LMFDB - Embedded newform 2432.2.c.c.1217.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (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:	$19.4196177716$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.4
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.c.1217.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -2.00000i q^{5} +2.46410 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -2.00000i q^{5} +2.46410 q^{7} +2.00000 q^{9} +4.00000i q^{11} +2.46410i q^{13} +2.00000 q^{15} +5.92820 q^{17} +1.00000i q^{19} +2.46410i q^{21} -0.464102 q^{23} +1.00000 q^{25} +5.00000i q^{27} -4.46410i q^{29} -8.92820 q^{31} -4.00000 q^{33} -4.92820i q^{35} -2.00000i q^{37} -2.46410 q^{39} -6.92820 q^{41} +8.92820i q^{43} -4.00000i q^{45} +6.92820 q^{47} -0.928203 q^{49} +5.92820i q^{51} +5.53590i q^{53} +8.00000 q^{55} -1.00000 q^{57} -3.92820i q^{59} -4.92820i q^{61} +4.92820 q^{63} +4.92820 q^{65} +7.92820i q^{67} -0.464102i q^{69} +14.0000 q^{71} +7.00000 q^{73} +1.00000i q^{75} +9.85641i q^{77} +2.00000 q^{79} +1.00000 q^{81} -10.9282i q^{83} -11.8564i q^{85} +4.46410 q^{87} -12.9282 q^{89} +6.07180i q^{91} -8.92820i q^{93} +2.00000 q^{95} -0.928203 q^{97} +8.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 + 8 * q^9	$ 4 q - 4 q^{7} + 8 q^{9} + 8 q^{15} - 4 q^{17} + 12 q^{23} + 4 q^{25} - 8 q^{31} - 16 q^{33} + 4 q^{39} + 24 q^{49} + 32 q^{55} - 4 q^{57} - 8 q^{63} - 8 q^{65} + 56 q^{71} + 28 q^{73} + 8 q^{79} + 4 q^{81} + 4 q^{87} - 24 q^{89} + 8 q^{95} + 24 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 8 * q^9 + 8 * q^15 - 4 * q^17 + 12 * q^23 + 4 * q^25 - 8 * q^31 - 16 * q^33 + 4 * q^39 + 24 * q^49 + 32 * q^55 - 4 * q^57 - 8 * q^63 - 8 * q^65 + 56 * q^71 + 28 * q^73 + 8 * q^79 + 4 * q^81 + 4 * q^87 - 24 * q^89 + 8 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.00000i	0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	− 2.00000i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 2.00000i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	2.46410	0.931343	0.465671	−	0.884958i	$-0.345813\pi$
$7$	2.46410	0.931343	0.465671	+	0.884958i	$0.345813\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	0	0
$13$	2.46410i	0.683419i	0.939806	+	0.341709i	$0.111006\pi$
$13$	2.46410i	0.683419i	−0.939806	+	0.341709i	$0.888994\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	5.92820	1.43780	0.718900	−	0.695113i	$-0.244646\pi$
$17$	5.92820	1.43780	0.718900	+	0.695113i	$0.244646\pi$
$18$	0	0
$19$	1.00000i	0.229416i
$19$	1.00000i	0.229416i
$20$	0	0
$21$	2.46410i	0.537711i
$22$	0	0
$23$	−0.464102	−0.0967719	−0.0483859	−	0.998829i	$-0.515408\pi$
$23$	−0.464102	−0.0967719	−0.0483859	+	0.998829i	$0.515408\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	− 4.46410i	− 0.828963i	−0.910058	−	0.414481i	$-0.863963\pi$
$29$	− 4.46410i	− 0.828963i	0.910058	−	0.414481i	$-0.136037\pi$
$30$	0	0
$31$	−8.92820	−1.60355	−0.801776	−	0.597624i	$-0.796111\pi$
$31$	−8.92820	−1.60355	−0.801776	+	0.597624i	$0.796111\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	− 4.92820i	− 0.833018i
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	−2.46410	−0.394572
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	8.92820i	1.36154i	0.732498	+	0.680769i	$0.238354\pi$
$43$	8.92820i	1.36154i	−0.732498	+	0.680769i	$0.761646\pi$
$44$	0	0
$45$	− 4.00000i	− 0.596285i
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	−0.928203	−0.132600
$50$	0	0
$51$	5.92820i	0.830114i
$52$	0	0
$53$	5.53590i	0.760414i	0.924901	+	0.380207i	$0.124147\pi$
$53$	5.53590i	0.760414i	−0.924901	+	0.380207i	$0.875853\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	− 3.92820i	− 0.511409i	−0.966755	−	0.255704i	$-0.917693\pi$
$59$	− 3.92820i	− 0.511409i	0.966755	−	0.255704i	$-0.0823073\pi$
$60$	0	0
$61$	− 4.92820i	− 0.630992i	−0.948927	−	0.315496i	$-0.897829\pi$
$61$	− 4.92820i	− 0.630992i	0.948927	−	0.315496i	$-0.102171\pi$
$62$	0	0
$63$	4.92820	0.620895
$64$	0	0
$65$	4.92820	0.611268
$66$	0	0
$67$	7.92820i	0.968584i	0.874906	+	0.484292i	$0.160923\pi$
$67$	7.92820i	0.968584i	−0.874906	+	0.484292i	$0.839077\pi$
$68$	0	0
$69$	− 0.464102i	− 0.0558713i
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	9.85641i	1.12324i
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 10.9282i	− 1.19953i	−0.800178	−	0.599763i	$-0.795261\pi$
$83$	− 10.9282i	− 1.19953i	0.800178	−	0.599763i	$-0.204739\pi$
$84$	0	0
$85$	− 11.8564i	− 1.28601i
$86$	0	0
$87$	4.46410	0.478602
$88$	0	0
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	6.07180i	0.636497i
$92$	0	0
$93$	− 8.92820i	− 0.925812i
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	−0.928203	−0.0942448	−0.0471224	−	0.998889i	$-0.515005\pi$
$97$	−0.928203	−0.0942448	−0.0471224	+	0.998889i	$0.515005\pi$
$98$	0	0
$99$	8.00000i	0.804030i
$100$	0	0
$101$	17.8564i	1.77678i	0.459091	+	0.888389i	$0.348175\pi$
$101$	17.8564i	1.77678i	−0.459091	+	0.888389i	$0.651825\pi$
$102$	0	0
$103$	10.9282	1.07679	0.538394	−	0.842693i	$-0.319031\pi$
$103$	10.9282	1.07679	0.538394	+	0.842693i	$0.319031\pi$
$104$	0	0
$105$	4.92820	0.480943
$106$	0	0
$107$	− 0.0717968i	− 0.00694086i	−0.999994	−	0.00347043i	$-0.998895\pi$
$107$	− 0.0717968i	− 0.00694086i	0.999994	−	0.00347043i	$-0.00110467\pi$
$108$	0	0
$109$	− 13.3923i	− 1.28275i	−0.767227	−	0.641375i	$-0.778364\pi$
$109$	− 13.3923i	− 1.28275i	0.767227	−	0.641375i	$-0.221636\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−0.928203	−0.0873180	−0.0436590	−	0.999046i	$-0.513902\pi$
$113$	−0.928203	−0.0873180	−0.0436590	+	0.999046i	$0.513902\pi$
$114$	0	0
$115$	0.928203i	0.0865554i
$116$	0	0
$117$	4.92820i	0.455613i
$118$	0	0
$119$	14.6077	1.33909
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 6.92820i	− 0.624695i
$124$	0	0
$125$	− 12.0000i	− 1.07331i
$126$	0	0
$127$	−12.9282	−1.14719	−0.573596	−	0.819138i	$-0.694452\pi$
$127$	−12.9282	−1.14719	−0.573596	+	0.819138i	$0.694452\pi$
$128$	0	0
$129$	−8.92820	−0.786084
$130$	0	0
$131$	20.9282i	1.82851i	0.405144	+	0.914253i	$0.367221\pi$
$131$	20.9282i	1.82851i	−0.405144	+	0.914253i	$0.632779\pi$
$132$	0	0
$133$	2.46410i	0.213665i
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	18.8564	1.61101	0.805506	−	0.592588i	$-0.201894\pi$
$137$	18.8564	1.61101	0.805506	+	0.592588i	$0.201894\pi$
$138$	0	0
$139$	− 11.8564i	− 1.00565i	−0.864389	−	0.502824i	$-0.832295\pi$
$139$	− 11.8564i	− 1.00565i	0.864389	−	0.502824i	$-0.167705\pi$
$140$	0	0
$141$	6.92820i	0.583460i
$142$	0	0
$143$	−9.85641	−0.824234
$144$	0	0
$145$	−8.92820	−0.741447
$146$	0	0
$147$	− 0.928203i	− 0.0765569i
$148$	0	0
$149$	− 2.92820i	− 0.239888i	−0.992781	−	0.119944i	$-0.961729\pi$
$149$	− 2.92820i	− 0.239888i	0.992781	−	0.119944i	$-0.0382715\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	11.8564	0.958534
$154$	0	0
$155$	17.8564i	1.43426i
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	−5.53590	−0.439025
$160$	0	0
$161$	−1.14359	−0.0901278
$162$	0	0
$163$	− 3.85641i	− 0.302057i	−0.988529	−	0.151029i	$-0.951741\pi$
$163$	− 3.85641i	− 0.302057i	0.988529	−	0.151029i	$-0.0482585\pi$
$164$	0	0
$165$	8.00000i	0.622799i
$166$	0	0
$167$	18.9282	1.46471	0.732354	−	0.680924i	$-0.238422\pi$
$167$	18.9282	1.46471	0.732354	+	0.680924i	$0.238422\pi$
$168$	0	0
$169$	6.92820	0.532939
$170$	0	0
$171$	2.00000i	0.152944i
$172$	0	0
$173$	10.0000i	0.760286i	0.924928	+	0.380143i	$0.124125\pi$
$173$	10.0000i	0.760286i	−0.924928	+	0.380143i	$0.875875\pi$
$174$	0	0
$175$	2.46410	0.186269
$176$	0	0
$177$	3.92820	0.295262
$178$	0	0
$179$	− 1.85641i	− 0.138754i	−0.997591	−	0.0693772i	$-0.977899\pi$
$179$	− 1.85641i	− 0.138754i	0.997591	−	0.0693772i	$-0.0221012\pi$
$180$	0	0
$181$	− 10.0000i	− 0.743294i	−0.928374	−	0.371647i	$-0.878793\pi$
$181$	− 10.0000i	− 0.743294i	0.928374	−	0.371647i	$-0.121207\pi$
$182$	0	0
$183$	4.92820	0.364303
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	23.7128i	1.73405i
$188$	0	0
$189$	12.3205i	0.896185i
$190$	0	0
$191$	−10.4641	−0.757156	−0.378578	−	0.925569i	$-0.623587\pi$
$191$	−10.4641	−0.757156	−0.378578	+	0.925569i	$0.623587\pi$
$192$	0	0
$193$	−17.8564	−1.28533	−0.642666	−	0.766146i	$-0.722172\pi$
$193$	−17.8564	−1.28533	−0.642666	+	0.766146i	$0.722172\pi$
$194$	0	0
$195$	4.92820i	0.352916i
$196$	0	0
$197$	6.92820i	0.493614i	0.969065	+	0.246807i	$0.0793814\pi$
$197$	6.92820i	0.493614i	−0.969065	+	0.246807i	$0.920619\pi$
$198$	0	0
$199$	−20.4641	−1.45066	−0.725331	−	0.688400i	$-0.758313\pi$
$199$	−20.4641	−1.45066	−0.725331	+	0.688400i	$0.758313\pi$
$200$	0	0
$201$	−7.92820	−0.559212
$202$	0	0
$203$	− 11.0000i	− 0.772049i
$204$	0	0
$205$	13.8564i	0.967773i
$206$	0	0
$207$	−0.928203	−0.0645146
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	2.85641i	0.196643i	0.995155	+	0.0983216i	$0.0313474\pi$
$211$	2.85641i	0.196643i	−0.995155	+	0.0983216i	$0.968653\pi$
$212$	0	0
$213$	14.0000i	0.959264i
$214$	0	0
$215$	17.8564	1.21780
$216$	0	0
$217$	−22.0000	−1.49346
$218$	0	0
$219$	7.00000i	0.473016i
$220$	0	0
$221$	14.6077i	0.982620i
$222$	0	0
$223$	−13.0718	−0.875352	−0.437676	−	0.899133i	$-0.644198\pi$
$223$	−13.0718	−0.875352	−0.437676	+	0.899133i	$0.644198\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	11.9282i	0.791703i	0.918315	+	0.395851i	$0.129550\pi$
$227$	11.9282i	0.791703i	−0.918315	+	0.395851i	$0.870450\pi$
$228$	0	0
$229$	− 16.9282i	− 1.11865i	−0.828949	−	0.559324i	$-0.811061\pi$
$229$	− 16.9282i	− 1.11865i	0.828949	−	0.559324i	$-0.188939\pi$
$230$	0	0
$231$	−9.85641	−0.648504
$232$	0	0
$233$	−11.8564	−0.776739	−0.388370	−	0.921504i	$-0.626962\pi$
$233$	−11.8564	−0.776739	−0.388370	+	0.921504i	$0.626962\pi$
$234$	0	0
$235$	− 13.8564i	− 0.903892i
$236$	0	0
$237$	2.00000i	0.129914i
$238$	0	0
$239$	18.3205	1.18506	0.592528	−	0.805550i	$-0.298130\pi$
$239$	18.3205	1.18506	0.592528	+	0.805550i	$0.298130\pi$
$240$	0	0
$241$	16.7846	1.08119	0.540596	−	0.841282i	$-0.318199\pi$
$241$	16.7846	1.08119	0.540596	+	0.841282i	$0.318199\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	1.85641i	0.118601i
$246$	0	0
$247$	−2.46410	−0.156787
$248$	0	0
$249$	10.9282	0.692547
$250$	0	0
$251$	0.928203i	0.0585877i	0.999571	+	0.0292938i	$0.00932585\pi$
$251$	0.928203i	0.0585877i	−0.999571	+	0.0292938i	$0.990674\pi$
$252$	0	0
$253$	− 1.85641i	− 0.116711i
$254$	0	0
$255$	11.8564	0.742477
$256$	0	0
$257$	12.9282	0.806439	0.403220	−	0.915103i	$-0.367891\pi$
$257$	12.9282	0.806439	0.403220	+	0.915103i	$0.367891\pi$
$258$	0	0
$259$	− 4.92820i	− 0.306224i
$260$	0	0
$261$	− 8.92820i	− 0.552642i
$262$	0	0
$263$	−10.9282	−0.673862	−0.336931	−	0.941529i	$-0.609389\pi$
$263$	−10.9282	−0.673862	−0.336931	+	0.941529i	$0.609389\pi$
$264$	0	0
$265$	11.0718	0.680135
$266$	0	0
$267$	− 12.9282i	− 0.791193i
$268$	0	0
$269$	− 19.8564i	− 1.21067i	−0.795972	−	0.605333i	$-0.793040\pi$
$269$	− 19.8564i	− 1.21067i	0.795972	−	0.605333i	$-0.206960\pi$
$270$	0	0
$271$	−25.3923	−1.54247	−0.771236	−	0.636549i	$-0.780361\pi$
$271$	−25.3923	−1.54247	−0.771236	+	0.636549i	$0.780361\pi$
$272$	0	0
$273$	−6.07180	−0.367482
$274$	0	0
$275$	4.00000i	0.241209i
$276$	0	0
$277$	− 11.0718i	− 0.665240i	−0.943061	−	0.332620i	$-0.892067\pi$
$277$	− 11.0718i	− 0.665240i	0.943061	−	0.332620i	$-0.107933\pi$
$278$	0	0
$279$	−17.8564	−1.06904
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	5.07180i	0.301487i	0.988573	+	0.150744i	$0.0481668\pi$
$283$	5.07180i	0.301487i	−0.988573	+	0.150744i	$0.951833\pi$
$284$	0	0
$285$	2.00000i	0.118470i
$286$	0	0
$287$	−17.0718	−1.00772
$288$	0	0
$289$	18.1436	1.06727
$290$	0	0
$291$	− 0.928203i	− 0.0544122i
$292$	0	0
$293$	− 26.3205i	− 1.53766i	−0.639453	−	0.768830i	$-0.720839\pi$
$293$	− 26.3205i	− 1.53766i	0.639453	−	0.768830i	$-0.279161\pi$
$294$	0	0
$295$	−7.85641	−0.457418
$296$	0	0
$297$	−20.0000	−1.16052
$298$	0	0
$299$	− 1.14359i	− 0.0661357i
$300$	0	0
$301$	22.0000i	1.26806i
$302$	0	0
$303$	−17.8564	−1.02582
$304$	0	0
$305$	−9.85641	−0.564376
$306$	0	0
$307$	− 25.8564i	− 1.47570i	−0.674963	−	0.737852i	$-0.735840\pi$
$307$	− 25.8564i	− 1.47570i	0.674963	−	0.737852i	$-0.264160\pi$
$308$	0	0
$309$	10.9282i	0.621684i
$310$	0	0
$311$	13.3923	0.759408	0.379704	−	0.925108i	$-0.376026\pi$
$311$	13.3923	0.759408	0.379704	+	0.925108i	$0.376026\pi$
$312$	0	0
$313$	−10.8564	−0.613640	−0.306820	−	0.951767i	$-0.599265\pi$
$313$	−10.8564	−0.613640	−0.306820	+	0.951767i	$0.599265\pi$
$314$	0	0
$315$	− 9.85641i	− 0.555346i
$316$	0	0
$317$	− 7.53590i	− 0.423258i	−0.977350	−	0.211629i	$-0.932123\pi$
$317$	− 7.53590i	− 0.423258i	0.977350	−	0.211629i	$-0.0678769\pi$
$318$	0	0
$319$	17.8564	0.999767
$320$	0	0
$321$	0.0717968	0.00400730
$322$	0	0
$323$	5.92820i	0.329854i
$324$	0	0
$325$	2.46410i	0.136684i
$326$	0	0
$327$	13.3923	0.740596
$328$	0	0
$329$	17.0718	0.941199
$330$	0	0
$331$	− 31.7846i	− 1.74704i	−0.486788	−	0.873520i	$-0.661832\pi$
$331$	− 31.7846i	− 1.74704i	0.486788	−	0.873520i	$-0.338168\pi$
$332$	0	0
$333$	− 4.00000i	− 0.219199i
$334$	0	0
$335$	15.8564	0.866328
$336$	0	0
$337$	−20.9282	−1.14003	−0.570016	−	0.821634i	$-0.693063\pi$
$337$	−20.9282	−1.14003	−0.570016	+	0.821634i	$0.693063\pi$
$338$	0	0
$339$	− 0.928203i	− 0.0504131i
$340$	0	0
$341$	− 35.7128i	− 1.93396i
$342$	0	0
$343$	−19.5359	−1.05484
$344$	0	0
$345$	−0.928203	−0.0499728
$346$	0	0
$347$	0.143594i	0.00770851i	0.999993	+	0.00385425i	$0.00122685\pi$
$347$	0.143594i	0.00770851i	−0.999993	+	0.00385425i	$0.998773\pi$
$348$	0	0
$349$	− 14.0000i	− 0.749403i	−0.927146	−	0.374701i	$-0.877745\pi$
$349$	− 14.0000i	− 0.749403i	0.927146	−	0.374701i	$-0.122255\pi$
$350$	0	0
$351$	−12.3205	−0.657620
$352$	0	0
$353$	23.9282	1.27357	0.636785	−	0.771042i	$-0.280264\pi$
$353$	23.9282	1.27357	0.636785	+	0.771042i	$0.280264\pi$
$354$	0	0
$355$	− 28.0000i	− 1.48609i
$356$	0	0
$357$	14.6077i	0.773121i
$358$	0	0
$359$	17.5359	0.925509	0.462755	−	0.886486i	$-0.346861\pi$
$359$	17.5359	0.925509	0.462755	+	0.886486i	$0.346861\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	− 5.00000i	− 0.262432i
$364$	0	0
$365$	− 14.0000i	− 0.732793i
$366$	0	0
$367$	−26.9282	−1.40564	−0.702820	−	0.711367i	$-0.748076\pi$
$367$	−26.9282	−1.40564	−0.702820	+	0.711367i	$0.748076\pi$
$368$	0	0
$369$	−13.8564	−0.721336
$370$	0	0
$371$	13.6410i	0.708206i
$372$	0	0
$373$	9.39230i	0.486315i	0.969987	+	0.243158i	$0.0781832\pi$
$373$	9.39230i	0.486315i	−0.969987	+	0.243158i	$0.921817\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	11.0000	0.566529
$378$	0	0
$379$	20.8564i	1.07132i	0.844433	+	0.535661i	$0.179937\pi$
$379$	20.8564i	1.07132i	−0.844433	+	0.535661i	$0.820063\pi$
$380$	0	0
$381$	− 12.9282i	− 0.662332i
$382$	0	0
$383$	−16.7846	−0.857653	−0.428827	−	0.903387i	$-0.641073\pi$
$383$	−16.7846	−0.857653	−0.428827	+	0.903387i	$0.641073\pi$
$384$	0	0
$385$	19.7128	1.00466
$386$	0	0
$387$	17.8564i	0.907692i
$388$	0	0
$389$	1.85641i	0.0941235i	0.998892	+	0.0470618i	$0.0149858\pi$
$389$	1.85641i	0.0941235i	−0.998892	+	0.0470618i	$0.985014\pi$
$390$	0	0
$391$	−2.75129	−0.139139
$392$	0	0
$393$	−20.9282	−1.05569
$394$	0	0
$395$	− 4.00000i	− 0.201262i
$396$	0	0
$397$	− 4.92820i	− 0.247339i	−0.992323	−	0.123670i	$-0.960534\pi$
$397$	− 4.92820i	− 0.247339i	0.992323	−	0.123670i	$-0.0394663\pi$
$398$	0	0
$399$	−2.46410	−0.123359
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	− 22.0000i	− 1.09590i
$404$	0	0
$405$	− 2.00000i	− 0.0993808i
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	18.8564i	0.930118i
$412$	0	0
$413$	− 9.67949i	− 0.476297i
$414$	0	0
$415$	−21.8564	−1.07289
$416$	0	0
$417$	11.8564	0.580611
$418$	0	0
$419$	10.7846i	0.526863i	0.964678	+	0.263431i	$0.0848542\pi$
$419$	10.7846i	0.526863i	−0.964678	+	0.263431i	$0.915146\pi$
$420$	0	0
$421$	− 18.4641i	− 0.899885i	−0.893057	−	0.449943i	$-0.851444\pi$
$421$	− 18.4641i	− 0.899885i	0.893057	−	0.449943i	$-0.148556\pi$
$422$	0	0
$423$	13.8564	0.673722
$424$	0	0
$425$	5.92820	0.287560
$426$	0	0
$427$	− 12.1436i	− 0.587670i
$428$	0	0
$429$	− 9.85641i	− 0.475872i
$430$	0	0
$431$	0.928203	0.0447100	0.0223550	−	0.999750i	$-0.492884\pi$
$431$	0.928203	0.0447100	0.0223550	+	0.999750i	$0.492884\pi$
$432$	0	0
$433$	16.7846	0.806617	0.403308	−	0.915064i	$-0.367860\pi$
$433$	16.7846	0.806617	0.403308	+	0.915064i	$0.367860\pi$
$434$	0	0
$435$	− 8.92820i	− 0.428075i
$436$	0	0
$437$	− 0.464102i	− 0.0222010i
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	−1.85641	−0.0884003
$442$	0	0
$443$	− 9.85641i	− 0.468292i	−0.972201	−	0.234146i	$-0.924771\pi$
$443$	− 9.85641i	− 0.468292i	0.972201	−	0.234146i	$-0.0752294\pi$
$444$	0	0
$445$	25.8564i	1.22571i
$446$	0	0
$447$	2.92820	0.138499
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	− 27.7128i	− 1.30495i
$452$	0	0
$453$	16.0000i	0.751746i
$454$	0	0
$455$	12.1436	0.569300
$456$	0	0
$457$	31.0000	1.45012	0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000	1.45012	0.725059	+	0.688686i	$0.241812\pi$
$458$	0	0
$459$	29.6410i	1.38352i
$460$	0	0
$461$	13.8564i	0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$	13.8564i	0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$	0	0
$463$	24.7846	1.15184	0.575919	−	0.817507i	$-0.304644\pi$
$463$	24.7846	1.15184	0.575919	+	0.817507i	$0.304644\pi$
$464$	0	0
$465$	−17.8564	−0.828071
$466$	0	0
$467$	− 38.7846i	− 1.79474i	−0.441281	−	0.897369i	$-0.645476\pi$
$467$	− 38.7846i	− 1.79474i	0.441281	−	0.897369i	$-0.354524\pi$
$468$	0	0
$469$	19.5359i	0.902084i
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	−35.7128	−1.64208
$474$	0	0
$475$	1.00000i	0.0458831i
$476$	0	0
$477$	11.0718i	0.506943i
$478$	0	0
$479$	−24.7846	−1.13244	−0.566219	−	0.824255i	$-0.691594\pi$
$479$	−24.7846	−1.13244	−0.566219	+	0.824255i	$0.691594\pi$
$480$	0	0
$481$	4.92820	0.224707
$482$	0	0
$483$	− 1.14359i	− 0.0520353i
$484$	0	0
$485$	1.85641i	0.0842951i
$486$	0	0
$487$	−34.6410	−1.56973	−0.784867	−	0.619664i	$-0.787269\pi$
$487$	−34.6410	−1.56973	−0.784867	+	0.619664i	$0.787269\pi$
$488$	0	0
$489$	3.85641	0.174393
$490$	0	0
$491$	− 41.8564i	− 1.88895i	−0.328579	−	0.944477i	$-0.606570\pi$
$491$	− 41.8564i	− 1.88895i	0.328579	−	0.944477i	$-0.393430\pi$
$492$	0	0
$493$	− 26.4641i	− 1.19188i
$494$	0	0
$495$	16.0000	0.719147
$496$	0	0
$497$	34.4974	1.54742
$498$	0	0
$499$	18.7846i	0.840915i	0.907312	+	0.420457i	$0.138130\pi$
$499$	18.7846i	0.840915i	−0.907312	+	0.420457i	$0.861870\pi$
$500$	0	0
$501$	18.9282i	0.845650i
$502$	0	0
$503$	12.3205	0.549344	0.274672	−	0.961538i	$-0.411431\pi$
$503$	12.3205	0.549344	0.274672	+	0.961538i	$0.411431\pi$
$504$	0	0
$505$	35.7128	1.58920
$506$	0	0
$507$	6.92820i	0.307692i
$508$	0	0
$509$	18.0000i	0.797836i	0.916987	+	0.398918i	$0.130614\pi$
$509$	18.0000i	0.797836i	−0.916987	+	0.398918i	$0.869386\pi$
$510$	0	0
$511$	17.2487	0.763038
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	− 21.8564i	− 0.963108i
$516$	0	0
$517$	27.7128i	1.21881i
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−26.9282	−1.17975	−0.589873	−	0.807496i	$-0.700822\pi$
$521$	−26.9282	−1.17975	−0.589873	+	0.807496i	$0.700822\pi$
$522$	0	0
$523$	− 31.0000i	− 1.35554i	−0.735276	−	0.677768i	$-0.762948\pi$
$523$	− 31.0000i	− 1.35554i	0.735276	−	0.677768i	$-0.237052\pi$
$524$	0	0
$525$	2.46410i	0.107542i
$526$	0	0
$527$	−52.9282	−2.30559
$528$	0	0
$529$	−22.7846	−0.990635
$530$	0	0
$531$	− 7.85641i	− 0.340939i
$532$	0	0
$533$	− 17.0718i	− 0.739462i
$534$	0	0
$535$	−0.143594	−0.00620809
$536$	0	0
$537$	1.85641	0.0801099
$538$	0	0
$539$	− 3.71281i	− 0.159922i
$540$	0	0
$541$	28.7846i	1.23755i	0.785569	+	0.618774i	$0.212370\pi$
$541$	28.7846i	1.23755i	−0.785569	+	0.618774i	$0.787630\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	−26.7846	−1.14733
$546$	0	0
$547$	− 33.8564i	− 1.44760i	−0.690012	−	0.723798i	$-0.742395\pi$
$547$	− 33.8564i	− 1.44760i	0.690012	−	0.723798i	$-0.257605\pi$
$548$	0	0
$549$	− 9.85641i	− 0.420661i
$550$	0	0
$551$	4.46410	0.190177
$552$	0	0
$553$	4.92820	0.209569
$554$	0	0
$555$	− 4.00000i	− 0.169791i
$556$	0	0
$557$	22.0000i	0.932170i	0.884740	+	0.466085i	$0.154336\pi$
$557$	22.0000i	0.932170i	−0.884740	+	0.466085i	$0.845664\pi$
$558$	0	0
$559$	−22.0000	−0.930501
$560$	0	0
$561$	−23.7128	−1.00116
$562$	0	0
$563$	− 23.7128i	− 0.999376i	−0.866205	−	0.499688i	$-0.833448\pi$
$563$	− 23.7128i	− 0.999376i	0.866205	−	0.499688i	$-0.166552\pi$
$564$	0	0
$565$	1.85641i	0.0780996i
$566$	0	0
$567$	2.46410	0.103483
$568$	0	0
$569$	32.7846	1.37440	0.687201	−	0.726467i	$-0.258839\pi$
$569$	32.7846	1.37440	0.687201	+	0.726467i	$0.258839\pi$
$570$	0	0
$571$	− 12.7846i	− 0.535019i	−0.963555	−	0.267510i	$-0.913799\pi$
$571$	− 12.7846i	− 0.535019i	0.963555	−	0.267510i	$-0.0862007\pi$
$572$	0	0
$573$	− 10.4641i	− 0.437144i
$574$	0	0
$575$	−0.464102	−0.0193544
$576$	0	0
$577$	−29.7846	−1.23995	−0.619975	−	0.784622i	$-0.712857\pi$
$577$	−29.7846	−1.23995	−0.619975	+	0.784622i	$0.712857\pi$
$578$	0	0
$579$	− 17.8564i	− 0.742087i
$580$	0	0
$581$	− 26.9282i	− 1.11717i
$582$	0	0
$583$	−22.1436	−0.917094
$584$	0	0
$585$	9.85641	0.407512
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	− 8.92820i	− 0.367880i
$590$	0	0
$591$	−6.92820	−0.284988
$592$	0	0
$593$	43.8564	1.80097	0.900483	−	0.434890i	$-0.143213\pi$
$593$	43.8564	1.80097	0.900483	+	0.434890i	$0.143213\pi$
$594$	0	0
$595$	− 29.2154i	− 1.19771i
$596$	0	0
$597$	− 20.4641i	− 0.837540i
$598$	0	0
$599$	−17.8564	−0.729593	−0.364796	−	0.931087i	$-0.618861\pi$
$599$	−17.8564	−0.729593	−0.364796	+	0.931087i	$0.618861\pi$
$600$	0	0
$601$	−0.143594	−0.00585730	−0.00292865	−	0.999996i	$-0.500932\pi$
$601$	−0.143594	−0.00585730	−0.00292865	+	0.999996i	$0.500932\pi$
$602$	0	0
$603$	15.8564i	0.645723i
$604$	0	0
$605$	10.0000i	0.406558i
$606$	0	0
$607$	−12.1436	−0.492893	−0.246447	−	0.969156i	$-0.579263\pi$
$607$	−12.1436	−0.492893	−0.246447	+	0.969156i	$0.579263\pi$
$608$	0	0
$609$	11.0000	0.445742
$610$	0	0
$611$	17.0718i	0.690651i
$612$	0	0
$613$	− 34.6410i	− 1.39914i	−0.714565	−	0.699569i	$-0.753375\pi$
$613$	− 34.6410i	− 1.39914i	0.714565	−	0.699569i	$-0.246625\pi$
$614$	0	0
$615$	−13.8564	−0.558744
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	− 0.143594i	− 0.00577151i	−0.999996	−	0.00288576i	$-0.999081\pi$
$619$	− 0.143594i	− 0.00577151i	0.999996	−	0.00288576i	$-0.000918566\pi$
$620$	0	0
$621$	− 2.32051i	− 0.0931188i
$622$	0	0
$623$	−31.8564	−1.27630
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	− 4.00000i	− 0.159745i
$628$	0	0
$629$	− 11.8564i	− 0.472746i
$630$	0	0
$631$	−30.6410	−1.21980	−0.609900	−	0.792479i	$-0.708790\pi$
$631$	−30.6410	−1.21980	−0.609900	+	0.792479i	$0.708790\pi$
$632$	0	0
$633$	−2.85641	−0.113532
$634$	0	0
$635$	25.8564i	1.02608i
$636$	0	0
$637$	− 2.28719i	− 0.0906217i
$638$	0	0
$639$	28.0000	1.10766
$640$	0	0
$641$	−32.7846	−1.29491	−0.647457	−	0.762102i	$-0.724168\pi$
$641$	−32.7846	−1.29491	−0.647457	+	0.762102i	$0.724168\pi$
$642$	0	0
$643$	− 19.0718i	− 0.752118i	−0.926596	−	0.376059i	$-0.877279\pi$
$643$	− 19.0718i	− 0.752118i	0.926596	−	0.376059i	$-0.122721\pi$
$644$	0	0
$645$	17.8564i	0.703095i
$646$	0	0
$647$	−20.4641	−0.804527	−0.402263	−	0.915524i	$-0.631776\pi$
$647$	−20.4641	−0.804527	−0.402263	+	0.915524i	$0.631776\pi$
$648$	0	0
$649$	15.7128	0.616782
$650$	0	0
$651$	− 22.0000i	− 0.862248i
$652$	0	0
$653$	− 34.7846i	− 1.36123i	−0.732643	−	0.680613i	$-0.761713\pi$
$653$	− 34.7846i	− 1.36123i	0.732643	−	0.680613i	$-0.238287\pi$
$654$	0	0
$655$	41.8564	1.63547
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	24.8564i	0.968268i	0.874994	+	0.484134i	$0.160865\pi$
$659$	24.8564i	0.968268i	−0.874994	+	0.484134i	$0.839135\pi$
$660$	0	0
$661$	23.5359i	0.915440i	0.889096	+	0.457720i	$0.151334\pi$
$661$	23.5359i	0.915440i	−0.889096	+	0.457720i	$0.848666\pi$
$662$	0	0
$663$	−14.6077	−0.567316
$664$	0	0
$665$	4.92820	0.191108
$666$	0	0
$667$	2.07180i	0.0802203i
$668$	0	0
$669$	− 13.0718i	− 0.505385i
$670$	0	0
$671$	19.7128	0.761005
$672$	0	0
$673$	22.9282	0.883817	0.441909	−	0.897060i	$-0.354302\pi$
$673$	22.9282	0.883817	0.441909	+	0.897060i	$0.354302\pi$
$674$	0	0
$675$	5.00000i	0.192450i
$676$	0	0
$677$	13.3923i	0.514708i	0.966317	+	0.257354i	$0.0828507\pi$
$677$	13.3923i	0.514708i	−0.966317	+	0.257354i	$0.917149\pi$
$678$	0	0
$679$	−2.28719	−0.0877742
$680$	0	0
$681$	−11.9282	−0.457090
$682$	0	0
$683$	9.85641i	0.377145i	0.982059	+	0.188572i	$0.0603860\pi$
$683$	9.85641i	0.377145i	−0.982059	+	0.188572i	$0.939614\pi$
$684$	0	0
$685$	− 37.7128i	− 1.44093i
$686$	0	0
$687$	16.9282	0.645851
$688$	0	0
$689$	−13.6410	−0.519681
$690$	0	0
$691$	19.8564i	0.755373i	0.925933	+	0.377687i	$0.123280\pi$
$691$	19.8564i	0.755373i	−0.925933	+	0.377687i	$0.876720\pi$
$692$	0	0
$693$	19.7128i	0.748828i
$694$	0	0
$695$	−23.7128	−0.899478
$696$	0	0
$697$	−41.0718	−1.55571
$698$	0	0
$699$	− 11.8564i	− 0.448450i
$700$	0	0
$701$	47.7128i	1.80209i	0.433728	+	0.901044i	$0.357198\pi$
$701$	47.7128i	1.80209i	−0.433728	+	0.901044i	$0.642802\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	13.8564	0.521862
$706$	0	0
$707$	44.0000i	1.65479i
$708$	0	0
$709$	− 30.7846i	− 1.15614i	−0.815987	−	0.578070i	$-0.803806\pi$
$709$	− 30.7846i	− 1.15614i	0.815987	−	0.578070i	$-0.196194\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	4.14359	0.155179
$714$	0	0
$715$	19.7128i	0.737217i
$716$	0	0
$717$	18.3205i	0.684192i
$718$	0	0
$719$	7.39230	0.275686	0.137843	−	0.990454i	$-0.455983\pi$
$719$	7.39230	0.275686	0.137843	+	0.990454i	$0.455983\pi$
$720$	0	0
$721$	26.9282	1.00286
$722$	0	0
$723$	16.7846i	0.624226i
$724$	0	0
$725$	− 4.46410i	− 0.165793i
$726$	0	0
$727$	−34.3205	−1.27288	−0.636439	−	0.771327i	$-0.719593\pi$
$727$	−34.3205	−1.27288	−0.636439	+	0.771327i	$0.719593\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	52.9282i	1.95762i
$732$	0	0
$733$	− 16.1436i	− 0.596277i	−0.954523	−	0.298139i	$-0.903634\pi$
$733$	− 16.1436i	− 0.596277i	0.954523	−	0.298139i	$-0.0963658\pi$
$734$	0	0
$735$	−1.85641	−0.0684746
$736$	0	0
$737$	−31.7128	−1.16816
$738$	0	0
$739$	17.8564i	0.656859i	0.944529	+	0.328429i	$0.106519\pi$
$739$	17.8564i	0.656859i	−0.944529	+	0.328429i	$0.893481\pi$
$740$	0	0
$741$	− 2.46410i	− 0.0905210i
$742$	0	0
$743$	−3.07180	−0.112693	−0.0563466	−	0.998411i	$-0.517945\pi$
$743$	−3.07180	−0.112693	−0.0563466	+	0.998411i	$0.517945\pi$
$744$	0	0
$745$	−5.85641	−0.214562
$746$	0	0
$747$	− 21.8564i	− 0.799684i
$748$	0	0
$749$	− 0.176915i	− 0.00646432i
$750$	0	0
$751$	−46.9282	−1.71243	−0.856217	−	0.516616i	$-0.827192\pi$
$751$	−46.9282	−1.71243	−0.856217	+	0.516616i	$0.827192\pi$
$752$	0	0
$753$	−0.928203	−0.0338256
$754$	0	0
$755$	− 32.0000i	− 1.16460i
$756$	0	0
$757$	42.7846i	1.55503i	0.628862	+	0.777517i	$0.283521\pi$
$757$	42.7846i	1.55503i	−0.628862	+	0.777517i	$0.716479\pi$
$758$	0	0
$759$	1.85641	0.0673833
$760$	0	0
$761$	−6.71281	−0.243339	−0.121670	−	0.992571i	$-0.538825\pi$
$761$	−6.71281	−0.243339	−0.121670	+	0.992571i	$0.538825\pi$
$762$	0	0
$763$	− 33.0000i	− 1.19468i
$764$	0	0
$765$	− 23.7128i	− 0.857339i
$766$	0	0
$767$	9.67949	0.349506
$768$	0	0
$769$	−47.0000	−1.69486	−0.847432	−	0.530904i	$-0.821852\pi$
$769$	−47.0000	−1.69486	−0.847432	+	0.530904i	$0.821852\pi$
$770$	0	0
$771$	12.9282i	0.465598i
$772$	0	0
$773$	47.5359i	1.70975i	0.518836	+	0.854874i	$0.326365\pi$
$773$	47.5359i	1.70975i	−0.518836	+	0.854874i	$0.673635\pi$
$774$	0	0
$775$	−8.92820	−0.320711
$776$	0	0
$777$	4.92820	0.176798
$778$	0	0
$779$	− 6.92820i	− 0.248229i
$780$	0	0
$781$	56.0000i	2.00384i
$782$	0	0
$783$	22.3205	0.797670
$784$	0	0
$785$	20.0000	0.713831
$786$	0	0
$787$	21.7846i	0.776537i	0.921546	+	0.388269i	$0.126927\pi$
$787$	21.7846i	0.776537i	−0.921546	+	0.388269i	$0.873073\pi$
$788$	0	0
$789$	− 10.9282i	− 0.389054i
$790$	0	0
$791$	−2.28719	−0.0813230
$792$	0	0
$793$	12.1436	0.431232
$794$	0	0
$795$	11.0718i	0.392676i
$796$	0	0
$797$	− 32.1769i	− 1.13976i	−0.821726	−	0.569882i	$-0.806989\pi$
$797$	− 32.1769i	− 1.13976i	0.821726	−	0.569882i	$-0.193011\pi$
$798$	0	0
$799$	41.0718	1.45302
$800$	0	0
$801$	−25.8564	−0.913591
$802$	0	0
$803$	28.0000i	0.988099i
$804$	0	0
$805$	2.28719i	0.0806128i
$806$	0	0
$807$	19.8564	0.698979
$808$	0	0
$809$	−31.9282	−1.12254	−0.561268	−	0.827634i	$-0.689686\pi$
$809$	−31.9282	−1.12254	−0.561268	+	0.827634i	$0.689686\pi$
$810$	0	0
$811$	− 28.8564i	− 1.01329i	−0.862156	−	0.506643i	$-0.830886\pi$
$811$	− 28.8564i	− 1.01329i	0.862156	−	0.506643i	$-0.169114\pi$
$812$	0	0
$813$	− 25.3923i	− 0.890547i
$814$	0	0
$815$	−7.71281	−0.270168
$816$	0	0
$817$	−8.92820	−0.312358
$818$	0	0
$819$	12.1436i	0.424331i
$820$	0	0
$821$	− 24.6410i	− 0.859977i	−0.902834	−	0.429989i	$-0.858518\pi$
$821$	− 24.6410i	− 0.859977i	0.902834	−	0.429989i	$-0.141482\pi$
$822$	0	0
$823$	−30.4641	−1.06191	−0.530956	−	0.847399i	$-0.678167\pi$
$823$	−30.4641	−1.06191	−0.530956	+	0.847399i	$0.678167\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	− 40.7128i	− 1.41572i	−0.706351	−	0.707862i	$-0.749660\pi$
$827$	− 40.7128i	− 1.41572i	0.706351	−	0.707862i	$-0.250340\pi$
$828$	0	0
$829$	22.3205i	0.775223i	0.921823	+	0.387612i	$0.126700\pi$
$829$	22.3205i	0.775223i	−0.921823	+	0.387612i	$0.873300\pi$
$830$	0	0
$831$	11.0718	0.384076
$832$	0	0
$833$	−5.50258	−0.190653
$834$	0	0
$835$	− 37.8564i	− 1.31007i
$836$	0	0
$837$	− 44.6410i	− 1.54302i
$838$	0	0
$839$	22.0000	0.759524	0.379762	−	0.925084i	$-0.376006\pi$
$839$	22.0000	0.759524	0.379762	+	0.925084i	$0.376006\pi$
$840$	0	0
$841$	9.07180	0.312821
$842$	0	0
$843$	4.00000i	0.137767i
$844$	0	0
$845$	− 13.8564i	− 0.476675i
$846$	0	0
$847$	−12.3205	−0.423338
$848$	0	0
$849$	−5.07180	−0.174064
$850$	0	0
$851$	0.928203i	0.0318184i
$852$	0	0
$853$	26.9282i	0.922004i	0.887399	+	0.461002i	$0.152510\pi$
$853$	26.9282i	0.922004i	−0.887399	+	0.461002i	$0.847490\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−22.1436	−0.756411	−0.378205	−	0.925722i	$-0.623459\pi$
$857$	−22.1436	−0.756411	−0.378205	+	0.925722i	$0.623459\pi$
$858$	0	0
$859$	34.9282i	1.19173i	0.803083	+	0.595867i	$0.203192\pi$
$859$	34.9282i	1.19173i	−0.803083	+	0.595867i	$0.796808\pi$
$860$	0	0
$861$	− 17.0718i	− 0.581805i
$862$	0	0
$863$	−26.7846	−0.911759	−0.455879	−	0.890042i	$-0.650675\pi$
$863$	−26.7846	−0.911759	−0.455879	+	0.890042i	$0.650675\pi$
$864$	0	0
$865$	20.0000	0.680020
$866$	0	0
$867$	18.1436i	0.616189i
$868$	0	0
$869$	8.00000i	0.271381i
$870$	0	0
$871$	−19.5359	−0.661949
$872$	0	0
$873$	−1.85641	−0.0628298
$874$	0	0
$875$	− 29.5692i	− 0.999622i
$876$	0	0
$877$	17.2487i	0.582448i	0.956655	+	0.291224i	$0.0940624\pi$
$877$	17.2487i	0.582448i	−0.956655	+	0.291224i	$0.905938\pi$
$878$	0	0
$879$	26.3205	0.887769
$880$	0	0
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	0	0
$883$	− 5.71281i	− 0.192251i	−0.995369	−	0.0961257i	$-0.969355\pi$
$883$	− 5.71281i	− 0.192251i	0.995369	−	0.0961257i	$-0.0306451\pi$
$884$	0	0
$885$	− 7.85641i	− 0.264090i
$886$	0	0
$887$	−0.928203	−0.0311660	−0.0155830	−	0.999879i	$-0.504960\pi$
$887$	−0.928203	−0.0311660	−0.0155830	+	0.999879i	$0.504960\pi$
$888$	0	0
$889$	−31.8564	−1.06843
$890$	0	0
$891$	4.00000i	0.134005i
$892$	0	0
$893$	6.92820i	0.231843i
$894$	0	0
$895$	−3.71281	−0.124106
$896$	0	0
$897$	1.14359	0.0381835
$898$	0	0
$899$	39.8564i	1.32929i
$900$	0	0
$901$	32.8179i	1.09332i
$902$	0	0
$903$	−22.0000	−0.732114
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	− 40.8564i	− 1.35661i	−0.734778	−	0.678307i	$-0.762714\pi$
$907$	− 40.8564i	− 1.35661i	0.734778	−	0.678307i	$-0.237286\pi$
$908$	0	0
$909$	35.7128i	1.18452i
$910$	0	0
$911$	19.8564	0.657872	0.328936	−	0.944352i	$-0.393310\pi$
$911$	19.8564	0.657872	0.328936	+	0.944352i	$0.393310\pi$
$912$	0	0
$913$	43.7128	1.44668
$914$	0	0
$915$	− 9.85641i	− 0.325843i
$916$	0	0
$917$	51.5692i	1.70297i
$918$	0	0
$919$	15.3923	0.507745	0.253873	−	0.967238i	$-0.418296\pi$
$919$	15.3923	0.507745	0.253873	+	0.967238i	$0.418296\pi$
$920$	0	0
$921$	25.8564	0.851998
$922$	0	0
$923$	34.4974i	1.13550i
$924$	0	0
$925$	− 2.00000i	− 0.0657596i
$926$	0	0
$927$	21.8564	0.717859
$928$	0	0
$929$	−46.7128	−1.53260	−0.766299	−	0.642484i	$-0.777904\pi$
$929$	−46.7128	−1.53260	−0.766299	+	0.642484i	$0.777904\pi$
$930$	0	0
$931$	− 0.928203i	− 0.0304206i
$932$	0	0
$933$	13.3923i	0.438444i
$934$	0	0
$935$	47.4256	1.55098
$936$	0	0
$937$	16.0718	0.525043	0.262521	−	0.964926i	$-0.415446\pi$
$937$	16.0718	0.525043	0.262521	+	0.964926i	$0.415446\pi$
$938$	0	0
$939$	− 10.8564i	− 0.354285i
$940$	0	0
$941$	− 35.1051i	− 1.14439i	−0.820116	−	0.572197i	$-0.806091\pi$
$941$	− 35.1051i	− 1.14439i	0.820116	−	0.572197i	$-0.193909\pi$
$942$	0	0
$943$	3.21539	0.104708
$944$	0	0
$945$	24.6410	0.801572
$946$	0	0
$947$	3.07180i	0.0998200i	0.998754	+	0.0499100i	$0.0158934\pi$
$947$	3.07180i	0.0998200i	−0.998754	+	0.0499100i	$0.984107\pi$
$948$	0	0
$949$	17.2487i	0.559917i
$950$	0	0
$951$	7.53590	0.244368
$952$	0	0
$953$	48.7846	1.58029	0.790144	−	0.612921i	$-0.210006\pi$
$953$	48.7846	1.58029	0.790144	+	0.612921i	$0.210006\pi$
$954$	0	0
$955$	20.9282i	0.677221i
$956$	0	0
$957$	17.8564i	0.577216i
$958$	0	0
$959$	46.4641	1.50040
$960$	0	0
$961$	48.7128	1.57138
$962$	0	0
$963$	− 0.143594i	− 0.00462724i
$964$	0	0
$965$	35.7128i	1.14964i
$966$	0	0
$967$	−16.7846	−0.539757	−0.269878	−	0.962894i	$-0.586983\pi$
$967$	−16.7846	−0.539757	−0.269878	+	0.962894i	$0.586983\pi$
$968$	0	0
$969$	−5.92820	−0.190441
$970$	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
$973$	− 29.2154i	− 0.936602i
$974$	0	0
$975$	−2.46410	−0.0789144
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	− 51.7128i	− 1.65275i
$980$	0	0
$981$	− 26.7846i	− 0.855167i
$982$	0	0
$983$	−48.6410	−1.55141	−0.775704	−	0.631097i	$-0.782605\pi$
$983$	−48.6410	−1.55141	−0.775704	+	0.631097i	$0.782605\pi$
$984$	0	0
$985$	13.8564	0.441502
$986$	0	0
$987$	17.0718i	0.543401i
$988$	0	0
$989$	− 4.14359i	− 0.131759i
$990$	0	0
$991$	39.5692	1.25696	0.628479	−	0.777827i	$-0.283678\pi$
$991$	39.5692	1.25696	0.628479	+	0.777827i	$0.283678\pi$
$992$	0	0
$993$	31.7846	1.00865
$994$	0	0
$995$	40.9282i	1.29751i
$996$	0	0
$997$	1.21539i	0.0384918i	0.999815	+	0.0192459i	$0.00612654\pi$
$997$	1.21539i	0.0384918i	−0.999815	+	0.0192459i	$0.993873\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.c.1217.4	yes	4
4.3	odd	2		2432.2.c.d.1217.1	yes	4
8.3	odd	2		2432.2.c.d.1217.3	yes	4
8.5	even	2	inner	2432.2.c.c.1217.2	✓	4
16.3	odd	4		4864.2.a.v.1.2		2
16.5	even	4		4864.2.a.x.1.1		2
16.11	odd	4		4864.2.a.u.1.2		2
16.13	even	4		4864.2.a.s.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.c.1217.2	✓	4	8.5	even	2	inner
2432.2.c.c.1217.4	yes	4	1.1	even	1	trivial
2432.2.c.d.1217.1	yes	4	4.3	odd	2
2432.2.c.d.1217.3	yes	4	8.3	odd	2
4864.2.a.s.1.1		2	16.13	even	4
4864.2.a.u.1.2		2	16.11	odd	4
4864.2.a.v.1.2		2	16.3	odd	4
4864.2.a.x.1.1		2	16.5	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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -2.00000i q^{5} +2.46410 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -2.00000i q^{5} +2.46410 q^{7} +2.00000 q^{9} +4.00000i q^{11} +2.46410i q^{13} +2.00000 q^{15} +5.92820 q^{17} +1.00000i q^{19} +2.46410i q^{21} -0.464102 q^{23} +1.00000 q^{25} +5.00000i q^{27} -4.46410i q^{29} -8.92820 q^{31} -4.00000 q^{33} -4.92820i q^{35} -2.00000i q^{37} -2.46410 q^{39} -6.92820 q^{41} +8.92820i q^{43} -4.00000i q^{45} +6.92820 q^{47} -0.928203 q^{49} +5.92820i q^{51} +5.53590i q^{53} +8.00000 q^{55} -1.00000 q^{57} -3.92820i q^{59} -4.92820i q^{61} +4.92820 q^{63} +4.92820 q^{65} +7.92820i q^{67} -0.464102i q^{69} +14.0000 q^{71} +7.00000 q^{73} +1.00000i q^{75} +9.85641i q^{77} +2.00000 q^{79} +1.00000 q^{81} -10.9282i q^{83} -11.8564i q^{85} +4.46410 q^{87} -12.9282 q^{89} +6.07180i q^{91} -8.92820i q^{93} +2.00000 q^{95} -0.928203 q^{97} +8.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 + 8 * q^9	\( 4 q - 4 q^{7} + 8 q^{9} + 8 q^{15} - 4 q^{17} + 12 q^{23} + 4 q^{25} - 8 q^{31} - 16 q^{33} + 4 q^{39} + 24 q^{49} + 32 q^{55} - 4 q^{57} - 8 q^{63} - 8 q^{65} + 56 q^{71} + 28 q^{73} + 8 q^{79} + 4 q^{81} + 4 q^{87} - 24 q^{89} + 8 q^{95} + 24 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 8 * q^9 + 8 * q^15 - 4 * q^17 + 12 * q^23 + 4 * q^25 - 8 * q^31 - 16 * q^33 + 4 * q^39 + 24 * q^49 + 32 * q^55 - 4 * q^57 - 8 * q^63 - 8 * q^65 + 56 * q^71 + 28 * q^73 + 8 * q^79 + 4 * q^81 + 4 * q^87 - 24 * q^89 + 8 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

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