LMFDB - Embedded newform 448.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,2,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

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:	yes
Analytic conductor:	$3.57729801055$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} -3.23607 q^{5} -1.00000 q^{7} -1.47214 q^{9} +O(q^{10})$	$q+1.23607 q^{3} -3.23607 q^{5} -1.00000 q^{7} -1.47214 q^{9} -6.47214 q^{11} -0.763932 q^{13} -4.00000 q^{15} +4.47214 q^{17} -1.23607 q^{19} -1.23607 q^{21} -4.00000 q^{23} +5.47214 q^{25} -5.52786 q^{27} +4.47214 q^{29} +2.47214 q^{31} -8.00000 q^{33} +3.23607 q^{35} +4.47214 q^{37} -0.944272 q^{39} -8.47214 q^{41} +6.47214 q^{43} +4.76393 q^{45} -10.4721 q^{47} +1.00000 q^{49} +5.52786 q^{51} +10.0000 q^{53} +20.9443 q^{55} -1.52786 q^{57} -9.23607 q^{59} -11.2361 q^{61} +1.47214 q^{63} +2.47214 q^{65} +4.00000 q^{67} -4.94427 q^{69} +4.94427 q^{71} -2.94427 q^{73} +6.76393 q^{75} +6.47214 q^{77} -12.9443 q^{79} -2.41641 q^{81} -9.23607 q^{83} -14.4721 q^{85} +5.52786 q^{87} -6.00000 q^{89} +0.763932 q^{91} +3.05573 q^{93} +4.00000 q^{95} +12.4721 q^{97} +9.52786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9} - 4 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} - 20 q^{27} - 4 q^{31} - 16 q^{33} + 2 q^{35} + 16 q^{39} - 8 q^{41} + 4 q^{43} + 14 q^{45} - 12 q^{47} + 2 q^{49} + 20 q^{51} + 20 q^{53} + 24 q^{55} - 12 q^{57} - 14 q^{59} - 18 q^{61} - 6 q^{63} - 4 q^{65} + 8 q^{67} + 8 q^{69} - 8 q^{71} + 12 q^{73} + 18 q^{75} + 4 q^{77} - 8 q^{79} + 22 q^{81} - 14 q^{83} - 20 q^{85} + 20 q^{87} - 12 q^{89} + 6 q^{91} + 24 q^{93} + 8 q^{95} + 16 q^{97} + 28 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9 - 4 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^19 + 2 * q^21 - 8 * q^23 + 2 * q^25 - 20 * q^27 - 4 * q^31 - 16 * q^33 + 2 * q^35 + 16 * q^39 - 8 * q^41 + 4 * q^43 + 14 * q^45 - 12 * q^47 + 2 * q^49 + 20 * q^51 + 20 * q^53 + 24 * q^55 - 12 * q^57 - 14 * q^59 - 18 * q^61 - 6 * q^63 - 4 * q^65 + 8 * q^67 + 8 * q^69 - 8 * q^71 + 12 * q^73 + 18 * q^75 + 4 * q^77 - 8 * q^79 + 22 * q^81 - 14 * q^83 - 20 * q^85 + 20 * q^87 - 12 * q^89 + 6 * q^91 + 24 * q^93 + 8 * q^95 + 16 * q^97 + 28 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	−6.47214	−1.95142	−0.975711	−	0.219061i	$-0.929701\pi$
$11$	−6.47214	−1.95142	−0.975711	+	0.219061i	$0.929701\pi$
$12$	0	0
$13$	−0.763932	−0.211877	−0.105938	−	0.994373i	$-0.533785\pi$
$13$	−0.763932	−0.211877	−0.105938	+	0.994373i	$0.533785\pi$
$14$	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
$19$	−1.23607	−0.283573	−0.141787	−	0.989897i	$-0.545285\pi$
$19$	−1.23607	−0.283573	−0.141787	+	0.989897i	$0.545285\pi$
$20$	0	0
$21$	−1.23607	−0.269732
$22$	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
$27$	−5.52786	−1.06384
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	0	0
$33$	−8.00000	−1.39262
$34$	0	0
$35$	3.23607	0.546995
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	0	0
$39$	−0.944272	−0.151205
$40$	0	0
$41$	−8.47214	−1.32313	−0.661563	−	0.749890i	$-0.730106\pi$
$41$	−8.47214	−1.32313	−0.661563	+	0.749890i	$0.730106\pi$
$42$	0	0
$43$	6.47214	0.986991	0.493496	−	0.869748i	$-0.335719\pi$
$43$	6.47214	0.986991	0.493496	+	0.869748i	$0.335719\pi$
$44$	0	0
$45$	4.76393	0.710165
$46$	0	0
$47$	−10.4721	−1.52752	−0.763759	−	0.645501i	$-0.776648\pi$
$47$	−10.4721	−1.52752	−0.763759	+	0.645501i	$0.776648\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.52786	0.774056
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	20.9443	2.82413
$56$	0	0
$57$	−1.52786	−0.202371
$58$	0	0
$59$	−9.23607	−1.20243	−0.601217	−	0.799086i	$-0.705317\pi$
$59$	−9.23607	−1.20243	−0.601217	+	0.799086i	$0.705317\pi$
$60$	0	0
$61$	−11.2361	−1.43863	−0.719316	−	0.694683i	$-0.755544\pi$
$61$	−11.2361	−1.43863	−0.719316	+	0.694683i	$0.755544\pi$
$62$	0	0
$63$	1.47214	0.185472
$64$	0	0
$65$	2.47214	0.306631
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	−4.94427	−0.595220
$70$	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	$-0.555120\pi$
$73$	−2.94427	−0.344601	−0.172300	+	0.985044i	$0.555120\pi$
$74$	0	0
$75$	6.76393	0.781032
$76$	0	0
$77$	6.47214	0.737568
$78$	0	0
$79$	−12.9443	−1.45634	−0.728172	−	0.685394i	$-0.759630\pi$
$79$	−12.9443	−1.45634	−0.728172	+	0.685394i	$0.759630\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	−9.23607	−1.01379	−0.506895	−	0.862008i	$-0.669207\pi$
$83$	−9.23607	−1.01379	−0.506895	+	0.862008i	$0.669207\pi$
$84$	0	0
$85$	−14.4721	−1.56972
$86$	0	0
$87$	5.52786	0.592649
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	0.763932	0.0800818
$92$	0	0
$93$	3.05573	0.316864
$94$	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
$99$	9.52786	0.957586
$100$	0	0
$101$	1.70820	0.169973	0.0849863	−	0.996382i	$-0.472915\pi$
$101$	1.70820	0.169973	0.0849863	+	0.996382i	$0.472915\pi$
$102$	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
$107$	8.94427	0.864675	0.432338	−	0.901712i	$-0.357689\pi$
$107$	8.94427	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$108$	0	0
$109$	−8.47214	−0.811483	−0.405742	−	0.913988i	$-0.632987\pi$
$109$	−8.47214	−0.811483	−0.405742	+	0.913988i	$0.632987\pi$
$110$	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
$115$	12.9443	1.20706
$116$	0	0
$117$	1.12461	0.103970
$118$	0	0
$119$	−4.47214	−0.409960
$120$	0	0
$121$	30.8885	2.80805
$122$	0	0
$123$	−10.4721	−0.944241
$124$	0	0
$125$	−1.52786	−0.136656
$126$	0	0
$127$	−8.94427	−0.793676	−0.396838	−	0.917889i	$-0.629892\pi$
$127$	−8.94427	−0.793676	−0.396838	+	0.917889i	$0.629892\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	11.7082	1.02295	0.511475	−	0.859298i	$-0.329099\pi$
$131$	11.7082	1.02295	0.511475	+	0.859298i	$0.329099\pi$
$132$	0	0
$133$	1.23607	0.107181
$134$	0	0
$135$	17.8885	1.53960
$136$	0	0
$137$	14.9443	1.27678	0.638388	−	0.769715i	$-0.279602\pi$
$137$	14.9443	1.27678	0.638388	+	0.769715i	$0.279602\pi$
$138$	0	0
$139$	1.23607	0.104842	0.0524210	−	0.998625i	$-0.483306\pi$
$139$	1.23607	0.104842	0.0524210	+	0.998625i	$0.483306\pi$
$140$	0	0
$141$	−12.9443	−1.09010
$142$	0	0
$143$	4.94427	0.413461
$144$	0	0
$145$	−14.4721	−1.20185
$146$	0	0
$147$	1.23607	0.101949
$148$	0	0
$149$	−2.94427	−0.241204	−0.120602	−	0.992701i	$-0.538483\pi$
$149$	−2.94427	−0.241204	−0.120602	+	0.992701i	$0.538483\pi$
$150$	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
$155$	−8.00000	−0.642575
$156$	0	0
$157$	−0.763932	−0.0609684	−0.0304842	−	0.999535i	$-0.509705\pi$
$157$	−0.763932	−0.0609684	−0.0304842	+	0.999535i	$0.509705\pi$
$158$	0	0
$159$	12.3607	0.980266
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−3.41641	−0.267594	−0.133797	−	0.991009i	$-0.542717\pi$
$163$	−3.41641	−0.267594	−0.133797	+	0.991009i	$0.542717\pi$
$164$	0	0
$165$	25.8885	2.01542
$166$	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
$171$	1.81966	0.139153
$172$	0	0
$173$	−5.70820	−0.433987	−0.216993	−	0.976173i	$-0.569625\pi$
$173$	−5.70820	−0.433987	−0.216993	+	0.976173i	$0.569625\pi$
$174$	0	0
$175$	−5.47214	−0.413655
$176$	0	0
$177$	−11.4164	−0.858110
$178$	0	0
$179$	−7.05573	−0.527370	−0.263685	−	0.964609i	$-0.584938\pi$
$179$	−7.05573	−0.527370	−0.263685	+	0.964609i	$0.584938\pi$
$180$	0	0
$181$	12.1803	0.905358	0.452679	−	0.891674i	$-0.350468\pi$
$181$	12.1803	0.905358	0.452679	+	0.891674i	$0.350468\pi$
$182$	0	0
$183$	−13.8885	−1.02667
$184$	0	0
$185$	−14.4721	−1.06401
$186$	0	0
$187$	−28.9443	−2.11661
$188$	0	0
$189$	5.52786	0.402093
$190$	0	0
$191$	−4.94427	−0.357755	−0.178877	−	0.983871i	$-0.557247\pi$
$191$	−4.94427	−0.357755	−0.178877	+	0.983871i	$0.557247\pi$
$192$	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
$195$	3.05573	0.218825
$196$	0	0
$197$	−10.9443	−0.779747	−0.389874	−	0.920868i	$-0.627481\pi$
$197$	−10.9443	−0.779747	−0.389874	+	0.920868i	$0.627481\pi$
$198$	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
$203$	−4.47214	−0.313882
$204$	0	0
$205$	27.4164	1.91484
$206$	0	0
$207$	5.88854	0.409282
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	6.11146	0.418750
$214$	0	0
$215$	−20.9443	−1.42839
$216$	0	0
$217$	−2.47214	−0.167820
$218$	0	0
$219$	−3.63932	−0.245922
$220$	0	0
$221$	−3.41641	−0.229812
$222$	0	0
$223$	−12.9443	−0.866813	−0.433406	−	0.901199i	$-0.642688\pi$
$223$	−12.9443	−0.866813	−0.433406	+	0.901199i	$0.642688\pi$
$224$	0	0
$225$	−8.05573	−0.537049
$226$	0	0
$227$	−17.2361	−1.14400	−0.571999	−	0.820254i	$-0.693832\pi$
$227$	−17.2361	−1.14400	−0.571999	+	0.820254i	$0.693832\pi$
$228$	0	0
$229$	−23.5967	−1.55932	−0.779658	−	0.626205i	$-0.784607\pi$
$229$	−23.5967	−1.55932	−0.779658	+	0.626205i	$0.784607\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	−15.8885	−1.04089	−0.520447	−	0.853894i	$-0.674235\pi$
$233$	−15.8885	−1.04089	−0.520447	+	0.853894i	$0.674235\pi$
$234$	0	0
$235$	33.8885	2.21064
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	13.8885	0.898375	0.449188	−	0.893437i	$-0.351713\pi$
$239$	13.8885	0.898375	0.449188	+	0.893437i	$0.351713\pi$
$240$	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
$243$	13.5967	0.872232
$244$	0	0
$245$	−3.23607	−0.206745
$246$	0	0
$247$	0.944272	0.0600826
$248$	0	0
$249$	−11.4164	−0.723485
$250$	0	0
$251$	4.29180	0.270896	0.135448	−	0.990784i	$-0.456753\pi$
$251$	4.29180	0.270896	0.135448	+	0.990784i	$0.456753\pi$
$252$	0	0
$253$	25.8885	1.62760
$254$	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
$259$	−4.47214	−0.277885
$260$	0	0
$261$	−6.58359	−0.407514
$262$	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
$267$	−7.41641	−0.453877
$268$	0	0
$269$	4.18034	0.254880	0.127440	−	0.991846i	$-0.459324\pi$
$269$	4.18034	0.254880	0.127440	+	0.991846i	$0.459324\pi$
$270$	0	0
$271$	24.0000	1.45790	0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000	1.45790	0.728948	+	0.684569i	$0.240010\pi$
$272$	0	0
$273$	0.944272	0.0571499
$274$	0	0
$275$	−35.4164	−2.13569
$276$	0	0
$277$	−7.88854	−0.473977	−0.236988	−	0.971512i	$-0.576160\pi$
$277$	−7.88854	−0.473977	−0.236988	+	0.971512i	$0.576160\pi$
$278$	0	0
$279$	−3.63932	−0.217880
$280$	0	0
$281$	26.0000	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	−6.18034	−0.367383	−0.183692	−	0.982984i	$-0.558805\pi$
$283$	−6.18034	−0.367383	−0.183692	+	0.982984i	$0.558805\pi$
$284$	0	0
$285$	4.94427	0.292873
$286$	0	0
$287$	8.47214	0.500094
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	15.4164	0.903726
$292$	0	0
$293$	12.7639	0.745677	0.372838	−	0.927896i	$-0.378385\pi$
$293$	12.7639	0.745677	0.372838	+	0.927896i	$0.378385\pi$
$294$	0	0
$295$	29.8885	1.74018
$296$	0	0
$297$	35.7771	2.07600
$298$	0	0
$299$	3.05573	0.176717
$300$	0	0
$301$	−6.47214	−0.373048
$302$	0	0
$303$	2.11146	0.121300
$304$	0	0
$305$	36.3607	2.08201
$306$	0	0
$307$	1.81966	0.103853	0.0519267	−	0.998651i	$-0.483464\pi$
$307$	1.81966	0.103853	0.0519267	+	0.998651i	$0.483464\pi$
$308$	0	0
$309$	6.83282	0.388705
$310$	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	$-0.576963\pi$
$313$	−8.47214	−0.478873	−0.239437	+	0.970912i	$0.576963\pi$
$314$	0	0
$315$	−4.76393	−0.268417
$316$	0	0
$317$	−9.05573	−0.508620	−0.254310	−	0.967123i	$-0.581848\pi$
$317$	−9.05573	−0.508620	−0.254310	+	0.967123i	$0.581848\pi$
$318$	0	0
$319$	−28.9443	−1.62057
$320$	0	0
$321$	11.0557	0.617071
$322$	0	0
$323$	−5.52786	−0.307579
$324$	0	0
$325$	−4.18034	−0.231884
$326$	0	0
$327$	−10.4721	−0.579110
$328$	0	0
$329$	10.4721	0.577348
$330$	0	0
$331$	22.4721	1.23518	0.617590	−	0.786500i	$-0.288109\pi$
$331$	22.4721	1.23518	0.617590	+	0.786500i	$0.288109\pi$
$332$	0	0
$333$	−6.58359	−0.360779
$334$	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
$339$	−15.4164	−0.837304
$340$	0	0
$341$	−16.0000	−0.866449
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	16.0000	0.861411
$346$	0	0
$347$	6.47214	0.347442	0.173721	−	0.984795i	$-0.444421\pi$
$347$	6.47214	0.347442	0.173721	+	0.984795i	$0.444421\pi$
$348$	0	0
$349$	−26.6525	−1.42667	−0.713337	−	0.700821i	$-0.752817\pi$
$349$	−26.6525	−1.42667	−0.713337	+	0.700821i	$0.752817\pi$
$350$	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
$355$	−16.0000	−0.849192
$356$	0	0
$357$	−5.52786	−0.292566
$358$	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
$363$	38.1803	2.00395
$364$	0	0
$365$	9.52786	0.498711
$366$	0	0
$367$	22.8328	1.19186	0.595932	−	0.803035i	$-0.296783\pi$
$367$	22.8328	1.19186	0.595932	+	0.803035i	$0.296783\pi$
$368$	0	0
$369$	12.4721	0.649273
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	−2.94427	−0.152449	−0.0762243	−	0.997091i	$-0.524287\pi$
$373$	−2.94427	−0.152449	−0.0762243	+	0.997091i	$0.524287\pi$
$374$	0	0
$375$	−1.88854	−0.0975240
$376$	0	0
$377$	−3.41641	−0.175954
$378$	0	0
$379$	−4.58359	−0.235443	−0.117722	−	0.993047i	$-0.537559\pi$
$379$	−4.58359	−0.235443	−0.117722	+	0.993047i	$0.537559\pi$
$380$	0	0
$381$	−11.0557	−0.566402
$382$	0	0
$383$	15.4164	0.787742	0.393871	−	0.919166i	$-0.371136\pi$
$383$	15.4164	0.787742	0.393871	+	0.919166i	$0.371136\pi$
$384$	0	0
$385$	−20.9443	−1.06742
$386$	0	0
$387$	−9.52786	−0.484329
$388$	0	0
$389$	4.47214	0.226746	0.113373	−	0.993552i	$-0.463834\pi$
$389$	4.47214	0.226746	0.113373	+	0.993552i	$0.463834\pi$
$390$	0	0
$391$	−17.8885	−0.904663
$392$	0	0
$393$	14.4721	0.730023
$394$	0	0
$395$	41.8885	2.10764
$396$	0	0
$397$	15.2361	0.764676	0.382338	−	0.924022i	$-0.375119\pi$
$397$	15.2361	0.764676	0.382338	+	0.924022i	$0.375119\pi$
$398$	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
$403$	−1.88854	−0.0940751
$404$	0	0
$405$	7.81966	0.388562
$406$	0	0
$407$	−28.9443	−1.43471
$408$	0	0
$409$	−21.4164	−1.05897	−0.529487	−	0.848318i	$-0.677615\pi$
$409$	−21.4164	−1.05897	−0.529487	+	0.848318i	$0.677615\pi$
$410$	0	0
$411$	18.4721	0.911163
$412$	0	0
$413$	9.23607	0.454477
$414$	0	0
$415$	29.8885	1.46717
$416$	0	0
$417$	1.52786	0.0748198
$418$	0	0
$419$	22.1803	1.08358	0.541790	−	0.840514i	$-0.317747\pi$
$419$	22.1803	1.08358	0.541790	+	0.840514i	$0.317747\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	15.4164	0.749571
$424$	0	0
$425$	24.4721	1.18707
$426$	0	0
$427$	11.2361	0.543751
$428$	0	0
$429$	6.11146	0.295064
$430$	0	0
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	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
$435$	−17.8885	−0.857690
$436$	0	0
$437$	4.94427	0.236517
$438$	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
$443$	−13.8885	−0.659865	−0.329932	−	0.944005i	$-0.607026\pi$
$443$	−13.8885	−0.659865	−0.329932	+	0.944005i	$0.607026\pi$
$444$	0	0
$445$	19.4164	0.920426
$446$	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
$451$	54.8328	2.58198
$452$	0	0
$453$	11.0557	0.519443
$454$	0	0
$455$	−2.47214	−0.115896
$456$	0	0
$457$	−7.52786	−0.352139	−0.176069	−	0.984378i	$-0.556338\pi$
$457$	−7.52786	−0.352139	−0.176069	+	0.984378i	$0.556338\pi$
$458$	0	0
$459$	−24.7214	−1.15389
$460$	0	0
$461$	−21.7082	−1.01105	−0.505526	−	0.862811i	$-0.668702\pi$
$461$	−21.7082	−1.01105	−0.505526	+	0.862811i	$0.668702\pi$
$462$	0	0
$463$	35.7771	1.66270	0.831351	−	0.555748i	$-0.187568\pi$
$463$	35.7771	1.66270	0.831351	+	0.555748i	$0.187568\pi$
$464$	0	0
$465$	−9.88854	−0.458570
$466$	0	0
$467$	−32.0689	−1.48397	−0.741985	−	0.670416i	$-0.766116\pi$
$467$	−32.0689	−1.48397	−0.741985	+	0.670416i	$0.766116\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	−0.944272	−0.0435098
$472$	0	0
$473$	−41.8885	−1.92604
$474$	0	0
$475$	−6.76393	−0.310350
$476$	0	0
$477$	−14.7214	−0.674045
$478$	0	0
$479$	8.58359	0.392194	0.196097	−	0.980584i	$-0.437173\pi$
$479$	8.58359	0.392194	0.196097	+	0.980584i	$0.437173\pi$
$480$	0	0
$481$	−3.41641	−0.155775
$482$	0	0
$483$	4.94427	0.224972
$484$	0	0
$485$	−40.3607	−1.83268
$486$	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
$491$	37.8885	1.70989	0.854943	−	0.518722i	$-0.173592\pi$
$491$	37.8885	1.70989	0.854943	+	0.518722i	$0.173592\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	−30.8328	−1.38583
$496$	0	0
$497$	−4.94427	−0.221781
$498$	0	0
$499$	21.8885	0.979866	0.489933	−	0.871760i	$-0.337021\pi$
$499$	21.8885	0.979866	0.489933	+	0.871760i	$0.337021\pi$
$500$	0	0
$501$	−28.9443	−1.29313
$502$	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
$507$	−15.3475	−0.681607
$508$	0	0
$509$	41.1246	1.82282	0.911408	−	0.411503i	$-0.134996\pi$
$509$	41.1246	1.82282	0.911408	+	0.411503i	$0.134996\pi$
$510$	0	0
$511$	2.94427	0.130247
$512$	0	0
$513$	6.83282	0.301676
$514$	0	0
$515$	−17.8885	−0.788263
$516$	0	0
$517$	67.7771	2.98083
$518$	0	0
$519$	−7.05573	−0.309712
$520$	0	0
$521$	−6.58359	−0.288432	−0.144216	−	0.989546i	$-0.546066\pi$
$521$	−6.58359	−0.288432	−0.144216	+	0.989546i	$0.546066\pi$
$522$	0	0
$523$	−4.29180	−0.187667	−0.0938336	−	0.995588i	$-0.529912\pi$
$523$	−4.29180	−0.187667	−0.0938336	+	0.995588i	$0.529912\pi$
$524$	0	0
$525$	−6.76393	−0.295202
$526$	0	0
$527$	11.0557	0.481595
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	13.5967	0.590049
$532$	0	0
$533$	6.47214	0.280339
$534$	0	0
$535$	−28.9443	−1.25137
$536$	0	0
$537$	−8.72136	−0.376354
$538$	0	0
$539$	−6.47214	−0.278775
$540$	0	0
$541$	5.05573	0.217363	0.108681	−	0.994077i	$-0.465337\pi$
$541$	5.05573	0.217363	0.108681	+	0.994077i	$0.465337\pi$
$542$	0	0
$543$	15.0557	0.646103
$544$	0	0
$545$	27.4164	1.17439
$546$	0	0
$547$	−4.58359	−0.195980	−0.0979901	−	0.995187i	$-0.531241\pi$
$547$	−4.58359	−0.195980	−0.0979901	+	0.995187i	$0.531241\pi$
$548$	0	0
$549$	16.5410	0.705954
$550$	0	0
$551$	−5.52786	−0.235495
$552$	0	0
$553$	12.9443	0.550446
$554$	0	0
$555$	−17.8885	−0.759326
$556$	0	0
$557$	−9.05573	−0.383704	−0.191852	−	0.981424i	$-0.561449\pi$
$557$	−9.05573	−0.383704	−0.191852	+	0.981424i	$0.561449\pi$
$558$	0	0
$559$	−4.94427	−0.209120
$560$	0	0
$561$	−35.7771	−1.51051
$562$	0	0
$563$	−17.8197	−0.751009	−0.375505	−	0.926821i	$-0.622531\pi$
$563$	−17.8197	−0.751009	−0.375505	+	0.926821i	$0.622531\pi$
$564$	0	0
$565$	40.3607	1.69799
$566$	0	0
$567$	2.41641	0.101480
$568$	0	0
$569$	26.3607	1.10510	0.552549	−	0.833481i	$-0.313655\pi$
$569$	26.3607	1.10510	0.552549	+	0.833481i	$0.313655\pi$
$570$	0	0
$571$	−14.4721	−0.605640	−0.302820	−	0.953048i	$-0.597928\pi$
$571$	−14.4721	−0.605640	−0.302820	+	0.953048i	$0.597928\pi$
$572$	0	0
$573$	−6.11146	−0.255310
$574$	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
$579$	0.583592	0.0242533
$580$	0	0
$581$	9.23607	0.383177
$582$	0	0
$583$	−64.7214	−2.68048
$584$	0	0
$585$	−3.63932	−0.150467
$586$	0	0
$587$	−3.70820	−0.153054	−0.0765270	−	0.997068i	$-0.524383\pi$
$587$	−3.70820	−0.153054	−0.0765270	+	0.997068i	$0.524383\pi$
$588$	0	0
$589$	−3.05573	−0.125909
$590$	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
$595$	14.4721	0.593300
$596$	0	0
$597$	−19.0557	−0.779899
$598$	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	$-0.292246\pi$
$601$	29.7771	1.21463	0.607316	+	0.794460i	$0.292246\pi$
$602$	0	0
$603$	−5.88854	−0.239800
$604$	0	0
$605$	−99.9574	−4.06385
$606$	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
$609$	−5.52786	−0.224000
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	−29.4164	−1.18812	−0.594059	−	0.804422i	$-0.702475\pi$
$613$	−29.4164	−1.18812	−0.594059	+	0.804422i	$0.702475\pi$
$614$	0	0
$615$	33.8885	1.36652
$616$	0	0
$617$	34.3607	1.38331	0.691654	−	0.722229i	$-0.256882\pi$
$617$	34.3607	1.38331	0.691654	+	0.722229i	$0.256882\pi$
$618$	0	0
$619$	−48.0689	−1.93205	−0.966026	−	0.258446i	$-0.916790\pi$
$619$	−48.0689	−1.93205	−0.966026	+	0.258446i	$0.916790\pi$
$620$	0	0
$621$	22.1115	0.887302
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	9.88854	0.394910
$628$	0	0
$629$	20.0000	0.797452
$630$	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
$635$	28.9443	1.14862
$636$	0	0
$637$	−0.763932	−0.0302681
$638$	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
$643$	−43.7082	−1.72368	−0.861842	−	0.507177i	$-0.830689\pi$
$643$	−43.7082	−1.72368	−0.861842	+	0.507177i	$0.830689\pi$
$644$	0	0
$645$	−25.8885	−1.01936
$646$	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
$651$	−3.05573	−0.119763
$652$	0	0
$653$	9.41641	0.368493	0.184246	−	0.982880i	$-0.441016\pi$
$653$	9.41641	0.368493	0.184246	+	0.982880i	$0.441016\pi$
$654$	0	0
$655$	−37.8885	−1.48043
$656$	0	0
$657$	4.33437	0.169100
$658$	0	0
$659$	37.3050	1.45319	0.726597	−	0.687064i	$-0.241101\pi$
$659$	37.3050	1.45319	0.726597	+	0.687064i	$0.241101\pi$
$660$	0	0
$661$	22.6525	0.881079	0.440540	−	0.897733i	$-0.354787\pi$
$661$	22.6525	0.881079	0.440540	+	0.897733i	$0.354787\pi$
$662$	0	0
$663$	−4.22291	−0.164004
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	−17.8885	−0.692647
$668$	0	0
$669$	−16.0000	−0.618596
$670$	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
$675$	−30.2492	−1.16429
$676$	0	0
$677$	−19.8197	−0.761731	−0.380866	−	0.924630i	$-0.624374\pi$
$677$	−19.8197	−0.761731	−0.380866	+	0.924630i	$0.624374\pi$
$678$	0	0
$679$	−12.4721	−0.478637
$680$	0	0
$681$	−21.3050	−0.816408
$682$	0	0
$683$	23.7771	0.909805	0.454902	−	0.890541i	$-0.349674\pi$
$683$	23.7771	0.909805	0.454902	+	0.890541i	$0.349674\pi$
$684$	0	0
$685$	−48.3607	−1.84777
$686$	0	0
$687$	−29.1672	−1.11280
$688$	0	0
$689$	−7.63932	−0.291035
$690$	0	0
$691$	−14.1803	−0.539446	−0.269723	−	0.962938i	$-0.586932\pi$
$691$	−14.1803	−0.539446	−0.269723	+	0.962938i	$0.586932\pi$
$692$	0	0
$693$	−9.52786	−0.361934
$694$	0	0
$695$	−4.00000	−0.151729
$696$	0	0
$697$	−37.8885	−1.43513
$698$	0	0
$699$	−19.6393	−0.742827
$700$	0	0
$701$	41.4164	1.56428	0.782138	−	0.623105i	$-0.214129\pi$
$701$	41.4164	1.56428	0.782138	+	0.623105i	$0.214129\pi$
$702$	0	0
$703$	−5.52786	−0.208487
$704$	0	0
$705$	41.8885	1.57761
$706$	0	0
$707$	−1.70820	−0.0642436
$708$	0	0
$709$	−1.63932	−0.0615660	−0.0307830	−	0.999526i	$-0.509800\pi$
$709$	−1.63932	−0.0615660	−0.0307830	+	0.999526i	$0.509800\pi$
$710$	0	0
$711$	19.0557	0.714646
$712$	0	0
$713$	−9.88854	−0.370329
$714$	0	0
$715$	−16.0000	−0.598366
$716$	0	0
$717$	17.1672	0.641120
$718$	0	0
$719$	−51.1935	−1.90920	−0.954598	−	0.297898i	$-0.903714\pi$
$719$	−51.1935	−1.90920	−0.954598	+	0.297898i	$0.903714\pi$
$720$	0	0
$721$	−5.52786	−0.205868
$722$	0	0
$723$	15.4164	0.573342
$724$	0	0
$725$	24.4721	0.908872
$726$	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
$731$	28.9443	1.07054
$732$	0	0
$733$	4.76393	0.175960	0.0879799	−	0.996122i	$-0.471959\pi$
$733$	4.76393	0.175960	0.0879799	+	0.996122i	$0.471959\pi$
$734$	0	0
$735$	−4.00000	−0.147542
$736$	0	0
$737$	−25.8885	−0.953617
$738$	0	0
$739$	−3.41641	−0.125675	−0.0628373	−	0.998024i	$-0.520015\pi$
$739$	−3.41641	−0.125675	−0.0628373	+	0.998024i	$0.520015\pi$
$740$	0	0
$741$	1.16718	0.0428776
$742$	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
$747$	13.5967	0.497479
$748$	0	0
$749$	−8.94427	−0.326817
$750$	0	0
$751$	−36.0000	−1.31366	−0.656829	−	0.754039i	$-0.728103\pi$
$751$	−36.0000	−1.31366	−0.656829	+	0.754039i	$0.728103\pi$
$752$	0	0
$753$	5.30495	0.193323
$754$	0	0
$755$	−28.9443	−1.05339
$756$	0	0
$757$	−39.3050	−1.42856	−0.714281	−	0.699859i	$-0.753246\pi$
$757$	−39.3050	−1.42856	−0.714281	+	0.699859i	$0.753246\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	−3.52786	−0.127885	−0.0639425	−	0.997954i	$-0.520367\pi$
$761$	−3.52786	−0.127885	−0.0639425	+	0.997954i	$0.520367\pi$
$762$	0	0
$763$	8.47214	0.306712
$764$	0	0
$765$	21.3050	0.770282
$766$	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
$771$	−17.3050	−0.623223
$772$	0	0
$773$	−40.1803	−1.44519	−0.722593	−	0.691274i	$-0.757050\pi$
$773$	−40.1803	−1.44519	−0.722593	+	0.691274i	$0.757050\pi$
$774$	0	0
$775$	13.5279	0.485935
$776$	0	0
$777$	−5.52786	−0.198311
$778$	0	0
$779$	10.4721	0.375203
$780$	0	0
$781$	−32.0000	−1.14505
$782$	0	0
$783$	−24.7214	−0.883469
$784$	0	0
$785$	2.47214	0.0882343
$786$	0	0
$787$	−12.2918	−0.438155	−0.219078	−	0.975707i	$-0.570305\pi$
$787$	−12.2918	−0.438155	−0.219078	+	0.975707i	$0.570305\pi$
$788$	0	0
$789$	−13.6656	−0.486509
$790$	0	0
$791$	12.4721	0.443458
$792$	0	0
$793$	8.58359	0.304812
$794$	0	0
$795$	−40.0000	−1.41865
$796$	0	0
$797$	52.1803	1.84832	0.924161	−	0.382003i	$-0.124765\pi$
$797$	52.1803	1.84832	0.924161	+	0.382003i	$0.124765\pi$
$798$	0	0
$799$	−46.8328	−1.65683
$800$	0	0
$801$	8.83282	0.312092
$802$	0	0
$803$	19.0557	0.672462
$804$	0	0
$805$	−12.9443	−0.456226
$806$	0	0
$807$	5.16718	0.181894
$808$	0	0
$809$	−17.4164	−0.612328	−0.306164	−	0.951979i	$-0.599046\pi$
$809$	−17.4164	−0.612328	−0.306164	+	0.951979i	$0.599046\pi$
$810$	0	0
$811$	−53.0132	−1.86154	−0.930772	−	0.365601i	$-0.880864\pi$
$811$	−53.0132	−1.86154	−0.930772	+	0.365601i	$0.880864\pi$
$812$	0	0
$813$	29.6656	1.04042
$814$	0	0
$815$	11.0557	0.387265
$816$	0	0
$817$	−8.00000	−0.279885
$818$	0	0
$819$	−1.12461	−0.0392971
$820$	0	0
$821$	−4.11146	−0.143491	−0.0717454	−	0.997423i	$-0.522857\pi$
$821$	−4.11146	−0.143491	−0.0717454	+	0.997423i	$0.522857\pi$
$822$	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
$827$	15.0557	0.523539	0.261769	−	0.965130i	$-0.415694\pi$
$827$	15.0557	0.523539	0.261769	+	0.965130i	$0.415694\pi$
$828$	0	0
$829$	−11.2361	−0.390245	−0.195122	−	0.980779i	$-0.562510\pi$
$829$	−11.2361	−0.390245	−0.195122	+	0.980779i	$0.562510\pi$
$830$	0	0
$831$	−9.75078	−0.338251
$832$	0	0
$833$	4.47214	0.154950
$834$	0	0
$835$	75.7771	2.62237
$836$	0	0
$837$	−13.6656	−0.472353
$838$	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
$843$	32.1378	1.10688
$844$	0	0
$845$	40.1803	1.38225
$846$	0	0
$847$	−30.8885	−1.06134
$848$	0	0
$849$	−7.63932	−0.262181
$850$	0	0
$851$	−17.8885	−0.613211
$852$	0	0
$853$	−13.7082	−0.469360	−0.234680	−	0.972073i	$-0.575404\pi$
$853$	−13.7082	−0.469360	−0.234680	+	0.972073i	$0.575404\pi$
$854$	0	0
$855$	−5.88854	−0.201384
$856$	0	0
$857$	−27.5279	−0.940334	−0.470167	−	0.882577i	$-0.655806\pi$
$857$	−27.5279	−0.940334	−0.470167	+	0.882577i	$0.655806\pi$
$858$	0	0
$859$	−33.8197	−1.15391	−0.576956	−	0.816775i	$-0.695760\pi$
$859$	−33.8197	−1.15391	−0.576956	+	0.816775i	$0.695760\pi$
$860$	0	0
$861$	10.4721	0.356889
$862$	0	0
$863$	20.9443	0.712951	0.356476	−	0.934305i	$-0.383978\pi$
$863$	20.9443	0.712951	0.356476	+	0.934305i	$0.383978\pi$
$864$	0	0
$865$	18.4721	0.628071
$866$	0	0
$867$	3.70820	0.125937
$868$	0	0
$869$	83.7771	2.84194
$870$	0	0
$871$	−3.05573	−0.103539
$872$	0	0
$873$	−18.3607	−0.621415
$874$	0	0
$875$	1.52786	0.0516512
$876$	0	0
$877$	−13.4164	−0.453040	−0.226520	−	0.974007i	$-0.572735\pi$
$877$	−13.4164	−0.453040	−0.226520	+	0.974007i	$0.572735\pi$
$878$	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
$883$	23.0557	0.775887	0.387944	−	0.921683i	$-0.373186\pi$
$883$	23.0557	0.775887	0.387944	+	0.921683i	$0.373186\pi$
$884$	0	0
$885$	36.9443	1.24187
$886$	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
$891$	15.6393	0.523937
$892$	0	0
$893$	12.9443	0.433164
$894$	0	0
$895$	22.8328	0.763217
$896$	0	0
$897$	3.77709	0.126113
$898$	0	0
$899$	11.0557	0.368729
$900$	0	0
$901$	44.7214	1.48988
$902$	0	0
$903$	−8.00000	−0.266223
$904$	0	0
$905$	−39.4164	−1.31025
$906$	0	0
$907$	0.944272	0.0313540	0.0156770	−	0.999877i	$-0.495010\pi$
$907$	0.944272	0.0313540	0.0156770	+	0.999877i	$0.495010\pi$
$908$	0	0
$909$	−2.51471	−0.0834076
$910$	0	0
$911$	34.8328	1.15406	0.577031	−	0.816722i	$-0.304211\pi$
$911$	34.8328	1.15406	0.577031	+	0.816722i	$0.304211\pi$
$912$	0	0
$913$	59.7771	1.97833
$914$	0	0
$915$	44.9443	1.48581
$916$	0	0
$917$	−11.7082	−0.386639
$918$	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
$923$	−3.77709	−0.124324
$924$	0	0
$925$	24.4721	0.804639
$926$	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
$931$	−1.23607	−0.0405105
$932$	0	0
$933$	−9.88854	−0.323736
$934$	0	0
$935$	93.6656	3.06319
$936$	0	0
$937$	−9.05573	−0.295838	−0.147919	−	0.988999i	$-0.547257\pi$
$937$	−9.05573	−0.295838	−0.147919	+	0.988999i	$0.547257\pi$
$938$	0	0
$939$	−10.4721	−0.341745
$940$	0	0
$941$	35.5967	1.16042	0.580210	−	0.814467i	$-0.302970\pi$
$941$	35.5967	1.16042	0.580210	+	0.814467i	$0.302970\pi$
$942$	0	0
$943$	33.8885	1.10356
$944$	0	0
$945$	−17.8885	−0.581914
$946$	0	0
$947$	−4.58359	−0.148947	−0.0744734	−	0.997223i	$-0.523728\pi$
$947$	−4.58359	−0.148947	−0.0744734	+	0.997223i	$0.523728\pi$
$948$	0	0
$949$	2.24922	0.0730129
$950$	0	0
$951$	−11.1935	−0.362974
$952$	0	0
$953$	51.8885	1.68083	0.840417	−	0.541940i	$-0.182310\pi$
$953$	51.8885	1.68083	0.840417	+	0.541940i	$0.182310\pi$
$954$	0	0
$955$	16.0000	0.517748
$956$	0	0
$957$	−35.7771	−1.15651
$958$	0	0
$959$	−14.9443	−0.482576
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	−13.1672	−0.424307
$964$	0	0
$965$	−1.52786	−0.0491837
$966$	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
$971$	22.7639	0.730529	0.365265	−	0.930904i	$-0.380978\pi$
$971$	22.7639	0.730529	0.365265	+	0.930904i	$0.380978\pi$
$972$	0	0
$973$	−1.23607	−0.0396265
$974$	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
$979$	38.8328	1.24110
$980$	0	0
$981$	12.4721	0.398205
$982$	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
$987$	12.9443	0.412021
$988$	0	0
$989$	−25.8885	−0.823208
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	0	0
$993$	27.7771	0.881479
$994$	0	0
$995$	49.8885	1.58157
$996$	0	0
$997$	−18.0689	−0.572247	−0.286124	−	0.958193i	$-0.592367\pi$
$997$	−18.0689	−0.572247	−0.286124	+	0.958193i	$0.592367\pi$
$998$	0	0
$999$	−24.7214	−0.782149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.2.a.i.1.2		2
3.2	odd	2		4032.2.a.bv.1.2		2
4.3	odd	2		448.2.a.j.1.1		2
7.6	odd	2		3136.2.a.by.1.1		2
8.3	odd	2		224.2.a.c.1.2	✓	2
8.5	even	2		224.2.a.d.1.1	yes	2
12.11	even	2		4032.2.a.bw.1.2		2
16.3	odd	4		1792.2.b.k.897.2		4
16.5	even	4		1792.2.b.m.897.2		4
16.11	odd	4		1792.2.b.k.897.3		4
16.13	even	4		1792.2.b.m.897.3		4
24.5	odd	2		2016.2.a.o.1.1		2
24.11	even	2		2016.2.a.r.1.1		2
28.27	even	2		3136.2.a.bf.1.2		2
40.19	odd	2		5600.2.a.bk.1.1		2
40.29	even	2		5600.2.a.z.1.2		2
56.3	even	6		1568.2.i.n.961.2		4
56.5	odd	6		1568.2.i.w.1537.1		4
56.11	odd	6		1568.2.i.v.961.1		4
56.13	odd	2		1568.2.a.k.1.2		2
56.19	even	6		1568.2.i.n.1537.2		4
56.27	even	2		1568.2.a.v.1.1		2
56.37	even	6		1568.2.i.m.1537.2		4
56.45	odd	6		1568.2.i.w.961.1		4
56.51	odd	6		1568.2.i.v.1537.1		4
56.53	even	6		1568.2.i.m.961.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.a.c.1.2	✓	2	8.3	odd	2
224.2.a.d.1.1	yes	2	8.5	even	2
448.2.a.i.1.2		2	1.1	even	1	trivial
448.2.a.j.1.1		2	4.3	odd	2
1568.2.a.k.1.2		2	56.13	odd	2
1568.2.a.v.1.1		2	56.27	even	2
1568.2.i.m.961.2		4	56.53	even	6
1568.2.i.m.1537.2		4	56.37	even	6
1568.2.i.n.961.2		4	56.3	even	6
1568.2.i.n.1537.2		4	56.19	even	6
1568.2.i.v.961.1		4	56.11	odd	6
1568.2.i.v.1537.1		4	56.51	odd	6
1568.2.i.w.961.1		4	56.45	odd	6
1568.2.i.w.1537.1		4	56.5	odd	6
1792.2.b.k.897.2		4	16.3	odd	4
1792.2.b.k.897.3		4	16.11	odd	4
1792.2.b.m.897.2		4	16.5	even	4
1792.2.b.m.897.3		4	16.13	even	4
2016.2.a.o.1.1		2	24.5	odd	2
2016.2.a.r.1.1		2	24.11	even	2
3136.2.a.bf.1.2		2	28.27	even	2
3136.2.a.by.1.1		2	7.6	odd	2
4032.2.a.bv.1.2		2	3.2	odd	2
4032.2.a.bw.1.2		2	12.11	even	2
5600.2.a.z.1.2		2	40.29	even	2
5600.2.a.bk.1.1		2	40.19	odd	2

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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -3.23607 q^{5} -1.00000 q^{7} -1.47214 q^{9} +O(q^{10})\)	\(q+1.23607 q^{3} -3.23607 q^{5} -1.00000 q^{7} -1.47214 q^{9} -6.47214 q^{11} -0.763932 q^{13} -4.00000 q^{15} +4.47214 q^{17} -1.23607 q^{19} -1.23607 q^{21} -4.00000 q^{23} +5.47214 q^{25} -5.52786 q^{27} +4.47214 q^{29} +2.47214 q^{31} -8.00000 q^{33} +3.23607 q^{35} +4.47214 q^{37} -0.944272 q^{39} -8.47214 q^{41} +6.47214 q^{43} +4.76393 q^{45} -10.4721 q^{47} +1.00000 q^{49} +5.52786 q^{51} +10.0000 q^{53} +20.9443 q^{55} -1.52786 q^{57} -9.23607 q^{59} -11.2361 q^{61} +1.47214 q^{63} +2.47214 q^{65} +4.00000 q^{67} -4.94427 q^{69} +4.94427 q^{71} -2.94427 q^{73} +6.76393 q^{75} +6.47214 q^{77} -12.9443 q^{79} -2.41641 q^{81} -9.23607 q^{83} -14.4721 q^{85} +5.52786 q^{87} -6.00000 q^{89} +0.763932 q^{91} +3.05573 q^{93} +4.00000 q^{95} +12.4721 q^{97} +9.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9} - 4 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{19} + 2 q^{21} - 8 q^{23} + 2 q^{25} - 20 q^{27} - 4 q^{31} - 16 q^{33} + 2 q^{35} + 16 q^{39} - 8 q^{41} + 4 q^{43} + 14 q^{45} - 12 q^{47} + 2 q^{49} + 20 q^{51} + 20 q^{53} + 24 q^{55} - 12 q^{57} - 14 q^{59} - 18 q^{61} - 6 q^{63} - 4 q^{65} + 8 q^{67} + 8 q^{69} - 8 q^{71} + 12 q^{73} + 18 q^{75} + 4 q^{77} - 8 q^{79} + 22 q^{81} - 14 q^{83} - 20 q^{85} + 20 q^{87} - 12 q^{89} + 6 q^{91} + 24 q^{93} + 8 q^{95} + 16 q^{97} + 28 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9 - 4 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^19 + 2 * q^21 - 8 * q^23 + 2 * q^25 - 20 * q^27 - 4 * q^31 - 16 * q^33 + 2 * q^35 + 16 * q^39 - 8 * q^41 + 4 * q^43 + 14 * q^45 - 12 * q^47 + 2 * q^49 + 20 * q^51 + 20 * q^53 + 24 * q^55 - 12 * q^57 - 14 * q^59 - 18 * q^61 - 6 * q^63 - 4 * q^65 + 8 * q^67 + 8 * q^69 - 8 * q^71 + 12 * q^73 + 18 * q^75 + 4 * q^77 - 8 * q^79 + 22 * q^81 - 14 * q^83 - 20 * q^85 + 20 * q^87 - 12 * q^89 + 6 * q^91 + 24 * q^93 + 8 * q^95 + 16 * q^97 + 28 * q^99

Introduction

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