LMFDB - Embedded newform 5376.2.c.bh.2689.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5376,2,Mod(2689,5376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5376, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5376.2689");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5376 = 2^{8} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5376.c (of order $2$, degree $1$, 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:	$42.9275761266$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 672)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2689.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	5376.2689
Dual form			5376.2.c.bh.2689.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -3.46410i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -3.46410i q^{5} -1.00000 q^{7} -1.00000 q^{9} -1.46410i q^{11} +2.00000i q^{13} -3.46410 q^{15} +0.535898 q^{17} -6.92820i q^{19} +1.00000i q^{21} -1.46410 q^{23} -7.00000 q^{25} +1.00000i q^{27} -4.92820i q^{29} +10.9282 q^{31} -1.46410 q^{33} +3.46410i q^{35} +2.00000i q^{37} +2.00000 q^{39} -11.4641 q^{41} -8.00000i q^{43} +3.46410i q^{45} -10.9282 q^{47} +1.00000 q^{49} -0.535898i q^{51} +2.00000i q^{53} -5.07180 q^{55} -6.92820 q^{57} -1.07180i q^{59} -8.92820i q^{61} +1.00000 q^{63} +6.92820 q^{65} +2.92820i q^{67} +1.46410i q^{69} +9.46410 q^{71} -12.9282 q^{73} +7.00000i q^{75} +1.46410i q^{77} +10.9282 q^{79} +1.00000 q^{81} +4.00000i q^{83} -1.85641i q^{85} -4.92820 q^{87} -3.46410 q^{89} -2.00000i q^{91} -10.9282i q^{93} -24.0000 q^{95} -8.92820 q^{97} +1.46410i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 - 4 * q^9	$ 4 q - 4 q^{7} - 4 q^{9} + 16 q^{17} + 8 q^{23} - 28 q^{25} + 16 q^{31} + 8 q^{33} + 8 q^{39} - 32 q^{41} - 16 q^{47} + 4 q^{49} - 48 q^{55} + 4 q^{63} + 24 q^{71} - 24 q^{73} + 16 q^{79} + 4 q^{81} + 8 q^{87} - 96 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^9 + 16 * q^17 + 8 * q^23 - 28 * q^25 + 16 * q^31 + 8 * q^33 + 8 * q^39 - 32 * q^41 - 16 * q^47 + 4 * q^49 - 48 * q^55 + 4 * q^63 + 24 * q^71 - 24 * q^73 + 16 * q^79 + 4 * q^81 + 8 * q^87 - 96 * q^95 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times$.

$n$	$1793$	$2815$	$4609$	$5125$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	− 3.46410i	− 1.54919i	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	− 3.46410i	− 1.54919i	0.632456	−	0.774597i	$-0.282047\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 1.46410i	− 0.441443i	−0.975337	−	0.220722i	$-0.929159\pi$
$11$	− 1.46410i	− 0.441443i	0.975337	−	0.220722i	$-0.0708412\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	0	0
$15$	−3.46410	−0.894427
$16$	0	0
$17$	0.535898	0.129974	0.0649872	−	0.997886i	$-0.479299\pi$
$17$	0.535898	0.129974	0.0649872	+	0.997886i	$0.479299\pi$
$18$	0	0
$19$	− 6.92820i	− 1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	− 6.92820i	− 1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$	0	0
$21$	1.00000i	0.218218i
$22$	0	0
$23$	−1.46410	−0.305286	−0.152643	−	0.988281i	$-0.548779\pi$
$23$	−1.46410	−0.305286	−0.152643	+	0.988281i	$0.548779\pi$
$24$	0	0
$25$	−7.00000	−1.40000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 4.92820i	− 0.915144i	−0.889172	−	0.457572i	$-0.848719\pi$
$29$	− 4.92820i	− 0.915144i	0.889172	−	0.457572i	$-0.151281\pi$
$30$	0	0
$31$	10.9282	1.96276	0.981382	−	0.192068i	$-0.0615194\pi$
$31$	10.9282	1.96276	0.981382	+	0.192068i	$0.0615194\pi$
$32$	0	0
$33$	−1.46410	−0.254867
$34$	0	0
$35$	3.46410i	0.585540i
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	2.00000	0.320256
$40$	0	0
$41$	−11.4641	−1.79039	−0.895196	−	0.445673i	$-0.852964\pi$
$41$	−11.4641	−1.79039	−0.895196	+	0.445673i	$0.852964\pi$
$42$	0	0
$43$	− 8.00000i	− 1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$	− 8.00000i	− 1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$	0	0
$45$	3.46410i	0.516398i
$46$	0	0
$47$	−10.9282	−1.59404	−0.797021	−	0.603951i	$-0.793592\pi$
$47$	−10.9282	−1.59404	−0.797021	+	0.603951i	$0.793592\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 0.535898i	− 0.0750408i
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	−5.07180	−0.683881
$56$	0	0
$57$	−6.92820	−0.917663
$58$	0	0
$59$	− 1.07180i	− 0.139536i	−0.997563	−	0.0697680i	$-0.977774\pi$
$59$	− 1.07180i	− 0.139536i	0.997563	−	0.0697680i	$-0.0222259\pi$
$60$	0	0
$61$	− 8.92820i	− 1.14314i	−0.820554	−	0.571570i	$-0.806335\pi$
$61$	− 8.92820i	− 1.14314i	0.820554	−	0.571570i	$-0.193665\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	6.92820	0.859338
$66$	0	0
$67$	2.92820i	0.357737i	0.983873	+	0.178868i	$0.0572437\pi$
$67$	2.92820i	0.357737i	−0.983873	+	0.178868i	$0.942756\pi$
$68$	0	0
$69$	1.46410i	0.176257i
$70$	0	0
$71$	9.46410	1.12318	0.561591	−	0.827415i	$-0.310189\pi$
$71$	9.46410	1.12318	0.561591	+	0.827415i	$0.310189\pi$
$72$	0	0
$73$	−12.9282	−1.51313	−0.756566	−	0.653917i	$-0.773124\pi$
$73$	−12.9282	−1.51313	−0.756566	+	0.653917i	$0.773124\pi$
$74$	0	0
$75$	7.00000i	0.808290i
$76$	0	0
$77$	1.46410i	0.166850i
$78$	0	0
$79$	10.9282	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	− 1.85641i	− 0.201356i
$86$	0	0
$87$	−4.92820	−0.528359
$88$	0	0
$89$	−3.46410	−0.367194	−0.183597	−	0.983002i	$-0.558774\pi$
$89$	−3.46410	−0.367194	−0.183597	+	0.983002i	$0.558774\pi$
$90$	0	0
$91$	− 2.00000i	− 0.209657i
$92$	0	0
$93$	− 10.9282i	− 1.13320i
$94$	0	0
$95$	−24.0000	−2.46235
$96$	0	0
$97$	−8.92820	−0.906522	−0.453261	−	0.891378i	$-0.649739\pi$
$97$	−8.92820	−0.906522	−0.453261	+	0.891378i	$0.649739\pi$
$98$	0	0
$99$	1.46410i	0.147148i
$100$	0	0
$101$	15.4641i	1.53874i	0.638806	+	0.769368i	$0.279429\pi$
$101$	15.4641i	1.53874i	−0.638806	+	0.769368i	$0.720571\pi$
$102$	0	0
$103$	−10.9282	−1.07679	−0.538394	−	0.842693i	$-0.680969\pi$
$103$	−10.9282	−1.07679	−0.538394	+	0.842693i	$0.680969\pi$
$104$	0	0
$105$	3.46410	0.338062
$106$	0	0
$107$	12.3923i	1.19801i	0.800746	+	0.599005i	$0.204437\pi$
$107$	12.3923i	1.19801i	−0.800746	+	0.599005i	$0.795563\pi$
$108$	0	0
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−19.8564	−1.86793	−0.933967	−	0.357360i	$-0.883677\pi$
$113$	−19.8564	−1.86793	−0.933967	+	0.357360i	$0.883677\pi$
$114$	0	0
$115$	5.07180i	0.472947i
$116$	0	0
$117$	− 2.00000i	− 0.184900i
$118$	0	0
$119$	−0.535898	−0.0491257
$120$	0	0
$121$	8.85641	0.805128
$122$	0	0
$123$	11.4641i	1.03368i
$124$	0	0
$125$	6.92820i	0.619677i
$126$	0	0
$127$	−2.92820	−0.259836	−0.129918	−	0.991525i	$-0.541471\pi$
$127$	−2.92820	−0.259836	−0.129918	+	0.991525i	$0.541471\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	− 12.0000i	− 1.04844i	−0.851581	−	0.524222i	$-0.824356\pi$
$131$	− 12.0000i	− 1.04844i	0.851581	−	0.524222i	$-0.175644\pi$
$132$	0	0
$133$	6.92820i	0.600751i
$134$	0	0
$135$	3.46410	0.298142
$136$	0	0
$137$	8.92820	0.762788	0.381394	−	0.924413i	$-0.375444\pi$
$137$	8.92820	0.762788	0.381394	+	0.924413i	$0.375444\pi$
$138$	0	0
$139$	− 17.8564i	− 1.51456i	−0.653090	−	0.757280i	$-0.726528\pi$
$139$	− 17.8564i	− 1.51456i	0.653090	−	0.757280i	$-0.273472\pi$
$140$	0	0
$141$	10.9282i	0.920321i
$142$	0	0
$143$	2.92820	0.244869
$144$	0	0
$145$	−17.0718	−1.41774
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	18.0000i	1.47462i	0.675556	+	0.737309i	$0.263904\pi$
$149$	18.0000i	1.47462i	−0.675556	+	0.737309i	$0.736096\pi$
$150$	0	0
$151$	21.8564	1.77865	0.889325	−	0.457277i	$-0.151175\pi$
$151$	21.8564	1.77865	0.889325	+	0.457277i	$0.151175\pi$
$152$	0	0
$153$	−0.535898	−0.0433248
$154$	0	0
$155$	− 37.8564i	− 3.04070i
$156$	0	0
$157$	− 0.928203i	− 0.0740787i	−0.999314	−	0.0370393i	$-0.988207\pi$
$157$	− 0.928203i	− 0.0740787i	0.999314	−	0.0370393i	$-0.0117927\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	1.46410	0.115387
$162$	0	0
$163$	− 10.9282i	− 0.855963i	−0.903788	−	0.427981i	$-0.859225\pi$
$163$	− 10.9282i	− 0.855963i	0.903788	−	0.427981i	$-0.140775\pi$
$164$	0	0
$165$	5.07180i	0.394839i
$166$	0	0
$167$	−10.9282	−0.845650	−0.422825	−	0.906211i	$-0.638961\pi$
$167$	−10.9282	−0.845650	−0.422825	+	0.906211i	$0.638961\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	6.92820i	0.529813i
$172$	0	0
$173$	22.3923i	1.70246i	0.524797	+	0.851228i	$0.324141\pi$
$173$	22.3923i	1.70246i	−0.524797	+	0.851228i	$0.675859\pi$
$174$	0	0
$175$	7.00000	0.529150
$176$	0	0
$177$	−1.07180	−0.0805612
$178$	0	0
$179$	− 9.46410i	− 0.707380i	−0.935363	−	0.353690i	$-0.884927\pi$
$179$	− 9.46410i	− 0.707380i	0.935363	−	0.353690i	$-0.115073\pi$
$180$	0	0
$181$	6.00000i	0.445976i	0.974821	+	0.222988i	$0.0715812\pi$
$181$	6.00000i	0.445976i	−0.974821	+	0.222988i	$0.928419\pi$
$182$	0	0
$183$	−8.92820	−0.659992
$184$	0	0
$185$	6.92820	0.509372
$186$	0	0
$187$	− 0.784610i	− 0.0573763i
$188$	0	0
$189$	− 1.00000i	− 0.0727393i
$190$	0	0
$191$	−9.46410	−0.684798	−0.342399	−	0.939555i	$-0.611240\pi$
$191$	−9.46410	−0.684798	−0.342399	+	0.939555i	$0.611240\pi$
$192$	0	0
$193$	15.8564	1.14137	0.570685	−	0.821169i	$-0.306678\pi$
$193$	15.8564	1.14137	0.570685	+	0.821169i	$0.306678\pi$
$194$	0	0
$195$	− 6.92820i	− 0.496139i
$196$	0	0
$197$	7.85641i	0.559746i	0.960037	+	0.279873i	$0.0902923\pi$
$197$	7.85641i	0.559746i	−0.960037	+	0.279873i	$0.909708\pi$
$198$	0	0
$199$	5.85641	0.415150	0.207575	−	0.978219i	$-0.433443\pi$
$199$	5.85641	0.415150	0.207575	+	0.978219i	$0.433443\pi$
$200$	0	0
$201$	2.92820	0.206540
$202$	0	0
$203$	4.92820i	0.345892i
$204$	0	0
$205$	39.7128i	2.77366i
$206$	0	0
$207$	1.46410	0.101762
$208$	0	0
$209$	−10.1436	−0.701647
$210$	0	0
$211$	16.0000i	1.10149i	0.834675	+	0.550743i	$0.185655\pi$
$211$	16.0000i	1.10149i	−0.834675	+	0.550743i	$0.814345\pi$
$212$	0	0
$213$	− 9.46410i	− 0.648470i
$214$	0	0
$215$	−27.7128	−1.89000
$216$	0	0
$217$	−10.9282	−0.741855
$218$	0	0
$219$	12.9282i	0.873607i
$220$	0	0
$221$	1.07180i	0.0720969i
$222$	0	0
$223$	24.0000	1.60716	0.803579	−	0.595198i	$-0.202926\pi$
$223$	24.0000	1.60716	0.803579	+	0.595198i	$0.202926\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	22.9282i	1.52180i	0.648870	+	0.760899i	$0.275242\pi$
$227$	22.9282i	1.52180i	−0.648870	+	0.760899i	$0.724758\pi$
$228$	0	0
$229$	27.8564i	1.84080i	0.390974	+	0.920402i	$0.372138\pi$
$229$	27.8564i	1.84080i	−0.390974	+	0.920402i	$0.627862\pi$
$230$	0	0
$231$	1.46410	0.0963308
$232$	0	0
$233$	−12.9282	−0.846955	−0.423477	−	0.905907i	$-0.639191\pi$
$233$	−12.9282	−0.846955	−0.423477	+	0.905907i	$0.639191\pi$
$234$	0	0
$235$	37.8564i	2.46948i
$236$	0	0
$237$	− 10.9282i	− 0.709863i
$238$	0	0
$239$	−14.5359	−0.940249	−0.470125	−	0.882600i	$-0.655791\pi$
$239$	−14.5359	−0.940249	−0.470125	+	0.882600i	$0.655791\pi$
$240$	0	0
$241$	−3.07180	−0.197872	−0.0989359	−	0.995094i	$-0.531544\pi$
$241$	−3.07180	−0.197872	−0.0989359	+	0.995094i	$0.531544\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 3.46410i	− 0.221313i
$246$	0	0
$247$	13.8564	0.881662
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	− 9.07180i	− 0.572607i	−0.958139	−	0.286303i	$-0.907573\pi$
$251$	− 9.07180i	− 0.572607i	0.958139	−	0.286303i	$-0.0924265\pi$
$252$	0	0
$253$	2.14359i	0.134767i
$254$	0	0
$255$	−1.85641	−0.116253
$256$	0	0
$257$	11.4641	0.715111	0.357556	−	0.933892i	$-0.383610\pi$
$257$	11.4641	0.715111	0.357556	+	0.933892i	$0.383610\pi$
$258$	0	0
$259$	− 2.00000i	− 0.124274i
$260$	0	0
$261$	4.92820i	0.305048i
$262$	0	0
$263$	12.3923	0.764142	0.382071	−	0.924133i	$-0.375211\pi$
$263$	12.3923	0.764142	0.382071	+	0.924133i	$0.375211\pi$
$264$	0	0
$265$	6.92820	0.425596
$266$	0	0
$267$	3.46410i	0.212000i
$268$	0	0
$269$	3.46410i	0.211210i	0.994408	+	0.105605i	$0.0336779\pi$
$269$	3.46410i	0.211210i	−0.994408	+	0.105605i	$0.966322\pi$
$270$	0	0
$271$	−2.92820	−0.177876	−0.0889378	−	0.996037i	$-0.528347\pi$
$271$	−2.92820	−0.177876	−0.0889378	+	0.996037i	$0.528347\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	10.2487i	0.618021i
$276$	0	0
$277$	− 0.143594i	− 0.00862770i	−0.999991	−	0.00431385i	$-0.998627\pi$
$277$	− 0.143594i	− 0.00862770i	0.999991	−	0.00431385i	$-0.00137315\pi$
$278$	0	0
$279$	−10.9282	−0.654254
$280$	0	0
$281$	−12.9282	−0.771232	−0.385616	−	0.922659i	$-0.626011\pi$
$281$	−12.9282	−0.771232	−0.385616	+	0.922659i	$0.626011\pi$
$282$	0	0
$283$	− 14.9282i	− 0.887390i	−0.896178	−	0.443695i	$-0.853667\pi$
$283$	− 14.9282i	− 0.887390i	0.896178	−	0.443695i	$-0.146333\pi$
$284$	0	0
$285$	24.0000i	1.42164i
$286$	0	0
$287$	11.4641	0.676705
$288$	0	0
$289$	−16.7128	−0.983107
$290$	0	0
$291$	8.92820i	0.523381i
$292$	0	0
$293$	− 16.5359i	− 0.966037i	−0.875610	−	0.483019i	$-0.839540\pi$
$293$	− 16.5359i	− 0.966037i	0.875610	−	0.483019i	$-0.160460\pi$
$294$	0	0
$295$	−3.71281	−0.216168
$296$	0	0
$297$	1.46410	0.0849558
$298$	0	0
$299$	− 2.92820i	− 0.169342i
$300$	0	0
$301$	8.00000i	0.461112i
$302$	0	0
$303$	15.4641	0.888389
$304$	0	0
$305$	−30.9282	−1.77094
$306$	0	0
$307$	− 22.9282i	− 1.30858i	−0.756243	−	0.654291i	$-0.772967\pi$
$307$	− 22.9282i	− 1.30858i	0.756243	−	0.654291i	$-0.227033\pi$
$308$	0	0
$309$	10.9282i	0.621684i
$310$	0	0
$311$	18.9282	1.07332	0.536660	−	0.843799i	$-0.319686\pi$
$311$	18.9282	1.07332	0.536660	+	0.843799i	$0.319686\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	− 3.46410i	− 0.195180i
$316$	0	0
$317$	0.143594i	0.00806502i	0.999992	+	0.00403251i	$0.00128359\pi$
$317$	0.143594i	0.00806502i	−0.999992	+	0.00403251i	$0.998716\pi$
$318$	0	0
$319$	−7.21539	−0.403984
$320$	0	0
$321$	12.3923	0.691671
$322$	0	0
$323$	− 3.71281i	− 0.206586i
$324$	0	0
$325$	− 14.0000i	− 0.776580i
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	10.9282	0.602491
$330$	0	0
$331$	24.0000i	1.31916i	0.751635	+	0.659580i	$0.229266\pi$
$331$	24.0000i	1.31916i	−0.751635	+	0.659580i	$0.770734\pi$
$332$	0	0
$333$	− 2.00000i	− 0.109599i
$334$	0	0
$335$	10.1436	0.554204
$336$	0	0
$337$	7.85641	0.427966	0.213983	−	0.976837i	$-0.431356\pi$
$337$	7.85641	0.427966	0.213983	+	0.976837i	$0.431356\pi$
$338$	0	0
$339$	19.8564i	1.07845i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.07180	0.273056
$346$	0	0
$347$	23.3205i	1.25191i	0.779859	+	0.625955i	$0.215291\pi$
$347$	23.3205i	1.25191i	−0.779859	+	0.625955i	$0.784709\pi$
$348$	0	0
$349$	10.7846i	0.577287i	0.957437	+	0.288643i	$0.0932042\pi$
$349$	10.7846i	0.577287i	−0.957437	+	0.288643i	$0.906796\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	8.53590	0.454320	0.227160	−	0.973857i	$-0.427056\pi$
$353$	8.53590	0.454320	0.227160	+	0.973857i	$0.427056\pi$
$354$	0	0
$355$	− 32.7846i	− 1.74003i
$356$	0	0
$357$	0.535898i	0.0283628i
$358$	0	0
$359$	7.32051	0.386362	0.193181	−	0.981163i	$-0.438120\pi$
$359$	7.32051	0.386362	0.193181	+	0.981163i	$0.438120\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	0	0
$363$	− 8.85641i	− 0.464841i
$364$	0	0
$365$	44.7846i	2.34413i
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	11.4641	0.596797
$370$	0	0
$371$	− 2.00000i	− 0.103835i
$372$	0	0
$373$	− 30.0000i	− 1.55334i	−0.629907	−	0.776671i	$-0.716907\pi$
$373$	− 30.0000i	− 1.55334i	0.629907	−	0.776671i	$-0.283093\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	9.85641	0.507631
$378$	0	0
$379$	− 8.00000i	− 0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$	− 8.00000i	− 0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$	0	0
$381$	2.92820i	0.150016i
$382$	0	0
$383$	−35.7128	−1.82484	−0.912420	−	0.409256i	$-0.865788\pi$
$383$	−35.7128	−1.82484	−0.912420	+	0.409256i	$0.865788\pi$
$384$	0	0
$385$	5.07180	0.258483
$386$	0	0
$387$	8.00000i	0.406663i
$388$	0	0
$389$	− 14.7846i	− 0.749609i	−0.927104	−	0.374805i	$-0.877710\pi$
$389$	− 14.7846i	− 0.749609i	0.927104	−	0.374805i	$-0.122290\pi$
$390$	0	0
$391$	−0.784610	−0.0396794
$392$	0	0
$393$	−12.0000	−0.605320
$394$	0	0
$395$	− 37.8564i	− 1.90476i
$396$	0	0
$397$	10.7846i	0.541264i	0.962683	+	0.270632i	$0.0872327\pi$
$397$	10.7846i	0.541264i	−0.962683	+	0.270632i	$0.912767\pi$
$398$	0	0
$399$	6.92820	0.346844
$400$	0	0
$401$	4.92820	0.246103	0.123051	−	0.992400i	$-0.460732\pi$
$401$	4.92820	0.246103	0.123051	+	0.992400i	$0.460732\pi$
$402$	0	0
$403$	21.8564i	1.08875i
$404$	0	0
$405$	− 3.46410i	− 0.172133i
$406$	0	0
$407$	2.92820	0.145146
$408$	0	0
$409$	−4.92820	−0.243684	−0.121842	−	0.992550i	$-0.538880\pi$
$409$	−4.92820	−0.243684	−0.121842	+	0.992550i	$0.538880\pi$
$410$	0	0
$411$	− 8.92820i	− 0.440396i
$412$	0	0
$413$	1.07180i	0.0527397i
$414$	0	0
$415$	13.8564	0.680184
$416$	0	0
$417$	−17.8564	−0.874432
$418$	0	0
$419$	− 17.0718i	− 0.834012i	−0.908904	−	0.417006i	$-0.863079\pi$
$419$	− 17.0718i	− 0.834012i	0.908904	−	0.417006i	$-0.136921\pi$
$420$	0	0
$421$	4.14359i	0.201946i	0.994889	+	0.100973i	$0.0321956\pi$
$421$	4.14359i	0.201946i	−0.994889	+	0.100973i	$0.967804\pi$
$422$	0	0
$423$	10.9282	0.531347
$424$	0	0
$425$	−3.75129	−0.181964
$426$	0	0
$427$	8.92820i	0.432066i
$428$	0	0
$429$	− 2.92820i	− 0.141375i
$430$	0	0
$431$	10.2487	0.493663	0.246832	−	0.969058i	$-0.420611\pi$
$431$	10.2487	0.493663	0.246832	+	0.969058i	$0.420611\pi$
$432$	0	0
$433$	−27.8564	−1.33869	−0.669347	−	0.742950i	$-0.733426\pi$
$433$	−27.8564	−1.33869	−0.669347	+	0.742950i	$0.733426\pi$
$434$	0	0
$435$	17.0718i	0.818530i
$436$	0	0
$437$	10.1436i	0.485234i
$438$	0	0
$439$	13.8564	0.661330	0.330665	−	0.943748i	$-0.392727\pi$
$439$	13.8564	0.661330	0.330665	+	0.943748i	$0.392727\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	− 28.3923i	− 1.34896i	−0.738294	−	0.674480i	$-0.764368\pi$
$443$	− 28.3923i	− 1.34896i	0.738294	−	0.674480i	$-0.235632\pi$
$444$	0	0
$445$	12.0000i	0.568855i
$446$	0	0
$447$	18.0000	0.851371
$448$	0	0
$449$	7.85641	0.370767	0.185383	−	0.982666i	$-0.440647\pi$
$449$	7.85641	0.370767	0.185383	+	0.982666i	$0.440647\pi$
$450$	0	0
$451$	16.7846i	0.790356i
$452$	0	0
$453$	− 21.8564i	− 1.02690i
$454$	0	0
$455$	−6.92820	−0.324799
$456$	0	0
$457$	−37.7128	−1.76413	−0.882065	−	0.471127i	$-0.843847\pi$
$457$	−37.7128	−1.76413	−0.882065	+	0.471127i	$0.843847\pi$
$458$	0	0
$459$	0.535898i	0.0250136i
$460$	0	0
$461$	− 27.1769i	− 1.26576i	−0.774252	−	0.632878i	$-0.781874\pi$
$461$	− 27.1769i	− 1.26576i	0.774252	−	0.632878i	$-0.218126\pi$
$462$	0	0
$463$	−32.7846	−1.52363	−0.761815	−	0.647795i	$-0.775691\pi$
$463$	−32.7846	−1.52363	−0.761815	+	0.647795i	$0.775691\pi$
$464$	0	0
$465$	−37.8564	−1.75555
$466$	0	0
$467$	− 20.7846i	− 0.961797i	−0.876776	−	0.480899i	$-0.840311\pi$
$467$	− 20.7846i	− 0.961797i	0.876776	−	0.480899i	$-0.159689\pi$
$468$	0	0
$469$	− 2.92820i	− 0.135212i
$470$	0	0
$471$	−0.928203	−0.0427693
$472$	0	0
$473$	−11.7128	−0.538556
$474$	0	0
$475$	48.4974i	2.22521i
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	5.07180	0.231736	0.115868	−	0.993265i	$-0.463035\pi$
$479$	5.07180	0.231736	0.115868	+	0.993265i	$0.463035\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	− 1.46410i	− 0.0666189i
$484$	0	0
$485$	30.9282i	1.40438i
$486$	0	0
$487$	−27.7128	−1.25579	−0.627894	−	0.778299i	$-0.716083\pi$
$487$	−27.7128	−1.25579	−0.627894	+	0.778299i	$0.716083\pi$
$488$	0	0
$489$	−10.9282	−0.494190
$490$	0	0
$491$	− 6.53590i	− 0.294961i	−0.989065	−	0.147480i	$-0.952884\pi$
$491$	− 6.53590i	− 0.294961i	0.989065	−	0.147480i	$-0.0471164\pi$
$492$	0	0
$493$	− 2.64102i	− 0.118945i
$494$	0	0
$495$	5.07180	0.227960
$496$	0	0
$497$	−9.46410	−0.424523
$498$	0	0
$499$	21.8564i	0.978427i	0.872164	+	0.489214i	$0.162716\pi$
$499$	21.8564i	0.978427i	−0.872164	+	0.489214i	$0.837284\pi$
$500$	0	0
$501$	10.9282i	0.488236i
$502$	0	0
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	53.5692	2.38380
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	− 40.2487i	− 1.78399i	−0.452043	−	0.891996i	$-0.649304\pi$
$509$	− 40.2487i	− 1.78399i	0.452043	−	0.891996i	$-0.350696\pi$
$510$	0	0
$511$	12.9282	0.571910
$512$	0	0
$513$	6.92820	0.305888
$514$	0	0
$515$	37.8564i	1.66815i
$516$	0	0
$517$	16.0000i	0.703679i
$518$	0	0
$519$	22.3923	0.982913
$520$	0	0
$521$	−30.3923	−1.33151	−0.665756	−	0.746170i	$-0.731891\pi$
$521$	−30.3923	−1.33151	−0.665756	+	0.746170i	$0.731891\pi$
$522$	0	0
$523$	28.0000i	1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$	28.0000i	1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$	0	0
$525$	− 7.00000i	− 0.305505i
$526$	0	0
$527$	5.85641	0.255109
$528$	0	0
$529$	−20.8564	−0.906800
$530$	0	0
$531$	1.07180i	0.0465120i
$532$	0	0
$533$	− 22.9282i	− 0.993131i
$534$	0	0
$535$	42.9282	1.85595
$536$	0	0
$537$	−9.46410	−0.408406
$538$	0	0
$539$	− 1.46410i	− 0.0630633i
$540$	0	0
$541$	− 7.85641i	− 0.337773i	−0.985635	−	0.168887i	$-0.945983\pi$
$541$	− 7.85641i	− 0.337773i	0.985635	−	0.168887i	$-0.0540172\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	0	0
$545$	−6.92820	−0.296772
$546$	0	0
$547$	− 10.9282i	− 0.467256i	−0.972326	−	0.233628i	$-0.924940\pi$
$547$	− 10.9282i	− 0.467256i	0.972326	−	0.233628i	$-0.0750598\pi$
$548$	0	0
$549$	8.92820i	0.381046i
$550$	0	0
$551$	−34.1436	−1.45457
$552$	0	0
$553$	−10.9282	−0.464714
$554$	0	0
$555$	− 6.92820i	− 0.294086i
$556$	0	0
$557$	− 4.14359i	− 0.175570i	−0.996139	−	0.0877848i	$-0.972021\pi$
$557$	− 4.14359i	− 0.175570i	0.996139	−	0.0877848i	$-0.0279788\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	−0.784610	−0.0331262
$562$	0	0
$563$	− 33.0718i	− 1.39381i	−0.717163	−	0.696905i	$-0.754560\pi$
$563$	− 33.0718i	− 1.39381i	0.717163	−	0.696905i	$-0.245440\pi$
$564$	0	0
$565$	68.7846i	2.89379i
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	24.9282	1.04504	0.522522	−	0.852626i	$-0.324991\pi$
$569$	24.9282	1.04504	0.522522	+	0.852626i	$0.324991\pi$
$570$	0	0
$571$	16.7846i	0.702414i	0.936298	+	0.351207i	$0.114229\pi$
$571$	16.7846i	0.702414i	−0.936298	+	0.351207i	$0.885771\pi$
$572$	0	0
$573$	9.46410i	0.395369i
$574$	0	0
$575$	10.2487	0.427401
$576$	0	0
$577$	23.8564	0.993155	0.496578	−	0.867992i	$-0.334590\pi$
$577$	23.8564	0.993155	0.496578	+	0.867992i	$0.334590\pi$
$578$	0	0
$579$	− 15.8564i	− 0.658970i
$580$	0	0
$581$	− 4.00000i	− 0.165948i
$582$	0	0
$583$	2.92820	0.121274
$584$	0	0
$585$	−6.92820	−0.286446
$586$	0	0
$587$	− 20.7846i	− 0.857873i	−0.903335	−	0.428936i	$-0.858888\pi$
$587$	− 20.7846i	− 0.857873i	0.903335	−	0.428936i	$-0.141112\pi$
$588$	0	0
$589$	− 75.7128i	− 3.11969i
$590$	0	0
$591$	7.85641	0.323169
$592$	0	0
$593$	8.53590	0.350527	0.175264	−	0.984522i	$-0.443922\pi$
$593$	8.53590	0.350527	0.175264	+	0.984522i	$0.443922\pi$
$594$	0	0
$595$	1.85641i	0.0761052i
$596$	0	0
$597$	− 5.85641i	− 0.239687i
$598$	0	0
$599$	−28.3923	−1.16008	−0.580039	−	0.814589i	$-0.696963\pi$
$599$	−28.3923	−1.16008	−0.580039	+	0.814589i	$0.696963\pi$
$600$	0	0
$601$	−43.5692	−1.77723	−0.888613	−	0.458658i	$-0.848330\pi$
$601$	−43.5692	−1.77723	−0.888613	+	0.458658i	$0.848330\pi$
$602$	0	0
$603$	− 2.92820i	− 0.119246i
$604$	0	0
$605$	− 30.6795i	− 1.24730i
$606$	0	0
$607$	−13.8564	−0.562414	−0.281207	−	0.959647i	$-0.590735\pi$
$607$	−13.8564	−0.562414	−0.281207	+	0.959647i	$0.590735\pi$
$608$	0	0
$609$	4.92820	0.199701
$610$	0	0
$611$	− 21.8564i	− 0.884216i
$612$	0	0
$613$	− 27.8564i	− 1.12511i	−0.826760	−	0.562555i	$-0.809819\pi$
$613$	− 27.8564i	− 1.12511i	0.826760	−	0.562555i	$-0.190181\pi$
$614$	0	0
$615$	39.7128	1.60138
$616$	0	0
$617$	−18.7846	−0.756240	−0.378120	−	0.925757i	$-0.623429\pi$
$617$	−18.7846	−0.756240	−0.378120	+	0.925757i	$0.623429\pi$
$618$	0	0
$619$	− 4.00000i	− 0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$	− 4.00000i	− 0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$	0	0
$621$	− 1.46410i	− 0.0587524i
$622$	0	0
$623$	3.46410	0.138786
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	10.1436i	0.405096i
$628$	0	0
$629$	1.07180i	0.0427353i
$630$	0	0
$631$	26.9282	1.07199	0.535997	−	0.844220i	$-0.319936\pi$
$631$	26.9282	1.07199	0.535997	+	0.844220i	$0.319936\pi$
$632$	0	0
$633$	16.0000	0.635943
$634$	0	0
$635$	10.1436i	0.402536i
$636$	0	0
$637$	2.00000i	0.0792429i
$638$	0	0
$639$	−9.46410	−0.374394
$640$	0	0
$641$	−6.78461	−0.267976	−0.133988	−	0.990983i	$-0.542778\pi$
$641$	−6.78461	−0.267976	−0.133988	+	0.990983i	$0.542778\pi$
$642$	0	0
$643$	12.7846i	0.504176i	0.967704	+	0.252088i	$0.0811172\pi$
$643$	12.7846i	0.504176i	−0.967704	+	0.252088i	$0.918883\pi$
$644$	0	0
$645$	27.7128i	1.09119i
$646$	0	0
$647$	−5.07180	−0.199393	−0.0996965	−	0.995018i	$-0.531787\pi$
$647$	−5.07180	−0.199393	−0.0996965	+	0.995018i	$0.531787\pi$
$648$	0	0
$649$	−1.56922	−0.0615972
$650$	0	0
$651$	10.9282i	0.428310i
$652$	0	0
$653$	− 20.9282i	− 0.818984i	−0.912314	−	0.409492i	$-0.865706\pi$
$653$	− 20.9282i	− 0.818984i	0.912314	−	0.409492i	$-0.134294\pi$
$654$	0	0
$655$	−41.5692	−1.62424
$656$	0	0
$657$	12.9282	0.504377
$658$	0	0
$659$	− 14.5359i	− 0.566238i	−0.959085	−	0.283119i	$-0.908631\pi$
$659$	− 14.5359i	− 0.566238i	0.959085	−	0.283119i	$-0.0913692\pi$
$660$	0	0
$661$	− 26.7846i	− 1.04180i	−0.853618	−	0.520900i	$-0.825596\pi$
$661$	− 26.7846i	− 1.04180i	0.853618	−	0.520900i	$-0.174404\pi$
$662$	0	0
$663$	1.07180	0.0416251
$664$	0	0
$665$	24.0000	0.930680
$666$	0	0
$667$	7.21539i	0.279381i
$668$	0	0
$669$	− 24.0000i	− 0.927894i
$670$	0	0
$671$	−13.0718	−0.504631
$672$	0	0
$673$	31.8564	1.22797	0.613987	−	0.789316i	$-0.289565\pi$
$673$	31.8564	1.22797	0.613987	+	0.789316i	$0.289565\pi$
$674$	0	0
$675$	− 7.00000i	− 0.269430i
$676$	0	0
$677$	1.60770i	0.0617887i	0.999523	+	0.0308944i	$0.00983555\pi$
$677$	1.60770i	0.0617887i	−0.999523	+	0.0308944i	$0.990164\pi$
$678$	0	0
$679$	8.92820	0.342633
$680$	0	0
$681$	22.9282	0.878611
$682$	0	0
$683$	− 18.2487i	− 0.698268i	−0.937073	−	0.349134i	$-0.886476\pi$
$683$	− 18.2487i	− 0.698268i	0.937073	−	0.349134i	$-0.113524\pi$
$684$	0	0
$685$	− 30.9282i	− 1.18171i
$686$	0	0
$687$	27.8564	1.06279
$688$	0	0
$689$	−4.00000	−0.152388
$690$	0	0
$691$	41.8564i	1.59229i	0.605104	+	0.796146i	$0.293131\pi$
$691$	41.8564i	1.59229i	−0.605104	+	0.796146i	$0.706869\pi$
$692$	0	0
$693$	− 1.46410i	− 0.0556166i
$694$	0	0
$695$	−61.8564	−2.34635
$696$	0	0
$697$	−6.14359	−0.232705
$698$	0	0
$699$	12.9282i	0.488990i
$700$	0	0
$701$	28.6410i	1.08176i	0.841101	+	0.540878i	$0.181908\pi$
$701$	28.6410i	1.08176i	−0.841101	+	0.540878i	$0.818092\pi$
$702$	0	0
$703$	13.8564	0.522604
$704$	0	0
$705$	37.8564	1.42575
$706$	0	0
$707$	− 15.4641i	− 0.581587i
$708$	0	0
$709$	− 17.7128i	− 0.665219i	−0.943065	−	0.332609i	$-0.892071\pi$
$709$	− 17.7128i	− 0.665219i	0.943065	−	0.332609i	$-0.107929\pi$
$710$	0	0
$711$	−10.9282	−0.409840
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	− 10.1436i	− 0.379349i
$716$	0	0
$717$	14.5359i	0.542853i
$718$	0	0
$719$	13.8564	0.516757	0.258378	−	0.966044i	$-0.416812\pi$
$719$	13.8564	0.516757	0.258378	+	0.966044i	$0.416812\pi$
$720$	0	0
$721$	10.9282	0.406988
$722$	0	0
$723$	3.07180i	0.114241i
$724$	0	0
$725$	34.4974i	1.28120i
$726$	0	0
$727$	21.0718	0.781510	0.390755	−	0.920495i	$-0.372214\pi$
$727$	21.0718	0.781510	0.390755	+	0.920495i	$0.372214\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	− 4.28719i	− 0.158567i
$732$	0	0
$733$	5.71281i	0.211008i	0.994419	+	0.105504i	$0.0336455\pi$
$733$	5.71281i	0.211008i	−0.994419	+	0.105504i	$0.966354\pi$
$734$	0	0
$735$	−3.46410	−0.127775
$736$	0	0
$737$	4.28719	0.157921
$738$	0	0
$739$	37.0718i	1.36371i	0.731488	+	0.681854i	$0.238826\pi$
$739$	37.0718i	1.36371i	−0.731488	+	0.681854i	$0.761174\pi$
$740$	0	0
$741$	− 13.8564i	− 0.509028i
$742$	0	0
$743$	11.6077	0.425845	0.212923	−	0.977069i	$-0.431702\pi$
$743$	11.6077	0.425845	0.212923	+	0.977069i	$0.431702\pi$
$744$	0	0
$745$	62.3538	2.28447
$746$	0	0
$747$	− 4.00000i	− 0.146352i
$748$	0	0
$749$	− 12.3923i	− 0.452805i
$750$	0	0
$751$	−27.7128	−1.01125	−0.505627	−	0.862752i	$-0.668739\pi$
$751$	−27.7128	−1.01125	−0.505627	+	0.862752i	$0.668739\pi$
$752$	0	0
$753$	−9.07180	−0.330595
$754$	0	0
$755$	− 75.7128i	− 2.75547i
$756$	0	0
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	0	0
$759$	2.14359	0.0778075
$760$	0	0
$761$	−27.4641	−0.995573	−0.497786	−	0.867300i	$-0.665854\pi$
$761$	−27.4641	−0.995573	−0.497786	+	0.867300i	$0.665854\pi$
$762$	0	0
$763$	2.00000i	0.0724049i
$764$	0	0
$765$	1.85641i	0.0671185i
$766$	0	0
$767$	2.14359	0.0774007
$768$	0	0
$769$	4.14359	0.149422	0.0747109	−	0.997205i	$-0.476197\pi$
$769$	4.14359	0.149422	0.0747109	+	0.997205i	$0.476197\pi$
$770$	0	0
$771$	− 11.4641i	− 0.412870i
$772$	0	0
$773$	5.32051i	0.191365i	0.995412	+	0.0956827i	$0.0305034\pi$
$773$	5.32051i	0.191365i	−0.995412	+	0.0956827i	$0.969497\pi$
$774$	0	0
$775$	−76.4974	−2.74787
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	0	0
$779$	79.4256i	2.84572i
$780$	0	0
$781$	− 13.8564i	− 0.495821i
$782$	0	0
$783$	4.92820	0.176120
$784$	0	0
$785$	−3.21539	−0.114762
$786$	0	0
$787$	− 23.7128i	− 0.845270i	−0.906300	−	0.422635i	$-0.861105\pi$
$787$	− 23.7128i	− 0.845270i	0.906300	−	0.422635i	$-0.138895\pi$
$788$	0	0
$789$	− 12.3923i	− 0.441178i
$790$	0	0
$791$	19.8564	0.706013
$792$	0	0
$793$	17.8564	0.634100
$794$	0	0
$795$	− 6.92820i	− 0.245718i
$796$	0	0
$797$	6.39230i	0.226427i	0.993571	+	0.113214i	$0.0361144\pi$
$797$	6.39230i	0.226427i	−0.993571	+	0.113214i	$0.963886\pi$
$798$	0	0
$799$	−5.85641	−0.207185
$800$	0	0
$801$	3.46410	0.122398
$802$	0	0
$803$	18.9282i	0.667962i
$804$	0	0
$805$	− 5.07180i	− 0.178757i
$806$	0	0
$807$	3.46410	0.121942
$808$	0	0
$809$	38.0000	1.33601	0.668004	−	0.744157i	$-0.267149\pi$
$809$	38.0000	1.33601	0.668004	+	0.744157i	$0.267149\pi$
$810$	0	0
$811$	− 37.5692i	− 1.31923i	−0.751602	−	0.659617i	$-0.770719\pi$
$811$	− 37.5692i	− 1.31923i	0.751602	−	0.659617i	$-0.229281\pi$
$812$	0	0
$813$	2.92820i	0.102697i
$814$	0	0
$815$	−37.8564	−1.32605
$816$	0	0
$817$	−55.4256	−1.93910
$818$	0	0
$819$	2.00000i	0.0698857i
$820$	0	0
$821$	12.1436i	0.423814i	0.977290	+	0.211907i	$0.0679674\pi$
$821$	12.1436i	0.423814i	−0.977290	+	0.211907i	$0.932033\pi$
$822$	0	0
$823$	−21.0718	−0.734517	−0.367258	−	0.930119i	$-0.619704\pi$
$823$	−21.0718	−0.734517	−0.367258	+	0.930119i	$0.619704\pi$
$824$	0	0
$825$	10.2487	0.356814
$826$	0	0
$827$	− 37.1769i	− 1.29277i	−0.763012	−	0.646384i	$-0.776280\pi$
$827$	− 37.1769i	− 1.29277i	0.763012	−	0.646384i	$-0.223720\pi$
$828$	0	0
$829$	21.7128i	0.754117i	0.926189	+	0.377059i	$0.123064\pi$
$829$	21.7128i	0.754117i	−0.926189	+	0.377059i	$0.876936\pi$
$830$	0	0
$831$	−0.143594	−0.00498120
$832$	0	0
$833$	0.535898	0.0185678
$834$	0	0
$835$	37.8564i	1.31007i
$836$	0	0
$837$	10.9282i	0.377734i
$838$	0	0
$839$	−10.9282	−0.377283	−0.188642	−	0.982046i	$-0.560408\pi$
$839$	−10.9282	−0.377283	−0.188642	+	0.982046i	$0.560408\pi$
$840$	0	0
$841$	4.71281	0.162511
$842$	0	0
$843$	12.9282i	0.445271i
$844$	0	0
$845$	− 31.1769i	− 1.07252i
$846$	0	0
$847$	−8.85641	−0.304310
$848$	0	0
$849$	−14.9282	−0.512335
$850$	0	0
$851$	− 2.92820i	− 0.100378i
$852$	0	0
$853$	− 23.0718i	− 0.789963i	−0.918689	−	0.394982i	$-0.870751\pi$
$853$	− 23.0718i	− 0.789963i	0.918689	−	0.394982i	$-0.129249\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	0	0
$857$	−55.1769	−1.88481	−0.942404	−	0.334477i	$-0.891440\pi$
$857$	−55.1769	−1.88481	−0.942404	+	0.334477i	$0.891440\pi$
$858$	0	0
$859$	− 38.9282i	− 1.32821i	−0.747638	−	0.664107i	$-0.768812\pi$
$859$	− 38.9282i	− 1.32821i	0.747638	−	0.664107i	$-0.231188\pi$
$860$	0	0
$861$	− 11.4641i	− 0.390696i
$862$	0	0
$863$	9.46410	0.322162	0.161081	−	0.986941i	$-0.448502\pi$
$863$	9.46410	0.322162	0.161081	+	0.986941i	$0.448502\pi$
$864$	0	0
$865$	77.5692	2.63743
$866$	0	0
$867$	16.7128i	0.567597i
$868$	0	0
$869$	− 16.0000i	− 0.542763i
$870$	0	0
$871$	−5.85641	−0.198437
$872$	0	0
$873$	8.92820	0.302174
$874$	0	0
$875$	− 6.92820i	− 0.234216i
$876$	0	0
$877$	8.14359i	0.274990i	0.990502	+	0.137495i	$0.0439050\pi$
$877$	8.14359i	0.274990i	−0.990502	+	0.137495i	$0.956095\pi$
$878$	0	0
$879$	−16.5359	−0.557742
$880$	0	0
$881$	−13.3205	−0.448779	−0.224390	−	0.974500i	$-0.572039\pi$
$881$	−13.3205	−0.448779	−0.224390	+	0.974500i	$0.572039\pi$
$882$	0	0
$883$	33.5692i	1.12969i	0.825196	+	0.564847i	$0.191065\pi$
$883$	33.5692i	1.12969i	−0.825196	+	0.564847i	$0.808935\pi$
$884$	0	0
$885$	3.71281i	0.124805i
$886$	0	0
$887$	−26.9282	−0.904161	−0.452080	−	0.891977i	$-0.649318\pi$
$887$	−26.9282	−0.904161	−0.452080	+	0.891977i	$0.649318\pi$
$888$	0	0
$889$	2.92820	0.0982088
$890$	0	0
$891$	− 1.46410i	− 0.0490492i
$892$	0	0
$893$	75.7128i	2.53363i
$894$	0	0
$895$	−32.7846	−1.09587
$896$	0	0
$897$	−2.92820	−0.0977699
$898$	0	0
$899$	− 53.8564i	− 1.79621i
$900$	0	0
$901$	1.07180i	0.0357067i
$902$	0	0
$903$	8.00000	0.266223
$904$	0	0
$905$	20.7846	0.690904
$906$	0	0
$907$	− 43.7128i	− 1.45146i	−0.687980	−	0.725730i	$-0.741502\pi$
$907$	− 43.7128i	− 1.45146i	0.687980	−	0.725730i	$-0.258498\pi$
$908$	0	0
$909$	− 15.4641i	− 0.512912i
$910$	0	0
$911$	−29.1769	−0.966674	−0.483337	−	0.875434i	$-0.660575\pi$
$911$	−29.1769	−0.966674	−0.483337	+	0.875434i	$0.660575\pi$
$912$	0	0
$913$	5.85641	0.193819
$914$	0	0
$915$	30.9282i	1.02245i
$916$	0	0
$917$	12.0000i	0.396275i
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	−22.9282	−0.755510
$922$	0	0
$923$	18.9282i	0.623029i
$924$	0	0
$925$	− 14.0000i	− 0.460317i
$926$	0	0
$927$	10.9282	0.358929
$928$	0	0
$929$	13.6077	0.446454	0.223227	−	0.974766i	$-0.428341\pi$
$929$	13.6077	0.446454	0.223227	+	0.974766i	$0.428341\pi$
$930$	0	0
$931$	− 6.92820i	− 0.227063i
$932$	0	0
$933$	− 18.9282i	− 0.619682i
$934$	0	0
$935$	−2.71797	−0.0888870
$936$	0	0
$937$	−26.0000	−0.849383	−0.424691	−	0.905338i	$-0.639617\pi$
$937$	−26.0000	−0.849383	−0.424691	+	0.905338i	$0.639617\pi$
$938$	0	0
$939$	− 14.0000i	− 0.456873i
$940$	0	0
$941$	− 36.5359i	− 1.19104i	−0.803342	−	0.595518i	$-0.796947\pi$
$941$	− 36.5359i	− 1.19104i	0.803342	−	0.595518i	$-0.203053\pi$
$942$	0	0
$943$	16.7846	0.546582
$944$	0	0
$945$	−3.46410	−0.112687
$946$	0	0
$947$	− 26.2487i	− 0.852969i	−0.904495	−	0.426484i	$-0.859752\pi$
$947$	− 26.2487i	− 0.852969i	0.904495	−	0.426484i	$-0.140248\pi$
$948$	0	0
$949$	− 25.8564i	− 0.839334i
$950$	0	0
$951$	0.143594	0.00465634
$952$	0	0
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	32.7846i	1.06089i
$956$	0	0
$957$	7.21539i	0.233240i
$958$	0	0
$959$	−8.92820	−0.288307
$960$	0	0
$961$	88.4256	2.85244
$962$	0	0
$963$	− 12.3923i	− 0.399336i
$964$	0	0
$965$	− 54.9282i	− 1.76820i
$966$	0	0
$967$	34.9282	1.12322	0.561608	−	0.827404i	$-0.310183\pi$
$967$	34.9282	1.12322	0.561608	+	0.827404i	$0.310183\pi$
$968$	0	0
$969$	−3.71281	−0.119273
$970$	0	0
$971$	1.85641i	0.0595749i	0.999556	+	0.0297875i	$0.00948305\pi$
$971$	1.85641i	0.0595749i	−0.999556	+	0.0297875i	$0.990517\pi$
$972$	0	0
$973$	17.8564i	0.572450i
$974$	0	0
$975$	−14.0000	−0.448359
$976$	0	0
$977$	38.4974	1.23164	0.615821	−	0.787886i	$-0.288825\pi$
$977$	38.4974	1.23164	0.615821	+	0.787886i	$0.288825\pi$
$978$	0	0
$979$	5.07180i	0.162095i
$980$	0	0
$981$	2.00000i	0.0638551i
$982$	0	0
$983$	−3.71281	−0.118420	−0.0592102	−	0.998246i	$-0.518858\pi$
$983$	−3.71281	−0.118420	−0.0592102	+	0.998246i	$0.518858\pi$
$984$	0	0
$985$	27.2154	0.867154
$986$	0	0
$987$	− 10.9282i	− 0.347849i
$988$	0	0
$989$	11.7128i	0.372446i
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	24.0000	0.761617
$994$	0	0
$995$	− 20.2872i	− 0.643147i
$996$	0	0
$997$	− 31.0718i	− 0.984054i	−0.870580	−	0.492027i	$-0.836256\pi$
$997$	− 31.0718i	− 0.984054i	0.870580	−	0.492027i	$-0.163744\pi$
$998$	0	0
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.c.bh.2689.1		4
4.3	odd	2		5376.2.c.bn.2689.3		4
8.3	odd	2		5376.2.c.bn.2689.2		4
8.5	even	2	inner	5376.2.c.bh.2689.4		4
16.3	odd	4		1344.2.a.u.1.1		2
16.5	even	4		672.2.a.i.1.2	✓	2
16.11	odd	4		672.2.a.j.1.2	yes	2
16.13	even	4		1344.2.a.v.1.1		2
48.5	odd	4		2016.2.a.t.1.1		2
48.11	even	4		2016.2.a.s.1.1		2
48.29	odd	4		4032.2.a.bs.1.2		2
48.35	even	4		4032.2.a.br.1.2		2
112.13	odd	4		9408.2.a.do.1.2		2
112.27	even	4		4704.2.a.bm.1.1		2
112.69	odd	4		4704.2.a.bn.1.1		2
112.83	even	4		9408.2.a.dx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
672.2.a.i.1.2	✓	2	16.5	even	4
672.2.a.j.1.2	yes	2	16.11	odd	4
1344.2.a.u.1.1		2	16.3	odd	4
1344.2.a.v.1.1		2	16.13	even	4
2016.2.a.s.1.1		2	48.11	even	4
2016.2.a.t.1.1		2	48.5	odd	4
4032.2.a.br.1.2		2	48.35	even	4
4032.2.a.bs.1.2		2	48.29	odd	4
4704.2.a.bm.1.1		2	112.27	even	4
4704.2.a.bn.1.1		2	112.69	odd	4
5376.2.c.bh.2689.1		4	1.1	even	1	trivial
5376.2.c.bh.2689.4		4	8.5	even	2	inner
5376.2.c.bn.2689.2		4	8.3	odd	2
5376.2.c.bn.2689.3		4	4.3	odd	2
9408.2.a.do.1.2		2	112.13	odd	4
9408.2.a.dx.1.2		2	112.83	even	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:	\( N \)	\(=\)	\( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5376.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -3.46410i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -3.46410i q^{5} -1.00000 q^{7} -1.00000 q^{9} -1.46410i q^{11} +2.00000i q^{13} -3.46410 q^{15} +0.535898 q^{17} -6.92820i q^{19} +1.00000i q^{21} -1.46410 q^{23} -7.00000 q^{25} +1.00000i q^{27} -4.92820i q^{29} +10.9282 q^{31} -1.46410 q^{33} +3.46410i q^{35} +2.00000i q^{37} +2.00000 q^{39} -11.4641 q^{41} -8.00000i q^{43} +3.46410i q^{45} -10.9282 q^{47} +1.00000 q^{49} -0.535898i q^{51} +2.00000i q^{53} -5.07180 q^{55} -6.92820 q^{57} -1.07180i q^{59} -8.92820i q^{61} +1.00000 q^{63} +6.92820 q^{65} +2.92820i q^{67} +1.46410i q^{69} +9.46410 q^{71} -12.9282 q^{73} +7.00000i q^{75} +1.46410i q^{77} +10.9282 q^{79} +1.00000 q^{81} +4.00000i q^{83} -1.85641i q^{85} -4.92820 q^{87} -3.46410 q^{89} -2.00000i q^{91} -10.9282i q^{93} -24.0000 q^{95} -8.92820 q^{97} +1.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 - 4 * q^9	\( 4 q - 4 q^{7} - 4 q^{9} + 16 q^{17} + 8 q^{23} - 28 q^{25} + 16 q^{31} + 8 q^{33} + 8 q^{39} - 32 q^{41} - 16 q^{47} + 4 q^{49} - 48 q^{55} + 4 q^{63} + 24 q^{71} - 24 q^{73} + 16 q^{79} + 4 q^{81} + 8 q^{87} - 96 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^9 + 16 * q^17 + 8 * q^23 - 28 * q^25 + 16 * q^31 + 8 * q^33 + 8 * q^39 - 32 * q^41 - 16 * q^47 + 4 * q^49 - 48 * q^55 + 4 * q^63 + 24 * q^71 - 24 * q^73 + 16 * q^79 + 4 * q^81 + 8 * q^87 - 96 * q^95 - 8 * q^97

Introduction

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