
$535$			486.415i			0.909188i
$536$		0			0
$537$			815.868i			1.51931i
$538$		0			0
$539$	88.5438	+	42.0000i	0.164274	+	0.0779221i
$540$		0			0
$541$		−	567.944i		−	1.04980i	−0.851163	−	0.524902i	$-0.824102\pi$
$541$		−	567.944i		−	1.04980i	0.851163	−	0.524902i	$-0.175898\pi$
$542$		0			0
$543$	414.126			0.762662
$544$		0			0
$545$			619.806i			1.13726i
$546$		0			0
$547$		−	410.000i		−	0.749543i	−0.927117	−	0.374771i	$-0.877721\pi$
$547$		−	410.000i		−	0.749543i	0.927117	−	0.374771i	$-0.122279\pi$
$548$		0			0
$549$	56.1249			0.102231
$550$		0			0
$551$		−	74.8331i		−	0.135813i
$552$		0			0
$553$	−70.0000	+	44.2719i	−0.126582	+	0.0800577i
$554$		0			0
$555$		−	560.000i		−	1.00901i
$556$		0			0
$557$			733.594i			1.31704i	0.752561	+	0.658522i	$0.228818\pi$
$557$			733.594i			1.31704i	−0.752561	+	0.658522i	$0.771182\pi$
$558$		0			0
$559$		−	112.250i		−	0.200805i
$560$		0			0
$561$	40.0000			0.0713012
$562$		0			0
$563$	173.925			0.308926			0.154463	−	0.987999i	$-0.450635\pi$
$563$	173.925			0.308926			0.154463	+	0.987999i	$0.450635\pi$
$564$		0			0
$565$	224.499			0.397344
$566$		0			0
$567$	526.531	−	333.008i	0.928626	−	0.587315i
$568$		0			0
$569$	−308.000			−0.541301			−0.270650	−	0.962678i	$-0.587239\pi$
$569$	−308.000			−0.541301			−0.270650	+	0.962678i	$0.587239\pi$
$570$		0			0
$571$		−	142.000i		−	0.248687i	−0.992239	−	0.124343i	$-0.960318\pi$
$571$		−	142.000i		−	0.248687i	0.992239	−	0.124343i	$-0.0396824\pi$
$572$		0			0
$573$	−37.4166			−0.0652994
$574$		0			0
$575$	260.308			0.452709
$576$		0			0
$577$		−	126.491i		−	0.219222i	−0.993975	−	0.109611i	$-0.965039\pi$
$577$		−	126.491i		−	0.219222i	0.993975	−	0.109611i	$-0.0349605\pi$
$578$		0			0
$579$	−1011.93			−1.74772
$580$		0			0
$581$	729.623	−	461.454i	1.25581	−	0.794241i
$582$		0			0
$583$	−141.986			−0.243544
$584$		0			0
$585$	42.0000			0.0717949
$586$		0			0
$587$	−989.793			−1.68619			−0.843094	−	0.537765i	$-0.819269\pi$
$587$	−989.793			−1.68619			−0.843094	+	0.537765i	$0.819269\pi$
$588$		0			0
$589$			118.322i			0.200886i
$590$		0			0
$591$			748.331i			1.26621i
$592$		0			0
$593$		−	518.614i		−	0.874559i	−0.899326	−	0.437280i	$-0.855942\pi$
$593$		−	518.614i		−	0.874559i	0.899326	−	0.437280i	$-0.144058\pi$
$594$		0			0
$595$	−88.5438	−	140.000i	−0.148813	−	0.235294i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	177.482			0.296298			0.148149	−	0.988965i	$-0.452669\pi$
$599$	177.482			0.296298			0.148149	+	0.988965i	$0.452669\pi$
$600$		0			0
$601$			499.640i			0.831348i	0.909514	+	0.415674i	$0.136454\pi$
$601$			499.640i			0.831348i	−0.909514	+	0.415674i	$0.863546\pi$
$602$		0			0
$603$			70.0000i			0.116086i
$604$		0			0
$605$	−437.774			−0.723593
$606$		0			0
$607$			44.8999i			0.0739702i	0.999316	+	0.0369851i	$0.0117754\pi$
$607$			44.8999i			0.0739702i	−0.999316	+	0.0369851i	$0.988225\pi$
$608$		0			0
$609$	−280.000	−	442.719i	−0.459770	−	0.726960i
$610$		0			0
$611$			924.000i			1.51227i
$612$		0			0
$613$		−	331.300i		−	0.540458i	−0.962796	−	0.270229i	$-0.912901\pi$
$613$		−	331.300i		−	0.540458i	0.962796	−	0.270229i	$-0.0870993\pi$
$614$		0			0
$615$		−	598.665i		−	0.973439i
$616$		0			0
$617$	−520.000			−0.842788			−0.421394	−	0.906878i	$-0.638459\pi$
$617$	−520.000			−0.842788			−0.421394	+	0.906878i	$0.638459\pi$
$618$		0			0
$619$	926.547			1.49685			0.748423	−	0.663222i	$-0.230811\pi$
$619$	926.547			1.49685			0.748423	+	0.663222i	$0.230811\pi$
$620$		0			0
$621$	598.665			0.964034
$622$		0			0
$623$	−212.979	−	336.749i	−0.341860	−	0.540528i
$624$		0			0
$625$	−229.000			−0.366400
$626$		0			0
$627$		−	20.0000i		−	0.0318979i
$628$		0			0
$629$	299.333			0.475886
$630$		0			0
$631$	1029.40			1.63138			0.815688	−	0.578492i	$-0.196359\pi$
$631$	1029.40			1.63138			0.815688	+	0.578492i	$0.196359\pi$
$632$		0			0
$633$		−	132.816i		−	0.209819i
$634$		0			0
$635$	−442.719			−0.697195
$636$		0			0
$637$	−235.724	+	496.951i	−0.370054	+	0.780142i
$638$		0			0
$639$	−35.4965			−0.0555500
$640$		0			0
$641$	448.000			0.698908			0.349454	−	0.936954i	$-0.386367\pi$
$641$	448.000			0.698908			0.349454	+	0.936954i	$0.386367\pi$
$642$		0			0
$643$	−812.705			−1.26393			−0.631964	−	0.774998i	$-0.717751\pi$
$643$	−812.705			−1.26393			−0.631964	+	0.774998i	$0.717751\pi$
$644$		0			0
$645$		−	118.322i		−	0.183444i
$646$		0			0
$647$		−	868.065i		−	1.34168i	−0.741604	−	0.670838i	$-0.765935\pi$
$647$		−	868.065i		−	1.34168i	0.741604	−	0.670838i	$-0.234065\pi$
$648$		0			0
$649$			120.167i			0.185156i
$650$		0			0
$651$	442.719	+	700.000i	0.680060	+	1.07527i
$652$		0			0
$653$			851.915i			1.30462i	0.757953	+	0.652309i	$0.226200\pi$
$653$			851.915i			1.30462i	−0.757953	+	0.652309i	$0.773800\pi$
$654$		0			0
$655$	35.4965			0.0541931
$656$		0			0
$657$		−	107.517i		−	0.163649i
$658$		0			0
$659$		−	318.000i		−	0.482549i	−0.970457	−	0.241275i	$-0.922435\pi$
$659$		−	318.000i		−	0.482549i	0.970457	−	0.241275i	$-0.0775655\pi$
$660$		0			0
$661$	617.373			0.933999			0.467000	−	0.884258i	$-0.345335\pi$
$661$	617.373			0.933999			0.467000	+	0.884258i	$0.345335\pi$
$662$		0			0
$663$			224.499i			0.338612i
$664$		0			0
$665$	−70.0000	+	44.2719i	−0.105263	+	0.0665743i
$666$		0			0
$667$		−	560.000i		−	0.839580i
$668$		0			0
$669$			331.300i			0.495217i
$670$		0			0
$671$			112.250i			0.167287i
$672$		0			0
$673$	690.000			1.02526			0.512630	−	0.858610i	$-0.328671\pi$
$673$	690.000			1.02526			0.512630	+	0.858610i	$0.328671\pi$
$674$		0			0
$675$	278.280			0.412267
$676$		0			0
$677$	−138.441			−0.204492			−0.102246	−	0.994759i	$-0.532603\pi$
$677$	−138.441			−0.204492			−0.102246	+	0.994759i	$0.532603\pi$
$678$		0			0
$679$	−544.279	−	860.581i	−0.801590	−	1.26742i
$680$		0			0
$681$	−290.000			−0.425844
$682$		0			0
$683$		−	770.000i		−	1.12738i	−0.825987	−	0.563690i	$-0.809381\pi$
$683$		−	770.000i		−	1.12738i	0.825987	−	0.563690i	$-0.190619\pi$
$684$		0			0
$685$	636.082			0.928587
$686$		0			0
$687$	−887.412			−1.29172
$688$		0			0
$689$		−	796.894i		−	1.15660i
$690$		0			0
$691$	585.021			0.846630			0.423315	−	0.905983i	$-0.360866\pi$
$691$	585.021			0.846630			0.423315	+	0.905983i	$0.360866\pi$
$692$		0			0
$693$	−7.48331	−	11.8322i	−0.0107984	−	0.0170738i
$694$		0			0
$695$	437.790			0.629914
$696$		0			0
$697$	320.000			0.459110
$698$		0			0
$699$	600.833			0.859560
$700$		0			0
$701$			331.300i			0.472611i	0.971679	+	0.236306i	$0.0759366\pi$
$701$			331.300i			0.472611i	−0.971679	+	0.236306i	$0.924063\pi$
$702$		0			0
$703$		−	149.666i		−	0.212897i
$704$		0			0
$705$			973.982i			1.38153i
$706$		0			0
$707$	332.039	−	210.000i	0.469645	−	0.297030i
$708$		0			0
$709$			496.951i			0.700918i	0.936578	+	0.350459i	$0.113974\pi$
$709$			496.951i			0.700918i	−0.936578	+	0.350459i	$0.886026\pi$
$710$		0			0
$711$	11.8322			0.0166416
$712$		0			0
$713$			885.438i			1.24185i
$714$		0			0
$715$			84.0000i			0.117483i
$716$		0			0
$717$	−1122.50			−1.56555
$718$		0			0
$719$		−	187.083i		−	0.260199i	−0.991501	−	0.130099i	$-0.958470\pi$
$719$		−	187.083i		−	0.260199i	0.991501	−	0.130099i	$-0.0415296\pi$
$720$		0			0
$721$	−168.000	−	265.631i	−0.233010	−	0.368421i
$722$		0			0
$723$			1180.00i			1.63209i
$724$		0			0
$725$		−	260.308i		−	0.359045i
$726$		0			0
$727$		−	493.899i		−	0.679366i	−0.940540	−	0.339683i	$-0.889680\pi$
$727$		−	493.899i		−	0.679366i	0.940540	−	0.339683i	$-0.110320\pi$
$728$		0			0
$729$	631.000			0.865569
$730$		0			0
$731$	63.2456			0.0865192
$732$		0			0
$733$	−1335.77			−1.82234			−0.911168	−	0.412036i	$-0.864818\pi$
$733$	−1335.77			−1.82234			−0.911168	+	0.412036i	$0.864818\pi$
$734$		0			0
$735$	−248.475	+	523.832i	−0.338062	+	0.712697i
$736$		0			0
$737$	−140.000			−0.189959
$738$		0			0
$739$		−	58.0000i		−	0.0784844i	−0.999230	−	0.0392422i	$-0.987506\pi$
$739$		−	58.0000i		−	0.0784844i	0.999230	−	0.0392422i	$-0.0124944\pi$
$740$		0			0
$741$	112.250			0.151484
$742$		0			0
$743$	−922.908			−1.24214			−0.621069	−	0.783756i	$-0.713301\pi$
$743$	−922.908			−1.24214			−0.621069	+	0.783756i	$0.713301\pi$
$744$		0			0
$745$		−	354.175i		−	0.475403i
$746$		0			0
$747$	−123.329			−0.165099
$748$		0			0
$749$	486.415	+	769.090i	0.649420	+	1.02682i
$750$		0			0
$751$	−402.293			−0.535677			−0.267838	−	0.963464i	$-0.586309\pi$
$751$	−402.293			−0.535677			−0.267838	+	0.963464i	$0.586309\pi$
$752$		0			0
$753$	650.000			0.863214
$754$		0			0
$755$	−796.894			−1.05549
$756$		0			0
$757$			615.272i			0.812777i	0.913700	+	0.406389i	$0.133212\pi$
$757$			615.272i			0.812777i	−0.913700	+	0.406389i	$0.866788\pi$
$758$		0			0
$759$		−	149.666i		−	0.197189i
$760$		0			0
$761$		−	645.105i		−	0.847706i	−0.905731	−	0.423853i	$-0.860677\pi$
$761$		−	645.105i		−	0.847706i	0.905731	−	0.423853i	$-0.139323\pi$
$762$		0			0
$763$	619.806	+	980.000i	0.812328	+	1.28440i
$764$		0			0
$765$			23.6643i			0.0309338i
$766$		0			0
$767$	−674.433			−0.879313
$768$		0			0
$769$			702.026i			0.912907i	0.889747	+	0.456454i	$0.150881\pi$
$769$			702.026i			0.912907i	−0.889747	+	0.456454i	$0.849119\pi$
$770$		0			0
$771$		−	1040.00i		−	1.34890i
$772$		0			0
$773$	86.0581			0.111330			0.0556650	−	0.998450i	$-0.482272\pi$
$773$	86.0581			0.111330			0.0556650	+	0.998450i	$0.482272\pi$
$774$		0			0
$775$			411.582i			0.531074i
$776$		0			0
$777$	−560.000	−	885.438i	−0.720721	−	1.13956i
$778$		0			0
$779$		−	160.000i		−	0.205392i
$780$		0			0
$781$		−	70.9930i		−	0.0909001i
$782$		0			0
$783$		−	598.665i		−	0.764579i
$784$		0			0
$785$	−1022.00			−1.30191
$786$		0			0
$787$	237.171			0.301361			0.150680	−	0.988583i	$-0.451854\pi$
$787$	237.171			0.301361			0.150680	+	0.988583i	$0.451854\pi$
$788$		0			0
$789$	−1459.25			−1.84949
$790$		0			0
$791$	354.965	−	224.499i	0.448754	−	0.283817i
$792$		0			0
$793$	−630.000			−0.794451
$794$		0			0
$795$		−	840.000i		−	1.05660i
$796$		0			0
$797$	−999.023			−1.25348			−0.626739	−	0.779229i	$-0.715611\pi$
$797$	−999.023			−1.25348			−0.626739	+	0.779229i	$0.715611\pi$
$798$		0			0
$799$	−520.615			−0.651583
$800$		0			0
$801$			56.9210i			0.0710624i
$802$		0			0
$803$	215.035			0.267789
$804$		0			0
$805$	−523.832	+	331.300i	−0.650723	+	0.411553i
$806$		0			0
$807$	−1597.34			−1.97936
$808$		0			0
$809$	1412.00			1.74536			0.872682	−	0.488288i	$-0.162379\pi$
$809$	1412.00			1.74536			0.872682	+	0.488288i	$0.162379\pi$
$810$		0			0
$811$	−1274.40			−1.57139			−0.785695	−	0.618614i	$-0.787695\pi$
$811$	−1274.40			−1.57139			−0.785695	+	0.618614i	$0.787695\pi$
$812$		0			0
$813$			473.286i			0.582148i
$814$		0			0
$815$			336.749i			0.413189i
$816$		0			0
$817$		−	31.6228i		−	0.0387060i
$818$		0			0
$819$	66.4078	−	42.0000i	0.0810840	−	0.0512821i
$820$		0			0
$821$		−	780.923i		−	0.951185i	−0.879666	−	0.475592i	$-0.842234\pi$
$821$		−	780.923i		−	0.951185i	0.879666	−	0.475592i	$-0.157766\pi$
$822$		0			0
$823$	−130.154			−0.158146			−0.0790728	−	0.996869i	$-0.525196\pi$
$823$	−130.154			−0.158146			−0.0790728	+	0.996869i	$0.525196\pi$
$824$		0			0
$825$		−	69.5701i		−	0.0843274i
$826$		0			0
$827$			1370.00i			1.65659i	0.560292	+	0.828295i	$0.310689\pi$
$827$			1370.00i			1.65659i	−0.560292	+	0.828295i	$0.689311\pi$
$828$		0			0
$829$	430.291			0.519048			0.259524	−	0.965737i	$-0.416434\pi$
$829$	430.291			0.519048			0.259524	+	0.965737i	$0.416434\pi$
$830$		0			0
$831$			673.498i			0.810467i
$832$		0			0
$833$	−280.000	−	132.816i	−0.336134	−	0.159443i
$834$		0			0
$835$			1008.00i			1.20719i
$836$		0			0
$837$			946.573i			1.13091i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	281.000			0.334126
$842$		0			0
$843$	−499.640			−0.592693
$844$		0			0
$845$	160.891			0.190404
$846$		0			0
$847$	−692.181	+	437.774i	−0.817215	+	0.516852i
$848$		0			0
$849$	−750.000			−0.883392
$850$		0			0
$851$		−	1120.00i		−	1.31610i
$852$		0			0
$853$	−475.190			−0.557081			−0.278541	−	0.960424i	$-0.589851\pi$
$853$	−475.190			−0.557081			−0.278541	+	0.960424i	$0.589851\pi$
$854$		0			0
$855$	11.8322			0.0138388
$856$		0			0
$857$			113.842i			0.132838i	0.997792	+	0.0664189i	$0.0211574\pi$
$857$			113.842i			0.132838i	−0.997792	+	0.0664189i	$0.978843\pi$
$858$		0			0
$859$	−755.784			−0.879842			−0.439921	−	0.898036i	$-0.644994\pi$
$859$	−755.784			−0.879842			−0.439921	+	0.898036i	$0.644994\pi$
$860$		0			0
$861$	−598.665	−	946.573i	−0.695314	−	1.09939i
$862$		0			0
$863$	224.811			0.260499			0.130250	−	0.991481i	$-0.458422\pi$
$863$	224.811			0.260499			0.130250	+	0.991481i	$0.458422\pi$
$864$		0			0
$865$	42.0000			0.0485549
$866$		0			0
$867$	787.407			0.908197
$868$		0			0
$869$			23.6643i			0.0272317i
$870$		0			0
$871$		−	785.748i		−	0.902122i
$872$		0			0
$873$			145.465i			0.166626i
$874$		0			0
$875$	−796.894	+	504.000i	−0.910736	+	0.576000i
$876$		0			0
$877$			851.915i			0.971397i	0.874126	+	0.485699i	$0.161435\pi$
$877$			851.915i			0.971397i	−0.874126	+	0.485699i	$0.838565\pi$
$878$		0			0
$879$	−1218.71			−1.38648
$880$		0			0
$881$			518.614i			0.588665i	0.955703	+	0.294332i	$0.0950972\pi$
$881$			518.614i			0.588665i	−0.955703	+	0.294332i	$0.904903\pi$
$882$		0			0
$883$		−	1150.00i		−	1.30238i	−0.758915	−	0.651189i	$-0.774270\pi$
$883$		−	1150.00i		−	1.30238i	0.758915	−	0.651189i	$-0.225730\pi$
$884$		0			0
$885$	−710.915			−0.803294
$886$		0			0
$887$			344.232i			0.388086i	0.980993	+	0.194043i	$0.0621602\pi$
$887$			344.232i			0.388086i	−0.980993	+	0.194043i	$0.937840\pi$
$888$		0			0
$889$	−700.000	+	442.719i	−0.787402	+	0.497996i
$890$		0			0
$891$		−	178.000i		−	0.199776i
$892$		0			0
$893$			260.308i			0.291498i
$894$		0			0
$895$		−	965.348i		−	1.07860i
$896$		0			0
$897$	840.000			0.936455
$898$		0			0
$899$	885.438			0.984914
$900$		0			0
$901$	448.999			0.498334
$902$		0			0
$903$	−118.322	−	187.083i	−0.131032	−	0.207179i
$904$		0			0
$905$	−490.000			−0.541436
$906$		0			0
$907$		−	1210.00i		−	1.33407i	−0.745027	−	0.667034i	$-0.767563\pi$
$907$		−	1210.00i		−	1.33407i	0.745027	−	0.667034i	$-0.232437\pi$
$908$		0			0
$909$	−56.1249			−0.0617435
$910$		0			0
$911$	1064.89			1.16893			0.584465	−	0.811419i	$-0.301305\pi$
$911$	1064.89			1.16893			0.584465	+	0.811419i	$0.301305\pi$
$912$		0			0
$913$		−	246.658i		−	0.270162i
$914$		0			0
$915$	−664.078			−0.725769
$916$		0			0
$917$	56.1249	−	35.4965i	0.0612049	−	0.0387094i
$918$		0			0
$919$	201.147			0.218876			0.109438	−	0.993994i	$-0.465095\pi$
$919$	201.147			0.218876			0.109438	+	0.993994i	$0.465095\pi$
$920$		0			0
$921$	610.000			0.662324
$922$		0			0
$923$	398.447			0.431687
$924$		0			0
$925$		−	520.615i		−	0.562827i
$926$		0			0
$927$			44.8999i			0.0484357i
$928$		0			0
$929$			638.780i			0.687600i	0.939043	+	0.343800i	$0.111714\pi$
$929$			638.780i			0.687600i	−0.939043	+	0.343800i	$0.888286\pi$
$930$		0			0
$931$	−66.4078	+	140.000i	−0.0713296	+	0.150376i
$932$		0			0
$933$			354.965i			0.380455i
$934$		0			0
$935$	−47.3286			−0.0506189
$936$		0			0
$937$			170.763i			0.182244i	0.995840	+	0.0911222i	$0.0290454\pi$
$937$			170.763i			0.182244i	−0.995840	+	0.0911222i	$0.970955\pi$
$938$		0			0
$939$		−	40.0000i		−	0.0425985i
$940$		0			0
$941$	841.873			0.894658			0.447329	−	0.894370i	$-0.352375\pi$
$941$	841.873			0.894658			0.447329	+	0.894370i	$0.352375\pi$
$942$		0			0
$943$		−	1197.33i		−	1.26970i
$944$		0			0
$945$	−560.000	+	354.175i	−0.592593	+	0.374788i
$946$		0			0
$947$			1490.00i			1.57339i	0.617342	+	0.786695i	$0.288209\pi$
$947$			1490.00i			1.57339i	−0.617342	+	0.786695i	$0.711791\pi$
$948$		0			0
$949$			1206.88i			1.27174i
$950$		0			0
$951$			1796.00i			1.88853i
$952$		0			0
$953$	−70.0000			−0.0734523			−0.0367261	−	0.999325i	$-0.511693\pi$
$953$	−70.0000			−0.0734523			−0.0367261	+	0.999325i	$0.511693\pi$
$954$		0			0
$955$	44.2719			0.0463580
$956$		0			0
$957$	−149.666			−0.156391
$958$		0			0
$959$	1005.73	−	636.082i	1.04873	−	0.663276i
$960$		0			0
$961$	−439.000			−0.456816
$962$		0			0
$963$		−	130.000i		−	0.134995i
$964$		0			0
$965$	1197.33			1.24076
$966$		0			0
$967$	−922.908			−0.954404			−0.477202	−	0.878794i	$-0.658349\pi$
$967$	−922.908			−0.954404			−0.477202	+	0.878794i	$0.658349\pi$
$968$		0			0
$969$			63.2456i			0.0652689i
$970$		0			0
$971$	1856.26			1.91170			0.955848	−	0.293861i	$-0.0949404\pi$
$971$	1856.26			1.91170			0.955848	+	0.293861i	$0.0949404\pi$
$972$		0			0
$973$	692.207	−	437.790i	0.711415	−	0.449938i
$974$		0			0
$975$	390.461			0.400473
$976$		0			0
$977$	1790.00			1.83214			0.916070	−	0.401019i	$-0.131344\pi$
$977$	1790.00			1.83214			0.916070	+	0.401019i	$0.131344\pi$
$978$		0			0
$979$	−113.842			−0.116284
$980$		0			0
$981$		−	165.650i		−	0.168859i
$982$		0			0
$983$			179.600i			0.182706i	0.995819	+	0.0913528i	$0.0291191\pi$
$983$			179.600i			0.182706i	−0.995819	+	0.0913528i	$0.970881\pi$
$984$		0			0
$985$		−	885.438i		−	0.898922i
$986$		0			0
$987$	973.982	+	1540.00i	0.986810	+	1.56028i
$988$		0			0
$989$		−	236.643i		−	0.239275i
$990$		0			0
$991$	1526.35			1.54021			0.770105	−	0.637917i	$-0.220204\pi$
$991$	1526.35			1.54021			0.770105	+	0.637917i	$0.220204\pi$
$992$		0			0
$993$		−	1574.81i		−	1.58592i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1096.31			−1.09960			−0.549802	−	0.835295i	$-0.685297\pi$
$997$	−1096.31			−1.09960			−0.549802	+	0.835295i	$0.685297\pi$
$998$		0			0
$999$		−	1197.33i		−	1.19853i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.h.a.97.6	yes	8
4.3	odd	2	inner	448.3.h.a.97.1	✓	8
7.6	odd	2	inner	448.3.h.a.97.3	yes	8
8.3	odd	2	inner	448.3.h.a.97.7	yes	8
8.5	even	2	inner	448.3.h.a.97.4	yes	8
28.27	even	2	inner	448.3.h.a.97.8	yes	8
56.13	odd	2	inner	448.3.h.a.97.5	yes	8
56.27	even	2	inner	448.3.h.a.97.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.3.h.a.97.1	✓	8	4.3	odd	2	inner
448.3.h.a.97.2	yes	8	56.27	even	2	inner
448.3.h.a.97.3	yes	8	7.6	odd	2	inner
448.3.h.a.97.4	yes	8	8.5	even	2	inner
448.3.h.a.97.5	yes	8	56.13	odd	2	inner
448.3.h.a.97.6	yes	8	1.1	even	1	trivial
448.3.h.a.97.7	yes	8	8.3	odd	2	inner
448.3.h.a.97.8	yes	8	28.27	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.16228 q^{3} -3.74166 q^{5} +(-5.91608 + 3.74166i) q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+3.16228 q^{3} -3.74166 q^{5} +(-5.91608 + 3.74166i) q^{7} +1.00000 q^{9} +2.00000i q^{11} -11.2250 q^{13} -11.8322 q^{15} -6.32456i q^{17} -3.16228 q^{19} +(-18.7083 + 11.8322i) q^{21} -23.6643 q^{23} -11.0000 q^{25} -25.2982 q^{27} +23.6643i q^{29} -37.4166i q^{31} +6.32456i q^{33} +(22.1359 - 14.0000i) q^{35} +47.3286i q^{37} -35.4965 q^{39} +50.5964i q^{41} +10.0000i q^{43} -3.74166 q^{45} -82.3165i q^{47} +(21.0000 - 44.2719i) q^{49} -20.0000i q^{51} +70.9930i q^{53} -7.48331i q^{55} -10.0000 q^{57} +60.0833 q^{59} +56.1249 q^{61} +(-5.91608 + 3.74166i) q^{63} +42.0000 q^{65} +70.0000i q^{67} -74.8331 q^{69} -35.4965 q^{71} -107.517i q^{73} -34.7851 q^{75} +(-7.48331 - 11.8322i) q^{77} +11.8322 q^{79} -89.0000 q^{81} -123.329 q^{83} +23.6643i q^{85} +74.8331i q^{87} +56.9210i q^{89} +(66.4078 - 42.0000i) q^{91} -118.322i q^{93} +11.8322 q^{95} +145.465i q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} - 88 q^{25} + 168 q^{49} - 80 q^{57} + 336 q^{65} - 712 q^{81}+O(q^{100}) \) 8 * q + 8 * q^9 - 88 * q^25 + 168 * q^49 - 80 * q^57 + 336 * q^65 - 712 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more