LMFDB - Embedded newform 896.2.b.e.449.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(449,896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(896, 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("896.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.b (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:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	896.449
Dual form			896.2.b.e.449.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607i q^{3} +1.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +O(q^{10})$	$q+1.23607i q^{3} +1.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +4.00000i q^{11} +1.23607i q^{13} -1.52786 q^{15} -2.00000 q^{17} -1.23607i q^{19} -1.23607i q^{21} -6.47214 q^{23} +3.47214 q^{25} +5.52786i q^{27} -1.52786i q^{29} -4.94427 q^{33} -1.23607i q^{35} +6.47214i q^{37} -1.52786 q^{39} +2.00000 q^{41} +8.94427i q^{43} +1.81966i q^{45} -12.9443 q^{47} +1.00000 q^{49} -2.47214i q^{51} -8.94427i q^{53} -4.94427 q^{55} +1.52786 q^{57} +9.23607i q^{59} -1.23607i q^{61} -1.47214 q^{63} -1.52786 q^{65} -1.52786i q^{67} -8.00000i q^{69} +4.94427 q^{71} -14.9443 q^{73} +4.29180i q^{75} -4.00000i q^{77} +4.94427 q^{79} -2.41641 q^{81} -9.23607i q^{83} -2.47214i q^{85} +1.88854 q^{87} -2.00000 q^{89} -1.23607i q^{91} +1.52786 q^{95} +10.9443 q^{97} +5.88854i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} - 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 - 12 * q^9	$ 4 q - 4 q^{7} - 12 q^{9} - 24 q^{15} - 8 q^{17} - 8 q^{23} - 4 q^{25} + 16 q^{33} - 24 q^{39} + 8 q^{41} - 16 q^{47} + 4 q^{49} + 16 q^{55} + 24 q^{57} + 12 q^{63} - 24 q^{65} - 16 q^{71} - 24 q^{73} - 16 q^{79} + 44 q^{81} - 64 q^{87} - 8 q^{89} + 24 q^{95} + 8 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 12 * q^9 - 24 * q^15 - 8 * q^17 - 8 * q^23 - 4 * q^25 + 16 * q^33 - 24 * q^39 + 8 * q^41 - 16 * q^47 + 4 * q^49 + 16 * q^55 + 24 * q^57 + 12 * q^63 - 24 * q^65 - 16 * q^71 - 24 * q^73 - 16 * q^79 + 44 * q^81 - 64 * q^87 - 8 * q^89 + 24 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/896\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$645$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	1.23607i	0.713644i	0.934172	+	0.356822i	$0.116140\pi$
$3$	1.23607i	0.713644i	−0.934172	+	0.356822i	$0.883860\pi$
$4$	0	0
$5$	1.23607i	0.552786i	0.961045	+	0.276393i	$0.0891392\pi$
$5$	1.23607i	0.552786i	−0.961045	+	0.276393i	$0.910861\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.47214	0.490712
$10$	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
$13$	1.23607i	0.342824i	0.985199	+	0.171412i	$0.0548329\pi$
$13$	1.23607i	0.342824i	−0.985199	+	0.171412i	$0.945167\pi$
$14$	0	0
$15$	−1.52786	−0.394493
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	− 1.23607i	− 0.283573i	−0.989897	−	0.141787i	$-0.954715\pi$
$19$	− 1.23607i	− 0.283573i	0.989897	−	0.141787i	$-0.0452847\pi$
$20$	0	0
$21$	− 1.23607i	− 0.269732i
$22$	0	0
$23$	−6.47214	−1.34953	−0.674767	−	0.738031i	$-0.735756\pi$
$23$	−6.47214	−1.34953	−0.674767	+	0.738031i	$0.735756\pi$
$24$	0	0
$25$	3.47214	0.694427
$26$	0	0
$27$	5.52786i	1.06384i
$28$	0	0
$29$	− 1.52786i	− 0.283717i	−0.989887	−	0.141859i	$-0.954692\pi$
$29$	− 1.52786i	− 0.283717i	0.989887	−	0.141859i	$-0.0453078\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−4.94427	−0.860687
$34$	0	0
$35$	− 1.23607i	− 0.208934i
$36$	0	0
$37$	6.47214i	1.06401i	0.846740	+	0.532006i	$0.178562\pi$
$37$	6.47214i	1.06401i	−0.846740	+	0.532006i	$0.821438\pi$
$38$	0	0
$39$	−1.52786	−0.244654
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	8.94427i	1.36399i	0.731357	+	0.681994i	$0.238887\pi$
$43$	8.94427i	1.36399i	−0.731357	+	0.681994i	$0.761113\pi$
$44$	0	0
$45$	1.81966i	0.271259i
$46$	0	0
$47$	−12.9443	−1.88812	−0.944058	−	0.329779i	$-0.893026\pi$
$47$	−12.9443	−1.88812	−0.944058	+	0.329779i	$0.893026\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	− 2.47214i	− 0.346168i
$52$	0	0
$53$	− 8.94427i	− 1.22859i	−0.789076	−	0.614295i	$-0.789440\pi$
$53$	− 8.94427i	− 1.22859i	0.789076	−	0.614295i	$-0.210560\pi$
$54$	0	0
$55$	−4.94427	−0.666685
$56$	0	0
$57$	1.52786	0.202371
$58$	0	0
$59$	9.23607i	1.20243i	0.799086	+	0.601217i	$0.205317\pi$
$59$	9.23607i	1.20243i	−0.799086	+	0.601217i	$0.794683\pi$
$60$	0	0
$61$	− 1.23607i	− 0.158262i	−0.996864	−	0.0791311i	$-0.974785\pi$
$61$	− 1.23607i	− 0.158262i	0.996864	−	0.0791311i	$-0.0252146\pi$
$62$	0	0
$63$	−1.47214	−0.185472
$64$	0	0
$65$	−1.52786	−0.189508
$66$	0	0
$67$	− 1.52786i	− 0.186658i	−0.995635	−	0.0933292i	$-0.970249\pi$
$67$	− 1.52786i	− 0.186658i	0.995635	−	0.0933292i	$-0.0297509\pi$
$68$	0	0
$69$	− 8.00000i	− 0.963087i
$70$	0	0
$71$	4.94427	0.586777	0.293389	−	0.955993i	$-0.405217\pi$
$71$	4.94427	0.586777	0.293389	+	0.955993i	$0.405217\pi$
$72$	0	0
$73$	−14.9443	−1.74909	−0.874547	−	0.484940i	$-0.838841\pi$
$73$	−14.9443	−1.74909	−0.874547	+	0.484940i	$0.838841\pi$
$74$	0	0
$75$	4.29180i	0.495574i
$76$	0	0
$77$	− 4.00000i	− 0.455842i
$78$	0	0
$79$	4.94427	0.556274	0.278137	−	0.960541i	$-0.410283\pi$
$79$	4.94427	0.556274	0.278137	+	0.960541i	$0.410283\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	− 9.23607i	− 1.01379i	−0.862008	−	0.506895i	$-0.830793\pi$
$83$	− 9.23607i	− 1.01379i	0.862008	−	0.506895i	$-0.169207\pi$
$84$	0	0
$85$	− 2.47214i	− 0.268141i
$86$	0	0
$87$	1.88854	0.202473
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	− 1.23607i	− 0.129575i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.52786	0.156756
$96$	0	0
$97$	10.9443	1.11122	0.555611	−	0.831442i	$-0.312484\pi$
$97$	10.9443	1.11122	0.555611	+	0.831442i	$0.312484\pi$
$98$	0	0
$99$	5.88854i	0.591821i
$100$	0	0
$101$	14.1803i	1.41100i	0.708712	+	0.705498i	$0.249277\pi$
$101$	14.1803i	1.41100i	−0.708712	+	0.705498i	$0.750723\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	1.52786	0.149104
$106$	0	0
$107$	9.52786i	0.921093i	0.887635	+	0.460547i	$0.152347\pi$
$107$	9.52786i	0.921093i	−0.887635	+	0.460547i	$0.847653\pi$
$108$	0	0
$109$	− 1.52786i	− 0.146343i	−0.997319	−	0.0731714i	$-0.976688\pi$
$109$	− 1.52786i	− 0.146343i	0.997319	−	0.0731714i	$-0.0233120\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	13.4164	1.26211	0.631055	−	0.775738i	$-0.282622\pi$
$113$	13.4164	1.26211	0.631055	+	0.775738i	$0.282622\pi$
$114$	0	0
$115$	− 8.00000i	− 0.746004i
$116$	0	0
$117$	1.81966i	0.168228i
$118$	0	0
$119$	2.00000	0.183340
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	2.47214i	0.222905i
$124$	0	0
$125$	10.4721i	0.936656i
$126$	0	0
$127$	19.4164	1.72293	0.861464	−	0.507819i	$-0.169548\pi$
$127$	19.4164	1.72293	0.861464	+	0.507819i	$0.169548\pi$
$128$	0	0
$129$	−11.0557	−0.973403
$130$	0	0
$131$	− 14.1803i	− 1.23894i	−0.785020	−	0.619471i	$-0.787347\pi$
$131$	− 14.1803i	− 1.23894i	0.785020	−	0.619471i	$-0.212653\pi$
$132$	0	0
$133$	1.23607i	0.107181i
$134$	0	0
$135$	−6.83282	−0.588075
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	− 17.2361i	− 1.46194i	−0.682407	−	0.730972i	$-0.739067\pi$
$139$	− 17.2361i	− 1.46194i	0.682407	−	0.730972i	$-0.260933\pi$
$140$	0	0
$141$	− 16.0000i	− 1.34744i
$142$	0	0
$143$	−4.94427	−0.413461
$144$	0	0
$145$	1.88854	0.156835
$146$	0	0
$147$	1.23607i	0.101949i
$148$	0	0
$149$	16.9443i	1.38813i	0.719913	+	0.694064i	$0.244182\pi$
$149$	16.9443i	1.38813i	−0.719913	+	0.694064i	$0.755818\pi$
$150$	0	0
$151$	6.47214	0.526695	0.263347	−	0.964701i	$-0.415173\pi$
$151$	6.47214	0.526695	0.263347	+	0.964701i	$0.415173\pi$
$152$	0	0
$153$	−2.94427	−0.238030
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.1803i	1.77018i	0.465416	+	0.885092i	$0.345905\pi$
$157$	22.1803i	1.77018i	−0.465416	+	0.885092i	$0.654095\pi$
$158$	0	0
$159$	11.0557	0.876776
$160$	0	0
$161$	6.47214	0.510076
$162$	0	0
$163$	− 12.0000i	− 0.939913i	−0.882690	−	0.469956i	$-0.844270\pi$
$163$	− 12.0000i	− 0.939913i	0.882690	−	0.469956i	$-0.155730\pi$
$164$	0	0
$165$	− 6.11146i	− 0.475776i
$166$	0	0
$167$	−4.94427	−0.382599	−0.191300	−	0.981532i	$-0.561270\pi$
$167$	−4.94427	−0.382599	−0.191300	+	0.981532i	$0.561270\pi$
$168$	0	0
$169$	11.4721	0.882472
$170$	0	0
$171$	− 1.81966i	− 0.139153i
$172$	0	0
$173$	− 3.70820i	− 0.281930i	−0.990015	−	0.140965i	$-0.954980\pi$
$173$	− 3.70820i	− 0.281930i	0.990015	−	0.140965i	$-0.0450204\pi$
$174$	0	0
$175$	−3.47214	−0.262469
$176$	0	0
$177$	−11.4164	−0.858110
$178$	0	0
$179$	− 6.47214i	− 0.483750i	−0.970307	−	0.241875i	$-0.922238\pi$
$179$	− 6.47214i	− 0.483750i	0.970307	−	0.241875i	$-0.0777624\pi$
$180$	0	0
$181$	− 25.2361i	− 1.87578i	−0.346930	−	0.937891i	$-0.612776\pi$
$181$	− 25.2361i	− 1.87578i	0.346930	−	0.937891i	$-0.387224\pi$
$182$	0	0
$183$	1.52786	0.112943
$184$	0	0
$185$	−8.00000	−0.588172
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	− 5.52786i	− 0.402093i
$190$	0	0
$191$	17.8885	1.29437	0.647185	−	0.762333i	$-0.275946\pi$
$191$	17.8885	1.29437	0.647185	+	0.762333i	$0.275946\pi$
$192$	0	0
$193$	12.4721	0.897764	0.448882	−	0.893591i	$-0.351822\pi$
$193$	12.4721	0.897764	0.448882	+	0.893591i	$0.351822\pi$
$194$	0	0
$195$	− 1.88854i	− 0.135241i
$196$	0	0
$197$	− 4.00000i	− 0.284988i	−0.989796	−	0.142494i	$-0.954488\pi$
$197$	− 4.00000i	− 0.284988i	0.989796	−	0.142494i	$-0.0455122\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	1.88854	0.133208
$202$	0	0
$203$	1.52786i	0.107235i
$204$	0	0
$205$	2.47214i	0.172661i
$206$	0	0
$207$	−9.52786	−0.662232
$208$	0	0
$209$	4.94427	0.342002
$210$	0	0
$211$	− 11.4164i	− 0.785938i	−0.919552	−	0.392969i	$-0.871448\pi$
$211$	− 11.4164i	− 0.785938i	0.919552	−	0.392969i	$-0.128552\pi$
$212$	0	0
$213$	6.11146i	0.418750i
$214$	0	0
$215$	−11.0557	−0.753994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 18.4721i	− 1.24823i
$220$	0	0
$221$	− 2.47214i	− 0.166294i
$222$	0	0
$223$	28.9443	1.93825	0.969126	−	0.246566i	$-0.0793023\pi$
$223$	28.9443	1.93825	0.969126	+	0.246566i	$0.0793023\pi$
$224$	0	0
$225$	5.11146	0.340764
$226$	0	0
$227$	3.70820i	0.246122i	0.992399	+	0.123061i	$0.0392711\pi$
$227$	3.70820i	0.246122i	−0.992399	+	0.123061i	$0.960729\pi$
$228$	0	0
$229$	24.6525i	1.62908i	0.580106	+	0.814541i	$0.303011\pi$
$229$	24.6525i	1.62908i	−0.580106	+	0.814541i	$0.696989\pi$
$230$	0	0
$231$	4.94427	0.325309
$232$	0	0
$233$	−19.8885	−1.30294	−0.651471	−	0.758674i	$-0.725848\pi$
$233$	−19.8885	−1.30294	−0.651471	+	0.758674i	$0.725848\pi$
$234$	0	0
$235$	− 16.0000i	− 1.04372i
$236$	0	0
$237$	6.11146i	0.396982i
$238$	0	0
$239$	−9.52786	−0.616306	−0.308153	−	0.951337i	$-0.599711\pi$
$239$	−9.52786	−0.616306	−0.308153	+	0.951337i	$0.599711\pi$
$240$	0	0
$241$	−14.9443	−0.962645	−0.481323	−	0.876544i	$-0.659843\pi$
$241$	−14.9443	−0.962645	−0.481323	+	0.876544i	$0.659843\pi$
$242$	0	0
$243$	13.5967i	0.872232i
$244$	0	0
$245$	1.23607i	0.0789695i
$246$	0	0
$247$	1.52786	0.0972157
$248$	0	0
$249$	11.4164	0.723485
$250$	0	0
$251$	16.6525i	1.05109i	0.850764	+	0.525547i	$0.176139\pi$
$251$	16.6525i	1.05109i	−0.850764	+	0.525547i	$0.823861\pi$
$252$	0	0
$253$	− 25.8885i	− 1.62760i
$254$	0	0
$255$	3.05573	0.191357
$256$	0	0
$257$	−7.88854	−0.492074	−0.246037	−	0.969260i	$-0.579128\pi$
$257$	−7.88854	−0.492074	−0.246037	+	0.969260i	$0.579128\pi$
$258$	0	0
$259$	− 6.47214i	− 0.402159i
$260$	0	0
$261$	− 2.24922i	− 0.139223i
$262$	0	0
$263$	20.9443	1.29148	0.645740	−	0.763558i	$-0.276549\pi$
$263$	20.9443	1.29148	0.645740	+	0.763558i	$0.276549\pi$
$264$	0	0
$265$	11.0557	0.679148
$266$	0	0
$267$	− 2.47214i	− 0.151292i
$268$	0	0
$269$	27.1246i	1.65382i	0.562337	+	0.826908i	$0.309903\pi$
$269$	27.1246i	1.65382i	−0.562337	+	0.826908i	$0.690097\pi$
$270$	0	0
$271$	3.05573	0.185622	0.0928111	−	0.995684i	$-0.470415\pi$
$271$	3.05573	0.185622	0.0928111	+	0.995684i	$0.470415\pi$
$272$	0	0
$273$	1.52786	0.0924705
$274$	0	0
$275$	13.8885i	0.837511i
$276$	0	0
$277$	− 20.0000i	− 1.20168i	−0.799368	−	0.600842i	$-0.794832\pi$
$277$	− 20.0000i	− 1.20168i	0.799368	−	0.600842i	$-0.205168\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	− 24.6525i	− 1.46544i	−0.680532	−	0.732719i	$-0.738251\pi$
$283$	− 24.6525i	− 1.46544i	0.680532	−	0.732719i	$-0.261749\pi$
$284$	0	0
$285$	1.88854i	0.111868i
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	13.5279i	0.793017i
$292$	0	0
$293$	− 27.7082i	− 1.61873i	−0.587306	−	0.809365i	$-0.699811\pi$
$293$	− 27.7082i	− 1.61873i	0.587306	−	0.809365i	$-0.300189\pi$
$294$	0	0
$295$	−11.4164	−0.664689
$296$	0	0
$297$	−22.1115	−1.28304
$298$	0	0
$299$	− 8.00000i	− 0.462652i
$300$	0	0
$301$	− 8.94427i	− 0.515539i
$302$	0	0
$303$	−17.5279	−1.00695
$304$	0	0
$305$	1.52786	0.0874852
$306$	0	0
$307$	21.5967i	1.23259i	0.787515	+	0.616296i	$0.211367\pi$
$307$	21.5967i	1.23259i	−0.787515	+	0.616296i	$0.788633\pi$
$308$	0	0
$309$	9.88854i	0.562540i
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−7.88854	−0.445887	−0.222943	−	0.974831i	$-0.571567\pi$
$313$	−7.88854	−0.445887	−0.222943	+	0.974831i	$0.571567\pi$
$314$	0	0
$315$	− 1.81966i	− 0.102526i
$316$	0	0
$317$	− 21.8885i	− 1.22938i	−0.788768	−	0.614692i	$-0.789280\pi$
$317$	− 21.8885i	− 1.22938i	0.788768	−	0.614692i	$-0.210720\pi$
$318$	0	0
$319$	6.11146	0.342176
$320$	0	0
$321$	−11.7771	−0.657333
$322$	0	0
$323$	2.47214i	0.137553i
$324$	0	0
$325$	4.29180i	0.238066i
$326$	0	0
$327$	1.88854	0.104437
$328$	0	0
$329$	12.9443	0.713641
$330$	0	0
$331$	− 7.05573i	− 0.387818i	−0.981020	−	0.193909i	$-0.937883\pi$
$331$	− 7.05573i	− 0.387818i	0.981020	−	0.193909i	$-0.0621166\pi$
$332$	0	0
$333$	9.52786i	0.522124i
$334$	0	0
$335$	1.88854	0.103182
$336$	0	0
$337$	−9.41641	−0.512944	−0.256472	−	0.966552i	$-0.582560\pi$
$337$	−9.41641	−0.512944	−0.256472	+	0.966552i	$0.582560\pi$
$338$	0	0
$339$	16.5836i	0.900697i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	9.88854	0.532381
$346$	0	0
$347$	15.0557i	0.808234i	0.914707	+	0.404117i	$0.132421\pi$
$347$	15.0557i	0.808234i	−0.914707	+	0.404117i	$0.867579\pi$
$348$	0	0
$349$	4.29180i	0.229735i	0.993381	+	0.114867i	$0.0366443\pi$
$349$	4.29180i	0.229735i	−0.993381	+	0.114867i	$0.963356\pi$
$350$	0	0
$351$	−6.83282	−0.364709
$352$	0	0
$353$	−17.0557	−0.907785	−0.453892	−	0.891056i	$-0.649965\pi$
$353$	−17.0557	−0.907785	−0.453892	+	0.891056i	$0.649965\pi$
$354$	0	0
$355$	6.11146i	0.324362i
$356$	0	0
$357$	2.47214i	0.130839i
$358$	0	0
$359$	3.41641	0.180311	0.0901556	−	0.995928i	$-0.471264\pi$
$359$	3.41641	0.180311	0.0901556	+	0.995928i	$0.471264\pi$
$360$	0	0
$361$	17.4721	0.919586
$362$	0	0
$363$	− 6.18034i	− 0.324384i
$364$	0	0
$365$	− 18.4721i	− 0.966876i
$366$	0	0
$367$	−28.9443	−1.51088	−0.755439	−	0.655219i	$-0.772577\pi$
$367$	−28.9443	−1.51088	−0.755439	+	0.655219i	$0.772577\pi$
$368$	0	0
$369$	2.94427	0.153273
$370$	0	0
$371$	8.94427i	0.464363i
$372$	0	0
$373$	32.9443i	1.70579i	0.522083	+	0.852895i	$0.325155\pi$
$373$	32.9443i	1.70579i	−0.522083	+	0.852895i	$0.674845\pi$
$374$	0	0
$375$	−12.9443	−0.668439
$376$	0	0
$377$	1.88854	0.0972650
$378$	0	0
$379$	− 37.8885i	− 1.94620i	−0.230375	−	0.973102i	$-0.573995\pi$
$379$	− 37.8885i	− 1.94620i	0.230375	−	0.973102i	$-0.426005\pi$
$380$	0	0
$381$	24.0000i	1.22956i
$382$	0	0
$383$	−3.05573	−0.156140	−0.0780702	−	0.996948i	$-0.524876\pi$
$383$	−3.05573	−0.156140	−0.0780702	+	0.996948i	$0.524876\pi$
$384$	0	0
$385$	4.94427	0.251983
$386$	0	0
$387$	13.1672i	0.669326i
$388$	0	0
$389$	− 19.4164i	− 0.984451i	−0.870468	−	0.492225i	$-0.836184\pi$
$389$	− 19.4164i	− 0.984451i	0.870468	−	0.492225i	$-0.163816\pi$
$390$	0	0
$391$	12.9443	0.654620
$392$	0	0
$393$	17.5279	0.884164
$394$	0	0
$395$	6.11146i	0.307501i
$396$	0	0
$397$	3.12461i	0.156820i	0.996921	+	0.0784099i	$0.0249843\pi$
$397$	3.12461i	0.156820i	−0.996921	+	0.0784099i	$0.975016\pi$
$398$	0	0
$399$	−1.52786	−0.0764889
$400$	0	0
$401$	−0.472136	−0.0235773	−0.0117887	−	0.999931i	$-0.503753\pi$
$401$	−0.472136	−0.0235773	−0.0117887	+	0.999931i	$0.503753\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 2.98684i	− 0.148417i
$406$	0	0
$407$	−25.8885	−1.28325
$408$	0	0
$409$	14.9443	0.738947	0.369473	−	0.929241i	$-0.379538\pi$
$409$	14.9443	0.738947	0.369473	+	0.929241i	$0.379538\pi$
$410$	0	0
$411$	12.3607i	0.609707i
$412$	0	0
$413$	− 9.23607i	− 0.454477i
$414$	0	0
$415$	11.4164	0.560409
$416$	0	0
$417$	21.3050	1.04331
$418$	0	0
$419$	6.18034i	0.301929i	0.988539	+	0.150965i	$0.0482380\pi$
$419$	6.18034i	0.301929i	−0.988539	+	0.150965i	$0.951762\pi$
$420$	0	0
$421$	12.0000i	0.584844i	0.956289	+	0.292422i	$0.0944612\pi$
$421$	12.0000i	0.584844i	−0.956289	+	0.292422i	$0.905539\pi$
$422$	0	0
$423$	−19.0557	−0.926521
$424$	0	0
$425$	−6.94427	−0.336847
$426$	0	0
$427$	1.23607i	0.0598175i
$428$	0	0
$429$	− 6.11146i	− 0.295064i
$430$	0	0
$431$	−32.3607	−1.55876	−0.779380	−	0.626552i	$-0.784466\pi$
$431$	−32.3607	−1.55876	−0.779380	+	0.626552i	$0.784466\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	0	0
$435$	2.33437i	0.111924i
$436$	0	0
$437$	8.00000i	0.382692i
$438$	0	0
$439$	20.9443	0.999616	0.499808	−	0.866136i	$-0.333404\pi$
$439$	20.9443	0.999616	0.499808	+	0.866136i	$0.333404\pi$
$440$	0	0
$441$	1.47214	0.0701017
$442$	0	0
$443$	24.3607i	1.15741i	0.815537	+	0.578705i	$0.196442\pi$
$443$	24.3607i	1.15741i	−0.815537	+	0.578705i	$0.803558\pi$
$444$	0	0
$445$	− 2.47214i	− 0.117190i
$446$	0	0
$447$	−20.9443	−0.990630
$448$	0	0
$449$	23.8885	1.12737	0.563685	−	0.825990i	$-0.309383\pi$
$449$	23.8885	1.12737	0.563685	+	0.825990i	$0.309383\pi$
$450$	0	0
$451$	8.00000i	0.376705i
$452$	0	0
$453$	8.00000i	0.375873i
$454$	0	0
$455$	1.52786	0.0716274
$456$	0	0
$457$	−20.4721	−0.957646	−0.478823	−	0.877911i	$-0.658936\pi$
$457$	−20.4721	−0.957646	−0.478823	+	0.877911i	$0.658936\pi$
$458$	0	0
$459$	− 11.0557i	− 0.516037i
$460$	0	0
$461$	40.0689i	1.86619i	0.359625	+	0.933097i	$0.382905\pi$
$461$	40.0689i	1.86619i	−0.359625	+	0.933097i	$0.617095\pi$
$462$	0	0
$463$	−1.88854	−0.0877681	−0.0438840	−	0.999037i	$-0.513973\pi$
$463$	−1.88854	−0.0877681	−0.0438840	+	0.999037i	$0.513973\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	24.6525i	1.14078i	0.821374	+	0.570390i	$0.193208\pi$
$467$	24.6525i	1.14078i	−0.821374	+	0.570390i	$0.806792\pi$
$468$	0	0
$469$	1.52786i	0.0705502i
$470$	0	0
$471$	−27.4164	−1.26328
$472$	0	0
$473$	−35.7771	−1.64503
$474$	0	0
$475$	− 4.29180i	− 0.196921i
$476$	0	0
$477$	− 13.1672i	− 0.602884i
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−8.00000	−0.364769
$482$	0	0
$483$	8.00000i	0.364013i
$484$	0	0
$485$	13.5279i	0.614269i
$486$	0	0
$487$	−22.4721	−1.01831	−0.509155	−	0.860675i	$-0.670042\pi$
$487$	−22.4721	−1.01831	−0.509155	+	0.860675i	$0.670042\pi$
$488$	0	0
$489$	14.8328	0.670763
$490$	0	0
$491$	14.4721i	0.653118i	0.945177	+	0.326559i	$0.105889\pi$
$491$	14.4721i	0.653118i	−0.945177	+	0.326559i	$0.894111\pi$
$492$	0	0
$493$	3.05573i	0.137623i
$494$	0	0
$495$	−7.27864	−0.327151
$496$	0	0
$497$	−4.94427	−0.221781
$498$	0	0
$499$	24.3607i	1.09053i	0.838262	+	0.545267i	$0.183572\pi$
$499$	24.3607i	1.09053i	−0.838262	+	0.545267i	$0.816428\pi$
$500$	0	0
$501$	− 6.11146i	− 0.273040i
$502$	0	0
$503$	−11.0557	−0.492951	−0.246475	−	0.969149i	$-0.579272\pi$
$503$	−11.0557	−0.492951	−0.246475	+	0.969149i	$0.579272\pi$
$504$	0	0
$505$	−17.5279	−0.779980
$506$	0	0
$507$	14.1803i	0.629771i
$508$	0	0
$509$	− 0.652476i	− 0.0289205i	−0.999895	−	0.0144602i	$-0.995397\pi$
$509$	− 0.652476i	− 0.0289205i	0.999895	−	0.0144602i	$-0.00460300\pi$
$510$	0	0
$511$	14.9443	0.661096
$512$	0	0
$513$	6.83282	0.301676
$514$	0	0
$515$	9.88854i	0.435741i
$516$	0	0
$517$	− 51.7771i	− 2.27715i
$518$	0	0
$519$	4.58359	0.201197
$520$	0	0
$521$	−26.9443	−1.18045	−0.590225	−	0.807239i	$-0.700961\pi$
$521$	−26.9443	−1.18045	−0.590225	+	0.807239i	$0.700961\pi$
$522$	0	0
$523$	− 1.81966i	− 0.0795682i	−0.999208	−	0.0397841i	$-0.987333\pi$
$523$	− 1.81966i	− 0.0795682i	0.999208	−	0.0397841i	$-0.0126670\pi$
$524$	0	0
$525$	− 4.29180i	− 0.187309i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	18.8885	0.821241
$530$	0	0
$531$	13.5967i	0.590049i
$532$	0	0
$533$	2.47214i	0.107080i
$534$	0	0
$535$	−11.7771	−0.509168
$536$	0	0
$537$	8.00000	0.345225
$538$	0	0
$539$	4.00000i	0.172292i
$540$	0	0
$541$	− 0.944272i	− 0.0405974i	−0.999794	−	0.0202987i	$-0.993538\pi$
$541$	− 0.944272i	− 0.0405974i	0.999794	−	0.0202987i	$-0.00646172\pi$
$542$	0	0
$543$	31.1935	1.33864
$544$	0	0
$545$	1.88854	0.0808963
$546$	0	0
$547$	34.8328i	1.48934i	0.667431	+	0.744672i	$0.267394\pi$
$547$	34.8328i	1.48934i	−0.667431	+	0.744672i	$0.732606\pi$
$548$	0	0
$549$	− 1.81966i	− 0.0776612i
$550$	0	0
$551$	−1.88854	−0.0804547
$552$	0	0
$553$	−4.94427	−0.210252
$554$	0	0
$555$	− 9.88854i	− 0.419745i
$556$	0	0
$557$	− 5.88854i	− 0.249506i	−0.992188	−	0.124753i	$-0.960186\pi$
$557$	− 5.88854i	− 0.249506i	0.992188	−	0.124753i	$-0.0398138\pi$
$558$	0	0
$559$	−11.0557	−0.467607
$560$	0	0
$561$	9.88854	0.417495
$562$	0	0
$563$	− 42.5410i	− 1.79289i	−0.443155	−	0.896445i	$-0.646141\pi$
$563$	− 42.5410i	− 1.79289i	0.443155	−	0.896445i	$-0.353859\pi$
$564$	0	0
$565$	16.5836i	0.697677i
$566$	0	0
$567$	2.41641	0.101480
$568$	0	0
$569$	14.5836	0.611376	0.305688	−	0.952132i	$-0.401114\pi$
$569$	14.5836	0.611376	0.305688	+	0.952132i	$0.401114\pi$
$570$	0	0
$571$	− 31.7771i	− 1.32983i	−0.746919	−	0.664915i	$-0.768468\pi$
$571$	− 31.7771i	− 1.32983i	0.746919	−	0.664915i	$-0.231532\pi$
$572$	0	0
$573$	22.1115i	0.923719i
$574$	0	0
$575$	−22.4721	−0.937153
$576$	0	0
$577$	−20.8328	−0.867281	−0.433641	−	0.901086i	$-0.642771\pi$
$577$	−20.8328	−0.867281	−0.433641	+	0.901086i	$0.642771\pi$
$578$	0	0
$579$	15.4164i	0.640684i
$580$	0	0
$581$	9.23607i	0.383177i
$582$	0	0
$583$	35.7771	1.48174
$584$	0	0
$585$	−2.24922	−0.0929940
$586$	0	0
$587$	14.7639i	0.609373i	0.952453	+	0.304686i	$0.0985516\pi$
$587$	14.7639i	0.609373i	−0.952453	+	0.304686i	$0.901448\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	4.94427	0.203380
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	2.47214i	0.101348i
$596$	0	0
$597$	29.6656i	1.21413i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	4.11146	0.167710	0.0838549	−	0.996478i	$-0.473277\pi$
$601$	4.11146	0.167710	0.0838549	+	0.996478i	$0.473277\pi$
$602$	0	0
$603$	− 2.24922i	− 0.0915955i
$604$	0	0
$605$	− 6.18034i	− 0.251267i
$606$	0	0
$607$	−9.88854	−0.401364	−0.200682	−	0.979656i	$-0.564316\pi$
$607$	−9.88854	−0.401364	−0.200682	+	0.979656i	$0.564316\pi$
$608$	0	0
$609$	−1.88854	−0.0765277
$610$	0	0
$611$	− 16.0000i	− 0.647291i
$612$	0	0
$613$	− 14.4721i	− 0.584524i	−0.956338	−	0.292262i	$-0.905592\pi$
$613$	− 14.4721i	− 0.584524i	0.956338	−	0.292262i	$-0.0944079\pi$
$614$	0	0
$615$	−3.05573	−0.123219
$616$	0	0
$617$	23.5279	0.947196	0.473598	−	0.880741i	$-0.342955\pi$
$617$	23.5279	0.947196	0.473598	+	0.880741i	$0.342955\pi$
$618$	0	0
$619$	− 9.81966i	− 0.394685i	−0.980335	−	0.197343i	$-0.936769\pi$
$619$	− 9.81966i	− 0.394685i	0.980335	−	0.197343i	$-0.0632312\pi$
$620$	0	0
$621$	− 35.7771i	− 1.43569i
$622$	0	0
$623$	2.00000	0.0801283
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	6.11146i	0.244068i
$628$	0	0
$629$	− 12.9443i	− 0.516122i
$630$	0	0
$631$	17.8885	0.712132	0.356066	−	0.934461i	$-0.384118\pi$
$631$	17.8885	0.712132	0.356066	+	0.934461i	$0.384118\pi$
$632$	0	0
$633$	14.1115	0.560880
$634$	0	0
$635$	24.0000i	0.952411i
$636$	0	0
$637$	1.23607i	0.0489748i
$638$	0	0
$639$	7.27864	0.287939
$640$	0	0
$641$	−10.3607	−0.409222	−0.204611	−	0.978843i	$-0.565593\pi$
$641$	−10.3607	−0.409222	−0.204611	+	0.978843i	$0.565593\pi$
$642$	0	0
$643$	− 26.5410i	− 1.04668i	−0.852125	−	0.523338i	$-0.824687\pi$
$643$	− 26.5410i	− 1.04668i	0.852125	−	0.523338i	$-0.175313\pi$
$644$	0	0
$645$	− 13.6656i	− 0.538084i
$646$	0	0
$647$	−17.8885	−0.703271	−0.351636	−	0.936137i	$-0.614374\pi$
$647$	−17.8885	−0.703271	−0.351636	+	0.936137i	$0.614374\pi$
$648$	0	0
$649$	−36.9443	−1.45019
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 12.5836i	− 0.492434i	−0.969215	−	0.246217i	$-0.920812\pi$
$653$	− 12.5836i	− 0.492434i	0.969215	−	0.246217i	$-0.0791876\pi$
$654$	0	0
$655$	17.5279	0.684870
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	− 42.8328i	− 1.66853i	−0.551364	−	0.834265i	$-0.685892\pi$
$659$	− 42.8328i	− 1.66853i	0.551364	−	0.834265i	$-0.314108\pi$
$660$	0	0
$661$	− 22.7639i	− 0.885414i	−0.896666	−	0.442707i	$-0.854018\pi$
$661$	− 22.7639i	− 0.885414i	0.896666	−	0.442707i	$-0.145982\pi$
$662$	0	0
$663$	3.05573	0.118675
$664$	0	0
$665$	−1.52786	−0.0592480
$666$	0	0
$667$	9.88854i	0.382886i
$668$	0	0
$669$	35.7771i	1.38322i
$670$	0	0
$671$	4.94427	0.190872
$672$	0	0
$673$	−11.8885	−0.458270	−0.229135	−	0.973395i	$-0.573590\pi$
$673$	−11.8885	−0.458270	−0.229135	+	0.973395i	$0.573590\pi$
$674$	0	0
$675$	19.1935i	0.738758i
$676$	0	0
$677$	− 19.1246i	− 0.735019i	−0.930020	−	0.367509i	$-0.880211\pi$
$677$	− 19.1246i	− 0.735019i	0.930020	−	0.367509i	$-0.119789\pi$
$678$	0	0
$679$	−10.9443	−0.420003
$680$	0	0
$681$	−4.58359	−0.175644
$682$	0	0
$683$	30.4721i	1.16598i	0.812478	+	0.582992i	$0.198118\pi$
$683$	30.4721i	1.16598i	−0.812478	+	0.582992i	$0.801882\pi$
$684$	0	0
$685$	12.3607i	0.472277i
$686$	0	0
$687$	−30.4721	−1.16258
$688$	0	0
$689$	11.0557	0.421190
$690$	0	0
$691$	− 20.2918i	− 0.771936i	−0.922512	−	0.385968i	$-0.873867\pi$
$691$	− 20.2918i	− 0.771936i	0.922512	−	0.385968i	$-0.126133\pi$
$692$	0	0
$693$	− 5.88854i	− 0.223687i
$694$	0	0
$695$	21.3050	0.808143
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	− 24.5836i	− 0.929837i
$700$	0	0
$701$	− 16.3607i	− 0.617934i	−0.951073	−	0.308967i	$-0.900017\pi$
$701$	− 16.3607i	− 0.617934i	0.951073	−	0.308967i	$-0.0999833\pi$
$702$	0	0
$703$	8.00000	0.301726
$704$	0	0
$705$	19.7771	0.744848
$706$	0	0
$707$	− 14.1803i	− 0.533307i
$708$	0	0
$709$	6.47214i	0.243066i	0.992587	+	0.121533i	$0.0387811\pi$
$709$	6.47214i	0.243066i	−0.992587	+	0.121533i	$0.961219\pi$
$710$	0	0
$711$	7.27864	0.272970
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 6.11146i	− 0.228556i
$716$	0	0
$717$	− 11.7771i	− 0.439823i
$718$	0	0
$719$	19.0557	0.710659	0.355329	−	0.934741i	$-0.384369\pi$
$719$	19.0557	0.710659	0.355329	+	0.934741i	$0.384369\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	− 18.4721i	− 0.686986i
$724$	0	0
$725$	− 5.30495i	− 0.197021i
$726$	0	0
$727$	14.8328	0.550119	0.275059	−	0.961427i	$-0.411302\pi$
$727$	14.8328	0.550119	0.275059	+	0.961427i	$0.411302\pi$
$728$	0	0
$729$	−24.0557	−0.890953
$730$	0	0
$731$	− 17.8885i	− 0.661632i
$732$	0	0
$733$	− 32.0689i	− 1.18449i	−0.805757	−	0.592246i	$-0.798242\pi$
$733$	− 32.0689i	− 1.18449i	0.805757	−	0.592246i	$-0.201758\pi$
$734$	0	0
$735$	−1.52786	−0.0563561
$736$	0	0
$737$	6.11146	0.225118
$738$	0	0
$739$	− 21.8885i	− 0.805183i	−0.915380	−	0.402592i	$-0.868110\pi$
$739$	− 21.8885i	− 0.805183i	0.915380	−	0.402592i	$-0.131890\pi$
$740$	0	0
$741$	1.88854i	0.0693774i
$742$	0	0
$743$	25.5279	0.936527	0.468263	−	0.883589i	$-0.344880\pi$
$743$	25.5279	0.936527	0.468263	+	0.883589i	$0.344880\pi$
$744$	0	0
$745$	−20.9443	−0.767339
$746$	0	0
$747$	− 13.5967i	− 0.497479i
$748$	0	0
$749$	− 9.52786i	− 0.348141i
$750$	0	0
$751$	−16.3607	−0.597010	−0.298505	−	0.954408i	$-0.596488\pi$
$751$	−16.3607	−0.597010	−0.298505	+	0.954408i	$0.596488\pi$
$752$	0	0
$753$	−20.5836	−0.750108
$754$	0	0
$755$	8.00000i	0.291150i
$756$	0	0
$757$	43.4164i	1.57800i	0.614396	+	0.788998i	$0.289400\pi$
$757$	43.4164i	1.57800i	−0.614396	+	0.788998i	$0.710600\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	27.8885	1.01096	0.505479	−	0.862839i	$-0.331316\pi$
$761$	27.8885	1.01096	0.505479	+	0.862839i	$0.331316\pi$
$762$	0	0
$763$	1.52786i	0.0553124i
$764$	0	0
$765$	− 3.63932i	− 0.131580i
$766$	0	0
$767$	−11.4164	−0.412223
$768$	0	0
$769$	17.0557	0.615045	0.307523	−	0.951541i	$-0.400500\pi$
$769$	17.0557	0.615045	0.307523	+	0.951541i	$0.400500\pi$
$770$	0	0
$771$	− 9.75078i	− 0.351166i
$772$	0	0
$773$	11.1246i	0.400124i	0.979783	+	0.200062i	$0.0641144\pi$
$773$	11.1246i	0.400124i	−0.979783	+	0.200062i	$0.935886\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.00000	0.286998
$778$	0	0
$779$	− 2.47214i	− 0.0885735i
$780$	0	0
$781$	19.7771i	0.707680i
$782$	0	0
$783$	8.44582	0.301829
$784$	0	0
$785$	−27.4164	−0.978534
$786$	0	0
$787$	4.87539i	0.173789i	0.996218	+	0.0868944i	$0.0276943\pi$
$787$	4.87539i	0.173789i	−0.996218	+	0.0868944i	$0.972306\pi$
$788$	0	0
$789$	25.8885i	0.921657i
$790$	0	0
$791$	−13.4164	−0.477033
$792$	0	0
$793$	1.52786	0.0542560
$794$	0	0
$795$	13.6656i	0.484670i
$796$	0	0
$797$	33.2361i	1.17728i	0.808395	+	0.588641i	$0.200337\pi$
$797$	33.2361i	1.17728i	−0.808395	+	0.588641i	$0.799663\pi$
$798$	0	0
$799$	25.8885	0.915871
$800$	0	0
$801$	−2.94427	−0.104031
$802$	0	0
$803$	− 59.7771i	− 2.10949i
$804$	0	0
$805$	8.00000i	0.281963i
$806$	0	0
$807$	−33.5279	−1.18024
$808$	0	0
$809$	5.41641	0.190431	0.0952154	−	0.995457i	$-0.469646\pi$
$809$	5.41641	0.190431	0.0952154	+	0.995457i	$0.469646\pi$
$810$	0	0
$811$	27.1246i	0.952474i	0.879317	+	0.476237i	$0.158000\pi$
$811$	27.1246i	0.952474i	−0.879317	+	0.476237i	$0.842000\pi$
$812$	0	0
$813$	3.77709i	0.132468i
$814$	0	0
$815$	14.8328	0.519571
$816$	0	0
$817$	11.0557	0.386791
$818$	0	0
$819$	− 1.81966i	− 0.0635841i
$820$	0	0
$821$	− 4.00000i	− 0.139601i	−0.997561	−	0.0698005i	$-0.977764\pi$
$821$	− 4.00000i	− 0.139601i	0.997561	−	0.0698005i	$-0.0222363\pi$
$822$	0	0
$823$	30.8328	1.07476	0.537382	−	0.843339i	$-0.319413\pi$
$823$	30.8328	1.07476	0.537382	+	0.843339i	$0.319413\pi$
$824$	0	0
$825$	−17.1672	−0.597685
$826$	0	0
$827$	35.4164i	1.23155i	0.787922	+	0.615775i	$0.211157\pi$
$827$	35.4164i	1.23155i	−0.787922	+	0.615775i	$0.788843\pi$
$828$	0	0
$829$	− 27.1246i	− 0.942077i	−0.882113	−	0.471038i	$-0.843879\pi$
$829$	− 27.1246i	− 0.942077i	0.882113	−	0.471038i	$-0.156121\pi$
$830$	0	0
$831$	24.7214	0.857574
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	− 6.11146i	− 0.211496i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.0000	1.38095	0.690477	−	0.723355i	$-0.257401\pi$
$839$	40.0000	1.38095	0.690477	+	0.723355i	$0.257401\pi$
$840$	0	0
$841$	26.6656	0.919505
$842$	0	0
$843$	7.41641i	0.255435i
$844$	0	0
$845$	14.1803i	0.487819i
$846$	0	0
$847$	5.00000	0.171802
$848$	0	0
$849$	30.4721	1.04580
$850$	0	0
$851$	− 41.8885i	− 1.43592i
$852$	0	0
$853$	− 33.2361i	− 1.13798i	−0.822344	−	0.568991i	$-0.807334\pi$
$853$	− 33.2361i	− 1.13798i	0.822344	−	0.568991i	$-0.192666\pi$
$854$	0	0
$855$	2.24922	0.0769218
$856$	0	0
$857$	56.8328	1.94137	0.970686	−	0.240351i	$-0.0772626\pi$
$857$	56.8328	1.94137	0.970686	+	0.240351i	$0.0772626\pi$
$858$	0	0
$859$	− 30.1803i	− 1.02974i	−0.857268	−	0.514870i	$-0.827840\pi$
$859$	− 30.1803i	− 1.02974i	0.857268	−	0.514870i	$-0.172160\pi$
$860$	0	0
$861$	− 2.47214i	− 0.0842502i
$862$	0	0
$863$	−33.8885	−1.15358	−0.576790	−	0.816893i	$-0.695695\pi$
$863$	−33.8885	−1.15358	−0.576790	+	0.816893i	$0.695695\pi$
$864$	0	0
$865$	4.58359	0.155847
$866$	0	0
$867$	− 16.0689i	− 0.545728i
$868$	0	0
$869$	19.7771i	0.670892i
$870$	0	0
$871$	1.88854	0.0639909
$872$	0	0
$873$	16.1115	0.545290
$874$	0	0
$875$	− 10.4721i	− 0.354023i
$876$	0	0
$877$	− 0.360680i	− 0.0121793i	−0.999981	−	0.00608965i	$-0.998062\pi$
$877$	− 0.360680i	− 0.0121793i	0.999981	−	0.00608965i	$-0.00193841\pi$
$878$	0	0
$879$	34.2492	1.15520
$880$	0	0
$881$	−20.8328	−0.701875	−0.350938	−	0.936399i	$-0.614137\pi$
$881$	−20.8328	−0.701875	−0.350938	+	0.936399i	$0.614137\pi$
$882$	0	0
$883$	− 38.4721i	− 1.29469i	−0.762197	−	0.647345i	$-0.775879\pi$
$883$	− 38.4721i	− 1.29469i	0.762197	−	0.647345i	$-0.224121\pi$
$884$	0	0
$885$	− 14.1115i	− 0.474351i
$886$	0	0
$887$	−14.8328	−0.498037	−0.249019	−	0.968499i	$-0.580108\pi$
$887$	−14.8328	−0.498037	−0.249019	+	0.968499i	$0.580108\pi$
$888$	0	0
$889$	−19.4164	−0.651205
$890$	0	0
$891$	− 9.66563i	− 0.323811i
$892$	0	0
$893$	16.0000i	0.535420i
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	9.88854	0.330169
$898$	0	0
$899$	0	0
$900$	0	0
$901$	17.8885i	0.595954i
$902$	0	0
$903$	11.0557	0.367912
$904$	0	0
$905$	31.1935	1.03691
$906$	0	0
$907$	35.4164i	1.17598i	0.808867	+	0.587991i	$0.200081\pi$
$907$	35.4164i	1.17598i	−0.808867	+	0.587991i	$0.799919\pi$
$908$	0	0
$909$	20.8754i	0.692393i
$910$	0	0
$911$	54.4721	1.80474	0.902371	−	0.430960i	$-0.141825\pi$
$911$	54.4721	1.80474	0.902371	+	0.430960i	$0.141825\pi$
$912$	0	0
$913$	36.9443	1.22268
$914$	0	0
$915$	1.88854i	0.0624333i
$916$	0	0
$917$	14.1803i	0.468276i
$918$	0	0
$919$	46.8328	1.54487	0.772436	−	0.635093i	$-0.219038\pi$
$919$	46.8328	1.54487	0.772436	+	0.635093i	$0.219038\pi$
$920$	0	0
$921$	−26.6950	−0.879632
$922$	0	0
$923$	6.11146i	0.201161i
$924$	0	0
$925$	22.4721i	0.738879i
$926$	0	0
$927$	11.7771	0.386810
$928$	0	0
$929$	−27.8885	−0.914993	−0.457497	−	0.889211i	$-0.651254\pi$
$929$	−27.8885	−0.914993	−0.457497	+	0.889211i	$0.651254\pi$
$930$	0	0
$931$	− 1.23607i	− 0.0405105i
$932$	0	0
$933$	− 9.88854i	− 0.323736i
$934$	0	0
$935$	9.88854	0.323390
$936$	0	0
$937$	26.9443	0.880231	0.440115	−	0.897941i	$-0.354937\pi$
$937$	26.9443	0.880231	0.440115	+	0.897941i	$0.354937\pi$
$938$	0	0
$939$	− 9.75078i	− 0.318205i
$940$	0	0
$941$	0.652476i	0.0212701i	0.999943	+	0.0106351i	$0.00338531\pi$
$941$	0.652476i	0.0212701i	−0.999943	+	0.0106351i	$0.996615\pi$
$942$	0	0
$943$	−12.9443	−0.421523
$944$	0	0
$945$	6.83282	0.222272
$946$	0	0
$947$	− 39.0557i	− 1.26914i	−0.772865	−	0.634570i	$-0.781177\pi$
$947$	− 39.0557i	− 1.26914i	0.772865	−	0.634570i	$-0.218823\pi$
$948$	0	0
$949$	− 18.4721i	− 0.599631i
$950$	0	0
$951$	27.0557	0.877342
$952$	0	0
$953$	−31.8885	−1.03297	−0.516486	−	0.856296i	$-0.672760\pi$
$953$	−31.8885	−1.03297	−0.516486	+	0.856296i	$0.672760\pi$
$954$	0	0
$955$	22.1115i	0.715510i
$956$	0	0
$957$	7.55418i	0.244192i
$958$	0	0
$959$	−10.0000	−0.322917
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	14.0263i	0.451992i
$964$	0	0
$965$	15.4164i	0.496272i
$966$	0	0
$967$	−12.5836	−0.404661	−0.202331	−	0.979317i	$-0.564852\pi$
$967$	−12.5836	−0.404661	−0.202331	+	0.979317i	$0.564852\pi$
$968$	0	0
$969$	−3.05573	−0.0981641
$970$	0	0
$971$	15.3475i	0.492525i	0.969203	+	0.246263i	$0.0792026\pi$
$971$	15.3475i	0.492525i	−0.969203	+	0.246263i	$0.920797\pi$
$972$	0	0
$973$	17.2361i	0.552563i
$974$	0	0
$975$	−5.30495	−0.169894
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	− 8.00000i	− 0.255681i
$980$	0	0
$981$	− 2.24922i	− 0.0718122i
$982$	0	0
$983$	−27.7771	−0.885952	−0.442976	−	0.896534i	$-0.646077\pi$
$983$	−27.7771	−0.885952	−0.442976	+	0.896534i	$0.646077\pi$
$984$	0	0
$985$	4.94427	0.157538
$986$	0	0
$987$	16.0000i	0.509286i
$988$	0	0
$989$	− 57.8885i	− 1.84075i
$990$	0	0
$991$	−30.8328	−0.979437	−0.489718	−	0.871881i	$-0.662900\pi$
$991$	−30.8328	−0.979437	−0.489718	+	0.871881i	$0.662900\pi$
$992$	0	0
$993$	8.72136	0.276764
$994$	0	0
$995$	29.6656i	0.940464i
$996$	0	0
$997$	− 30.7639i	− 0.974304i	−0.873317	−	0.487152i	$-0.838036\pi$
$997$	− 30.7639i	− 0.974304i	0.873317	−	0.487152i	$-0.161964\pi$
$998$	0	0
$999$	−35.7771	−1.13194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.b.e.449.3	yes	4
4.3	odd	2		896.2.b.g.449.2	yes	4
8.3	odd	2		896.2.b.g.449.3	yes	4
8.5	even	2	inner	896.2.b.e.449.2	✓	4
16.3	odd	4		1792.2.a.j.1.2		2
16.5	even	4		1792.2.a.l.1.2		2
16.11	odd	4		1792.2.a.t.1.1		2
16.13	even	4		1792.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.b.e.449.2	✓	4	8.5	even	2	inner
896.2.b.e.449.3	yes	4	1.1	even	1	trivial
896.2.b.g.449.2	yes	4	4.3	odd	2
896.2.b.g.449.3	yes	4	8.3	odd	2
1792.2.a.j.1.2		2	16.3	odd	4
1792.2.a.l.1.2		2	16.5	even	4
1792.2.a.r.1.1		2	16.13	even	4
1792.2.a.t.1.1		2	16.11	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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.23607i q^{3} +1.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +O(q^{10})\)	\(q+1.23607i q^{3} +1.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +4.00000i q^{11} +1.23607i q^{13} -1.52786 q^{15} -2.00000 q^{17} -1.23607i q^{19} -1.23607i q^{21} -6.47214 q^{23} +3.47214 q^{25} +5.52786i q^{27} -1.52786i q^{29} -4.94427 q^{33} -1.23607i q^{35} +6.47214i q^{37} -1.52786 q^{39} +2.00000 q^{41} +8.94427i q^{43} +1.81966i q^{45} -12.9443 q^{47} +1.00000 q^{49} -2.47214i q^{51} -8.94427i q^{53} -4.94427 q^{55} +1.52786 q^{57} +9.23607i q^{59} -1.23607i q^{61} -1.47214 q^{63} -1.52786 q^{65} -1.52786i q^{67} -8.00000i q^{69} +4.94427 q^{71} -14.9443 q^{73} +4.29180i q^{75} -4.00000i q^{77} +4.94427 q^{79} -2.41641 q^{81} -9.23607i q^{83} -2.47214i q^{85} +1.88854 q^{87} -2.00000 q^{89} -1.23607i q^{91} +1.52786 q^{95} +10.9443 q^{97} +5.88854i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 - 12 * q^9	\( 4 q - 4 q^{7} - 12 q^{9} - 24 q^{15} - 8 q^{17} - 8 q^{23} - 4 q^{25} + 16 q^{33} - 24 q^{39} + 8 q^{41} - 16 q^{47} + 4 q^{49} + 16 q^{55} + 24 q^{57} + 12 q^{63} - 24 q^{65} - 16 q^{71} - 24 q^{73} - 16 q^{79} + 44 q^{81} - 64 q^{87} - 8 q^{89} + 24 q^{95} + 8 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 12 * q^9 - 24 * q^15 - 8 * q^17 - 8 * q^23 - 4 * q^25 + 16 * q^33 - 24 * q^39 + 8 * q^41 - 16 * q^47 + 4 * q^49 + 16 * q^55 + 24 * q^57 + 12 * q^63 - 24 * q^65 - 16 * q^71 - 24 * q^73 - 16 * q^79 + 44 * q^81 - 64 * q^87 - 8 * q^89 + 24 * q^95 + 8 * q^97

Introduction

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Modular forms

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