LMFDB - Embedded newform 1792.2.b.o.897.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,2,Mod(897,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, 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("1792.897");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1792.b (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:	$14.3091920422$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 896)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			897.2
Root			$0.264658 - 1.38923i$ of defining polynomial
Character	$\chi$	$=$	1792.897
Dual form			1792.2.b.o.897.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.24914i q^{3} +3.30777i q^{5} -1.00000 q^{7} -2.05863 q^{9} +O(q^{10})$	$q-2.24914i q^{3} +3.30777i q^{5} -1.00000 q^{7} -2.05863 q^{9} -0.941367i q^{11} -5.19051i q^{13} +7.43965 q^{15} -6.49828 q^{17} +1.75086i q^{19} +2.24914i q^{21} +5.55691 q^{23} -5.94137 q^{25} -2.11727i q^{27} +2.49828i q^{29} +6.61555 q^{31} -2.11727 q^{33} -3.30777i q^{35} -4.61555i q^{37} -11.6742 q^{39} -10.4983 q^{41} -12.0552i q^{43} -6.80949i q^{45} -4.49828 q^{47} +1.00000 q^{49} +14.6155i q^{51} -2.00000i q^{53} +3.11383 q^{55} +3.93793 q^{57} -12.6302i q^{59} -11.3078i q^{61} +2.05863 q^{63} +17.1690 q^{65} -0.443086i q^{67} -12.4983i q^{69} +0.117266 q^{73} +13.3630i q^{75} +0.941367i q^{77} -7.11383 q^{79} -10.9379 q^{81} -12.8647i q^{83} -21.4948i q^{85} +5.61899 q^{87} -10.9966 q^{89} +5.19051i q^{91} -14.8793i q^{93} -5.79145 q^{95} +13.6121 q^{97} +1.93793i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{7} - 14 q^{9}+O(q^{10}) $ 6 * q - 6 * q^7 - 14 * q^9	$ 6 q - 6 q^{7} - 14 q^{9} + 8 q^{15} - 4 q^{17} - 34 q^{25} + 8 q^{31} - 16 q^{33} - 40 q^{39} - 28 q^{41} + 8 q^{47} + 6 q^{49} - 48 q^{55} - 48 q^{57} + 14 q^{63} - 32 q^{65} + 4 q^{73} + 24 q^{79} + 6 q^{81} + 72 q^{87} + 4 q^{89} - 8 q^{95} - 20 q^{97}+O(q^{100}) $ 6 * q - 6 * q^7 - 14 * q^9 + 8 * q^15 - 4 * q^17 - 34 * q^25 + 8 * q^31 - 16 * q^33 - 40 * q^39 - 28 * q^41 + 8 * q^47 + 6 * q^49 - 48 * q^55 - 48 * q^57 + 14 * q^63 - 32 * q^65 + 4 * q^73 + 24 * q^79 + 6 * q^81 + 72 * q^87 + 4 * q^89 - 8 * q^95 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 2.24914i	− 1.29854i	−0.760557	−	0.649271i	$-0.775074\pi$
$3$	− 2.24914i	− 1.29854i	0.760557	−	0.649271i	$-0.224926\pi$
$4$	0	0
$5$	3.30777i	1.47928i	0.673002	+	0.739641i	$0.265005\pi$
$5$	3.30777i	1.47928i	−0.673002	+	0.739641i	$0.734995\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.05863	−0.686211
$10$	0	0
$11$	− 0.941367i	− 0.283833i	−0.989879	−	0.141916i	$-0.954674\pi$
$11$	− 0.941367i	− 0.283833i	0.989879	−	0.141916i	$-0.0453264\pi$
$12$	0	0
$13$	− 5.19051i	− 1.43959i	−0.694188	−	0.719794i	$-0.744236\pi$
$13$	− 5.19051i	− 1.43959i	0.694188	−	0.719794i	$-0.255764\pi$
$14$	0	0
$15$	7.43965	1.92091
$16$	0	0
$17$	−6.49828	−1.57606	−0.788032	−	0.615634i	$-0.788900\pi$
$17$	−6.49828	−1.57606	−0.788032	+	0.615634i	$0.788900\pi$
$18$	0	0
$19$	1.75086i	0.401675i	0.979625	+	0.200837i	$0.0643663\pi$
$19$	1.75086i	0.401675i	−0.979625	+	0.200837i	$0.935634\pi$
$20$	0	0
$21$	2.24914i	0.490803i
$22$	0	0
$23$	5.55691	1.15870	0.579348	−	0.815080i	$-0.303307\pi$
$23$	5.55691	1.15870	0.579348	+	0.815080i	$0.303307\pi$
$24$	0	0
$25$	−5.94137	−1.18827
$26$	0	0
$27$	− 2.11727i	− 0.407468i
$28$	0	0
$29$	2.49828i	0.463919i	0.972725	+	0.231960i	$0.0745137\pi$
$29$	2.49828i	0.463919i	−0.972725	+	0.231960i	$0.925486\pi$
$30$	0	0
$31$	6.61555	1.18819	0.594094	−	0.804396i	$-0.297511\pi$
$31$	6.61555	1.18819	0.594094	+	0.804396i	$0.297511\pi$
$32$	0	0
$33$	−2.11727	−0.368569
$34$	0	0
$35$	− 3.30777i	− 0.559116i
$36$	0	0
$37$	− 4.61555i	− 0.758791i	−0.925235	−	0.379396i	$-0.876132\pi$
$37$	− 4.61555i	− 0.758791i	0.925235	−	0.379396i	$-0.123868\pi$
$38$	0	0
$39$	−11.6742	−1.86936
$40$	0	0
$41$	−10.4983	−1.63956	−0.819778	−	0.572681i	$-0.805903\pi$
$41$	−10.4983	−1.63956	−0.819778	+	0.572681i	$0.805903\pi$
$42$	0	0
$43$	− 12.0552i	− 1.83840i	−0.393791	−	0.919200i	$-0.628837\pi$
$43$	− 12.0552i	− 1.83840i	0.393791	−	0.919200i	$-0.371163\pi$
$44$	0	0
$45$	− 6.80949i	− 1.01510i
$46$	0	0
$47$	−4.49828	−0.656142	−0.328071	−	0.944653i	$-0.606398\pi$
$47$	−4.49828	−0.656142	−0.328071	+	0.944653i	$0.606398\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	14.6155i	2.04659i
$52$	0	0
$53$	− 2.00000i	− 0.274721i	−0.990521	−	0.137361i	$-0.956138\pi$
$53$	− 2.00000i	− 0.274721i	0.990521	−	0.137361i	$-0.0438619\pi$
$54$	0	0
$55$	3.11383	0.419869
$56$	0	0
$57$	3.93793	0.521591
$58$	0	0
$59$	− 12.6302i	− 1.64431i	−0.569266	−	0.822153i	$-0.692773\pi$
$59$	− 12.6302i	− 1.64431i	0.569266	−	0.822153i	$-0.307227\pi$
$60$	0	0
$61$	− 11.3078i	− 1.44781i	−0.689899	−	0.723906i	$-0.742345\pi$
$61$	− 11.3078i	− 1.44781i	0.689899	−	0.723906i	$-0.257655\pi$
$62$	0	0
$63$	2.05863	0.259363
$64$	0	0
$65$	17.1690	2.12956
$66$	0	0
$67$	− 0.443086i	− 0.0541315i	−0.999634	−	0.0270658i	$-0.991384\pi$
$67$	− 0.443086i	− 0.0541315i	0.999634	−	0.0270658i	$-0.00861635\pi$
$68$	0	0
$69$	− 12.4983i	− 1.50462i
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	0.117266	0.0137250	0.00686249	−	0.999976i	$-0.497816\pi$
$73$	0.117266	0.0137250	0.00686249	+	0.999976i	$0.497816\pi$
$74$	0	0
$75$	13.3630i	1.54302i
$76$	0	0
$77$	0.941367i	0.107279i
$78$	0	0
$79$	−7.11383	−0.800368	−0.400184	−	0.916435i	$-0.631054\pi$
$79$	−7.11383	−0.800368	−0.400184	+	0.916435i	$0.631054\pi$
$80$	0	0
$81$	−10.9379	−1.21533
$82$	0	0
$83$	− 12.8647i	− 1.41208i	−0.708170	−	0.706041i	$-0.750479\pi$
$83$	− 12.8647i	− 1.41208i	0.708170	−	0.706041i	$-0.249521\pi$
$84$	0	0
$85$	− 21.4948i	− 2.33144i
$86$	0	0
$87$	5.61899	0.602418
$88$	0	0
$89$	−10.9966	−1.16563	−0.582817	−	0.812604i	$-0.698049\pi$
$89$	−10.9966	−1.16563	−0.582817	+	0.812604i	$0.698049\pi$
$90$	0	0
$91$	5.19051i	0.544113i
$92$	0	0
$93$	− 14.8793i	− 1.54291i
$94$	0	0
$95$	−5.79145	−0.594190
$96$	0	0
$97$	13.6121	1.38210	0.691050	−	0.722807i	$-0.257148\pi$
$97$	13.6121	1.38210	0.691050	+	0.722807i	$0.257148\pi$
$98$	0	0
$99$	1.93793i	0.194769i
$100$	0	0
$101$	15.8061i	1.57276i	0.617742	+	0.786381i	$0.288047\pi$
$101$	15.8061i	1.57276i	−0.617742	+	0.786381i	$0.711953\pi$
$102$	0	0
$103$	−6.61555	−0.651849	−0.325925	−	0.945396i	$-0.605676\pi$
$103$	−6.61555	−0.651849	−0.325925	+	0.945396i	$0.605676\pi$
$104$	0	0
$105$	−7.43965	−0.726035
$106$	0	0
$107$	5.43965i	0.525871i	0.964813	+	0.262935i	$0.0846906\pi$
$107$	5.43965i	0.525871i	−0.964813	+	0.262935i	$0.915309\pi$
$108$	0	0
$109$	0.381015i	0.0364946i	0.999834	+	0.0182473i	$0.00580862\pi$
$109$	0.381015i	0.0364946i	−0.999834	+	0.0182473i	$0.994191\pi$
$110$	0	0
$111$	−10.3810	−0.985322
$112$	0	0
$113$	−8.05520	−0.757769	−0.378885	−	0.925444i	$-0.623692\pi$
$113$	−8.05520	−0.757769	−0.378885	+	0.925444i	$0.623692\pi$
$114$	0	0
$115$	18.3810i	1.71404i
$116$	0	0
$117$	10.6854i	0.987861i
$118$	0	0
$119$	6.49828	0.595696
$120$	0	0
$121$	10.1138	0.919439
$122$	0	0
$123$	23.6121i	2.12903i
$124$	0	0
$125$	− 3.11383i	− 0.278509i
$126$	0	0
$127$	11.6742	1.03592	0.517958	−	0.855406i	$-0.326692\pi$
$127$	11.6742	1.03592	0.517958	+	0.855406i	$0.326692\pi$
$128$	0	0
$129$	−27.1138	−2.38724
$130$	0	0
$131$	0.366407i	0.0320131i	0.999872	+	0.0160066i	$0.00509526\pi$
$131$	0.366407i	0.0320131i	−0.999872	+	0.0160066i	$0.994905\pi$
$132$	0	0
$133$	− 1.75086i	− 0.151819i
$134$	0	0
$135$	7.00344	0.602760
$136$	0	0
$137$	3.88273	0.331724	0.165862	−	0.986149i	$-0.446959\pi$
$137$	3.88273	0.331724	0.165862	+	0.986149i	$0.446959\pi$
$138$	0	0
$139$	− 14.2491i	− 1.20860i	−0.796758	−	0.604298i	$-0.793454\pi$
$139$	− 14.2491i	− 1.20860i	0.796758	−	0.604298i	$-0.206546\pi$
$140$	0	0
$141$	10.1173i	0.852028i
$142$	0	0
$143$	−4.88617	−0.408602
$144$	0	0
$145$	−8.26375	−0.686267
$146$	0	0
$147$	− 2.24914i	− 0.185506i
$148$	0	0
$149$	− 8.87930i	− 0.727420i	−0.931512	−	0.363710i	$-0.881510\pi$
$149$	− 8.87930i	− 0.727420i	0.931512	−	0.363710i	$-0.118490\pi$
$150$	0	0
$151$	−22.5535	−1.83538	−0.917688	−	0.397302i	$-0.869947\pi$
$151$	−22.5535	−1.83538	−0.917688	+	0.397302i	$0.869947\pi$
$152$	0	0
$153$	13.3776	1.08151
$154$	0	0
$155$	21.8827i	1.75766i
$156$	0	0
$157$	12.8026i	1.02176i	0.859652	+	0.510880i	$0.170680\pi$
$157$	12.8026i	1.02176i	−0.859652	+	0.510880i	$0.829320\pi$
$158$	0	0
$159$	−4.49828	−0.356737
$160$	0	0
$161$	−5.55691	−0.437946
$162$	0	0
$163$	− 9.93793i	− 0.778399i	−0.921154	−	0.389199i	$-0.872752\pi$
$163$	− 9.93793i	− 0.778399i	0.921154	−	0.389199i	$-0.127248\pi$
$164$	0	0
$165$	− 7.00344i	− 0.545217i
$166$	0	0
$167$	−12.4983	−0.967146	−0.483573	−	0.875304i	$-0.660661\pi$
$167$	−12.4983	−0.967146	−0.483573	+	0.875304i	$0.660661\pi$
$168$	0	0
$169$	−13.9414	−1.07241
$170$	0	0
$171$	− 3.60438i	− 0.275634i
$172$	0	0
$173$	16.0406i	1.21954i	0.792577	+	0.609772i	$0.208739\pi$
$173$	16.0406i	1.21954i	−0.792577	+	0.609772i	$0.791261\pi$
$174$	0	0
$175$	5.94137	0.449125
$176$	0	0
$177$	−28.4070	−2.13520
$178$	0	0
$179$	12.7880i	0.955821i	0.878409	+	0.477910i	$0.158606\pi$
$179$	12.7880i	0.955821i	−0.878409	+	0.477910i	$0.841394\pi$
$180$	0	0
$181$	9.68879i	0.720162i	0.932921	+	0.360081i	$0.117251\pi$
$181$	9.68879i	0.720162i	−0.932921	+	0.360081i	$0.882749\pi$
$182$	0	0
$183$	−25.4328	−1.88004
$184$	0	0
$185$	15.2672	1.12247
$186$	0	0
$187$	6.11727i	0.447339i
$188$	0	0
$189$	2.11727i	0.154008i
$190$	0	0
$191$	−9.88273	−0.715090	−0.357545	−	0.933896i	$-0.616386\pi$
$191$	−9.88273	−0.715090	−0.357545	+	0.933896i	$0.616386\pi$
$192$	0	0
$193$	8.70683	0.626732	0.313366	−	0.949632i	$-0.398543\pi$
$193$	8.70683	0.626732	0.313366	+	0.949632i	$0.398543\pi$
$194$	0	0
$195$	− 38.6155i	− 2.76532i
$196$	0	0
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	0	0
$199$	23.6121	1.67382	0.836909	−	0.547342i	$-0.184360\pi$
$199$	23.6121	1.67382	0.836909	+	0.547342i	$0.184360\pi$
$200$	0	0
$201$	−0.996562	−0.0702921
$202$	0	0
$203$	− 2.49828i	− 0.175345i
$204$	0	0
$205$	− 34.7259i	− 2.42536i
$206$	0	0
$207$	−11.4396	−0.795110
$208$	0	0
$209$	1.64820	0.114008
$210$	0	0
$211$	2.32582i	0.160116i	0.996790	+	0.0800580i	$0.0255106\pi$
$211$	2.32582i	0.160116i	−0.996790	+	0.0800580i	$0.974489\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	39.8759	2.71951
$216$	0	0
$217$	−6.61555	−0.449093
$218$	0	0
$219$	− 0.263748i	− 0.0178225i
$220$	0	0
$221$	33.7294i	2.26888i
$222$	0	0
$223$	6.87930	0.460672	0.230336	−	0.973111i	$-0.426018\pi$
$223$	6.87930	0.460672	0.230336	+	0.973111i	$0.426018\pi$
$224$	0	0
$225$	12.2311	0.815406
$226$	0	0
$227$	− 14.2491i	− 0.945749i	−0.881130	−	0.472874i	$-0.843216\pi$
$227$	− 14.2491i	− 0.945749i	0.881130	−	0.472874i	$-0.156784\pi$
$228$	0	0
$229$	− 3.80605i	− 0.251511i	−0.992061	−	0.125756i	$-0.959864\pi$
$229$	− 3.80605i	− 0.251511i	0.992061	−	0.125756i	$-0.0401355\pi$
$230$	0	0
$231$	2.11727	0.139306
$232$	0	0
$233$	24.8793	1.62990	0.814948	−	0.579533i	$-0.196765\pi$
$233$	24.8793	1.62990	0.814948	+	0.579533i	$0.196765\pi$
$234$	0	0
$235$	− 14.8793i	− 0.970618i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	17.5569	1.13566	0.567831	−	0.823145i	$-0.307783\pi$
$239$	17.5569	1.13566	0.567831	+	0.823145i	$0.307783\pi$
$240$	0	0
$241$	0.850080	0.0547585	0.0273792	−	0.999625i	$-0.491284\pi$
$241$	0.850080	0.0547585	0.0273792	+	0.999625i	$0.491284\pi$
$242$	0	0
$243$	18.2491i	1.17068i
$244$	0	0
$245$	3.30777i	0.211326i
$246$	0	0
$247$	9.08785	0.578246
$248$	0	0
$249$	−28.9345	−1.83365
$250$	0	0
$251$	8.36641i	0.528083i	0.964511	+	0.264041i	$0.0850556\pi$
$251$	8.36641i	0.528083i	−0.964511	+	0.264041i	$0.914944\pi$
$252$	0	0
$253$	− 5.23109i	− 0.328876i
$254$	0	0
$255$	−48.3449	−3.02748
$256$	0	0
$257$	−14.9966	−0.935460	−0.467730	−	0.883871i	$-0.654928\pi$
$257$	−14.9966	−0.935460	−0.467730	+	0.883871i	$0.654928\pi$
$258$	0	0
$259$	4.61555i	0.286796i
$260$	0	0
$261$	− 5.14304i	− 0.318346i
$262$	0	0
$263$	14.2277	0.877315	0.438657	−	0.898654i	$-0.355454\pi$
$263$	14.2277	0.877315	0.438657	+	0.898654i	$0.355454\pi$
$264$	0	0
$265$	6.61555	0.406390
$266$	0	0
$267$	24.7328i	1.51362i
$268$	0	0
$269$	− 4.19395i	− 0.255709i	−0.991793	−	0.127855i	$-0.959191\pi$
$269$	− 4.19395i	− 0.255709i	0.991793	−	0.127855i	$-0.0408091\pi$
$270$	0	0
$271$	1.64820	0.100121	0.0500605	−	0.998746i	$-0.484059\pi$
$271$	1.64820	0.100121	0.0500605	+	0.998746i	$0.484059\pi$
$272$	0	0
$273$	11.6742	0.706554
$274$	0	0
$275$	5.59301i	0.337271i
$276$	0	0
$277$	8.11727i	0.487719i	0.969811	+	0.243860i	$0.0784136\pi$
$277$	8.11727i	0.487719i	−0.969811	+	0.243860i	$0.921586\pi$
$278$	0	0
$279$	−13.6190	−0.815347
$280$	0	0
$281$	14.2345	0.849161	0.424581	−	0.905390i	$-0.360422\pi$
$281$	14.2345	0.849161	0.424581	+	0.905390i	$0.360422\pi$
$282$	0	0
$283$	− 31.4802i	− 1.87131i	−0.352922	−	0.935653i	$-0.614812\pi$
$283$	− 31.4802i	− 1.87131i	0.352922	−	0.935653i	$-0.385188\pi$
$284$	0	0
$285$	13.0258i	0.771580i
$286$	0	0
$287$	10.4983	0.619694
$288$	0	0
$289$	25.2277	1.48398
$290$	0	0
$291$	− 30.6155i	− 1.79472i
$292$	0	0
$293$	− 1.19051i	− 0.0695502i	−0.999395	−	0.0347751i	$-0.988929\pi$
$293$	− 1.19051i	− 0.0695502i	0.999395	−	0.0347751i	$-0.0110715\pi$
$294$	0	0
$295$	41.7777	2.43239
$296$	0	0
$297$	−1.99312	−0.115653
$298$	0	0
$299$	− 28.8432i	− 1.66805i
$300$	0	0
$301$	12.0552i	0.694850i
$302$	0	0
$303$	35.5500	2.04230
$304$	0	0
$305$	37.4036	2.14172
$306$	0	0
$307$	4.60094i	0.262589i	0.991343	+	0.131295i	$0.0419134\pi$
$307$	4.60094i	0.262589i	−0.991343	+	0.131295i	$0.958087\pi$
$308$	0	0
$309$	14.8793i	0.846454i
$310$	0	0
$311$	−13.2311	−0.750267	−0.375133	−	0.926971i	$-0.622403\pi$
$311$	−13.2311	−0.750267	−0.375133	+	0.926971i	$0.622403\pi$
$312$	0	0
$313$	−10.4983	−0.593398	−0.296699	−	0.954971i	$-0.595886\pi$
$313$	−10.4983	−0.593398	−0.296699	+	0.954971i	$0.595886\pi$
$314$	0	0
$315$	6.80949i	0.383671i
$316$	0	0
$317$	9.00344i	0.505683i	0.967508	+	0.252842i	$0.0813652\pi$
$317$	9.00344i	0.505683i	−0.967508	+	0.252842i	$0.918635\pi$
$318$	0	0
$319$	2.35180	0.131675
$320$	0	0
$321$	12.2345	0.682865
$322$	0	0
$323$	− 11.3776i	− 0.633065i
$324$	0	0
$325$	30.8387i	1.71062i
$326$	0	0
$327$	0.856956	0.0473898
$328$	0	0
$329$	4.49828	0.247998
$330$	0	0
$331$	10.1725i	0.559129i	0.960127	+	0.279565i	$0.0901901\pi$
$331$	10.1725i	0.559129i	−0.960127	+	0.279565i	$0.909810\pi$
$332$	0	0
$333$	9.50172i	0.520691i
$334$	0	0
$335$	1.46563	0.0800758
$336$	0	0
$337$	23.8207	1.29759	0.648797	−	0.760961i	$-0.275272\pi$
$337$	23.8207	1.29759	0.648797	+	0.760961i	$0.275272\pi$
$338$	0	0
$339$	18.1173i	0.983995i
$340$	0	0
$341$	− 6.22766i	− 0.337247i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	41.3415	2.22575
$346$	0	0
$347$	− 17.2863i	− 0.927977i	−0.885841	−	0.463988i	$-0.846418\pi$
$347$	− 17.2863i	− 0.927977i	0.885841	−	0.463988i	$-0.153582\pi$
$348$	0	0
$349$	− 16.6922i	− 0.893514i	−0.894655	−	0.446757i	$-0.852579\pi$
$349$	− 16.6922i	− 0.893514i	0.894655	−	0.446757i	$-0.147421\pi$
$350$	0	0
$351$	−10.9897	−0.586586
$352$	0	0
$353$	−14.3449	−0.763503	−0.381752	−	0.924265i	$-0.624679\pi$
$353$	−14.3449	−0.763503	−0.381752	+	0.924265i	$0.624679\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 14.6155i	− 0.773537i
$358$	0	0
$359$	−3.43965	−0.181538	−0.0907688	−	0.995872i	$-0.528932\pi$
$359$	−3.43965	−0.181538	−0.0907688	+	0.995872i	$0.528932\pi$
$360$	0	0
$361$	15.9345	0.838657
$362$	0	0
$363$	− 22.7474i	− 1.19393i
$364$	0	0
$365$	0.387890i	0.0203031i
$366$	0	0
$367$	−10.1173	−0.528117	−0.264059	−	0.964507i	$-0.585061\pi$
$367$	−10.1173	−0.528117	−0.264059	+	0.964507i	$0.585061\pi$
$368$	0	0
$369$	21.6121	1.12508
$370$	0	0
$371$	2.00000i	0.103835i
$372$	0	0
$373$	20.8793i	1.08109i	0.841315	+	0.540544i	$0.181782\pi$
$373$	20.8793i	1.08109i	−0.841315	+	0.540544i	$0.818218\pi$
$374$	0	0
$375$	−7.00344	−0.361656
$376$	0	0
$377$	12.9673	0.667852
$378$	0	0
$379$	13.9379i	0.715943i	0.933732	+	0.357972i	$0.116532\pi$
$379$	13.9379i	0.715943i	−0.933732	+	0.357972i	$0.883468\pi$
$380$	0	0
$381$	− 26.2569i	− 1.34518i
$382$	0	0
$383$	22.4914	1.14926	0.574629	−	0.818414i	$-0.305147\pi$
$383$	22.4914	1.14926	0.574629	+	0.818414i	$0.305147\pi$
$384$	0	0
$385$	−3.11383	−0.158695
$386$	0	0
$387$	24.8172i	1.26153i
$388$	0	0
$389$	− 17.8466i	− 0.904861i	−0.891800	−	0.452430i	$-0.850557\pi$
$389$	− 17.8466i	− 0.904861i	0.891800	−	0.452430i	$-0.149443\pi$
$390$	0	0
$391$	−36.1104	−1.82618
$392$	0	0
$393$	0.824101	0.0415704
$394$	0	0
$395$	− 23.5309i	− 1.18397i
$396$	0	0
$397$	− 0.692226i	− 0.0347418i	−0.999849	−	0.0173709i	$-0.994470\pi$
$397$	− 0.692226i	− 0.0347418i	0.999849	−	0.0173709i	$-0.00552961\pi$
$398$	0	0
$399$	−3.93793	−0.197143
$400$	0	0
$401$	−2.40699	−0.120200	−0.0600998	−	0.998192i	$-0.519142\pi$
$401$	−2.40699	−0.120200	−0.0600998	+	0.998192i	$0.519142\pi$
$402$	0	0
$403$	− 34.3380i	− 1.71050i
$404$	0	0
$405$	− 36.1802i	− 1.79781i
$406$	0	0
$407$	−4.34492	−0.215370
$408$	0	0
$409$	−21.6121	−1.06865	−0.534325	−	0.845279i	$-0.679434\pi$
$409$	−21.6121	−1.06865	−0.534325	+	0.845279i	$0.679434\pi$
$410$	0	0
$411$	− 8.73281i	− 0.430758i
$412$	0	0
$413$	12.6302i	0.621489i
$414$	0	0
$415$	42.5535	2.08887
$416$	0	0
$417$	−32.0483	−1.56941
$418$	0	0
$419$	− 9.25258i	− 0.452018i	−0.974125	−	0.226009i	$-0.927432\pi$
$419$	− 9.25258i	− 0.452018i	0.974125	−	0.226009i	$-0.0725679\pi$
$420$	0	0
$421$	14.3449i	0.699129i	0.936912	+	0.349564i	$0.113670\pi$
$421$	14.3449i	0.699129i	−0.936912	+	0.349564i	$0.886330\pi$
$422$	0	0
$423$	9.26031	0.450252
$424$	0	0
$425$	38.6087	1.87280
$426$	0	0
$427$	11.3078i	0.547222i
$428$	0	0
$429$	10.9897i	0.530587i
$430$	0	0
$431$	24.4362	1.17705	0.588525	−	0.808479i	$-0.299709\pi$
$431$	24.4362	1.17705	0.588525	+	0.808479i	$0.299709\pi$
$432$	0	0
$433$	−16.4914	−0.792526	−0.396263	−	0.918137i	$-0.629693\pi$
$433$	−16.4914	−0.792526	−0.396263	+	0.918137i	$0.629693\pi$
$434$	0	0
$435$	18.5863i	0.891146i
$436$	0	0
$437$	9.72938i	0.465419i
$438$	0	0
$439$	−38.8793	−1.85561	−0.927804	−	0.373069i	$-0.878306\pi$
$439$	−38.8793	−1.85561	−0.927804	+	0.373069i	$0.878306\pi$
$440$	0	0
$441$	−2.05863	−0.0980302
$442$	0	0
$443$	15.0878i	0.716845i	0.933559	+	0.358423i	$0.116685\pi$
$443$	15.0878i	0.716845i	−0.933559	+	0.358423i	$0.883315\pi$
$444$	0	0
$445$	− 36.3741i	− 1.72430i
$446$	0	0
$447$	−19.9708	−0.944586
$448$	0	0
$449$	28.8793	1.36290	0.681449	−	0.731865i	$-0.261350\pi$
$449$	28.8793	1.36290	0.681449	+	0.731865i	$0.261350\pi$
$450$	0	0
$451$	9.88273i	0.465360i
$452$	0	0
$453$	50.7259i	2.38331i
$454$	0	0
$455$	−17.1690	−0.804896
$456$	0	0
$457$	−24.2897	−1.13623	−0.568113	−	0.822951i	$-0.692326\pi$
$457$	−24.2897	−1.13623	−0.568113	+	0.822951i	$0.692326\pi$
$458$	0	0
$459$	13.7586i	0.642196i
$460$	0	0
$461$	10.0767i	0.469318i	0.972078	+	0.234659i	$0.0753973\pi$
$461$	10.0767i	0.469318i	−0.972078	+	0.234659i	$0.924603\pi$
$462$	0	0
$463$	34.8793	1.62098	0.810489	−	0.585754i	$-0.199201\pi$
$463$	34.8793	1.62098	0.810489	+	0.585754i	$0.199201\pi$
$464$	0	0
$465$	49.2173	2.28240
$466$	0	0
$467$	26.7474i	1.23772i	0.785500	+	0.618862i	$0.212406\pi$
$467$	26.7474i	1.23772i	−0.785500	+	0.618862i	$0.787594\pi$
$468$	0	0
$469$	0.443086i	0.0204598i
$470$	0	0
$471$	28.7949	1.32680
$472$	0	0
$473$	−11.3484	−0.521798
$474$	0	0
$475$	− 10.4025i	− 0.477299i
$476$	0	0
$477$	4.11727i	0.188517i
$478$	0	0
$479$	1.38445	0.0632573	0.0316286	−	0.999500i	$-0.489931\pi$
$479$	1.38445	0.0632573	0.0316286	+	0.999500i	$0.489931\pi$
$480$	0	0
$481$	−23.9570	−1.09235
$482$	0	0
$483$	12.4983i	0.568691i
$484$	0	0
$485$	45.0258i	2.04452i
$486$	0	0
$487$	−6.20855	−0.281336	−0.140668	−	0.990057i	$-0.544925\pi$
$487$	−6.20855	−0.281336	−0.140668	+	0.990057i	$0.544925\pi$
$488$	0	0
$489$	−22.3518	−1.01078
$490$	0	0
$491$	30.7811i	1.38913i	0.719428	+	0.694567i	$0.244404\pi$
$491$	30.7811i	1.38913i	−0.719428	+	0.694567i	$0.755596\pi$
$492$	0	0
$493$	− 16.2345i	− 0.731167i
$494$	0	0
$495$	−6.41023	−0.288118
$496$	0	0
$497$	0	0
$498$	0	0
$499$	33.6742i	1.50746i	0.657182	+	0.753732i	$0.271748\pi$
$499$	33.6742i	1.50746i	−0.657182	+	0.753732i	$0.728252\pi$
$500$	0	0
$501$	28.1104i	1.25588i
$502$	0	0
$503$	−29.8827	−1.33241	−0.666203	−	0.745771i	$-0.732081\pi$
$503$	−29.8827	−1.33241	−0.666203	+	0.745771i	$0.732081\pi$
$504$	0	0
$505$	−52.2829	−2.32656
$506$	0	0
$507$	31.3561i	1.39257i
$508$	0	0
$509$	3.54231i	0.157010i	0.996914	+	0.0785050i	$0.0250147\pi$
$509$	3.54231i	0.157010i	−0.996914	+	0.0785050i	$0.974985\pi$
$510$	0	0
$511$	−0.117266	−0.00518756
$512$	0	0
$513$	3.70704	0.163670
$514$	0	0
$515$	− 21.8827i	− 0.964268i
$516$	0	0
$517$	4.23453i	0.186235i
$518$	0	0
$519$	36.0775	1.58363
$520$	0	0
$521$	38.8432	1.70175	0.850876	−	0.525367i	$-0.176072\pi$
$521$	38.8432	1.70175	0.850876	+	0.525367i	$0.176072\pi$
$522$	0	0
$523$	− 3.63359i	− 0.158886i	−0.996839	−	0.0794430i	$-0.974686\pi$
$523$	− 3.63359i	− 0.158886i	0.996839	−	0.0794430i	$-0.0253142\pi$
$524$	0	0
$525$	− 13.3630i	− 0.583208i
$526$	0	0
$527$	−42.9897	−1.87266
$528$	0	0
$529$	7.87930	0.342578
$530$	0	0
$531$	26.0009i	1.12834i
$532$	0	0
$533$	54.4914i	2.36028i
$534$	0	0
$535$	−17.9931	−0.777911
$536$	0	0
$537$	28.7620	1.24117
$538$	0	0
$539$	− 0.941367i	− 0.0405475i
$540$	0	0
$541$	22.1104i	0.950600i	0.879824	+	0.475300i	$0.157660\pi$
$541$	22.1104i	0.950600i	−0.879824	+	0.475300i	$0.842340\pi$
$542$	0	0
$543$	21.7914	0.935160
$544$	0	0
$545$	−1.26031	−0.0539858
$546$	0	0
$547$	− 39.1690i	− 1.67475i	−0.546632	−	0.837373i	$-0.684090\pi$
$547$	− 39.1690i	− 1.67475i	0.546632	−	0.837373i	$-0.315910\pi$
$548$	0	0
$549$	23.2786i	0.993505i
$550$	0	0
$551$	−4.37414	−0.186345
$552$	0	0
$553$	7.11383	0.302511
$554$	0	0
$555$	− 34.3380i	− 1.45757i
$556$	0	0
$557$	13.7655i	0.583262i	0.956531	+	0.291631i	$0.0941979\pi$
$557$	13.7655i	0.583262i	−0.956531	+	0.291631i	$0.905802\pi$
$558$	0	0
$559$	−62.5726	−2.64654
$560$	0	0
$561$	13.7586	0.580888
$562$	0	0
$563$	− 21.6267i	− 0.911457i	−0.890119	−	0.455729i	$-0.849379\pi$
$563$	− 21.6267i	− 0.911457i	0.890119	−	0.455729i	$-0.150621\pi$
$564$	0	0
$565$	− 26.6448i	− 1.12095i
$566$	0	0
$567$	10.9379	0.459350
$568$	0	0
$569$	−7.16902	−0.300541	−0.150271	−	0.988645i	$-0.548014\pi$
$569$	−7.16902	−0.300541	−0.150271	+	0.988645i	$0.548014\pi$
$570$	0	0
$571$	− 36.0552i	− 1.50886i	−0.656379	−	0.754431i	$-0.727913\pi$
$571$	− 36.0552i	− 1.50886i	0.656379	−	0.754431i	$-0.272087\pi$
$572$	0	0
$573$	22.2277i	0.928574i
$574$	0	0
$575$	−33.0157	−1.37685
$576$	0	0
$577$	−34.1104	−1.42003	−0.710017	−	0.704184i	$-0.751313\pi$
$577$	−34.1104	−1.42003	−0.710017	+	0.704184i	$0.751313\pi$
$578$	0	0
$579$	− 19.5829i	− 0.813837i
$580$	0	0
$581$	12.8647i	0.533717i
$582$	0	0
$583$	−1.88273	−0.0779749
$584$	0	0
$585$	−35.3447	−1.46132
$586$	0	0
$587$	2.74742i	0.113398i	0.998391	+	0.0566991i	$0.0180576\pi$
$587$	2.74742i	0.113398i	−0.998391	+	0.0566991i	$0.981942\pi$
$588$	0	0
$589$	11.5829i	0.477265i
$590$	0	0
$591$	−22.4914	−0.925173
$592$	0	0
$593$	32.2277	1.32343	0.661716	−	0.749755i	$-0.269829\pi$
$593$	32.2277	1.32343	0.661716	+	0.749755i	$0.269829\pi$
$594$	0	0
$595$	21.4948i	0.881203i
$596$	0	0
$597$	− 53.1070i	− 2.17352i
$598$	0	0
$599$	0.469065	0.0191655	0.00958274	−	0.999954i	$-0.496950\pi$
$599$	0.469065	0.0191655	0.00958274	+	0.999954i	$0.496950\pi$
$600$	0	0
$601$	−37.4588	−1.52797	−0.763987	−	0.645231i	$-0.776761\pi$
$601$	−37.4588	−1.52797	−0.763987	+	0.645231i	$0.776761\pi$
$602$	0	0
$603$	0.912151i	0.0371457i
$604$	0	0
$605$	33.4543i	1.36011i
$606$	0	0
$607$	25.4656	1.03362	0.516809	−	0.856101i	$-0.327120\pi$
$607$	25.4656	1.03362	0.516809	+	0.856101i	$0.327120\pi$
$608$	0	0
$609$	−5.61899	−0.227693
$610$	0	0
$611$	23.3484i	0.944574i
$612$	0	0
$613$	− 1.03265i	− 0.0417085i	−0.999783	−	0.0208542i	$-0.993361\pi$
$613$	− 1.03265i	− 0.0417085i	0.999783	−	0.0208542i	$-0.00663859\pi$
$614$	0	0
$615$	−78.1035	−3.14944
$616$	0	0
$617$	−6.93449	−0.279172	−0.139586	−	0.990210i	$-0.544577\pi$
$617$	−6.93449	−0.279172	−0.139586	+	0.990210i	$0.544577\pi$
$618$	0	0
$619$	2.98195i	0.119855i	0.998203	+	0.0599274i	$0.0190869\pi$
$619$	2.98195i	0.119855i	−0.998203	+	0.0599274i	$0.980913\pi$
$620$	0	0
$621$	− 11.7655i	− 0.472132i
$622$	0	0
$623$	10.9966	0.440568
$624$	0	0
$625$	−19.4070	−0.776280
$626$	0	0
$627$	− 3.70704i	− 0.148045i
$628$	0	0
$629$	29.9931i	1.19590i
$630$	0	0
$631$	−38.7552	−1.54282	−0.771409	−	0.636339i	$-0.780448\pi$
$631$	−38.7552	−1.54282	−0.771409	+	0.636339i	$0.780448\pi$
$632$	0	0
$633$	5.23109	0.207917
$634$	0	0
$635$	38.6155i	1.53241i
$636$	0	0
$637$	− 5.19051i	− 0.205655i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0552	0.950123	0.475062	−	0.879953i	$-0.342426\pi$
$641$	24.0552	0.950123	0.475062	+	0.879953i	$0.342426\pi$
$642$	0	0
$643$	22.8286i	0.900272i	0.892960	+	0.450136i	$0.148624\pi$
$643$	22.8286i	0.900272i	−0.892960	+	0.450136i	$0.851376\pi$
$644$	0	0
$645$	− 89.6864i	− 3.53140i
$646$	0	0
$647$	5.14992	0.202464	0.101232	−	0.994863i	$-0.467722\pi$
$647$	5.14992	0.202464	0.101232	+	0.994863i	$0.467722\pi$
$648$	0	0
$649$	−11.8896	−0.466708
$650$	0	0
$651$	14.8793i	0.583166i
$652$	0	0
$653$	6.73281i	0.263475i	0.991285	+	0.131738i	$0.0420557\pi$
$653$	6.73281i	0.263475i	−0.991285	+	0.131738i	$0.957944\pi$
$654$	0	0
$655$	−1.21199	−0.0473564
$656$	0	0
$657$	−0.241408	−0.00941824
$658$	0	0
$659$	36.8172i	1.43420i	0.696973	+	0.717098i	$0.254530\pi$
$659$	36.8172i	1.43420i	−0.696973	+	0.717098i	$0.745470\pi$
$660$	0	0
$661$	− 21.4250i	− 0.833337i	−0.909058	−	0.416669i	$-0.863198\pi$
$661$	− 21.4250i	− 0.833337i	0.909058	−	0.416669i	$-0.136802\pi$
$662$	0	0
$663$	75.8621	2.94624
$664$	0	0
$665$	5.79145	0.224583
$666$	0	0
$667$	13.8827i	0.537542i
$668$	0	0
$669$	− 15.4725i	− 0.598202i
$670$	0	0
$671$	−10.6448	−0.410937
$672$	0	0
$673$	−6.76203	−0.260657	−0.130329	−	0.991471i	$-0.541603\pi$
$673$	−6.76203	−0.260657	−0.130329	+	0.991471i	$0.541603\pi$
$674$	0	0
$675$	12.5795i	0.484183i
$676$	0	0
$677$	− 42.2975i	− 1.62562i	−0.582526	−	0.812812i	$-0.697936\pi$
$677$	− 42.2975i	− 1.62562i	0.582526	−	0.812812i	$-0.302064\pi$
$678$	0	0
$679$	−13.6121	−0.522385
$680$	0	0
$681$	−32.0483	−1.22809
$682$	0	0
$683$	− 32.7880i	− 1.25460i	−0.778778	−	0.627299i	$-0.784160\pi$
$683$	− 32.7880i	− 1.25460i	0.778778	−	0.627299i	$-0.215840\pi$
$684$	0	0
$685$	12.8432i	0.490714i
$686$	0	0
$687$	−8.56035	−0.326598
$688$	0	0
$689$	−10.3810	−0.395485
$690$	0	0
$691$	22.1250i	0.841675i	0.907136	+	0.420837i	$0.138264\pi$
$691$	22.1250i	0.841675i	−0.907136	+	0.420837i	$0.861736\pi$
$692$	0	0
$693$	− 1.93793i	− 0.0736158i
$694$	0	0
$695$	47.1329	1.78785
$696$	0	0
$697$	68.2208	2.58405
$698$	0	0
$699$	− 55.9570i	− 2.11649i
$700$	0	0
$701$	− 36.7259i	− 1.38712i	−0.720399	−	0.693560i	$-0.756041\pi$
$701$	− 36.7259i	− 1.38712i	0.720399	−	0.693560i	$-0.243959\pi$
$702$	0	0
$703$	8.08117	0.304787
$704$	0	0
$705$	−33.4656	−1.26039
$706$	0	0
$707$	− 15.8061i	− 0.594448i
$708$	0	0
$709$	− 6.38789i	− 0.239902i	−0.992780	−	0.119951i	$-0.961726\pi$
$709$	− 6.38789i	− 0.239902i	0.992780	−	0.119951i	$-0.0382738\pi$
$710$	0	0
$711$	14.6448	0.549222
$712$	0	0
$713$	36.7620	1.37675
$714$	0	0
$715$	− 16.1623i	− 0.604438i
$716$	0	0
$717$	− 39.4880i	− 1.47471i
$718$	0	0
$719$	45.4948	1.69667	0.848336	−	0.529459i	$-0.177605\pi$
$719$	45.4948	1.69667	0.848336	+	0.529459i	$0.177605\pi$
$720$	0	0
$721$	6.61555	0.246376
$722$	0	0
$723$	− 1.91195i	− 0.0711062i
$724$	0	0
$725$	− 14.8432i	− 0.551263i
$726$	0	0
$727$	50.2569	1.86392	0.931962	−	0.362556i	$-0.118096\pi$
$727$	50.2569	1.86392	0.931962	+	0.362556i	$0.118096\pi$
$728$	0	0
$729$	8.23109	0.304855
$730$	0	0
$731$	78.3380i	2.89744i
$732$	0	0
$733$	− 13.3009i	− 0.491280i	−0.969361	−	0.245640i	$-0.921002\pi$
$733$	− 13.3009i	− 0.491280i	0.969361	−	0.245640i	$-0.0789981\pi$
$734$	0	0
$735$	7.43965	0.274416
$736$	0	0
$737$	−0.417106	−0.0153643
$738$	0	0
$739$	− 2.40699i	− 0.0885427i	−0.999020	−	0.0442714i	$-0.985903\pi$
$739$	− 2.40699i	− 0.0885427i	0.999020	−	0.0442714i	$-0.0140966\pi$
$740$	0	0
$741$	− 20.4398i	− 0.750877i
$742$	0	0
$743$	14.0844	0.516707	0.258353	−	0.966050i	$-0.416820\pi$
$743$	14.0844	0.516707	0.258353	+	0.966050i	$0.416820\pi$
$744$	0	0
$745$	29.3707	1.07606
$746$	0	0
$747$	26.4837i	0.968987i
$748$	0	0
$749$	− 5.43965i	− 0.198760i
$750$	0	0
$751$	−7.78457	−0.284063	−0.142032	−	0.989862i	$-0.545363\pi$
$751$	−7.78457	−0.284063	−0.142032	+	0.989862i	$0.545363\pi$
$752$	0	0
$753$	18.8172	0.685738
$754$	0	0
$755$	− 74.6018i	− 2.71504i
$756$	0	0
$757$	24.4914i	0.890155i	0.895492	+	0.445078i	$0.146824\pi$
$757$	24.4914i	0.890155i	−0.895492	+	0.445078i	$0.853176\pi$
$758$	0	0
$759$	−11.7655	−0.427059
$760$	0	0
$761$	−10.9673	−0.397566	−0.198783	−	0.980044i	$-0.563699\pi$
$761$	−10.9673	−0.397566	−0.198783	+	0.980044i	$0.563699\pi$
$762$	0	0
$763$	− 0.381015i	− 0.0137937i
$764$	0	0
$765$	44.2500i	1.59986i
$766$	0	0
$767$	−65.5569	−2.36712
$768$	0	0
$769$	−6.62242	−0.238811	−0.119405	−	0.992846i	$-0.538099\pi$
$769$	−6.62242	−0.238811	−0.119405	+	0.992846i	$0.538099\pi$
$770$	0	0
$771$	33.7294i	1.21473i
$772$	0	0
$773$	− 3.91645i	− 0.140865i	−0.997517	−	0.0704324i	$-0.977562\pi$
$773$	− 3.91645i	− 0.140865i	0.997517	−	0.0704324i	$-0.0224379\pi$
$774$	0	0
$775$	−39.3054	−1.41189
$776$	0	0
$777$	10.3810	0.372417
$778$	0	0
$779$	− 18.3810i	− 0.658568i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	5.28953	0.189032
$784$	0	0
$785$	−42.3482	−1.51147
$786$	0	0
$787$	− 9.01805i	− 0.321459i	−0.986999	−	0.160729i	$-0.948615\pi$
$787$	− 9.01805i	− 0.321459i	0.986999	−	0.160729i	$-0.0513846\pi$
$788$	0	0
$789$	− 32.0000i	− 1.13923i
$790$	0	0
$791$	8.05520	0.286410
$792$	0	0
$793$	−58.6931	−2.08425
$794$	0	0
$795$	− 14.8793i	− 0.527714i
$796$	0	0
$797$	23.5715i	0.834946i	0.908689	+	0.417473i	$0.137084\pi$
$797$	23.5715i	0.834946i	−0.908689	+	0.417473i	$0.862916\pi$
$798$	0	0
$799$	29.2311	1.03412
$800$	0	0
$801$	22.6379	0.799870
$802$	0	0
$803$	− 0.110391i	− 0.00389560i
$804$	0	0
$805$	− 18.3810i	− 0.647846i
$806$	0	0
$807$	−9.43277	−0.332049
$808$	0	0
$809$	20.1656	0.708984	0.354492	−	0.935059i	$-0.384654\pi$
$809$	20.1656	0.708984	0.354492	+	0.935059i	$0.384654\pi$
$810$	0	0
$811$	21.9716i	0.771529i	0.922597	+	0.385764i	$0.126062\pi$
$811$	21.9716i	0.771529i	−0.922597	+	0.385764i	$0.873938\pi$
$812$	0	0
$813$	− 3.70704i	− 0.130011i
$814$	0	0
$815$	32.8724	1.15147
$816$	0	0
$817$	21.1070	0.738439
$818$	0	0
$819$	− 10.6854i	− 0.373376i
$820$	0	0
$821$	3.10695i	0.108433i	0.998529	+	0.0542167i	$0.0172662\pi$
$821$	3.10695i	0.108433i	−0.998529	+	0.0542167i	$0.982734\pi$
$822$	0	0
$823$	12.7620	0.444856	0.222428	−	0.974949i	$-0.428602\pi$
$823$	12.7620	0.444856	0.222428	+	0.974949i	$0.428602\pi$
$824$	0	0
$825$	12.5795	0.437960
$826$	0	0
$827$	4.91215i	0.170812i	0.996346	+	0.0854061i	$0.0272188\pi$
$827$	4.91215i	0.170812i	−0.996346	+	0.0854061i	$0.972781\pi$
$828$	0	0
$829$	− 33.5354i	− 1.16473i	−0.812926	−	0.582367i	$-0.802127\pi$
$829$	− 33.5354i	− 1.16473i	0.812926	−	0.582367i	$-0.197873\pi$
$830$	0	0
$831$	18.2569	0.633324
$832$	0	0
$833$	−6.49828	−0.225152
$834$	0	0
$835$	− 41.3415i	− 1.43068i
$836$	0	0
$837$	− 14.0069i	− 0.484148i
$838$	0	0
$839$	6.61555	0.228394	0.114197	−	0.993458i	$-0.463570\pi$
$839$	6.61555	0.228394	0.114197	+	0.993458i	$0.463570\pi$
$840$	0	0
$841$	22.7586	0.784779
$842$	0	0
$843$	− 32.0155i	− 1.10267i
$844$	0	0
$845$	− 46.1149i	− 1.58640i
$846$	0	0
$847$	−10.1138	−0.347515
$848$	0	0
$849$	−70.8035	−2.42997
$850$	0	0
$851$	− 25.6482i	− 0.879209i
$852$	0	0
$853$	30.1871i	1.03359i	0.856111	+	0.516793i	$0.172874\pi$
$853$	30.1871i	1.03359i	−0.856111	+	0.516793i	$0.827126\pi$
$854$	0	0
$855$	11.9225	0.407740
$856$	0	0
$857$	−16.3810	−0.559565	−0.279782	−	0.960063i	$-0.590262\pi$
$857$	−16.3810	−0.559565	−0.279782	+	0.960063i	$0.590262\pi$
$858$	0	0
$859$	0.366407i	0.0125016i	0.999980	+	0.00625082i	$0.00198971\pi$
$859$	0.366407i	0.0125016i	−0.999980	+	0.00625082i	$0.998010\pi$
$860$	0	0
$861$	− 23.6121i	− 0.804699i
$862$	0	0
$863$	13.6482	0.464590	0.232295	−	0.972645i	$-0.425376\pi$
$863$	13.6482	0.464590	0.232295	+	0.972645i	$0.425376\pi$
$864$	0	0
$865$	−53.0586	−1.80405
$866$	0	0
$867$	− 56.7405i	− 1.92701i
$868$	0	0
$869$	6.69672i	0.227171i
$870$	0	0
$871$	−2.29984	−0.0779271
$872$	0	0
$873$	−28.0223	−0.948413
$874$	0	0
$875$	3.11383i	0.105267i
$876$	0	0
$877$	15.8535i	0.535335i	0.963511	+	0.267668i	$0.0862529\pi$
$877$	15.8535i	0.535335i	−0.963511	+	0.267668i	$0.913747\pi$
$878$	0	0
$879$	−2.67762	−0.0903138
$880$	0	0
$881$	7.35524	0.247804	0.123902	−	0.992294i	$-0.460459\pi$
$881$	7.35524	0.247804	0.123902	+	0.992294i	$0.460459\pi$
$882$	0	0
$883$	− 25.2571i	− 0.849968i	−0.905201	−	0.424984i	$-0.860280\pi$
$883$	− 25.2571i	− 0.849968i	0.905201	−	0.424984i	$-0.139720\pi$
$884$	0	0
$885$	− 93.9639i	− 3.15856i
$886$	0	0
$887$	43.1950	1.45035	0.725173	−	0.688567i	$-0.241760\pi$
$887$	43.1950	1.45035	0.725173	+	0.688567i	$0.241760\pi$
$888$	0	0
$889$	−11.6742	−0.391539
$890$	0	0
$891$	10.2966i	0.344949i
$892$	0	0
$893$	− 7.87586i	− 0.263556i
$894$	0	0
$895$	−42.2998	−1.41393
$896$	0	0
$897$	−64.8724	−2.16603
$898$	0	0
$899$	16.5275i	0.551223i
$900$	0	0
$901$	12.9966i	0.432978i
$902$	0	0
$903$	27.1138	0.902292
$904$	0	0
$905$	−32.0483	−1.06532
$906$	0	0
$907$	41.0810i	1.36407i	0.731319	+	0.682036i	$0.238905\pi$
$907$	41.0810i	1.36407i	−0.731319	+	0.682036i	$0.761095\pi$
$908$	0	0
$909$	− 32.5389i	− 1.07925i
$910$	0	0
$911$	−21.4465	−0.710555	−0.355278	−	0.934761i	$-0.615614\pi$
$911$	−21.4465	−0.710555	−0.355278	+	0.934761i	$0.615614\pi$
$912$	0	0
$913$	−12.1104	−0.400795
$914$	0	0
$915$	− 84.1259i	− 2.78111i
$916$	0	0
$917$	− 0.366407i	− 0.0120998i
$918$	0	0
$919$	4.76203	0.157085	0.0785424	−	0.996911i	$-0.474973\pi$
$919$	4.76203	0.157085	0.0785424	+	0.996911i	$0.474973\pi$
$920$	0	0
$921$	10.3482	0.340983
$922$	0	0
$923$	0	0
$924$	0	0
$925$	27.4227i	0.901652i
$926$	0	0
$927$	13.6190	0.447306
$928$	0	0
$929$	20.9605	0.687691	0.343845	−	0.939026i	$-0.388270\pi$
$929$	20.9605	0.687691	0.343845	+	0.939026i	$0.388270\pi$
$930$	0	0
$931$	1.75086i	0.0573821i
$932$	0	0
$933$	29.7586i	0.974253i
$934$	0	0
$935$	−20.2345	−0.661740
$936$	0	0
$937$	−7.10695	−0.232174	−0.116087	−	0.993239i	$-0.537035\pi$
$937$	−7.10695	−0.232174	−0.116087	+	0.993239i	$0.537035\pi$
$938$	0	0
$939$	23.6121i	0.770552i
$940$	0	0
$941$	− 3.57152i	− 0.116428i	−0.998304	−	0.0582141i	$-0.981459\pi$
$941$	− 3.57152i	− 0.116428i	0.998304	−	0.0582141i	$-0.0185406\pi$
$942$	0	0
$943$	−58.3380	−1.89975
$944$	0	0
$945$	−7.00344	−0.227822
$946$	0	0
$947$	− 25.4104i	− 0.825728i	−0.910793	−	0.412864i	$-0.864528\pi$
$947$	− 25.4104i	− 0.825728i	0.910793	−	0.412864i	$-0.135472\pi$
$948$	0	0
$949$	− 0.608672i	− 0.0197583i
$950$	0	0
$951$	20.2500	0.656651
$952$	0	0
$953$	28.2277	0.914383	0.457192	−	0.889368i	$-0.348855\pi$
$953$	28.2277	0.914383	0.457192	+	0.889368i	$0.348855\pi$
$954$	0	0
$955$	− 32.6898i	− 1.05782i
$956$	0	0
$957$	− 5.28953i	− 0.170986i
$958$	0	0
$959$	−3.88273	−0.125380
$960$	0	0
$961$	12.7655	0.411789
$962$	0	0
$963$	− 11.1982i	− 0.360858i
$964$	0	0
$965$	28.8002i	0.927112i
$966$	0	0
$967$	40.0191	1.28693	0.643464	−	0.765477i	$-0.277497\pi$
$967$	40.0191	1.28693	0.643464	+	0.765477i	$0.277497\pi$
$968$	0	0
$969$	−25.5898	−0.822062
$970$	0	0
$971$	− 16.8647i	− 0.541214i	−0.962690	−	0.270607i	$-0.912776\pi$
$971$	− 16.8647i	− 0.541214i	0.962690	−	0.270607i	$-0.0872243\pi$
$972$	0	0
$973$	14.2491i	0.456806i
$974$	0	0
$975$	69.3606	2.22132
$976$	0	0
$977$	−12.1173	−0.387666	−0.193833	−	0.981035i	$-0.562092\pi$
$977$	−12.1173	−0.387666	−0.193833	+	0.981035i	$0.562092\pi$
$978$	0	0
$979$	10.3518i	0.330845i
$980$	0	0
$981$	− 0.784370i	− 0.0250430i
$982$	0	0
$983$	−49.6052	−1.58216	−0.791081	−	0.611712i	$-0.790481\pi$
$983$	−49.6052	−1.58216	−0.791081	+	0.611712i	$0.790481\pi$
$984$	0	0
$985$	33.0777	1.05394
$986$	0	0
$987$	− 10.1173i	− 0.322036i
$988$	0	0
$989$	− 66.9897i	− 2.13015i
$990$	0	0
$991$	35.8759	1.13963	0.569817	−	0.821772i	$-0.307014\pi$
$991$	35.8759	1.13963	0.569817	+	0.821772i	$0.307014\pi$
$992$	0	0
$993$	22.8793	0.726053
$994$	0	0
$995$	78.1035i	2.47605i
$996$	0	0
$997$	− 9.45426i	− 0.299419i	−0.988730	−	0.149710i	$-0.952166\pi$
$997$	− 9.45426i	− 0.299419i	0.988730	−	0.149710i	$-0.0478339\pi$
$998$	0	0
$999$	−9.77234	−0.309183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.b.o.897.2		6
4.3	odd	2		1792.2.b.p.897.5		6
8.3	odd	2		1792.2.b.p.897.2		6
8.5	even	2	inner	1792.2.b.o.897.5		6
16.3	odd	4		896.2.a.l.1.1	yes	3
16.5	even	4		896.2.a.k.1.1	yes	3
16.11	odd	4		896.2.a.i.1.3	✓	3
16.13	even	4		896.2.a.j.1.3	yes	3
48.5	odd	4		8064.2.a.ch.1.3		3
48.11	even	4		8064.2.a.ce.1.3		3
48.29	odd	4		8064.2.a.cb.1.1		3
48.35	even	4		8064.2.a.bu.1.1		3
112.13	odd	4		6272.2.a.w.1.1		3
112.27	even	4		6272.2.a.x.1.1		3
112.69	odd	4		6272.2.a.v.1.3		3
112.83	even	4		6272.2.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.a.i.1.3	✓	3	16.11	odd	4
896.2.a.j.1.3	yes	3	16.13	even	4
896.2.a.k.1.1	yes	3	16.5	even	4
896.2.a.l.1.1	yes	3	16.3	odd	4
1792.2.b.o.897.2		6	1.1	even	1	trivial
1792.2.b.o.897.5		6	8.5	even	2	inner
1792.2.b.p.897.2		6	8.3	odd	2
1792.2.b.p.897.5		6	4.3	odd	2
6272.2.a.u.1.3		3	112.83	even	4
6272.2.a.v.1.3		3	112.69	odd	4
6272.2.a.w.1.1		3	112.13	odd	4
6272.2.a.x.1.1		3	112.27	even	4
8064.2.a.bu.1.1		3	48.35	even	4
8064.2.a.cb.1.1		3	48.29	odd	4
8064.2.a.ce.1.3		3	48.11	even	4
8064.2.a.ch.1.3		3	48.5	odd	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.24914i q^{3} +3.30777i q^{5} -1.00000 q^{7} -2.05863 q^{9} +O(q^{10})\)	\(q-2.24914i q^{3} +3.30777i q^{5} -1.00000 q^{7} -2.05863 q^{9} -0.941367i q^{11} -5.19051i q^{13} +7.43965 q^{15} -6.49828 q^{17} +1.75086i q^{19} +2.24914i q^{21} +5.55691 q^{23} -5.94137 q^{25} -2.11727i q^{27} +2.49828i q^{29} +6.61555 q^{31} -2.11727 q^{33} -3.30777i q^{35} -4.61555i q^{37} -11.6742 q^{39} -10.4983 q^{41} -12.0552i q^{43} -6.80949i q^{45} -4.49828 q^{47} +1.00000 q^{49} +14.6155i q^{51} -2.00000i q^{53} +3.11383 q^{55} +3.93793 q^{57} -12.6302i q^{59} -11.3078i q^{61} +2.05863 q^{63} +17.1690 q^{65} -0.443086i q^{67} -12.4983i q^{69} +0.117266 q^{73} +13.3630i q^{75} +0.941367i q^{77} -7.11383 q^{79} -10.9379 q^{81} -12.8647i q^{83} -21.4948i q^{85} +5.61899 q^{87} -10.9966 q^{89} +5.19051i q^{91} -14.8793i q^{93} -5.79145 q^{95} +13.6121 q^{97} +1.93793i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{7} - 14 q^{9}+O(q^{10}) \) 6 * q - 6 * q^7 - 14 * q^9	\( 6 q - 6 q^{7} - 14 q^{9} + 8 q^{15} - 4 q^{17} - 34 q^{25} + 8 q^{31} - 16 q^{33} - 40 q^{39} - 28 q^{41} + 8 q^{47} + 6 q^{49} - 48 q^{55} - 48 q^{57} + 14 q^{63} - 32 q^{65} + 4 q^{73} + 24 q^{79} + 6 q^{81} + 72 q^{87} + 4 q^{89} - 8 q^{95} - 20 q^{97}+O(q^{100}) \) 6 * q - 6 * q^7 - 14 * q^9 + 8 * q^15 - 4 * q^17 - 34 * q^25 + 8 * q^31 - 16 * q^33 - 40 * q^39 - 28 * q^41 + 8 * q^47 + 6 * q^49 - 48 * q^55 - 48 * q^57 + 14 * q^63 - 32 * q^65 + 4 * q^73 + 24 * q^79 + 6 * q^81 + 72 * q^87 + 4 * q^89 - 8 * q^95 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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