LMFDB - Embedded newform 672.2.c.b.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(337,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, 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("672.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.386672896.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{6} - 2x^{5} + 2x^{4} - 4x^{3} - 4x^{2} + 16 $ x^8 - x^6 - 2x^5 + 2x^4 - 4x^3 - 4x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-0.835949 - 1.14070i$ of defining polynomial
Character	$\chi$	$=$	672.337
Dual form			672.2.c.b.337.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +0.467138i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +0.467138i q^{5} -1.00000 q^{7} -1.00000 q^{9} +4.87666i q^{11} +4.56279i q^{13} +0.467138 q^{15} +6.09565 q^{17} +1.34379i q^{19} +1.00000i q^{21} +4.09565 q^{23} +4.78178 q^{25} +1.00000i q^{27} -7.78178i q^{29} -4.40952 q^{31} +4.87666 q^{33} -0.467138i q^{35} +4.40952i q^{37} +4.56279 q^{39} -6.09565 q^{41} +4.15327i q^{43} -0.467138i q^{45} +6.68759 q^{47} +1.00000 q^{49} -6.09565i q^{51} +1.34379i q^{53} -2.27807 q^{55} +1.34379 q^{57} +4.00000i q^{59} +5.49706i q^{61} +1.00000 q^{63} -2.13145 q^{65} +5.90658i q^{67} -4.09565i q^{69} +4.72339 q^{71} -12.0599 q^{73} -4.78178i q^{75} -4.87666i q^{77} +16.1913 q^{79} +1.00000 q^{81} -13.7533i q^{83} +2.84751i q^{85} -7.78178 q^{87} +7.96420 q^{89} -4.56279i q^{91} +4.40952i q^{93} -0.627737 q^{95} -12.8789 q^{97} -4.87666i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^7 - 8 * q^9	$ 8 q - 8 q^{7} - 8 q^{9} + 4 q^{15} + 4 q^{17} - 12 q^{23} - 24 q^{25} - 8 q^{31} + 12 q^{33} - 8 q^{39} - 4 q^{41} + 8 q^{49} + 8 q^{55} - 16 q^{57} + 8 q^{63} - 16 q^{65} + 28 q^{71} - 8 q^{73} + 40 q^{79} + 8 q^{81} + 20 q^{89} - 40 q^{95} + 40 q^{97}+O(q^{100}) $ 8 * q - 8 * q^7 - 8 * q^9 + 4 * q^15 + 4 * q^17 - 12 * q^23 - 24 * q^25 - 8 * q^31 + 12 * q^33 - 8 * q^39 - 4 * q^41 + 8 * q^49 + 8 * q^55 - 16 * q^57 + 8 * q^63 - 16 * q^65 + 28 * q^71 - 8 * q^73 + 40 * q^79 + 8 * q^81 + 20 * q^89 - 40 * q^95 + 40 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	0.467138i	0.208910i	0.994530	+	0.104455i	$0.0333099\pi$
$5$	0.467138i	0.208910i	−0.994530	+	0.104455i	$0.966690\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.87666i	1.47037i	0.677868	+	0.735184i	$0.262904\pi$
$11$	4.87666i	1.47037i	−0.677868	+	0.735184i	$0.737096\pi$
$12$	0	0
$13$	4.56279i	1.26549i	0.774360	+	0.632745i	$0.218072\pi$
$13$	4.56279i	1.26549i	−0.774360	+	0.632745i	$0.781928\pi$
$14$	0	0
$15$	0.467138	0.120614
$16$	0	0
$17$	6.09565	1.47841	0.739206	−	0.673479i	$-0.235201\pi$
$17$	6.09565	1.47841	0.739206	+	0.673479i	$0.235201\pi$
$18$	0	0
$19$	1.34379i	0.308288i	0.988048	+	0.154144i	$0.0492619\pi$
$19$	1.34379i	0.308288i	−0.988048	+	0.154144i	$0.950738\pi$
$20$	0	0
$21$	1.00000i	0.218218i
$22$	0	0
$23$	4.09565	0.854002	0.427001	−	0.904251i	$-0.359570\pi$
$23$	4.09565	0.854002	0.427001	+	0.904251i	$0.359570\pi$
$24$	0	0
$25$	4.78178	0.956357
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 7.78178i	− 1.44504i	−0.691350	−	0.722520i	$-0.742984\pi$
$29$	− 7.78178i	− 1.44504i	0.691350	−	0.722520i	$-0.257016\pi$
$30$	0	0
$31$	−4.40952	−0.791973	−0.395987	−	0.918256i	$-0.629597\pi$
$31$	−4.40952	−0.791973	−0.395987	+	0.918256i	$0.629597\pi$
$32$	0	0
$33$	4.87666	0.848917
$34$	0	0
$35$	− 0.467138i	− 0.0789607i
$36$	0	0
$37$	4.40952i	0.724921i	0.931999	+	0.362460i	$0.118063\pi$
$37$	4.40952i	0.724921i	−0.931999	+	0.362460i	$0.881937\pi$
$38$	0	0
$39$	4.56279	0.730631
$40$	0	0
$41$	−6.09565	−0.951981	−0.475990	−	0.879450i	$-0.657910\pi$
$41$	−6.09565	−0.951981	−0.475990	+	0.879450i	$0.657910\pi$
$42$	0	0
$43$	4.15327i	0.633368i	0.948531	+	0.316684i	$0.102569\pi$
$43$	4.15327i	0.633368i	−0.948531	+	0.316684i	$0.897431\pi$
$44$	0	0
$45$	− 0.467138i	− 0.0696368i
$46$	0	0
$47$	6.68759	0.975485	0.487743	−	0.872988i	$-0.337820\pi$
$47$	6.68759	0.975485	0.487743	+	0.872988i	$0.337820\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 6.09565i	− 0.853562i
$52$	0	0
$53$	1.34379i	0.184584i	0.995732	+	0.0922922i	$0.0294194\pi$
$53$	1.34379i	0.184584i	−0.995732	+	0.0922922i	$0.970581\pi$
$54$	0	0
$55$	−2.27807	−0.307175
$56$	0	0
$57$	1.34379	0.177990
$58$	0	0
$59$	4.00000i	0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$	4.00000i	0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$	0	0
$61$	5.49706i	0.703827i	0.936033	+	0.351913i	$0.114469\pi$
$61$	5.49706i	0.703827i	−0.936033	+	0.351913i	$0.885531\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−2.13145	−0.264374
$66$	0	0
$67$	5.90658i	0.721604i	0.932642	+	0.360802i	$0.117497\pi$
$67$	5.90658i	0.721604i	−0.932642	+	0.360802i	$0.882503\pi$
$68$	0	0
$69$	− 4.09565i	− 0.493058i
$70$	0	0
$71$	4.72339	0.560563	0.280281	−	0.959918i	$-0.409572\pi$
$71$	4.72339	0.560563	0.280281	+	0.959918i	$0.409572\pi$
$72$	0	0
$73$	−12.0599	−1.41150	−0.705749	−	0.708462i	$-0.749390\pi$
$73$	−12.0599	−1.41150	−0.705749	+	0.708462i	$0.749390\pi$
$74$	0	0
$75$	− 4.78178i	− 0.552153i
$76$	0	0
$77$	− 4.87666i	− 0.555747i
$78$	0	0
$79$	16.1913	1.82166	0.910832	−	0.412778i	$-0.135441\pi$
$79$	16.1913	1.82166	0.910832	+	0.412778i	$0.135441\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 13.7533i	− 1.50962i	−0.655942	−	0.754811i	$-0.727728\pi$
$83$	− 13.7533i	− 1.50962i	0.655942	−	0.754811i	$-0.272272\pi$
$84$	0	0
$85$	2.84751i	0.308856i
$86$	0	0
$87$	−7.78178	−0.834295
$88$	0	0
$89$	7.96420	0.844204	0.422102	−	0.906548i	$-0.361292\pi$
$89$	7.96420	0.844204	0.422102	+	0.906548i	$0.361292\pi$
$90$	0	0
$91$	− 4.56279i	− 0.478310i
$92$	0	0
$93$	4.40952i	0.457246i
$94$	0	0
$95$	−0.627737	−0.0644044
$96$	0	0
$97$	−12.8789	−1.30765	−0.653827	−	0.756644i	$-0.726837\pi$
$97$	−12.8789	−1.30765	−0.653827	+	0.756644i	$0.726837\pi$
$98$	0	0
$99$	− 4.87666i	− 0.490122i
$100$	0	0
$101$	− 2.22045i	− 0.220943i	−0.993879	−	0.110472i	$-0.964764\pi$
$101$	− 2.22045i	− 0.220943i	0.993879	−	0.110472i	$-0.0352361\pi$
$102$	0	0
$103$	−11.0971	−1.09343	−0.546715	−	0.837319i	$-0.684122\pi$
$103$	−11.0971	−1.09343	−0.546715	+	0.837319i	$0.684122\pi$
$104$	0	0
$105$	−0.467138	−0.0455880
$106$	0	0
$107$	3.12334i	0.301945i	0.988538	+	0.150972i	$0.0482405\pi$
$107$	3.12334i	0.301945i	−0.988538	+	0.150972i	$0.951760\pi$
$108$	0	0
$109$	10.6876i	1.02369i	0.859079	+	0.511843i	$0.171037\pi$
$109$	10.6876i	1.02369i	−0.859079	+	0.511843i	$0.828963\pi$
$110$	0	0
$111$	4.40952	0.418533
$112$	0	0
$113$	−12.0599	−1.13450	−0.567248	−	0.823547i	$-0.691992\pi$
$113$	−12.0599	−1.13450	−0.567248	+	0.823547i	$0.691992\pi$
$114$	0	0
$115$	1.91323i	0.178410i
$116$	0	0
$117$	− 4.56279i	− 0.421830i
$118$	0	0
$119$	−6.09565	−0.558787
$120$	0	0
$121$	−12.7818	−1.16198
$122$	0	0
$123$	6.09565i	0.549626i
$124$	0	0
$125$	4.56944i	0.408703i
$126$	0	0
$127$	−18.8789	−1.67523	−0.837615	−	0.546261i	$-0.816051\pi$
$127$	−18.8789	−1.67523	−0.837615	+	0.546261i	$0.816051\pi$
$128$	0	0
$129$	4.15327	0.365675
$130$	0	0
$131$	− 4.93428i	− 0.431110i	−0.976492	−	0.215555i	$-0.930844\pi$
$131$	− 4.93428i	− 0.431110i	0.976492	−	0.215555i	$-0.0691560\pi$
$132$	0	0
$133$	− 1.34379i	− 0.116522i
$134$	0	0
$135$	−0.467138	−0.0402048
$136$	0	0
$137$	19.0103	1.62416	0.812082	−	0.583544i	$-0.198334\pi$
$137$	19.0103	1.62416	0.812082	+	0.583544i	$0.198334\pi$
$138$	0	0
$139$	10.4380i	0.885339i	0.896685	+	0.442669i	$0.145968\pi$
$139$	10.4380i	0.885339i	−0.896685	+	0.442669i	$0.854032\pi$
$140$	0	0
$141$	− 6.68759i	− 0.563197i
$142$	0	0
$143$	−22.2512	−1.86073
$144$	0	0
$145$	3.63516	0.301884
$146$	0	0
$147$	− 1.00000i	− 0.0824786i
$148$	0	0
$149$	− 6.65621i	− 0.545298i	−0.962114	−	0.272649i	$-0.912100\pi$
$149$	− 6.65621i	− 0.545298i	0.962114	−	0.272649i	$-0.0878997\pi$
$150$	0	0
$151$	2.68759	0.218713	0.109356	−	0.994003i	$-0.465121\pi$
$151$	2.68759	0.218713	0.109356	+	0.994003i	$0.465121\pi$
$152$	0	0
$153$	−6.09565	−0.492804
$154$	0	0
$155$	− 2.05985i	− 0.165451i
$156$	0	0
$157$	− 23.1351i	− 1.84639i	−0.384338	−	0.923193i	$-0.625570\pi$
$157$	− 23.1351i	− 1.84639i	0.384338	−	0.923193i	$-0.374430\pi$
$158$	0	0
$159$	1.34379	0.106570
$160$	0	0
$161$	−4.09565	−0.322783
$162$	0	0
$163$	− 12.0380i	− 0.942891i	−0.881895	−	0.471446i	$-0.843732\pi$
$163$	− 12.0380i	− 0.942891i	0.881895	−	0.471446i	$-0.156268\pi$
$164$	0	0
$165$	2.27807i	0.177347i
$166$	0	0
$167$	−9.01034	−0.697241	−0.348621	−	0.937264i	$-0.613350\pi$
$167$	−9.01034	−0.697241	−0.348621	+	0.937264i	$0.613350\pi$
$168$	0	0
$169$	−7.81904	−0.601465
$170$	0	0
$171$	− 1.34379i	− 0.102763i
$172$	0	0
$173$	− 9.59271i	− 0.729321i	−0.931141	−	0.364660i	$-0.881185\pi$
$173$	− 9.59271i	− 0.729321i	0.931141	−	0.364660i	$-0.118815\pi$
$174$	0	0
$175$	−4.78178	−0.361469
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	− 7.69278i	− 0.574985i	−0.957783	−	0.287493i	$-0.907178\pi$
$179$	− 7.69278i	− 0.574985i	0.957783	−	0.287493i	$-0.0928217\pi$
$180$	0	0
$181$	− 15.2504i	− 1.13355i	−0.823872	−	0.566776i	$-0.808191\pi$
$181$	− 15.2504i	− 1.13355i	0.823872	−	0.566776i	$-0.191809\pi$
$182$	0	0
$183$	5.49706	0.406355
$184$	0	0
$185$	−2.05985	−0.151443
$186$	0	0
$187$	29.7264i	2.17381i
$188$	0	0
$189$	− 1.00000i	− 0.0727393i
$190$	0	0
$191$	4.72339	0.341772	0.170886	−	0.985291i	$-0.445337\pi$
$191$	4.72339	0.341772	0.170886	+	0.985291i	$0.445337\pi$
$192$	0	0
$193$	22.3379	1.60792	0.803959	−	0.594684i	$-0.202723\pi$
$193$	22.3379	1.60792	0.803959	+	0.594684i	$0.202723\pi$
$194$	0	0
$195$	2.13145i	0.152636i
$196$	0	0
$197$	− 1.53510i	− 0.109371i	−0.998504	−	0.0546855i	$-0.982584\pi$
$197$	− 1.53510i	− 0.109371i	0.998504	−	0.0546855i	$-0.0174156\pi$
$198$	0	0
$199$	14.6876	1.04118	0.520588	−	0.853808i	$-0.325713\pi$
$199$	14.6876	1.04118	0.520588	+	0.853808i	$0.325713\pi$
$200$	0	0
$201$	5.90658	0.416618
$202$	0	0
$203$	7.78178i	0.546174i
$204$	0	0
$205$	− 2.84751i	− 0.198879i
$206$	0	0
$207$	−4.09565	−0.284667
$208$	0	0
$209$	−6.55322	−0.453296
$210$	0	0
$211$	− 22.8409i	− 1.57243i	−0.617953	−	0.786215i	$-0.712038\pi$
$211$	− 22.8409i	− 1.57243i	0.617953	−	0.786215i	$-0.287962\pi$
$212$	0	0
$213$	− 4.72339i	− 0.323641i
$214$	0	0
$215$	−1.94015	−0.132317
$216$	0	0
$217$	4.40952	0.299338
$218$	0	0
$219$	12.0599i	0.814929i
$220$	0	0
$221$	27.8132i	1.87092i
$222$	0	0
$223$	−21.4321	−1.43520	−0.717600	−	0.696455i	$-0.754760\pi$
$223$	−21.4321	−1.43520	−0.717600	+	0.696455i	$0.754760\pi$
$224$	0	0
$225$	−4.78178	−0.318786
$226$	0	0
$227$	− 29.3169i	− 1.94583i	−0.231164	−	0.972915i	$-0.574253\pi$
$227$	− 29.3169i	− 1.94583i	0.231164	−	0.972915i	$-0.425747\pi$
$228$	0	0
$229$	22.5074i	1.48733i	0.668552	+	0.743666i	$0.266914\pi$
$229$	22.5074i	1.48733i	−0.668552	+	0.743666i	$0.733086\pi$
$230$	0	0
$231$	−4.87666	−0.320860
$232$	0	0
$233$	−0.687589	−0.0450454	−0.0225227	−	0.999746i	$-0.507170\pi$
$233$	−0.687589	−0.0450454	−0.0225227	+	0.999746i	$0.507170\pi$
$234$	0	0
$235$	3.12402i	0.203789i
$236$	0	0
$237$	− 16.1913i	− 1.05174i
$238$	0	0
$239$	9.59936	0.620931	0.310466	−	0.950585i	$-0.399515\pi$
$239$	9.59936	0.620931	0.310466	+	0.950585i	$0.399515\pi$
$240$	0	0
$241$	0.496287	0.0319687	0.0159843	−	0.999872i	$-0.494912\pi$
$241$	0.496287	0.0319687	0.0159843	+	0.999872i	$0.494912\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0.467138i	0.0298443i
$246$	0	0
$247$	−6.13145	−0.390135
$248$	0	0
$249$	−13.7533	−0.871581
$250$	0	0
$251$	− 22.1359i	− 1.39721i	−0.715509	−	0.698603i	$-0.753805\pi$
$251$	− 22.1359i	− 1.39721i	0.715509	−	0.698603i	$-0.246195\pi$
$252$	0	0
$253$	19.9731i	1.25570i
$254$	0	0
$255$	2.84751	0.178318
$256$	0	0
$257$	2.28695	0.142656	0.0713281	−	0.997453i	$-0.477276\pi$
$257$	2.28695	0.142656	0.0713281	+	0.997453i	$0.477276\pi$
$258$	0	0
$259$	− 4.40952i	− 0.273994i
$260$	0	0
$261$	7.78178i	0.481680i
$262$	0	0
$263$	30.1555	1.85947	0.929734	−	0.368232i	$-0.120037\pi$
$263$	30.1555	1.85947	0.929734	+	0.368232i	$0.120037\pi$
$264$	0	0
$265$	−0.627737	−0.0385616
$266$	0	0
$267$	− 7.96420i	− 0.487401i
$268$	0	0
$269$	9.47748i	0.577852i	0.957351	+	0.288926i	$0.0932982\pi$
$269$	9.47748i	0.577852i	−0.957351	+	0.288926i	$0.906702\pi$
$270$	0	0
$271$	13.2855	0.807036	0.403518	−	0.914972i	$-0.367787\pi$
$271$	13.2855	0.807036	0.403518	+	0.914972i	$0.367787\pi$
$272$	0	0
$273$	−4.56279	−0.276153
$274$	0	0
$275$	23.3191i	1.40620i
$276$	0	0
$277$	− 26.8475i	− 1.61311i	−0.591159	−	0.806555i	$-0.701329\pi$
$277$	− 26.8475i	− 1.61311i	0.591159	−	0.806555i	$-0.298671\pi$
$278$	0	0
$279$	4.40952	0.263991
$280$	0	0
$281$	11.8686	0.708018	0.354009	−	0.935242i	$-0.384818\pi$
$281$	11.8686	0.708018	0.354009	+	0.935242i	$0.384818\pi$
$282$	0	0
$283$	− 2.46937i	− 0.146789i	−0.997303	−	0.0733944i	$-0.976617\pi$
$283$	− 2.46937i	− 0.146789i	0.997303	−	0.0733944i	$-0.0233832\pi$
$284$	0	0
$285$	0.627737i	0.0371839i
$286$	0	0
$287$	6.09565	0.359815
$288$	0	0
$289$	20.1570	1.18570
$290$	0	0
$291$	12.8789i	0.754974i
$292$	0	0
$293$	22.7899i	1.33140i	0.746220	+	0.665700i	$0.231867\pi$
$293$	22.7899i	1.33140i	−0.746220	+	0.665700i	$0.768133\pi$
$294$	0	0
$295$	−1.86855	−0.108791
$296$	0	0
$297$	−4.87666	−0.282972
$298$	0	0
$299$	18.6876i	1.08073i
$300$	0	0
$301$	− 4.15327i	− 0.239390i
$302$	0	0
$303$	−2.22045	−0.127562
$304$	0	0
$305$	−2.56788	−0.147037
$306$	0	0
$307$	− 9.34379i	− 0.533279i	−0.963796	−	0.266639i	$-0.914087\pi$
$307$	− 9.34379i	− 0.533279i	0.963796	−	0.266639i	$-0.0859132\pi$
$308$	0	0
$309$	11.0971i	0.631292i
$310$	0	0
$311$	−6.68759	−0.379218	−0.189609	−	0.981860i	$-0.560722\pi$
$311$	−6.68759	−0.379218	−0.189609	+	0.981860i	$0.560722\pi$
$312$	0	0
$313$	−15.6381	−0.883916	−0.441958	−	0.897036i	$-0.645716\pi$
$313$	−15.6381	−0.883916	−0.441958	+	0.897036i	$0.645716\pi$
$314$	0	0
$315$	0.467138i	0.0263202i
$316$	0	0
$317$	9.34379i	0.524800i	0.964959	+	0.262400i	$0.0845139\pi$
$317$	9.34379i	0.524800i	−0.964959	+	0.262400i	$0.915486\pi$
$318$	0	0
$319$	37.9491	2.12474
$320$	0	0
$321$	3.12334	0.174328
$322$	0	0
$323$	8.19130i	0.455776i
$324$	0	0
$325$	21.8183i	1.21026i
$326$	0	0
$327$	10.6876	0.591025
$328$	0	0
$329$	−6.68759	−0.368699
$330$	0	0
$331$	5.40874i	0.297291i	0.988891	+	0.148646i	$0.0474914\pi$
$331$	5.40874i	0.297291i	−0.988891	+	0.148646i	$0.952509\pi$
$332$	0	0
$333$	− 4.40952i	− 0.241640i
$334$	0	0
$335$	−2.75919	−0.150750
$336$	0	0
$337$	−6.55614	−0.357136	−0.178568	−	0.983928i	$-0.557146\pi$
$337$	−6.55614	−0.357136	−0.178568	+	0.983928i	$0.557146\pi$
$338$	0	0
$339$	12.0599i	0.655001i
$340$	0	0
$341$	− 21.5037i	− 1.16449i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.91323	0.103005
$346$	0	0
$347$	24.9964i	1.34187i	0.741514	+	0.670937i	$0.234108\pi$
$347$	24.9964i	1.34187i	−0.741514	+	0.670937i	$0.765892\pi$
$348$	0	0
$349$	− 27.1921i	− 1.45556i	−0.685811	−	0.727779i	$-0.740553\pi$
$349$	− 27.1921i	− 1.45556i	0.685811	−	0.727779i	$-0.259447\pi$
$350$	0	0
$351$	−4.56279	−0.243544
$352$	0	0
$353$	−30.2870	−1.61201	−0.806006	−	0.591907i	$-0.798375\pi$
$353$	−30.2870	−1.61201	−0.806006	+	0.591907i	$0.798375\pi$
$354$	0	0
$355$	2.20647i	0.117107i
$356$	0	0
$357$	6.09565i	0.322616i
$358$	0	0
$359$	−6.59194	−0.347909	−0.173955	−	0.984754i	$-0.555655\pi$
$359$	−6.59194	−0.347909	−0.173955	+	0.984754i	$0.555655\pi$
$360$	0	0
$361$	17.1942	0.904959
$362$	0	0
$363$	12.7818i	0.670870i
$364$	0	0
$365$	− 5.63361i	− 0.294877i
$366$	0	0
$367$	6.68759	0.349089	0.174545	−	0.984649i	$-0.444155\pi$
$367$	6.68759	0.349089	0.174545	+	0.984649i	$0.444155\pi$
$368$	0	0
$369$	6.09565	0.317327
$370$	0	0
$371$	− 1.34379i	− 0.0697663i
$372$	0	0
$373$	− 17.1256i	− 0.886729i	−0.896341	−	0.443364i	$-0.853785\pi$
$373$	− 17.1256i	− 0.886729i	0.896341	−	0.443364i	$-0.146215\pi$
$374$	0	0
$375$	4.56944	0.235965
$376$	0	0
$377$	35.5066	1.82868
$378$	0	0
$379$	16.7094i	0.858305i	0.903232	+	0.429152i	$0.141188\pi$
$379$	16.7094i	0.858305i	−0.903232	+	0.429152i	$0.858812\pi$
$380$	0	0
$381$	18.8789i	0.967195i
$382$	0	0
$383$	9.94015	0.507918	0.253959	−	0.967215i	$-0.418267\pi$
$383$	9.94015	0.507918	0.253959	+	0.967215i	$0.418267\pi$
$384$	0	0
$385$	2.27807	0.116101
$386$	0	0
$387$	− 4.15327i	− 0.211123i
$388$	0	0
$389$	19.2884i	0.977961i	0.872295	+	0.488981i	$0.162631\pi$
$389$	19.2884i	0.977961i	−0.872295	+	0.488981i	$0.837369\pi$
$390$	0	0
$391$	24.9657	1.26257
$392$	0	0
$393$	−4.93428	−0.248901
$394$	0	0
$395$	7.56357i	0.380564i
$396$	0	0
$397$	23.4417i	1.17650i	0.808678	+	0.588252i	$0.200184\pi$
$397$	23.4417i	1.17650i	−0.808678	+	0.588252i	$0.799816\pi$
$398$	0	0
$399$	−1.34379	−0.0672739
$400$	0	0
$401$	−16.0029	−0.799147	−0.399574	−	0.916701i	$-0.630842\pi$
$401$	−16.0029	−0.799147	−0.399574	+	0.916701i	$0.630842\pi$
$402$	0	0
$403$	− 20.1197i	− 1.00223i
$404$	0	0
$405$	0.467138i	0.0232123i
$406$	0	0
$407$	−21.5037	−1.06590
$408$	0	0
$409$	15.5665	0.769713	0.384856	−	0.922976i	$-0.374251\pi$
$409$	15.5665	0.769713	0.384856	+	0.922976i	$0.374251\pi$
$410$	0	0
$411$	− 19.0103i	− 0.937711i
$412$	0	0
$413$	− 4.00000i	− 0.196827i
$414$	0	0
$415$	6.42469	0.315376
$416$	0	0
$417$	10.4380	0.511150
$418$	0	0
$419$	− 1.31241i	− 0.0641155i	−0.999486	−	0.0320577i	$-0.989794\pi$
$419$	− 1.31241i	− 0.0641155i	0.999486	−	0.0320577i	$-0.0102060\pi$
$420$	0	0
$421$	− 3.66655i	− 0.178697i	−0.996000	−	0.0893483i	$-0.971522\pi$
$421$	− 3.66655i	− 0.178697i	0.996000	−	0.0893483i	$-0.0284784\pi$
$422$	0	0
$423$	−6.68759	−0.325162
$424$	0	0
$425$	29.1481	1.41389
$426$	0	0
$427$	− 5.49706i	− 0.266022i
$428$	0	0
$429$	22.2512i	1.07430i
$430$	0	0
$431$	−20.0957	−0.967973	−0.483987	−	0.875075i	$-0.660812\pi$
$431$	−20.0957	−0.967973	−0.483987	+	0.875075i	$0.660812\pi$
$432$	0	0
$433$	8.68759	0.417499	0.208749	−	0.977969i	$-0.433061\pi$
$433$	8.68759	0.417499	0.208749	+	0.977969i	$0.433061\pi$
$434$	0	0
$435$	− 3.63516i	− 0.174293i
$436$	0	0
$437$	5.50371i	0.263278i
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	4.57012i	0.217133i	0.994089	+	0.108566i	$0.0346260\pi$
$443$	4.57012i	0.217133i	−0.994089	+	0.108566i	$0.965374\pi$
$444$	0	0
$445$	3.72038i	0.176363i
$446$	0	0
$447$	−6.65621	−0.314828
$448$	0	0
$449$	−3.94015	−0.185947	−0.0929735	−	0.995669i	$-0.529637\pi$
$449$	−3.94015	−0.185947	−0.0929735	+	0.995669i	$0.529637\pi$
$450$	0	0
$451$	− 29.7264i	− 1.39976i
$452$	0	0
$453$	− 2.68759i	− 0.126274i
$454$	0	0
$455$	2.13145	0.0999239
$456$	0	0
$457$	40.0207	1.87209	0.936044	−	0.351882i	$-0.114458\pi$
$457$	40.0207	1.87209	0.936044	+	0.351882i	$0.114458\pi$
$458$	0	0
$459$	6.09565i	0.284521i
$460$	0	0
$461$	− 34.6031i	− 1.61162i	−0.592171	−	0.805812i	$-0.701729\pi$
$461$	− 34.6031i	− 1.61162i	0.592171	−	0.805812i	$-0.298271\pi$
$462$	0	0
$463$	−0.191302	−0.00889055	−0.00444528	−	0.999990i	$-0.501415\pi$
$463$	−0.191302	−0.00889055	−0.00444528	+	0.999990i	$0.501415\pi$
$464$	0	0
$465$	−2.05985	−0.0955234
$466$	0	0
$467$	− 4.49784i	− 0.208135i	−0.994570	−	0.104068i	$-0.966814\pi$
$467$	− 4.49784i	− 0.208135i	0.994570	−	0.104068i	$-0.0331858\pi$
$468$	0	0
$469$	− 5.90658i	− 0.272741i
$470$	0	0
$471$	−23.1351	−1.06601
$472$	0	0
$473$	−20.2541	−0.931283
$474$	0	0
$475$	6.42573i	0.294833i
$476$	0	0
$477$	− 1.34379i	− 0.0615281i
$478$	0	0
$479$	22.2512	1.01668	0.508341	−	0.861156i	$-0.330259\pi$
$479$	22.2512	1.01668	0.508341	+	0.861156i	$0.330259\pi$
$480$	0	0
$481$	−20.1197	−0.917380
$482$	0	0
$483$	4.09565i	0.186359i
$484$	0	0
$485$	− 6.01621i	− 0.273182i
$486$	0	0
$487$	1.86855	0.0846721	0.0423360	−	0.999103i	$-0.486520\pi$
$487$	1.86855	0.0846721	0.0423360	+	0.999103i	$0.486520\pi$
$488$	0	0
$489$	−12.0380	−0.544379
$490$	0	0
$491$	17.6374i	0.795965i	0.917393	+	0.397982i	$0.130289\pi$
$491$	17.6374i	0.795965i	−0.917393	+	0.397982i	$0.869711\pi$
$492$	0	0
$493$	− 47.4350i	− 2.13637i
$494$	0	0
$495$	2.27807	0.102392
$496$	0	0
$497$	−4.72339	−0.211873
$498$	0	0
$499$	− 21.5854i	− 0.966295i	−0.875539	−	0.483147i	$-0.839494\pi$
$499$	− 21.5854i	− 0.966295i	0.875539	−	0.483147i	$-0.160506\pi$
$500$	0	0
$501$	9.01034i	0.402552i
$502$	0	0
$503$	−5.24081	−0.233676	−0.116838	−	0.993151i	$-0.537276\pi$
$503$	−5.24081	−0.233676	−0.116838	+	0.993151i	$0.537276\pi$
$504$	0	0
$505$	1.03726	0.0461573
$506$	0	0
$507$	7.81904i	0.347256i
$508$	0	0
$509$	18.3357i	0.812715i	0.913714	+	0.406358i	$0.133201\pi$
$509$	18.3357i	0.812715i	−0.913714	+	0.406358i	$0.866799\pi$
$510$	0	0
$511$	12.0599	0.533496
$512$	0	0
$513$	−1.34379	−0.0593300
$514$	0	0
$515$	− 5.18388i	− 0.228429i
$516$	0	0
$517$	32.6131i	1.43432i
$518$	0	0
$519$	−9.59271	−0.421073
$520$	0	0
$521$	6.78033	0.297051	0.148526	−	0.988909i	$-0.452547\pi$
$521$	6.78033	0.297051	0.148526	+	0.988909i	$0.452547\pi$
$522$	0	0
$523$	− 23.5066i	− 1.02787i	−0.857828	−	0.513937i	$-0.828187\pi$
$523$	− 23.5066i	− 1.02787i	0.857828	−	0.513937i	$-0.171813\pi$
$524$	0	0
$525$	4.78178i	0.208694i
$526$	0	0
$527$	−26.8789	−1.17086
$528$	0	0
$529$	−6.22564	−0.270680
$530$	0	0
$531$	− 4.00000i	− 0.173585i
$532$	0	0
$533$	− 27.8132i	− 1.20472i
$534$	0	0
$535$	−1.45903	−0.0630794
$536$	0	0
$537$	−7.69278	−0.331968
$538$	0	0
$539$	4.87666i	0.210052i
$540$	0	0
$541$	− 5.21526i	− 0.224222i	−0.993696	−	0.112111i	$-0.964239\pi$
$541$	− 5.21526i	− 0.224222i	0.993696	−	0.112111i	$-0.0357611\pi$
$542$	0	0
$543$	−15.2504	−0.654456
$544$	0	0
$545$	−4.99257	−0.213858
$546$	0	0
$547$	27.2951i	1.16705i	0.812094	+	0.583526i	$0.198327\pi$
$547$	27.2951i	1.16705i	−0.812094	+	0.583526i	$0.801673\pi$
$548$	0	0
$549$	− 5.49706i	− 0.234609i
$550$	0	0
$551$	10.4571	0.445488
$552$	0	0
$553$	−16.1913	−0.688524
$554$	0	0
$555$	2.05985i	0.0874359i
$556$	0	0
$557$	4.78766i	0.202859i	0.994843	+	0.101430i	$0.0323417\pi$
$557$	4.78766i	0.202859i	−0.994843	+	0.101430i	$0.967658\pi$
$558$	0	0
$559$	−18.9505	−0.801520
$560$	0	0
$561$	29.7264	1.25505
$562$	0	0
$563$	18.7445i	0.789988i	0.918684	+	0.394994i	$0.129253\pi$
$563$	18.7445i	0.789988i	−0.918684	+	0.394994i	$0.870747\pi$
$564$	0	0
$565$	− 5.63361i	− 0.237008i
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	27.6950	1.16104	0.580518	−	0.814248i	$-0.302850\pi$
$569$	27.6950	1.16104	0.580518	+	0.814248i	$0.302850\pi$
$570$	0	0
$571$	− 17.4132i	− 0.728720i	−0.931258	−	0.364360i	$-0.881288\pi$
$571$	− 17.4132i	− 0.728720i	0.931258	−	0.364360i	$-0.118712\pi$
$572$	0	0
$573$	− 4.72339i	− 0.197322i
$574$	0	0
$575$	19.5845	0.816731
$576$	0	0
$577$	−6.81904	−0.283880	−0.141940	−	0.989875i	$-0.545334\pi$
$577$	−6.81904	−0.283880	−0.141940	+	0.989875i	$0.545334\pi$
$578$	0	0
$579$	− 22.3379i	− 0.928332i
$580$	0	0
$581$	13.7533i	0.570584i
$582$	0	0
$583$	−6.55322	−0.271407
$584$	0	0
$585$	2.13145	0.0881246
$586$	0	0
$587$	− 9.81025i	− 0.404912i	−0.979291	−	0.202456i	$-0.935108\pi$
$587$	− 9.81025i	− 0.404912i	0.979291	−	0.202456i	$-0.0648924\pi$
$588$	0	0
$589$	− 5.92549i	− 0.244155i
$590$	0	0
$591$	−1.53510	−0.0631454
$592$	0	0
$593$	−39.0913	−1.60529	−0.802644	−	0.596458i	$-0.796574\pi$
$593$	−39.0913	−1.60529	−0.802644	+	0.596458i	$0.796574\pi$
$594$	0	0
$595$	− 2.84751i	− 0.116736i
$596$	0	0
$597$	− 14.6876i	− 0.601123i
$598$	0	0
$599$	−22.0270	−0.899998	−0.449999	−	0.893029i	$-0.648576\pi$
$599$	−22.0270	−0.899998	−0.449999	+	0.893029i	$0.648576\pi$
$600$	0	0
$601$	−20.2512	−0.826062	−0.413031	−	0.910717i	$-0.635530\pi$
$601$	−20.2512	−0.826062	−0.413031	+	0.910717i	$0.635530\pi$
$602$	0	0
$603$	− 5.90658i	− 0.240535i
$604$	0	0
$605$	− 5.97085i	− 0.242750i
$606$	0	0
$607$	−44.4454	−1.80398	−0.901991	−	0.431755i	$-0.857895\pi$
$607$	−44.4454	−1.80398	−0.901991	+	0.431755i	$0.857895\pi$
$608$	0	0
$609$	7.78178	0.315334
$610$	0	0
$611$	30.5141i	1.23447i
$612$	0	0
$613$	− 47.6263i	− 1.92361i	−0.273737	−	0.961805i	$-0.588260\pi$
$613$	− 47.6263i	− 1.92361i	0.273737	−	0.961805i	$-0.411740\pi$
$614$	0	0
$615$	−2.84751	−0.114823
$616$	0	0
$617$	−3.01034	−0.121192	−0.0605959	−	0.998162i	$-0.519300\pi$
$617$	−3.01034	−0.121192	−0.0605959	+	0.998162i	$0.519300\pi$
$618$	0	0
$619$	− 18.5141i	− 0.744143i	−0.928204	−	0.372071i	$-0.878648\pi$
$619$	− 18.5141i	− 0.744143i	0.928204	−	0.372071i	$-0.121352\pi$
$620$	0	0
$621$	4.09565i	0.164353i
$622$	0	0
$623$	−7.96420	−0.319079
$624$	0	0
$625$	21.7744	0.870974
$626$	0	0
$627$	6.55322i	0.261711i
$628$	0	0
$629$	26.8789i	1.07173i
$630$	0	0
$631$	−10.2658	−0.408676	−0.204338	−	0.978900i	$-0.565504\pi$
$631$	−10.2658	−0.408676	−0.204338	+	0.978900i	$0.565504\pi$
$632$	0	0
$633$	−22.8409	−0.907843
$634$	0	0
$635$	− 8.81904i	− 0.349973i
$636$	0	0
$637$	4.56279i	0.180784i
$638$	0	0
$639$	−4.72339	−0.186854
$640$	0	0
$641$	−16.9358	−0.668925	−0.334462	−	0.942409i	$-0.608555\pi$
$641$	−16.9358	−0.668925	−0.334462	+	0.942409i	$0.608555\pi$
$642$	0	0
$643$	19.5189i	0.769750i	0.922969	+	0.384875i	$0.125755\pi$
$643$	19.5189i	0.769750i	−0.922969	+	0.384875i	$0.874245\pi$
$644$	0	0
$645$	1.94015i	0.0763933i
$646$	0	0
$647$	23.3723	0.918858	0.459429	−	0.888214i	$-0.348054\pi$
$647$	23.3723	0.918858	0.459429	+	0.888214i	$0.348054\pi$
$648$	0	0
$649$	−19.5066	−0.765702
$650$	0	0
$651$	− 4.40952i	− 0.172823i
$652$	0	0
$653$	1.45903i	0.0570963i	0.999592	+	0.0285481i	$0.00908839\pi$
$653$	1.45903i	0.0570963i	−0.999592	+	0.0285481i	$0.990912\pi$
$654$	0	0
$655$	2.30499	0.0900632
$656$	0	0
$657$	12.0599	0.470500
$658$	0	0
$659$	− 10.8929i	− 0.424326i	−0.977234	−	0.212163i	$-0.931949\pi$
$659$	− 10.8929i	− 0.424326i	0.977234	−	0.212163i	$-0.0680508\pi$
$660$	0	0
$661$	8.69715i	0.338280i	0.985592	+	0.169140i	$0.0540990\pi$
$661$	8.69715i	0.338280i	−0.985592	+	0.169140i	$0.945901\pi$
$662$	0	0
$663$	27.8132	1.08017
$664$	0	0
$665$	0.627737	0.0243426
$666$	0	0
$667$	− 31.8715i	− 1.23407i
$668$	0	0
$669$	21.4321i	0.828613i
$670$	0	0
$671$	−26.8073	−1.03488
$672$	0	0
$673$	18.0447	0.695571	0.347786	−	0.937574i	$-0.386934\pi$
$673$	18.0447	0.695571	0.347786	+	0.937574i	$0.386934\pi$
$674$	0	0
$675$	4.78178i	0.184051i
$676$	0	0
$677$	− 28.7781i	− 1.10603i	−0.833170	−	0.553017i	$-0.813476\pi$
$677$	− 28.7781i	− 1.10603i	0.833170	−	0.553017i	$-0.186524\pi$
$678$	0	0
$679$	12.8789	0.494246
$680$	0	0
$681$	−29.3169	−1.12343
$682$	0	0
$683$	− 34.4431i	− 1.31793i	−0.752174	−	0.658965i	$-0.770995\pi$
$683$	− 34.4431i	− 1.31793i	0.752174	−	0.658965i	$-0.229005\pi$
$684$	0	0
$685$	8.88044i	0.339304i
$686$	0	0
$687$	22.5074	0.858711
$688$	0	0
$689$	−6.13145	−0.233590
$690$	0	0
$691$	41.5651i	1.58121i	0.612325	+	0.790606i	$0.290234\pi$
$691$	41.5651i	1.58121i	−0.612325	+	0.790606i	$0.709766\pi$
$692$	0	0
$693$	4.87666i	0.185249i
$694$	0	0
$695$	−4.87598	−0.184956
$696$	0	0
$697$	−37.1570	−1.40742
$698$	0	0
$699$	0.687589i	0.0260070i
$700$	0	0
$701$	− 46.7804i	− 1.76687i	−0.468553	−	0.883435i	$-0.655225\pi$
$701$	− 46.7804i	− 1.76687i	0.468553	−	0.883435i	$-0.344775\pi$
$702$	0	0
$703$	−5.92549	−0.223484
$704$	0	0
$705$	3.12402	0.117658
$706$	0	0
$707$	2.22045i	0.0835087i
$708$	0	0
$709$	8.43643i	0.316837i	0.987372	+	0.158418i	$0.0506395\pi$
$709$	8.43643i	0.316837i	−0.987372	+	0.158418i	$0.949360\pi$
$710$	0	0
$711$	−16.1913	−0.607221
$712$	0	0
$713$	−18.0599	−0.676347
$714$	0	0
$715$	− 10.3943i	− 0.388727i
$716$	0	0
$717$	− 9.59936i	− 0.358495i
$718$	0	0
$719$	38.4425	1.43366	0.716831	−	0.697247i	$-0.245592\pi$
$719$	38.4425	1.43366	0.716831	+	0.697247i	$0.245592\pi$
$720$	0	0
$721$	11.0971	0.413278
$722$	0	0
$723$	− 0.496287i	− 0.0184571i
$724$	0	0
$725$	− 37.2108i	− 1.38197i
$726$	0	0
$727$	17.5484	0.650834	0.325417	−	0.945571i	$-0.394495\pi$
$727$	17.5484	0.650834	0.325417	+	0.945571i	$0.394495\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	25.3169i	0.936379i
$732$	0	0
$733$	38.8272i	1.43412i	0.697013	+	0.717058i	$0.254512\pi$
$733$	38.8272i	1.43412i	−0.697013	+	0.717058i	$0.745488\pi$
$734$	0	0
$735$	0.467138	0.0172306
$736$	0	0
$737$	−28.8044	−1.06102
$738$	0	0
$739$	− 13.9066i	− 0.511562i	−0.966735	−	0.255781i	$-0.917667\pi$
$739$	− 13.9066i	− 0.511562i	0.966735	−	0.255781i	$-0.0823326\pi$
$740$	0	0
$741$	6.13145i	0.225244i
$742$	0	0
$743$	−40.2300	−1.47590	−0.737948	−	0.674858i	$-0.764205\pi$
$743$	−40.2300	−1.47590	−0.737948	+	0.674858i	$0.764205\pi$
$744$	0	0
$745$	3.10936	0.113918
$746$	0	0
$747$	13.7533i	0.503207i
$748$	0	0
$749$	− 3.12334i	− 0.114124i
$750$	0	0
$751$	−16.9957	−0.620181	−0.310091	−	0.950707i	$-0.600359\pi$
$751$	−16.9957	−0.620181	−0.310091	+	0.950707i	$0.600359\pi$
$752$	0	0
$753$	−22.1359	−0.806678
$754$	0	0
$755$	1.25547i	0.0456914i
$756$	0	0
$757$	9.43212i	0.342816i	0.985200	+	0.171408i	$0.0548316\pi$
$757$	9.43212i	0.342816i	−0.985200	+	0.171408i	$0.945168\pi$
$758$	0	0
$759$	19.9731	0.724977
$760$	0	0
$761$	−9.41098	−0.341148	−0.170574	−	0.985345i	$-0.554562\pi$
$761$	−9.41098	−0.341148	−0.170574	+	0.985345i	$0.554562\pi$
$762$	0	0
$763$	− 10.6876i	− 0.386917i
$764$	0	0
$765$	− 2.84751i	− 0.102952i
$766$	0	0
$767$	−18.2512	−0.659011
$768$	0	0
$769$	27.6950	0.998708	0.499354	−	0.866398i	$-0.333571\pi$
$769$	27.6950	0.998708	0.499354	+	0.866398i	$0.333571\pi$
$770$	0	0
$771$	− 2.28695i	− 0.0823626i
$772$	0	0
$773$	33.8993i	1.21927i	0.792682	+	0.609636i	$0.208684\pi$
$773$	33.8993i	1.21927i	−0.792682	+	0.609636i	$0.791316\pi$
$774$	0	0
$775$	−21.0854	−0.757409
$776$	0	0
$777$	−4.40952	−0.158191
$778$	0	0
$779$	− 8.19130i	− 0.293484i
$780$	0	0
$781$	23.0343i	0.824234i
$782$	0	0
$783$	7.78178	0.278098
$784$	0	0
$785$	10.8073	0.385729
$786$	0	0
$787$	31.2001i	1.11216i	0.831128	+	0.556082i	$0.187696\pi$
$787$	31.2001i	1.11216i	−0.831128	+	0.556082i	$0.812304\pi$
$788$	0	0
$789$	− 30.1555i	− 1.07356i
$790$	0	0
$791$	12.0599	0.428799
$792$	0	0
$793$	−25.0819	−0.890686
$794$	0	0
$795$	0.627737i	0.0222635i
$796$	0	0
$797$	32.2848i	1.14359i	0.820398	+	0.571793i	$0.193752\pi$
$797$	32.2848i	1.14359i	−0.820398	+	0.571793i	$0.806248\pi$
$798$	0	0
$799$	40.7652	1.44217
$800$	0	0
$801$	−7.96420	−0.281401
$802$	0	0
$803$	− 58.8118i	− 2.07542i
$804$	0	0
$805$	− 1.91323i	− 0.0674326i
$806$	0	0
$807$	9.47748	0.333623
$808$	0	0
$809$	8.49629	0.298714	0.149357	−	0.988783i	$-0.452280\pi$
$809$	8.49629	0.298714	0.149357	+	0.988783i	$0.452280\pi$
$810$	0	0
$811$	12.3826i	0.434812i	0.976081	+	0.217406i	$0.0697596\pi$
$811$	12.3826i	0.434812i	−0.976081	+	0.217406i	$0.930240\pi$
$812$	0	0
$813$	− 13.2855i	− 0.465943i
$814$	0	0
$815$	5.62342	0.196980
$816$	0	0
$817$	−5.58114	−0.195259
$818$	0	0
$819$	4.56279i	0.159437i
$820$	0	0
$821$	− 21.8999i	− 0.764313i	−0.924098	−	0.382156i	$-0.875182\pi$
$821$	− 21.8999i	− 0.764313i	0.924098	−	0.382156i	$-0.124818\pi$
$822$	0	0
$823$	1.88321	0.0656446	0.0328223	−	0.999461i	$-0.489550\pi$
$823$	1.88321	0.0656446	0.0328223	+	0.999461i	$0.489550\pi$
$824$	0	0
$825$	23.3191	0.811867
$826$	0	0
$827$	− 32.8051i	− 1.14074i	−0.821387	−	0.570372i	$-0.806799\pi$
$827$	− 32.8051i	− 1.14074i	0.821387	−	0.570372i	$-0.193201\pi$
$828$	0	0
$829$	9.63143i	0.334513i	0.985913	+	0.167257i	$0.0534909\pi$
$829$	9.63143i	0.334513i	−0.985913	+	0.167257i	$0.946509\pi$
$830$	0	0
$831$	−26.8475	−0.931330
$832$	0	0
$833$	6.09565	0.211202
$834$	0	0
$835$	− 4.20907i	− 0.145661i
$836$	0	0
$837$	− 4.40952i	− 0.152415i
$838$	0	0
$839$	48.0481	1.65880	0.829402	−	0.558652i	$-0.188681\pi$
$839$	48.0481	1.65880	0.829402	+	0.558652i	$0.188681\pi$
$840$	0	0
$841$	−31.5561	−1.08814
$842$	0	0
$843$	− 11.8686i	− 0.408775i
$844$	0	0
$845$	− 3.65257i	− 0.125652i
$846$	0	0
$847$	12.7818	0.439187
$848$	0	0
$849$	−2.46937	−0.0847486
$850$	0	0
$851$	18.0599i	0.619084i
$852$	0	0
$853$	1.44013i	0.0493090i	0.999696	+	0.0246545i	$0.00784856\pi$
$853$	1.44013i	0.0493090i	−0.999696	+	0.0246545i	$0.992151\pi$
$854$	0	0
$855$	0.627737	0.0214681
$856$	0	0
$857$	34.9775	1.19481	0.597404	−	0.801941i	$-0.296199\pi$
$857$	34.9775	1.19481	0.597404	+	0.801941i	$0.296199\pi$
$858$	0	0
$859$	− 6.45024i	− 0.220079i	−0.993927	−	0.110040i	$-0.964902\pi$
$859$	− 6.45024i	− 0.220079i	0.993927	−	0.110040i	$-0.0350978\pi$
$860$	0	0
$861$	− 6.09565i	− 0.207739i
$862$	0	0
$863$	34.4037	1.17112	0.585559	−	0.810630i	$-0.300875\pi$
$863$	34.4037	1.17112	0.585559	+	0.810630i	$0.300875\pi$
$864$	0	0
$865$	4.48112	0.152363
$866$	0	0
$867$	− 20.1570i	− 0.684566i
$868$	0	0
$869$	78.9594i	2.67852i
$870$	0	0
$871$	−26.9505	−0.913182
$872$	0	0
$873$	12.8789	0.435884
$874$	0	0
$875$	− 4.56944i	− 0.154475i
$876$	0	0
$877$	− 40.3826i	− 1.36362i	−0.731528	−	0.681812i	$-0.761192\pi$
$877$	− 40.3826i	− 1.36362i	0.731528	−	0.681812i	$-0.238808\pi$
$878$	0	0
$879$	22.7899	0.768684
$880$	0	0
$881$	15.8926	0.535435	0.267718	−	0.963497i	$-0.413731\pi$
$881$	15.8926	0.535435	0.267718	+	0.963497i	$0.413731\pi$
$882$	0	0
$883$	25.3113i	0.851792i	0.904772	+	0.425896i	$0.140041\pi$
$883$	25.3113i	0.851792i	−0.904772	+	0.425896i	$0.859959\pi$
$884$	0	0
$885$	1.86855i	0.0628106i
$886$	0	0
$887$	−19.6352	−0.659284	−0.329642	−	0.944106i	$-0.606928\pi$
$887$	−19.6352	−0.659284	−0.329642	+	0.944106i	$0.606928\pi$
$888$	0	0
$889$	18.8789	0.633178
$890$	0	0
$891$	4.87666i	0.163374i
$892$	0	0
$893$	8.98674i	0.300730i
$894$	0	0
$895$	3.59359	0.120120
$896$	0	0
$897$	18.6876	0.623960
$898$	0	0
$899$	34.3139i	1.14443i
$900$	0	0
$901$	8.19130i	0.272892i
$902$	0	0
$903$	−4.15327	−0.138212
$904$	0	0
$905$	7.12402	0.236811
$906$	0	0
$907$	4.84241i	0.160790i	0.996763	+	0.0803948i	$0.0256181\pi$
$907$	4.84241i	0.160790i	−0.996763	+	0.0803948i	$0.974382\pi$
$908$	0	0
$909$	2.22045i	0.0736477i
$910$	0	0
$911$	18.9176	0.626768	0.313384	−	0.949626i	$-0.398537\pi$
$911$	18.9176	0.626768	0.313384	+	0.949626i	$0.398537\pi$
$912$	0	0
$913$	67.0702	2.21970
$914$	0	0
$915$	2.56788i	0.0848917i
$916$	0	0
$917$	4.93428i	0.162944i
$918$	0	0
$919$	9.11228	0.300586	0.150293	−	0.988641i	$-0.451978\pi$
$919$	9.11228	0.300586	0.150293	+	0.988641i	$0.451978\pi$
$920$	0	0
$921$	−9.34379	−0.307888
$922$	0	0
$923$	21.5518i	0.709387i
$924$	0	0
$925$	21.0854i	0.693282i
$926$	0	0
$927$	11.0971	0.364477
$928$	0	0
$929$	32.7294	1.07382	0.536909	−	0.843640i	$-0.319592\pi$
$929$	32.7294	1.07382	0.536909	+	0.843640i	$0.319592\pi$
$930$	0	0
$931$	1.34379i	0.0440411i
$932$	0	0
$933$	6.68759i	0.218942i
$934$	0	0
$935$	−13.8863	−0.454131
$936$	0	0
$937$	−21.5066	−0.702591	−0.351295	−	0.936265i	$-0.614259\pi$
$937$	−21.5066	−0.702591	−0.351295	+	0.936265i	$0.614259\pi$
$938$	0	0
$939$	15.6381i	0.510329i
$940$	0	0
$941$	− 1.85115i	− 0.0603456i	−0.999545	−	0.0301728i	$-0.990394\pi$
$941$	− 1.85115i	− 0.0603456i	0.999545	−	0.0301728i	$-0.00960577\pi$
$942$	0	0
$943$	−24.9657	−0.812994
$944$	0	0
$945$	0.467138	0.0151960
$946$	0	0
$947$	− 45.3353i	− 1.47320i	−0.676329	−	0.736600i	$-0.736430\pi$
$947$	− 45.3353i	− 1.47320i	0.676329	−	0.736600i	$-0.263570\pi$
$948$	0	0
$949$	− 55.0266i	− 1.78624i
$950$	0	0
$951$	9.34379	0.302993
$952$	0	0
$953$	40.5768	1.31441	0.657206	−	0.753711i	$-0.271738\pi$
$953$	40.5768	1.31441	0.657206	+	0.753711i	$0.271738\pi$
$954$	0	0
$955$	2.20647i	0.0713997i
$956$	0	0
$957$	− 37.9491i	− 1.22672i
$958$	0	0
$959$	−19.0103	−0.613876
$960$	0	0
$961$	−11.5561	−0.372779
$962$	0	0
$963$	− 3.12334i	− 0.100648i
$964$	0	0
$965$	10.4349i	0.335911i
$966$	0	0
$967$	−15.8715	−0.510392	−0.255196	−	0.966889i	$-0.582140\pi$
$967$	−15.8715	−0.510392	−0.255196	+	0.966889i	$0.582140\pi$
$968$	0	0
$969$	8.19130	0.263143
$970$	0	0
$971$	− 14.6876i	− 0.471347i	−0.971832	−	0.235674i	$-0.924270\pi$
$971$	− 14.6876i	− 0.471347i	0.971832	−	0.235674i	$-0.0757296\pi$
$972$	0	0
$973$	− 10.4380i	− 0.334627i
$974$	0	0
$975$	21.8183	0.698744
$976$	0	0
$977$	−21.2437	−0.679647	−0.339824	−	0.940489i	$-0.610367\pi$
$977$	−21.2437	−0.679647	−0.339824	+	0.940489i	$0.610367\pi$
$978$	0	0
$979$	38.8387i	1.24129i
$980$	0	0
$981$	− 10.6876i	− 0.341228i
$982$	0	0
$983$	−11.8265	−0.377206	−0.188603	−	0.982053i	$-0.560396\pi$
$983$	−11.8265	−0.377206	−0.188603	+	0.982053i	$0.560396\pi$
$984$	0	0
$985$	0.717101	0.0228487
$986$	0	0
$987$	6.68759i	0.212868i
$988$	0	0
$989$	17.0103i	0.540897i
$990$	0	0
$991$	−28.9387	−0.919269	−0.459635	−	0.888108i	$-0.652020\pi$
$991$	−28.9387	−0.919269	−0.459635	+	0.888108i	$0.652020\pi$
$992$	0	0
$993$	5.40874	0.171641
$994$	0	0
$995$	6.86112i	0.217512i
$996$	0	0
$997$	56.0548i	1.77527i	0.460546	+	0.887636i	$0.347654\pi$
$997$	56.0548i	1.77527i	−0.460546	+	0.887636i	$0.652346\pi$
$998$	0	0
$999$	−4.40952	−0.139511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.c.b.337.2		8
3.2	odd	2		2016.2.c.e.1009.4		8
4.3	odd	2		168.2.c.b.85.6	yes	8
7.6	odd	2		4704.2.c.c.2353.7		8
8.3	odd	2		168.2.c.b.85.5	✓	8
8.5	even	2	inner	672.2.c.b.337.7		8
12.11	even	2		504.2.c.f.253.3		8
16.3	odd	4		5376.2.a.bm.1.2		4
16.5	even	4		5376.2.a.bl.1.3		4
16.11	odd	4		5376.2.a.bp.1.3		4
16.13	even	4		5376.2.a.bq.1.2		4
24.5	odd	2		2016.2.c.e.1009.5		8
24.11	even	2		504.2.c.f.253.4		8
28.27	even	2		1176.2.c.c.589.6		8
56.13	odd	2		4704.2.c.c.2353.2		8
56.27	even	2		1176.2.c.c.589.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.b.85.5	✓	8	8.3	odd	2
168.2.c.b.85.6	yes	8	4.3	odd	2
504.2.c.f.253.3		8	12.11	even	2
504.2.c.f.253.4		8	24.11	even	2
672.2.c.b.337.2		8	1.1	even	1	trivial
672.2.c.b.337.7		8	8.5	even	2	inner
1176.2.c.c.589.5		8	56.27	even	2
1176.2.c.c.589.6		8	28.27	even	2
2016.2.c.e.1009.4		8	3.2	odd	2
2016.2.c.e.1009.5		8	24.5	odd	2
4704.2.c.c.2353.2		8	56.13	odd	2
4704.2.c.c.2353.7		8	7.6	odd	2
5376.2.a.bl.1.3		4	16.5	even	4
5376.2.a.bm.1.2		4	16.3	odd	4
5376.2.a.bp.1.3		4	16.11	odd	4
5376.2.a.bq.1.2		4	16.13	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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +0.467138i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +0.467138i q^{5} -1.00000 q^{7} -1.00000 q^{9} +4.87666i q^{11} +4.56279i q^{13} +0.467138 q^{15} +6.09565 q^{17} +1.34379i q^{19} +1.00000i q^{21} +4.09565 q^{23} +4.78178 q^{25} +1.00000i q^{27} -7.78178i q^{29} -4.40952 q^{31} +4.87666 q^{33} -0.467138i q^{35} +4.40952i q^{37} +4.56279 q^{39} -6.09565 q^{41} +4.15327i q^{43} -0.467138i q^{45} +6.68759 q^{47} +1.00000 q^{49} -6.09565i q^{51} +1.34379i q^{53} -2.27807 q^{55} +1.34379 q^{57} +4.00000i q^{59} +5.49706i q^{61} +1.00000 q^{63} -2.13145 q^{65} +5.90658i q^{67} -4.09565i q^{69} +4.72339 q^{71} -12.0599 q^{73} -4.78178i q^{75} -4.87666i q^{77} +16.1913 q^{79} +1.00000 q^{81} -13.7533i q^{83} +2.84751i q^{85} -7.78178 q^{87} +7.96420 q^{89} -4.56279i q^{91} +4.40952i q^{93} -0.627737 q^{95} -12.8789 q^{97} -4.87666i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{7} - 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^7 - 8 * q^9	\( 8 q - 8 q^{7} - 8 q^{9} + 4 q^{15} + 4 q^{17} - 12 q^{23} - 24 q^{25} - 8 q^{31} + 12 q^{33} - 8 q^{39} - 4 q^{41} + 8 q^{49} + 8 q^{55} - 16 q^{57} + 8 q^{63} - 16 q^{65} + 28 q^{71} - 8 q^{73} + 40 q^{79} + 8 q^{81} + 20 q^{89} - 40 q^{95} + 40 q^{97}+O(q^{100}) \) 8 * q - 8 * q^7 - 8 * q^9 + 4 * q^15 + 4 * q^17 - 12 * q^23 - 24 * q^25 - 8 * q^31 + 12 * q^33 - 8 * q^39 - 4 * q^41 + 8 * q^49 + 8 * q^55 - 16 * q^57 + 8 * q^63 - 16 * q^65 + 28 * q^71 - 8 * q^73 + 40 * q^79 + 8 * q^81 + 20 * q^89 - 40 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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