LMFDB - Embedded newform 1792.2.b.k.897.3

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:	$4$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			897.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1792.897
Dual form			1792.2.b.k.897.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607i q^{3} -3.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +O(q^{10})$	$q+1.23607i q^{3} -3.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +6.47214i q^{11} +0.763932i q^{13} +4.00000 q^{15} +4.47214 q^{17} -1.23607i q^{19} -1.23607i q^{21} -4.00000 q^{23} -5.47214 q^{25} +5.52786i q^{27} -4.47214i q^{29} -2.47214 q^{31} -8.00000 q^{33} +3.23607i q^{35} +4.47214i q^{37} -0.944272 q^{39} +8.47214 q^{41} -6.47214i q^{43} -4.76393i q^{45} +10.4721 q^{47} +1.00000 q^{49} +5.52786i q^{51} +10.0000i q^{53} +20.9443 q^{55} +1.52786 q^{57} +9.23607i q^{59} +11.2361i q^{61} -1.47214 q^{63} +2.47214 q^{65} +4.00000i q^{67} -4.94427i q^{69} +4.94427 q^{71} +2.94427 q^{73} -6.76393i q^{75} -6.47214i q^{77} +12.9443 q^{79} -2.41641 q^{81} -9.23607i q^{83} -14.4721i q^{85} +5.52786 q^{87} +6.00000 q^{89} -0.763932i q^{91} -3.05573i q^{93} -4.00000 q^{95} +12.4721 q^{97} +9.52786i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 16 q^{15} - 16 q^{23} - 4 q^{25} + 8 q^{31} - 32 q^{33} + 32 q^{39} + 16 q^{41} + 24 q^{47} + 4 q^{49} + 48 q^{55} + 24 q^{57} + 12 q^{63} - 8 q^{65} - 16 q^{71} - 24 q^{73} + 16 q^{79} + 44 q^{81} + 40 q^{87} + 24 q^{89} - 16 q^{95} + 32 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 12 * q^9 + 16 * q^15 - 16 * q^23 - 4 * q^25 + 8 * q^31 - 32 * q^33 + 32 * q^39 + 16 * q^41 + 24 * q^47 + 4 * q^49 + 48 * q^55 + 24 * q^57 + 12 * q^63 - 8 * q^65 - 16 * q^71 - 24 * q^73 + 16 * q^79 + 44 * q^81 + 40 * q^87 + 24 * q^89 - 16 * q^95 + 32 * 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
$3$	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
$5$	− 3.23607i	− 1.44721i	−0.690212	−	0.723607i	$-0.742483\pi$
$5$	− 3.23607i	− 1.44721i	0.690212	−	0.723607i	$-0.257517\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.47214	0.490712
$10$	0	0
$11$	6.47214i	1.95142i	0.219061	+	0.975711i	$0.429701\pi$
$11$	6.47214i	1.95142i	−0.219061	+	0.975711i	$0.570299\pi$
$12$	0	0
$13$	0.763932i	0.211877i	0.994373	+	0.105938i	$0.0337846\pi$
$13$	0.763932i	0.211877i	−0.994373	+	0.105938i	$0.966215\pi$
$14$	0	0
$15$	4.00000	1.03280
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	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
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−5.47214	−1.09443
$26$	0	0
$27$	5.52786i	1.06384i
$28$	0	0
$29$	− 4.47214i	− 0.830455i	−0.909718	−	0.415227i	$-0.863702\pi$
$29$	− 4.47214i	− 0.830455i	0.909718	−	0.415227i	$-0.136298\pi$
$30$	0	0
$31$	−2.47214	−0.444009	−0.222004	−	0.975046i	$-0.571260\pi$
$31$	−2.47214	−0.444009	−0.222004	+	0.975046i	$0.571260\pi$
$32$	0	0
$33$	−8.00000	−1.39262
$34$	0	0
$35$	3.23607i	0.546995i
$36$	0	0
$37$	4.47214i	0.735215i	0.929981	+	0.367607i	$0.119823\pi$
$37$	4.47214i	0.735215i	−0.929981	+	0.367607i	$0.880177\pi$
$38$	0	0
$39$	−0.944272	−0.151205
$40$	0	0
$41$	8.47214	1.32313	0.661563	−	0.749890i	$-0.269894\pi$
$41$	8.47214	1.32313	0.661563	+	0.749890i	$0.269894\pi$
$42$	0	0
$43$	− 6.47214i	− 0.986991i	−0.869748	−	0.493496i	$-0.835719\pi$
$43$	− 6.47214i	− 0.986991i	0.869748	−	0.493496i	$-0.164281\pi$
$44$	0	0
$45$	− 4.76393i	− 0.710165i
$46$	0	0
$47$	10.4721	1.52752	0.763759	−	0.645501i	$-0.223352\pi$
$47$	10.4721	1.52752	0.763759	+	0.645501i	$0.223352\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.52786i	0.774056i
$52$	0	0
$53$	10.0000i	1.37361i	0.726844	+	0.686803i	$0.240986\pi$
$53$	10.0000i	1.37361i	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	20.9443	2.82413
$56$	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
$61$	11.2361i	1.43863i	0.694683	+	0.719316i	$0.255544\pi$
$61$	11.2361i	1.43863i	−0.694683	+	0.719316i	$0.744456\pi$
$62$	0	0
$63$	−1.47214	−0.185472
$64$	0	0
$65$	2.47214	0.306631
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	− 4.94427i	− 0.595220i
$70$	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
$73$	2.94427	0.344601	0.172300	−	0.985044i	$-0.444880\pi$
$73$	2.94427	0.344601	0.172300	+	0.985044i	$0.444880\pi$
$74$	0	0
$75$	− 6.76393i	− 0.781032i
$76$	0	0
$77$	− 6.47214i	− 0.737568i
$78$	0	0
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	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
$85$	− 14.4721i	− 1.56972i
$86$	0	0
$87$	5.52786	0.592649
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	− 0.763932i	− 0.0800818i
$92$	0	0
$93$	− 3.05573i	− 0.316864i
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	12.4721	1.26635	0.633177	−	0.774007i	$-0.281751\pi$
$97$	12.4721	1.26635	0.633177	+	0.774007i	$0.281751\pi$
$98$	0	0
$99$	9.52786i	0.957586i
$100$	0	0
$101$	1.70820i	0.169973i	0.996382	+	0.0849863i	$0.0270847\pi$
$101$	1.70820i	0.169973i	−0.996382	+	0.0849863i	$0.972915\pi$
$102$	0	0
$103$	5.52786	0.544677	0.272338	−	0.962202i	$-0.412203\pi$
$103$	5.52786	0.544677	0.272338	+	0.962202i	$0.412203\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	− 8.94427i	− 0.864675i	−0.901712	−	0.432338i	$-0.857689\pi$
$107$	− 8.94427i	− 0.864675i	0.901712	−	0.432338i	$-0.142311\pi$
$108$	0	0
$109$	8.47214i	0.811483i	0.913988	+	0.405742i	$0.132987\pi$
$109$	8.47214i	0.811483i	−0.913988	+	0.405742i	$0.867013\pi$
$110$	0	0
$111$	−5.52786	−0.524682
$112$	0	0
$113$	−12.4721	−1.17328	−0.586640	−	0.809848i	$-0.699550\pi$
$113$	−12.4721	−1.17328	−0.586640	+	0.809848i	$0.699550\pi$
$114$	0	0
$115$	12.9443i	1.20706i
$116$	0	0
$117$	1.12461i	0.103970i
$118$	0	0
$119$	−4.47214	−0.409960
$120$	0	0
$121$	−30.8885	−2.80805
$122$	0	0
$123$	10.4721i	0.944241i
$124$	0	0
$125$	1.52786i	0.136656i
$126$	0	0
$127$	8.94427	0.793676	0.396838	−	0.917889i	$-0.370108\pi$
$127$	8.94427	0.793676	0.396838	+	0.917889i	$0.370108\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	11.7082i	1.02295i	0.859298	+	0.511475i	$0.170901\pi$
$131$	11.7082i	1.02295i	−0.859298	+	0.511475i	$0.829099\pi$
$132$	0	0
$133$	1.23607i	0.107181i
$134$	0	0
$135$	17.8885	1.53960
$136$	0	0
$137$	−14.9443	−1.27678	−0.638388	−	0.769715i	$-0.720398\pi$
$137$	−14.9443	−1.27678	−0.638388	+	0.769715i	$0.720398\pi$
$138$	0	0
$139$	− 1.23607i	− 0.104842i	−0.998625	−	0.0524210i	$-0.983306\pi$
$139$	− 1.23607i	− 0.104842i	0.998625	−	0.0524210i	$-0.0166938\pi$
$140$	0	0
$141$	12.9443i	1.09010i
$142$	0	0
$143$	−4.94427	−0.413461
$144$	0	0
$145$	−14.4721	−1.20185
$146$	0	0
$147$	1.23607i	0.101949i
$148$	0	0
$149$	− 2.94427i	− 0.241204i	−0.992701	−	0.120602i	$-0.961517\pi$
$149$	− 2.94427i	− 0.241204i	0.992701	−	0.120602i	$-0.0384825\pi$
$150$	0	0
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	0	0
$153$	6.58359	0.532252
$154$	0	0
$155$	8.00000i	0.642575i
$156$	0	0
$157$	0.763932i	0.0609684i	0.999535	+	0.0304842i	$0.00970493\pi$
$157$	0.763932i	0.0609684i	−0.999535	+	0.0304842i	$0.990295\pi$
$158$	0	0
$159$	−12.3607	−0.980266
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	− 3.41641i	− 0.267594i	−0.991009	−	0.133797i	$-0.957283\pi$
$163$	− 3.41641i	− 0.267594i	0.991009	−	0.133797i	$-0.0427170\pi$
$164$	0	0
$165$	25.8885i	2.01542i
$166$	0	0
$167$	−23.4164	−1.81202	−0.906008	−	0.423261i	$-0.860886\pi$
$167$	−23.4164	−1.81202	−0.906008	+	0.423261i	$0.860886\pi$
$168$	0	0
$169$	12.4164	0.955108
$170$	0	0
$171$	− 1.81966i	− 0.139153i
$172$	0	0
$173$	5.70820i	0.433987i	0.976173	+	0.216993i	$0.0696250\pi$
$173$	5.70820i	0.433987i	−0.976173	+	0.216993i	$0.930375\pi$
$174$	0	0
$175$	5.47214	0.413655
$176$	0	0
$177$	−11.4164	−0.858110
$178$	0	0
$179$	− 7.05573i	− 0.527370i	−0.964609	−	0.263685i	$-0.915062\pi$
$179$	− 7.05573i	− 0.527370i	0.964609	−	0.263685i	$-0.0849379\pi$
$180$	0	0
$181$	12.1803i	0.905358i	0.891674	+	0.452679i	$0.149532\pi$
$181$	12.1803i	0.905358i	−0.891674	+	0.452679i	$0.850468\pi$
$182$	0	0
$183$	−13.8885	−1.02667
$184$	0	0
$185$	14.4721	1.06401
$186$	0	0
$187$	28.9443i	2.11661i
$188$	0	0
$189$	− 5.52786i	− 0.402093i
$190$	0	0
$191$	4.94427	0.357755	0.178877	−	0.983871i	$-0.442753\pi$
$191$	4.94427	0.357755	0.178877	+	0.983871i	$0.442753\pi$
$192$	0	0
$193$	0.472136	0.0339851	0.0169925	−	0.999856i	$-0.494591\pi$
$193$	0.472136	0.0339851	0.0169925	+	0.999856i	$0.494591\pi$
$194$	0	0
$195$	3.05573i	0.218825i
$196$	0	0
$197$	− 10.9443i	− 0.779747i	−0.920868	−	0.389874i	$-0.872519\pi$
$197$	− 10.9443i	− 0.779747i	0.920868	−	0.389874i	$-0.127481\pi$
$198$	0	0
$199$	−15.4164	−1.09284	−0.546420	−	0.837511i	$-0.684010\pi$
$199$	−15.4164	−1.09284	−0.546420	+	0.837511i	$0.684010\pi$
$200$	0	0
$201$	−4.94427	−0.348742
$202$	0	0
$203$	4.47214i	0.313882i
$204$	0	0
$205$	− 27.4164i	− 1.91484i
$206$	0	0
$207$	−5.88854	−0.409282
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	− 12.0000i	− 0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$	− 12.0000i	− 0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$	0	0
$213$	6.11146i	0.418750i
$214$	0	0
$215$	−20.9443	−1.42839
$216$	0	0
$217$	2.47214	0.167820
$218$	0	0
$219$	3.63932i	0.245922i
$220$	0	0
$221$	3.41641i	0.229812i
$222$	0	0
$223$	12.9443	0.866813	0.433406	−	0.901199i	$-0.357312\pi$
$223$	12.9443	0.866813	0.433406	+	0.901199i	$0.357312\pi$
$224$	0	0
$225$	−8.05573	−0.537049
$226$	0	0
$227$	− 17.2361i	− 1.14400i	−0.820254	−	0.571999i	$-0.806168\pi$
$227$	− 17.2361i	− 1.14400i	0.820254	−	0.571999i	$-0.193832\pi$
$228$	0	0
$229$	− 23.5967i	− 1.55932i	−0.626205	−	0.779658i	$-0.715393\pi$
$229$	− 23.5967i	− 1.55932i	0.626205	−	0.779658i	$-0.284607\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	15.8885	1.04089	0.520447	−	0.853894i	$-0.325765\pi$
$233$	15.8885	1.04089	0.520447	+	0.853894i	$0.325765\pi$
$234$	0	0
$235$	− 33.8885i	− 2.21064i
$236$	0	0
$237$	16.0000i	1.03931i
$238$	0	0
$239$	−13.8885	−0.898375	−0.449188	−	0.893437i	$-0.648287\pi$
$239$	−13.8885	−0.898375	−0.449188	+	0.893437i	$0.648287\pi$
$240$	0	0
$241$	12.4721	0.803401	0.401700	−	0.915771i	$-0.368419\pi$
$241$	12.4721	0.803401	0.401700	+	0.915771i	$0.368419\pi$
$242$	0	0
$243$	13.5967i	0.872232i
$244$	0	0
$245$	− 3.23607i	− 0.206745i
$246$	0	0
$247$	0.944272	0.0600826
$248$	0	0
$249$	11.4164	0.723485
$250$	0	0
$251$	− 4.29180i	− 0.270896i	−0.990784	−	0.135448i	$-0.956753\pi$
$251$	− 4.29180i	− 0.270896i	0.990784	−	0.135448i	$-0.0432473\pi$
$252$	0	0
$253$	− 25.8885i	− 1.62760i
$254$	0	0
$255$	17.8885	1.12022
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	− 4.47214i	− 0.277885i
$260$	0	0
$261$	− 6.58359i	− 0.407514i
$262$	0	0
$263$	−11.0557	−0.681725	−0.340863	−	0.940113i	$-0.610719\pi$
$263$	−11.0557	−0.681725	−0.340863	+	0.940113i	$0.610719\pi$
$264$	0	0
$265$	32.3607	1.98790
$266$	0	0
$267$	7.41641i	0.453877i
$268$	0	0
$269$	− 4.18034i	− 0.254880i	−0.991846	−	0.127440i	$-0.959324\pi$
$269$	− 4.18034i	− 0.254880i	0.991846	−	0.127440i	$-0.0406760\pi$
$270$	0	0
$271$	−24.0000	−1.45790	−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000	−1.45790	−0.728948	+	0.684569i	$0.759990\pi$
$272$	0	0
$273$	0.944272	0.0571499
$274$	0	0
$275$	− 35.4164i	− 2.13569i
$276$	0	0
$277$	− 7.88854i	− 0.473977i	−0.971512	−	0.236988i	$-0.923840\pi$
$277$	− 7.88854i	− 0.473977i	0.971512	−	0.236988i	$-0.0761603\pi$
$278$	0	0
$279$	−3.63932	−0.217880
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	6.18034i	0.367383i	0.982984	+	0.183692i	$0.0588048\pi$
$283$	6.18034i	0.367383i	−0.982984	+	0.183692i	$0.941195\pi$
$284$	0	0
$285$	− 4.94427i	− 0.292873i
$286$	0	0
$287$	−8.47214	−0.500094
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	15.4164i	0.903726i
$292$	0	0
$293$	12.7639i	0.745677i	0.927896	+	0.372838i	$0.121615\pi$
$293$	12.7639i	0.745677i	−0.927896	+	0.372838i	$0.878385\pi$
$294$	0	0
$295$	29.8885	1.74018
$296$	0	0
$297$	−35.7771	−2.07600
$298$	0	0
$299$	− 3.05573i	− 0.176717i
$300$	0	0
$301$	6.47214i	0.373048i
$302$	0	0
$303$	−2.11146	−0.121300
$304$	0	0
$305$	36.3607	2.08201
$306$	0	0
$307$	1.81966i	0.103853i	0.998651	+	0.0519267i	$0.0165362\pi$
$307$	1.81966i	0.103853i	−0.998651	+	0.0519267i	$0.983464\pi$
$308$	0	0
$309$	6.83282i	0.388705i
$310$	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
$313$	8.47214	0.478873	0.239437	−	0.970912i	$-0.423037\pi$
$313$	8.47214	0.478873	0.239437	+	0.970912i	$0.423037\pi$
$314$	0	0
$315$	4.76393i	0.268417i
$316$	0	0
$317$	9.05573i	0.508620i	0.967123	+	0.254310i	$0.0818484\pi$
$317$	9.05573i	0.508620i	−0.967123	+	0.254310i	$0.918152\pi$
$318$	0	0
$319$	28.9443	1.62057
$320$	0	0
$321$	11.0557	0.617071
$322$	0	0
$323$	− 5.52786i	− 0.307579i
$324$	0	0
$325$	− 4.18034i	− 0.231884i
$326$	0	0
$327$	−10.4721	−0.579110
$328$	0	0
$329$	−10.4721	−0.577348
$330$	0	0
$331$	− 22.4721i	− 1.23518i	−0.786500	−	0.617590i	$-0.788109\pi$
$331$	− 22.4721i	− 1.23518i	0.786500	−	0.617590i	$-0.211891\pi$
$332$	0	0
$333$	6.58359i	0.360779i
$334$	0	0
$335$	12.9443	0.707221
$336$	0	0
$337$	10.3607	0.564382	0.282191	−	0.959358i	$-0.408939\pi$
$337$	10.3607	0.564382	0.282191	+	0.959358i	$0.408939\pi$
$338$	0	0
$339$	− 15.4164i	− 0.837304i
$340$	0	0
$341$	− 16.0000i	− 0.866449i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−16.0000	−0.861411
$346$	0	0
$347$	− 6.47214i	− 0.347442i	−0.984795	−	0.173721i	$-0.944421\pi$
$347$	− 6.47214i	− 0.347442i	0.984795	−	0.173721i	$-0.0555792\pi$
$348$	0	0
$349$	26.6525i	1.42667i	0.700821	+	0.713337i	$0.252817\pi$
$349$	26.6525i	1.42667i	−0.700821	+	0.713337i	$0.747183\pi$
$350$	0	0
$351$	−4.22291	−0.225402
$352$	0	0
$353$	−15.8885	−0.845662	−0.422831	−	0.906209i	$-0.638964\pi$
$353$	−15.8885	−0.845662	−0.422831	+	0.906209i	$0.638964\pi$
$354$	0	0
$355$	− 16.0000i	− 0.849192i
$356$	0	0
$357$	− 5.52786i	− 0.292566i
$358$	0	0
$359$	16.9443	0.894284	0.447142	−	0.894463i	$-0.352442\pi$
$359$	16.9443	0.894284	0.447142	+	0.894463i	$0.352442\pi$
$360$	0	0
$361$	17.4721	0.919586
$362$	0	0
$363$	− 38.1803i	− 2.00395i
$364$	0	0
$365$	− 9.52786i	− 0.498711i
$366$	0	0
$367$	−22.8328	−1.19186	−0.595932	−	0.803035i	$-0.703217\pi$
$367$	−22.8328	−1.19186	−0.595932	+	0.803035i	$0.703217\pi$
$368$	0	0
$369$	12.4721	0.649273
$370$	0	0
$371$	− 10.0000i	− 0.519174i
$372$	0	0
$373$	− 2.94427i	− 0.152449i	−0.997091	−	0.0762243i	$-0.975713\pi$
$373$	− 2.94427i	− 0.152449i	0.997091	−	0.0762243i	$-0.0242865\pi$
$374$	0	0
$375$	−1.88854	−0.0975240
$376$	0	0
$377$	3.41641	0.175954
$378$	0	0
$379$	4.58359i	0.235443i	0.993047	+	0.117722i	$0.0375591\pi$
$379$	4.58359i	0.235443i	−0.993047	+	0.117722i	$0.962441\pi$
$380$	0	0
$381$	11.0557i	0.566402i
$382$	0	0
$383$	−15.4164	−0.787742	−0.393871	−	0.919166i	$-0.628864\pi$
$383$	−15.4164	−0.787742	−0.393871	+	0.919166i	$0.628864\pi$
$384$	0	0
$385$	−20.9443	−1.06742
$386$	0	0
$387$	− 9.52786i	− 0.484329i
$388$	0	0
$389$	4.47214i	0.226746i	0.993552	+	0.113373i	$0.0361656\pi$
$389$	4.47214i	0.226746i	−0.993552	+	0.113373i	$0.963834\pi$
$390$	0	0
$391$	−17.8885	−0.904663
$392$	0	0
$393$	−14.4721	−0.730023
$394$	0	0
$395$	− 41.8885i	− 2.10764i
$396$	0	0
$397$	− 15.2361i	− 0.764676i	−0.924022	−	0.382338i	$-0.875119\pi$
$397$	− 15.2361i	− 0.764676i	0.924022	−	0.382338i	$-0.124881\pi$
$398$	0	0
$399$	−1.52786	−0.0764889
$400$	0	0
$401$	−23.5279	−1.17493	−0.587463	−	0.809251i	$-0.699873\pi$
$401$	−23.5279	−1.17493	−0.587463	+	0.809251i	$0.699873\pi$
$402$	0	0
$403$	− 1.88854i	− 0.0940751i
$404$	0	0
$405$	7.81966i	0.388562i
$406$	0	0
$407$	−28.9443	−1.43471
$408$	0	0
$409$	21.4164	1.05897	0.529487	−	0.848318i	$-0.322385\pi$
$409$	21.4164	1.05897	0.529487	+	0.848318i	$0.322385\pi$
$410$	0	0
$411$	− 18.4721i	− 0.911163i
$412$	0	0
$413$	− 9.23607i	− 0.454477i
$414$	0	0
$415$	−29.8885	−1.46717
$416$	0	0
$417$	1.52786	0.0748198
$418$	0	0
$419$	22.1803i	1.08358i	0.840514	+	0.541790i	$0.182253\pi$
$419$	22.1803i	1.08358i	−0.840514	+	0.541790i	$0.817747\pi$
$420$	0	0
$421$	2.00000i	0.0974740i	0.998812	+	0.0487370i	$0.0155196\pi$
$421$	2.00000i	0.0974740i	−0.998812	+	0.0487370i	$0.984480\pi$
$422$	0	0
$423$	15.4164	0.749571
$424$	0	0
$425$	−24.4721	−1.18707
$426$	0	0
$427$	− 11.2361i	− 0.543751i
$428$	0	0
$429$	− 6.11146i	− 0.295064i
$430$	0	0
$431$	−28.0000	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$431$	−28.0000	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$432$	0	0
$433$	9.41641	0.452524	0.226262	−	0.974067i	$-0.427349\pi$
$433$	9.41641	0.452524	0.226262	+	0.974067i	$0.427349\pi$
$434$	0	0
$435$	− 17.8885i	− 0.857690i
$436$	0	0
$437$	4.94427i	0.236517i
$438$	0	0
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	0	0
$441$	1.47214	0.0701017
$442$	0	0
$443$	13.8885i	0.659865i	0.944005	+	0.329932i	$0.107026\pi$
$443$	13.8885i	0.659865i	−0.944005	+	0.329932i	$0.892974\pi$
$444$	0	0
$445$	− 19.4164i	− 0.920426i
$446$	0	0
$447$	3.63932	0.172134
$448$	0	0
$449$	−7.88854	−0.372283	−0.186142	−	0.982523i	$-0.559598\pi$
$449$	−7.88854	−0.372283	−0.186142	+	0.982523i	$0.559598\pi$
$450$	0	0
$451$	54.8328i	2.58198i
$452$	0	0
$453$	11.0557i	0.519443i
$454$	0	0
$455$	−2.47214	−0.115896
$456$	0	0
$457$	7.52786	0.352139	0.176069	−	0.984378i	$-0.443662\pi$
$457$	7.52786	0.352139	0.176069	+	0.984378i	$0.443662\pi$
$458$	0	0
$459$	24.7214i	1.15389i
$460$	0	0
$461$	21.7082i	1.01105i	0.862811	+	0.505526i	$0.168702\pi$
$461$	21.7082i	1.01105i	−0.862811	+	0.505526i	$0.831298\pi$
$462$	0	0
$463$	−35.7771	−1.66270	−0.831351	−	0.555748i	$-0.812432\pi$
$463$	−35.7771	−1.66270	−0.831351	+	0.555748i	$0.812432\pi$
$464$	0	0
$465$	−9.88854	−0.458570
$466$	0	0
$467$	− 32.0689i	− 1.48397i	−0.670416	−	0.741985i	$-0.733884\pi$
$467$	− 32.0689i	− 1.48397i	0.670416	−	0.741985i	$-0.266116\pi$
$468$	0	0
$469$	− 4.00000i	− 0.184703i
$470$	0	0
$471$	−0.944272	−0.0435098
$472$	0	0
$473$	41.8885	1.92604
$474$	0	0
$475$	6.76393i	0.310350i
$476$	0	0
$477$	14.7214i	0.674045i
$478$	0	0
$479$	−8.58359	−0.392194	−0.196097	−	0.980584i	$-0.562827\pi$
$479$	−8.58359	−0.392194	−0.196097	+	0.980584i	$0.562827\pi$
$480$	0	0
$481$	−3.41641	−0.155775
$482$	0	0
$483$	4.94427i	0.224972i
$484$	0	0
$485$	− 40.3607i	− 1.83268i
$486$	0	0
$487$	−20.0000	−0.906287	−0.453143	−	0.891438i	$-0.649697\pi$
$487$	−20.0000	−0.906287	−0.453143	+	0.891438i	$0.649697\pi$
$488$	0	0
$489$	4.22291	0.190967
$490$	0	0
$491$	− 37.8885i	− 1.70989i	−0.518722	−	0.854943i	$-0.673592\pi$
$491$	− 37.8885i	− 1.70989i	0.518722	−	0.854943i	$-0.326408\pi$
$492$	0	0
$493$	− 20.0000i	− 0.900755i
$494$	0	0
$495$	30.8328	1.38583
$496$	0	0
$497$	−4.94427	−0.221781
$498$	0	0
$499$	21.8885i	0.979866i	0.871760	+	0.489933i	$0.162979\pi$
$499$	21.8885i	0.979866i	−0.871760	+	0.489933i	$0.837021\pi$
$500$	0	0
$501$	− 28.9443i	− 1.29313i
$502$	0	0
$503$	4.94427	0.220454	0.110227	−	0.993906i	$-0.464842\pi$
$503$	4.94427	0.220454	0.110227	+	0.993906i	$0.464842\pi$
$504$	0	0
$505$	5.52786	0.245987
$506$	0	0
$507$	15.3475i	0.681607i
$508$	0	0
$509$	− 41.1246i	− 1.82282i	−0.411503	−	0.911408i	$-0.634996\pi$
$509$	− 41.1246i	− 1.82282i	0.411503	−	0.911408i	$-0.365004\pi$
$510$	0	0
$511$	−2.94427	−0.130247
$512$	0	0
$513$	6.83282	0.301676
$514$	0	0
$515$	− 17.8885i	− 0.788263i
$516$	0	0
$517$	67.7771i	2.98083i
$518$	0	0
$519$	−7.05573	−0.309712
$520$	0	0
$521$	6.58359	0.288432	0.144216	−	0.989546i	$-0.453934\pi$
$521$	6.58359	0.288432	0.144216	+	0.989546i	$0.453934\pi$
$522$	0	0
$523$	4.29180i	0.187667i	0.995588	+	0.0938336i	$0.0299122\pi$
$523$	4.29180i	0.187667i	−0.995588	+	0.0938336i	$0.970088\pi$
$524$	0	0
$525$	6.76393i	0.295202i
$526$	0	0
$527$	−11.0557	−0.481595
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	13.5967i	0.590049i
$532$	0	0
$533$	6.47214i	0.280339i
$534$	0	0
$535$	−28.9443	−1.25137
$536$	0	0
$537$	8.72136	0.376354
$538$	0	0
$539$	6.47214i	0.278775i
$540$	0	0
$541$	− 5.05573i	− 0.217363i	−0.994077	−	0.108681i	$-0.965337\pi$
$541$	− 5.05573i	− 0.217363i	0.994077	−	0.108681i	$-0.0346628\pi$
$542$	0	0
$543$	−15.0557	−0.646103
$544$	0	0
$545$	27.4164	1.17439
$546$	0	0
$547$	− 4.58359i	− 0.195980i	−0.995187	−	0.0979901i	$-0.968759\pi$
$547$	− 4.58359i	− 0.195980i	0.995187	−	0.0979901i	$-0.0312414\pi$
$548$	0	0
$549$	16.5410i	0.705954i
$550$	0	0
$551$	−5.52786	−0.235495
$552$	0	0
$553$	−12.9443	−0.550446
$554$	0	0
$555$	17.8885i	0.759326i
$556$	0	0
$557$	9.05573i	0.383704i	0.981424	+	0.191852i	$0.0614493\pi$
$557$	9.05573i	0.383704i	−0.981424	+	0.191852i	$0.938551\pi$
$558$	0	0
$559$	4.94427	0.209120
$560$	0	0
$561$	−35.7771	−1.51051
$562$	0	0
$563$	− 17.8197i	− 0.751009i	−0.926821	−	0.375505i	$-0.877469\pi$
$563$	− 17.8197i	− 0.751009i	0.926821	−	0.375505i	$-0.122531\pi$
$564$	0	0
$565$	40.3607i	1.69799i
$566$	0	0
$567$	2.41641	0.101480
$568$	0	0
$569$	−26.3607	−1.10510	−0.552549	−	0.833481i	$-0.686345\pi$
$569$	−26.3607	−1.10510	−0.552549	+	0.833481i	$0.686345\pi$
$570$	0	0
$571$	14.4721i	0.605640i	0.953048	+	0.302820i	$0.0979281\pi$
$571$	14.4721i	0.605640i	−0.953048	+	0.302820i	$0.902072\pi$
$572$	0	0
$573$	6.11146i	0.255310i
$574$	0	0
$575$	21.8885	0.912815
$576$	0	0
$577$	−6.00000	−0.249783	−0.124892	−	0.992170i	$-0.539858\pi$
$577$	−6.00000	−0.249783	−0.124892	+	0.992170i	$0.539858\pi$
$578$	0	0
$579$	0.583592i	0.0242533i
$580$	0	0
$581$	9.23607i	0.383177i
$582$	0	0
$583$	−64.7214	−2.68048
$584$	0	0
$585$	3.63932	0.150467
$586$	0	0
$587$	3.70820i	0.153054i	0.997068	+	0.0765270i	$0.0243831\pi$
$587$	3.70820i	0.153054i	−0.997068	+	0.0765270i	$0.975617\pi$
$588$	0	0
$589$	3.05573i	0.125909i
$590$	0	0
$591$	13.5279	0.556462
$592$	0	0
$593$	32.8328	1.34828	0.674141	−	0.738603i	$-0.264514\pi$
$593$	32.8328	1.34828	0.674141	+	0.738603i	$0.264514\pi$
$594$	0	0
$595$	14.4721i	0.593300i
$596$	0	0
$597$	− 19.0557i	− 0.779899i
$598$	0	0
$599$	−17.8885	−0.730906	−0.365453	−	0.930830i	$-0.619086\pi$
$599$	−17.8885	−0.730906	−0.365453	+	0.930830i	$0.619086\pi$
$600$	0	0
$601$	−29.7771	−1.21463	−0.607316	−	0.794460i	$-0.707754\pi$
$601$	−29.7771	−1.21463	−0.607316	+	0.794460i	$0.707754\pi$
$602$	0	0
$603$	5.88854i	0.239800i
$604$	0	0
$605$	99.9574i	4.06385i
$606$	0	0
$607$	9.88854	0.401364	0.200682	−	0.979656i	$-0.435684\pi$
$607$	9.88854	0.401364	0.200682	+	0.979656i	$0.435684\pi$
$608$	0	0
$609$	−5.52786	−0.224000
$610$	0	0
$611$	8.00000i	0.323645i
$612$	0	0
$613$	− 29.4164i	− 1.18812i	−0.804422	−	0.594059i	$-0.797525\pi$
$613$	− 29.4164i	− 1.18812i	0.804422	−	0.594059i	$-0.202475\pi$
$614$	0	0
$615$	33.8885	1.36652
$616$	0	0
$617$	−34.3607	−1.38331	−0.691654	−	0.722229i	$-0.743118\pi$
$617$	−34.3607	−1.38331	−0.691654	+	0.722229i	$0.743118\pi$
$618$	0	0
$619$	48.0689i	1.93205i	0.258446	+	0.966026i	$0.416790\pi$
$619$	48.0689i	1.93205i	−0.258446	+	0.966026i	$0.583210\pi$
$620$	0	0
$621$	− 22.1115i	− 0.887302i
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	9.88854i	0.394910i
$628$	0	0
$629$	20.0000i	0.797452i
$630$	0	0
$631$	44.9443	1.78920	0.894602	−	0.446865i	$-0.147459\pi$
$631$	44.9443	1.78920	0.894602	+	0.446865i	$0.147459\pi$
$632$	0	0
$633$	14.8328	0.589551
$634$	0	0
$635$	− 28.9443i	− 1.14862i
$636$	0	0
$637$	0.763932i	0.0302681i
$638$	0	0
$639$	7.27864	0.287939
$640$	0	0
$641$	14.5836	0.576017	0.288009	−	0.957628i	$-0.407007\pi$
$641$	14.5836	0.576017	0.288009	+	0.957628i	$0.407007\pi$
$642$	0	0
$643$	− 43.7082i	− 1.72368i	−0.507177	−	0.861842i	$-0.669311\pi$
$643$	− 43.7082i	− 1.72368i	0.507177	−	0.861842i	$-0.330689\pi$
$644$	0	0
$645$	− 25.8885i	− 1.01936i
$646$	0	0
$647$	20.3607	0.800461	0.400230	−	0.916415i	$-0.368930\pi$
$647$	20.3607	0.800461	0.400230	+	0.916415i	$0.368930\pi$
$648$	0	0
$649$	−59.7771	−2.34646
$650$	0	0
$651$	3.05573i	0.119763i
$652$	0	0
$653$	− 9.41641i	− 0.368493i	−0.982880	−	0.184246i	$-0.941016\pi$
$653$	− 9.41641i	− 0.368493i	0.982880	−	0.184246i	$-0.0589844\pi$
$654$	0	0
$655$	37.8885	1.48043
$656$	0	0
$657$	4.33437	0.169100
$658$	0	0
$659$	37.3050i	1.45319i	0.687064	+	0.726597i	$0.258899\pi$
$659$	37.3050i	1.45319i	−0.687064	+	0.726597i	$0.741101\pi$
$660$	0	0
$661$	22.6525i	0.881079i	0.897733	+	0.440540i	$0.145213\pi$
$661$	22.6525i	0.881079i	−0.897733	+	0.440540i	$0.854787\pi$
$662$	0	0
$663$	−4.22291	−0.164004
$664$	0	0
$665$	4.00000	0.155113
$666$	0	0
$667$	17.8885i	0.692647i
$668$	0	0
$669$	16.0000i	0.618596i
$670$	0	0
$671$	−72.7214	−2.80738
$672$	0	0
$673$	−2.94427	−0.113493	−0.0567467	−	0.998389i	$-0.518073\pi$
$673$	−2.94427	−0.113493	−0.0567467	+	0.998389i	$0.518073\pi$
$674$	0	0
$675$	− 30.2492i	− 1.16429i
$676$	0	0
$677$	− 19.8197i	− 0.761731i	−0.924630	−	0.380866i	$-0.875626\pi$
$677$	− 19.8197i	− 0.761731i	0.924630	−	0.380866i	$-0.124374\pi$
$678$	0	0
$679$	−12.4721	−0.478637
$680$	0	0
$681$	21.3050	0.816408
$682$	0	0
$683$	− 23.7771i	− 0.909805i	−0.890541	−	0.454902i	$-0.849674\pi$
$683$	− 23.7771i	− 0.909805i	0.890541	−	0.454902i	$-0.150326\pi$
$684$	0	0
$685$	48.3607i	1.84777i
$686$	0	0
$687$	29.1672	1.11280
$688$	0	0
$689$	−7.63932	−0.291035
$690$	0	0
$691$	− 14.1803i	− 0.539446i	−0.962938	−	0.269723i	$-0.913068\pi$
$691$	− 14.1803i	− 0.539446i	0.962938	−	0.269723i	$-0.0869321\pi$
$692$	0	0
$693$	− 9.52786i	− 0.361934i
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	37.8885	1.43513
$698$	0	0
$699$	19.6393i	0.742827i
$700$	0	0
$701$	− 41.4164i	− 1.56428i	−0.623105	−	0.782138i	$-0.714129\pi$
$701$	− 41.4164i	− 1.56428i	0.623105	−	0.782138i	$-0.285871\pi$
$702$	0	0
$703$	5.52786	0.208487
$704$	0	0
$705$	41.8885	1.57761
$706$	0	0
$707$	− 1.70820i	− 0.0642436i
$708$	0	0
$709$	− 1.63932i	− 0.0615660i	−0.999526	−	0.0307830i	$-0.990200\pi$
$709$	− 1.63932i	− 0.0615660i	0.999526	−	0.0307830i	$-0.00980008\pi$
$710$	0	0
$711$	19.0557	0.714646
$712$	0	0
$713$	9.88854	0.370329
$714$	0	0
$715$	16.0000i	0.598366i
$716$	0	0
$717$	− 17.1672i	− 0.641120i
$718$	0	0
$719$	51.1935	1.90920	0.954598	−	0.297898i	$-0.0962856\pi$
$719$	51.1935	1.90920	0.954598	+	0.297898i	$0.0962856\pi$
$720$	0	0
$721$	−5.52786	−0.205868
$722$	0	0
$723$	15.4164i	0.573342i
$724$	0	0
$725$	24.4721i	0.908872i
$726$	0	0
$727$	−12.3607	−0.458432	−0.229216	−	0.973376i	$-0.573616\pi$
$727$	−12.3607	−0.458432	−0.229216	+	0.973376i	$0.573616\pi$
$728$	0	0
$729$	−24.0557	−0.890953
$730$	0	0
$731$	− 28.9443i	− 1.07054i
$732$	0	0
$733$	− 4.76393i	− 0.175960i	−0.996122	−	0.0879799i	$-0.971959\pi$
$733$	− 4.76393i	− 0.175960i	0.996122	−	0.0879799i	$-0.0280411\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	0	0
$737$	−25.8885	−0.953617
$738$	0	0
$739$	− 3.41641i	− 0.125675i	−0.998024	−	0.0628373i	$-0.979985\pi$
$739$	− 3.41641i	− 0.125675i	0.998024	−	0.0628373i	$-0.0200149\pi$
$740$	0	0
$741$	1.16718i	0.0428776i
$742$	0	0
$743$	−24.9443	−0.915117	−0.457558	−	0.889180i	$-0.651276\pi$
$743$	−24.9443	−0.915117	−0.457558	+	0.889180i	$0.651276\pi$
$744$	0	0
$745$	−9.52786	−0.349074
$746$	0	0
$747$	− 13.5967i	− 0.497479i
$748$	0	0
$749$	8.94427i	0.326817i
$750$	0	0
$751$	36.0000	1.31366	0.656829	−	0.754039i	$-0.271897\pi$
$751$	36.0000	1.31366	0.656829	+	0.754039i	$0.271897\pi$
$752$	0	0
$753$	5.30495	0.193323
$754$	0	0
$755$	− 28.9443i	− 1.05339i
$756$	0	0
$757$	− 39.3050i	− 1.42856i	−0.699859	−	0.714281i	$-0.746754\pi$
$757$	− 39.3050i	− 1.42856i	0.699859	−	0.714281i	$-0.253246\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	3.52786	0.127885	0.0639425	−	0.997954i	$-0.479633\pi$
$761$	3.52786	0.127885	0.0639425	+	0.997954i	$0.479633\pi$
$762$	0	0
$763$	− 8.47214i	− 0.306712i
$764$	0	0
$765$	− 21.3050i	− 0.770282i
$766$	0	0
$767$	−7.05573	−0.254768
$768$	0	0
$769$	−18.3607	−0.662103	−0.331052	−	0.943613i	$-0.607403\pi$
$769$	−18.3607	−0.662103	−0.331052	+	0.943613i	$0.607403\pi$
$770$	0	0
$771$	− 17.3050i	− 0.623223i
$772$	0	0
$773$	− 40.1803i	− 1.44519i	−0.691274	−	0.722593i	$-0.742950\pi$
$773$	− 40.1803i	− 1.44519i	0.691274	−	0.722593i	$-0.257050\pi$
$774$	0	0
$775$	13.5279	0.485935
$776$	0	0
$777$	5.52786	0.198311
$778$	0	0
$779$	− 10.4721i	− 0.375203i
$780$	0	0
$781$	32.0000i	1.14505i
$782$	0	0
$783$	24.7214	0.883469
$784$	0	0
$785$	2.47214	0.0882343
$786$	0	0
$787$	− 12.2918i	− 0.438155i	−0.975707	−	0.219078i	$-0.929695\pi$
$787$	− 12.2918i	− 0.438155i	0.975707	−	0.219078i	$-0.0703048\pi$
$788$	0	0
$789$	− 13.6656i	− 0.486509i
$790$	0	0
$791$	12.4721	0.443458
$792$	0	0
$793$	−8.58359	−0.304812
$794$	0	0
$795$	40.0000i	1.41865i
$796$	0	0
$797$	− 52.1803i	− 1.84832i	−0.382003	−	0.924161i	$-0.624765\pi$
$797$	− 52.1803i	− 1.84832i	0.382003	−	0.924161i	$-0.375235\pi$
$798$	0	0
$799$	46.8328	1.65683
$800$	0	0
$801$	8.83282	0.312092
$802$	0	0
$803$	19.0557i	0.672462i
$804$	0	0
$805$	− 12.9443i	− 0.456226i
$806$	0	0
$807$	5.16718	0.181894
$808$	0	0
$809$	17.4164	0.612328	0.306164	−	0.951979i	$-0.400954\pi$
$809$	17.4164	0.612328	0.306164	+	0.951979i	$0.400954\pi$
$810$	0	0
$811$	53.0132i	1.86154i	0.365601	+	0.930772i	$0.380864\pi$
$811$	53.0132i	1.86154i	−0.365601	+	0.930772i	$0.619136\pi$
$812$	0	0
$813$	− 29.6656i	− 1.04042i
$814$	0	0
$815$	−11.0557	−0.387265
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	− 1.12461i	− 0.0392971i
$820$	0	0
$821$	− 4.11146i	− 0.143491i	−0.997423	−	0.0717454i	$-0.977143\pi$
$821$	− 4.11146i	− 0.143491i	0.997423	−	0.0717454i	$-0.0228569\pi$
$822$	0	0
$823$	19.7771	0.689386	0.344693	−	0.938715i	$-0.387983\pi$
$823$	19.7771	0.689386	0.344693	+	0.938715i	$0.387983\pi$
$824$	0	0
$825$	43.7771	1.52412
$826$	0	0
$827$	− 15.0557i	− 0.523539i	−0.965130	−	0.261769i	$-0.915694\pi$
$827$	− 15.0557i	− 0.523539i	0.965130	−	0.261769i	$-0.0843060\pi$
$828$	0	0
$829$	11.2361i	0.390245i	0.980779	+	0.195122i	$0.0625104\pi$
$829$	11.2361i	0.390245i	−0.980779	+	0.195122i	$0.937490\pi$
$830$	0	0
$831$	9.75078	0.338251
$832$	0	0
$833$	4.47214	0.154950
$834$	0	0
$835$	75.7771i	2.62237i
$836$	0	0
$837$	− 13.6656i	− 0.472353i
$838$	0	0
$839$	21.5279	0.743224	0.371612	−	0.928388i	$-0.378805\pi$
$839$	21.5279	0.743224	0.371612	+	0.928388i	$0.378805\pi$
$840$	0	0
$841$	9.00000	0.310345
$842$	0	0
$843$	− 32.1378i	− 1.10688i
$844$	0	0
$845$	− 40.1803i	− 1.38225i
$846$	0	0
$847$	30.8885	1.06134
$848$	0	0
$849$	−7.63932	−0.262181
$850$	0	0
$851$	− 17.8885i	− 0.613211i
$852$	0	0
$853$	− 13.7082i	− 0.469360i	−0.972073	−	0.234680i	$-0.924596\pi$
$853$	− 13.7082i	− 0.469360i	0.972073	−	0.234680i	$-0.0754042\pi$
$854$	0	0
$855$	−5.88854	−0.201384
$856$	0	0
$857$	27.5279	0.940334	0.470167	−	0.882577i	$-0.344194\pi$
$857$	27.5279	0.940334	0.470167	+	0.882577i	$0.344194\pi$
$858$	0	0
$859$	33.8197i	1.15391i	0.816775	+	0.576956i	$0.195760\pi$
$859$	33.8197i	1.15391i	−0.816775	+	0.576956i	$0.804240\pi$
$860$	0	0
$861$	− 10.4721i	− 0.356889i
$862$	0	0
$863$	−20.9443	−0.712951	−0.356476	−	0.934305i	$-0.616022\pi$
$863$	−20.9443	−0.712951	−0.356476	+	0.934305i	$0.616022\pi$
$864$	0	0
$865$	18.4721	0.628071
$866$	0	0
$867$	3.70820i	0.125937i
$868$	0	0
$869$	83.7771i	2.84194i
$870$	0	0
$871$	−3.05573	−0.103539
$872$	0	0
$873$	18.3607	0.621415
$874$	0	0
$875$	− 1.52786i	− 0.0516512i
$876$	0	0
$877$	13.4164i	0.453040i	0.974007	+	0.226520i	$0.0727348\pi$
$877$	13.4164i	0.453040i	−0.974007	+	0.226520i	$0.927265\pi$
$878$	0	0
$879$	−15.7771	−0.532148
$880$	0	0
$881$	24.8328	0.836639	0.418319	−	0.908300i	$-0.362619\pi$
$881$	24.8328	0.836639	0.418319	+	0.908300i	$0.362619\pi$
$882$	0	0
$883$	23.0557i	0.775887i	0.921683	+	0.387944i	$0.126814\pi$
$883$	23.0557i	0.775887i	−0.921683	+	0.387944i	$0.873186\pi$
$884$	0	0
$885$	36.9443i	1.24187i
$886$	0	0
$887$	33.3050	1.11827	0.559135	−	0.829076i	$-0.311133\pi$
$887$	33.3050	1.11827	0.559135	+	0.829076i	$0.311133\pi$
$888$	0	0
$889$	−8.94427	−0.299981
$890$	0	0
$891$	− 15.6393i	− 0.523937i
$892$	0	0
$893$	− 12.9443i	− 0.433164i
$894$	0	0
$895$	−22.8328	−0.763217
$896$	0	0
$897$	3.77709	0.126113
$898$	0	0
$899$	11.0557i	0.368729i
$900$	0	0
$901$	44.7214i	1.48988i
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	39.4164	1.31025
$906$	0	0
$907$	− 0.944272i	− 0.0313540i	−0.999877	−	0.0156770i	$-0.995010\pi$
$907$	− 0.944272i	− 0.0313540i	0.999877	−	0.0156770i	$-0.00499035\pi$
$908$	0	0
$909$	2.51471i	0.0834076i
$910$	0	0
$911$	−34.8328	−1.15406	−0.577031	−	0.816722i	$-0.695789\pi$
$911$	−34.8328	−1.15406	−0.577031	+	0.816722i	$0.695789\pi$
$912$	0	0
$913$	59.7771	1.97833
$914$	0	0
$915$	44.9443i	1.48581i
$916$	0	0
$917$	− 11.7082i	− 0.386639i
$918$	0	0
$919$	35.7771	1.18018	0.590089	−	0.807338i	$-0.299093\pi$
$919$	35.7771	1.18018	0.590089	+	0.807338i	$0.299093\pi$
$920$	0	0
$921$	−2.24922	−0.0741144
$922$	0	0
$923$	3.77709i	0.124324i
$924$	0	0
$925$	− 24.4721i	− 0.804639i
$926$	0	0
$927$	8.13777	0.267279
$928$	0	0
$929$	−47.3050	−1.55203	−0.776013	−	0.630717i	$-0.782761\pi$
$929$	−47.3050	−1.55203	−0.776013	+	0.630717i	$0.782761\pi$
$930$	0	0
$931$	− 1.23607i	− 0.0405105i
$932$	0	0
$933$	− 9.88854i	− 0.323736i
$934$	0	0
$935$	93.6656	3.06319
$936$	0	0
$937$	9.05573	0.295838	0.147919	−	0.988999i	$-0.452743\pi$
$937$	9.05573	0.295838	0.147919	+	0.988999i	$0.452743\pi$
$938$	0	0
$939$	10.4721i	0.341745i
$940$	0	0
$941$	− 35.5967i	− 1.16042i	−0.814467	−	0.580210i	$-0.802970\pi$
$941$	− 35.5967i	− 1.16042i	0.814467	−	0.580210i	$-0.197030\pi$
$942$	0	0
$943$	−33.8885	−1.10356
$944$	0	0
$945$	−17.8885	−0.581914
$946$	0	0
$947$	− 4.58359i	− 0.148947i	−0.997223	−	0.0744734i	$-0.976272\pi$
$947$	− 4.58359i	− 0.148947i	0.997223	−	0.0744734i	$-0.0237276\pi$
$948$	0	0
$949$	2.24922i	0.0730129i
$950$	0	0
$951$	−11.1935	−0.362974
$952$	0	0
$953$	−51.8885	−1.68083	−0.840417	−	0.541940i	$-0.817690\pi$
$953$	−51.8885	−1.68083	−0.840417	+	0.541940i	$0.817690\pi$
$954$	0	0
$955$	− 16.0000i	− 0.517748i
$956$	0	0
$957$	35.7771i	1.15651i
$958$	0	0
$959$	14.9443	0.482576
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	− 13.1672i	− 0.424307i
$964$	0	0
$965$	− 1.52786i	− 0.0491837i
$966$	0	0
$967$	−29.8885	−0.961151	−0.480575	−	0.876953i	$-0.659572\pi$
$967$	−29.8885	−0.961151	−0.480575	+	0.876953i	$0.659572\pi$
$968$	0	0
$969$	6.83282	0.219502
$970$	0	0
$971$	− 22.7639i	− 0.730529i	−0.930904	−	0.365265i	$-0.880978\pi$
$971$	− 22.7639i	− 0.730529i	0.930904	−	0.365265i	$-0.119022\pi$
$972$	0	0
$973$	1.23607i	0.0396265i
$974$	0	0
$975$	5.16718	0.165482
$976$	0	0
$977$	−12.8328	−0.410558	−0.205279	−	0.978703i	$-0.565810\pi$
$977$	−12.8328	−0.410558	−0.205279	+	0.978703i	$0.565810\pi$
$978$	0	0
$979$	38.8328i	1.24110i
$980$	0	0
$981$	12.4721i	0.398205i
$982$	0	0
$983$	5.52786	0.176311	0.0881557	−	0.996107i	$-0.471903\pi$
$983$	5.52786	0.176311	0.0881557	+	0.996107i	$0.471903\pi$
$984$	0	0
$985$	−35.4164	−1.12846
$986$	0	0
$987$	− 12.9443i	− 0.412021i
$988$	0	0
$989$	25.8885i	0.823208i
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	27.7771	0.881479
$994$	0	0
$995$	49.8885i	1.58157i
$996$	0	0
$997$	− 18.0689i	− 0.572247i	−0.958193	−	0.286124i	$-0.907633\pi$
$997$	− 18.0689i	− 0.572247i	0.958193	−	0.286124i	$-0.0923668\pi$
$998$	0	0
$999$	−24.7214	−0.782149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.2.b.k.897.3		4
4.3	odd	2		1792.2.b.m.897.2		4
8.3	odd	2		1792.2.b.m.897.3		4
8.5	even	2	inner	1792.2.b.k.897.2		4
16.3	odd	4		448.2.a.i.1.2		2
16.5	even	4		224.2.a.c.1.2	✓	2
16.11	odd	4		224.2.a.d.1.1	yes	2
16.13	even	4		448.2.a.j.1.1		2
48.5	odd	4		2016.2.a.r.1.1		2
48.11	even	4		2016.2.a.o.1.1		2
48.29	odd	4		4032.2.a.bw.1.2		2
48.35	even	4		4032.2.a.bv.1.2		2
80.59	odd	4		5600.2.a.z.1.2		2
80.69	even	4		5600.2.a.bk.1.1		2
112.5	odd	12		1568.2.i.n.1537.2		4
112.11	odd	12		1568.2.i.m.961.2		4
112.13	odd	4		3136.2.a.bf.1.2		2
112.27	even	4		1568.2.a.k.1.2		2
112.37	even	12		1568.2.i.v.1537.1		4
112.53	even	12		1568.2.i.v.961.1		4
112.59	even	12		1568.2.i.w.961.1		4
112.69	odd	4		1568.2.a.v.1.1		2
112.75	even	12		1568.2.i.w.1537.1		4
112.83	even	4		3136.2.a.by.1.1		2
112.101	odd	12		1568.2.i.n.961.2		4
112.107	odd	12		1568.2.i.m.1537.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.a.c.1.2	✓	2	16.5	even	4
224.2.a.d.1.1	yes	2	16.11	odd	4
448.2.a.i.1.2		2	16.3	odd	4
448.2.a.j.1.1		2	16.13	even	4
1568.2.a.k.1.2		2	112.27	even	4
1568.2.a.v.1.1		2	112.69	odd	4
1568.2.i.m.961.2		4	112.11	odd	12
1568.2.i.m.1537.2		4	112.107	odd	12
1568.2.i.n.961.2		4	112.101	odd	12
1568.2.i.n.1537.2		4	112.5	odd	12
1568.2.i.v.961.1		4	112.53	even	12
1568.2.i.v.1537.1		4	112.37	even	12
1568.2.i.w.961.1		4	112.59	even	12
1568.2.i.w.1537.1		4	112.75	even	12
1792.2.b.k.897.2		4	8.5	even	2	inner
1792.2.b.k.897.3		4	1.1	even	1	trivial
1792.2.b.m.897.2		4	4.3	odd	2
1792.2.b.m.897.3		4	8.3	odd	2
2016.2.a.o.1.1		2	48.11	even	4
2016.2.a.r.1.1		2	48.5	odd	4
3136.2.a.bf.1.2		2	112.13	odd	4
3136.2.a.by.1.1		2	112.83	even	4
4032.2.a.bv.1.2		2	48.35	even	4
4032.2.a.bw.1.2		2	48.29	odd	4
5600.2.a.z.1.2		2	80.59	odd	4
5600.2.a.bk.1.1		2	80.69	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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1792.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.23607i q^{3} -3.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +O(q^{10})\)	\(q+1.23607i q^{3} -3.23607i q^{5} -1.00000 q^{7} +1.47214 q^{9} +6.47214i q^{11} +0.763932i q^{13} +4.00000 q^{15} +4.47214 q^{17} -1.23607i q^{19} -1.23607i q^{21} -4.00000 q^{23} -5.47214 q^{25} +5.52786i q^{27} -4.47214i q^{29} -2.47214 q^{31} -8.00000 q^{33} +3.23607i q^{35} +4.47214i q^{37} -0.944272 q^{39} +8.47214 q^{41} -6.47214i q^{43} -4.76393i q^{45} +10.4721 q^{47} +1.00000 q^{49} +5.52786i q^{51} +10.0000i q^{53} +20.9443 q^{55} +1.52786 q^{57} +9.23607i q^{59} +11.2361i q^{61} -1.47214 q^{63} +2.47214 q^{65} +4.00000i q^{67} -4.94427i q^{69} +4.94427 q^{71} +2.94427 q^{73} -6.76393i q^{75} -6.47214i q^{77} +12.9443 q^{79} -2.41641 q^{81} -9.23607i q^{83} -14.4721i q^{85} +5.52786 q^{87} +6.00000 q^{89} -0.763932i q^{91} -3.05573i q^{93} -4.00000 q^{95} +12.4721 q^{97} +9.52786i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 - 12 * q^9	\( 4 q - 4 q^{7} - 12 q^{9} + 16 q^{15} - 16 q^{23} - 4 q^{25} + 8 q^{31} - 32 q^{33} + 32 q^{39} + 16 q^{41} + 24 q^{47} + 4 q^{49} + 48 q^{55} + 24 q^{57} + 12 q^{63} - 8 q^{65} - 16 q^{71} - 24 q^{73} + 16 q^{79} + 44 q^{81} + 40 q^{87} + 24 q^{89} - 16 q^{95} + 32 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 12 * q^9 + 16 * q^15 - 16 * q^23 - 4 * q^25 + 8 * q^31 - 32 * q^33 + 32 * q^39 + 16 * q^41 + 24 * q^47 + 4 * q^49 + 48 * q^55 + 24 * q^57 + 12 * q^63 - 8 * q^65 - 16 * q^71 - 24 * q^73 + 16 * q^79 + 44 * q^81 + 40 * q^87 + 24 * q^89 - 16 * q^95 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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