LMFDB - Embedded newform 896.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [896,2,Mod(1,896)]

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, 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("896.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 896 = 2^{7} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	896.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:	$7.15459602111$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	896.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.14637 q^{3} -3.83221 q^{5} -1.00000 q^{7} -1.68585 q^{9} +O(q^{10})$	$q+1.14637 q^{3} -3.83221 q^{5} -1.00000 q^{7} -1.68585 q^{9} +4.68585 q^{11} +5.53948 q^{13} -4.39312 q^{15} +0.292731 q^{17} +5.14637 q^{19} -1.14637 q^{21} -4.97858 q^{23} +9.68585 q^{25} -5.37169 q^{27} +4.29273 q^{29} +7.66442 q^{31} +5.37169 q^{33} +3.83221 q^{35} +9.66442 q^{37} +6.35027 q^{39} +3.70727 q^{41} -5.27131 q^{43} +6.46052 q^{45} -2.29273 q^{47} +1.00000 q^{49} +0.335577 q^{51} -2.00000 q^{53} -17.9572 q^{55} +5.89962 q^{57} +9.93260 q^{59} +4.16779 q^{61} +1.68585 q^{63} -21.2285 q^{65} -10.9786 q^{67} -5.70727 q^{69} +7.37169 q^{73} +11.1035 q^{75} -4.68585 q^{77} -13.9572 q^{79} -1.10038 q^{81} +4.81079 q^{83} -1.12181 q^{85} +4.92104 q^{87} -2.58546 q^{89} -5.53948 q^{91} +8.78623 q^{93} -19.7220 q^{95} -14.2499 q^{97} -7.89962 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} + 2 q^{5} - 3 q^{7} + 7 q^{9}+O(q^{10}) $ 3 * q + 2 * q^3 + 2 * q^5 - 3 * q^7 + 7 * q^9	$ 3 q + 2 q^{3} + 2 q^{5} - 3 q^{7} + 7 q^{9} + 2 q^{11} + 6 q^{13} - 4 q^{15} - 2 q^{17} + 14 q^{19} - 2 q^{21} + 17 q^{25} + 8 q^{27} + 10 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} + 2 q^{37} - 20 q^{39} + 14 q^{41} + 2 q^{43} + 30 q^{45} - 4 q^{47} + 3 q^{49} + 28 q^{51} - 6 q^{53} - 24 q^{55} + 24 q^{57} + 10 q^{59} + 26 q^{61} - 7 q^{63} - 16 q^{65} - 18 q^{67} - 20 q^{69} - 2 q^{73} + 2 q^{75} - 2 q^{77} - 12 q^{79} + 3 q^{81} - 14 q^{83} - 12 q^{85} + 36 q^{87} - 2 q^{89} - 6 q^{91} + 8 q^{93} + 4 q^{95} - 10 q^{97} - 30 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 + 2 * q^5 - 3 * q^7 + 7 * q^9 + 2 * q^11 + 6 * q^13 - 4 * q^15 - 2 * q^17 + 14 * q^19 - 2 * q^21 + 17 * q^25 + 8 * q^27 + 10 * q^29 - 4 * q^31 - 8 * q^33 - 2 * q^35 + 2 * q^37 - 20 * q^39 + 14 * q^41 + 2 * q^43 + 30 * q^45 - 4 * q^47 + 3 * q^49 + 28 * q^51 - 6 * q^53 - 24 * q^55 + 24 * q^57 + 10 * q^59 + 26 * q^61 - 7 * q^63 - 16 * q^65 - 18 * q^67 - 20 * q^69 - 2 * q^73 + 2 * q^75 - 2 * q^77 - 12 * q^79 + 3 * q^81 - 14 * q^83 - 12 * q^85 + 36 * q^87 - 2 * q^89 - 6 * q^91 + 8 * q^93 + 4 * q^95 - 10 * q^97 - 30 * 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
$3$	1.14637	0.661854	0.330927	−	0.943656i	$-0.392639\pi$
$3$	1.14637	0.661854	0.330927	+	0.943656i	$0.392639\pi$
$4$	0	0
$5$	−3.83221	−1.71382	−0.856909	−	0.515468i	$-0.827618\pi$
$5$	−3.83221	−1.71382	−0.856909	+	0.515468i	$0.827618\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.68585	−0.561949
$10$	0	0
$11$	4.68585	1.41284	0.706418	−	0.707795i	$-0.250310\pi$
$11$	4.68585	1.41284	0.706418	+	0.707795i	$0.250310\pi$
$12$	0	0
$13$	5.53948	1.53638	0.768188	−	0.640225i	$-0.221159\pi$
$13$	5.53948	1.53638	0.768188	+	0.640225i	$0.221159\pi$
$14$	0	0
$15$	−4.39312	−1.13430
$16$	0	0
$17$	0.292731	0.0709977	0.0354988	−	0.999370i	$-0.488698\pi$
$17$	0.292731	0.0709977	0.0354988	+	0.999370i	$0.488698\pi$
$18$	0	0
$19$	5.14637	1.18066	0.590329	−	0.807163i	$-0.298998\pi$
$19$	5.14637	1.18066	0.590329	+	0.807163i	$0.298998\pi$
$20$	0	0
$21$	−1.14637	−0.250157
$22$	0	0
$23$	−4.97858	−1.03811	−0.519053	−	0.854742i	$-0.673715\pi$
$23$	−4.97858	−1.03811	−0.519053	+	0.854742i	$0.673715\pi$
$24$	0	0
$25$	9.68585	1.93717
$26$	0	0
$27$	−5.37169	−1.03378
$28$	0	0
$29$	4.29273	0.797140	0.398570	−	0.917138i	$-0.369507\pi$
$29$	4.29273	0.797140	0.398570	+	0.917138i	$0.369507\pi$
$30$	0	0
$31$	7.66442	1.37657	0.688286	−	0.725440i	$-0.258364\pi$
$31$	7.66442	1.37657	0.688286	+	0.725440i	$0.258364\pi$
$32$	0	0
$33$	5.37169	0.935092
$34$	0	0
$35$	3.83221	0.647762
$36$	0	0
$37$	9.66442	1.58882	0.794411	−	0.607381i	$-0.207780\pi$
$37$	9.66442	1.58882	0.794411	+	0.607381i	$0.207780\pi$
$38$	0	0
$39$	6.35027	1.01686
$40$	0	0
$41$	3.70727	0.578978	0.289489	−	0.957181i	$-0.406515\pi$
$41$	3.70727	0.578978	0.289489	+	0.957181i	$0.406515\pi$
$42$	0	0
$43$	−5.27131	−0.803867	−0.401933	−	0.915669i	$-0.631662\pi$
$43$	−5.27131	−0.803867	−0.401933	+	0.915669i	$0.631662\pi$
$44$	0	0
$45$	6.46052	0.963077
$46$	0	0
$47$	−2.29273	−0.334429	−0.167215	−	0.985921i	$-0.553477\pi$
$47$	−2.29273	−0.334429	−0.167215	+	0.985921i	$0.553477\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.335577	0.0469901
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−17.9572	−2.42134
$56$	0	0
$57$	5.89962	0.781423
$58$	0	0
$59$	9.93260	1.29311	0.646557	−	0.762866i	$-0.276208\pi$
$59$	9.93260	1.29311	0.646557	+	0.762866i	$0.276208\pi$
$60$	0	0
$61$	4.16779	0.533631	0.266815	−	0.963748i	$-0.414029\pi$
$61$	4.16779	0.533631	0.266815	+	0.963748i	$0.414029\pi$
$62$	0	0
$63$	1.68585	0.212397
$64$	0	0
$65$	−21.2285	−2.63307
$66$	0	0
$67$	−10.9786	−1.34125	−0.670623	−	0.741798i	$-0.733973\pi$
$67$	−10.9786	−1.34125	−0.670623	+	0.741798i	$0.733973\pi$
$68$	0	0
$69$	−5.70727	−0.687074
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	7.37169	0.862791	0.431396	−	0.902163i	$-0.358021\pi$
$73$	7.37169	0.862791	0.431396	+	0.902163i	$0.358021\pi$
$74$	0	0
$75$	11.1035	1.28212
$76$	0	0
$77$	−4.68585	−0.534002
$78$	0	0
$79$	−13.9572	−1.57030	−0.785151	−	0.619304i	$-0.787415\pi$
$79$	−13.9572	−1.57030	−0.785151	+	0.619304i	$0.787415\pi$
$80$	0	0
$81$	−1.10038	−0.122265
$82$	0	0
$83$	4.81079	0.528053	0.264026	−	0.964515i	$-0.414949\pi$
$83$	4.81079	0.528053	0.264026	+	0.964515i	$0.414949\pi$
$84$	0	0
$85$	−1.12181	−0.121677
$86$	0	0
$87$	4.92104	0.527591
$88$	0	0
$89$	−2.58546	−0.274058	−0.137029	−	0.990567i	$-0.543755\pi$
$89$	−2.58546	−0.274058	−0.137029	+	0.990567i	$0.543755\pi$
$90$	0	0
$91$	−5.53948	−0.580695
$92$	0	0
$93$	8.78623	0.911090
$94$	0	0
$95$	−19.7220	−2.02343
$96$	0	0
$97$	−14.2499	−1.44686	−0.723428	−	0.690399i	$-0.757435\pi$
$97$	−14.2499	−1.44686	−0.723428	+	0.690399i	$0.757435\pi$
$98$	0	0
$99$	−7.89962	−0.793941
$100$	0	0
$101$	1.87506	0.186575	0.0932876	−	0.995639i	$-0.470262\pi$
$101$	1.87506	0.186575	0.0932876	+	0.995639i	$0.470262\pi$
$102$	0	0
$103$	7.66442	0.755198	0.377599	−	0.925969i	$-0.376750\pi$
$103$	7.66442	0.755198	0.377599	+	0.925969i	$0.376750\pi$
$104$	0	0
$105$	4.39312	0.428724
$106$	0	0
$107$	−2.39312	−0.231351	−0.115676	−	0.993287i	$-0.536903\pi$
$107$	−2.39312	−0.231351	−0.115676	+	0.993287i	$0.536903\pi$
$108$	0	0
$109$	−1.07896	−0.103346	−0.0516729	−	0.998664i	$-0.516455\pi$
$109$	−1.07896	−0.103346	−0.0516729	+	0.998664i	$0.516455\pi$
$110$	0	0
$111$	11.0790	1.05157
$112$	0	0
$113$	9.27131	0.872171	0.436086	−	0.899905i	$-0.356364\pi$
$113$	9.27131	0.872171	0.436086	+	0.899905i	$0.356364\pi$
$114$	0	0
$115$	19.0790	1.77912
$116$	0	0
$117$	−9.33871	−0.863364
$118$	0	0
$119$	−0.292731	−0.0268346
$120$	0	0
$121$	10.9572	0.996105
$122$	0	0
$123$	4.24989	0.383199
$124$	0	0
$125$	−17.9572	−1.60614
$126$	0	0
$127$	6.35027	0.563495	0.281748	−	0.959489i	$-0.409086\pi$
$127$	6.35027	0.563495	0.281748	+	0.959489i	$0.409086\pi$
$128$	0	0
$129$	−6.04285	−0.532043
$130$	0	0
$131$	−10.5181	−0.918967	−0.459483	−	0.888186i	$-0.651965\pi$
$131$	−10.5181	−0.918967	−0.459483	+	0.888186i	$0.651965\pi$
$132$	0	0
$133$	−5.14637	−0.446246
$134$	0	0
$135$	20.5855	1.77171
$136$	0	0
$137$	−11.3717	−0.971549	−0.485775	−	0.874084i	$-0.661462\pi$
$137$	−11.3717	−0.971549	−0.485775	+	0.874084i	$0.661462\pi$
$138$	0	0
$139$	10.8536	0.920593	0.460297	−	0.887765i	$-0.347743\pi$
$139$	10.8536	0.920593	0.460297	+	0.887765i	$0.347743\pi$
$140$	0	0
$141$	−2.62831	−0.221343
$142$	0	0
$143$	25.9572	2.17065
$144$	0	0
$145$	−16.4507	−1.36615
$146$	0	0
$147$	1.14637	0.0945506
$148$	0	0
$149$	−2.78623	−0.228257	−0.114128	−	0.993466i	$-0.536408\pi$
$149$	−2.78623	−0.228257	−0.114128	+	0.993466i	$0.536408\pi$
$150$	0	0
$151$	1.56404	0.127280	0.0636398	−	0.997973i	$-0.479729\pi$
$151$	1.56404	0.127280	0.0636398	+	0.997973i	$0.479729\pi$
$152$	0	0
$153$	−0.493499	−0.0398971
$154$	0	0
$155$	−29.3717	−2.35919
$156$	0	0
$157$	14.7104	1.17402	0.587009	−	0.809580i	$-0.300305\pi$
$157$	14.7104	1.17402	0.587009	+	0.809580i	$0.300305\pi$
$158$	0	0
$159$	−2.29273	−0.181825
$160$	0	0
$161$	4.97858	0.392367
$162$	0	0
$163$	−0.100384	−0.00786270	−0.00393135	−	0.999992i	$-0.501251\pi$
$163$	−0.100384	−0.00786270	−0.00393135	+	0.999992i	$0.501251\pi$
$164$	0	0
$165$	−20.5855	−1.60258
$166$	0	0
$167$	−5.70727	−0.441642	−0.220821	−	0.975314i	$-0.570874\pi$
$167$	−5.70727	−0.441642	−0.220821	+	0.975314i	$0.570874\pi$
$168$	0	0
$169$	17.6858	1.36045
$170$	0	0
$171$	−8.67598	−0.663469
$172$	0	0
$173$	12.8683	0.978361	0.489180	−	0.872183i	$-0.337296\pi$
$173$	12.8683	0.978361	0.489180	+	0.872183i	$0.337296\pi$
$174$	0	0
$175$	−9.68585	−0.732181
$176$	0	0
$177$	11.3864	0.855853
$178$	0	0
$179$	−26.3074	−1.96631	−0.983155	−	0.182776i	$-0.941492\pi$
$179$	−26.3074	−1.96631	−0.983155	+	0.182776i	$0.941492\pi$
$180$	0	0
$181$	3.24675	0.241329	0.120665	−	0.992693i	$-0.461497\pi$
$181$	3.24675	0.241329	0.120665	+	0.992693i	$0.461497\pi$
$182$	0	0
$183$	4.77781	0.353186
$184$	0	0
$185$	−37.0361	−2.72295
$186$	0	0
$187$	1.37169	0.100308
$188$	0	0
$189$	5.37169	0.390733
$190$	0	0
$191$	17.3717	1.25697	0.628486	−	0.777821i	$-0.283675\pi$
$191$	17.3717	1.25697	0.628486	+	0.777821i	$0.283675\pi$
$192$	0	0
$193$	27.4292	1.97440	0.987200	−	0.159490i	$-0.0509848\pi$
$193$	27.4292	1.97440	0.987200	+	0.159490i	$0.0509848\pi$
$194$	0	0
$195$	−24.3356	−1.74271
$196$	0	0
$197$	−10.0000	−0.712470	−0.356235	−	0.934396i	$-0.615940\pi$
$197$	−10.0000	−0.712470	−0.356235	+	0.934396i	$0.615940\pi$
$198$	0	0
$199$	−4.24989	−0.301266	−0.150633	−	0.988590i	$-0.548131\pi$
$199$	−4.24989	−0.301266	−0.150633	+	0.988590i	$0.548131\pi$
$200$	0	0
$201$	−12.5855	−0.887710
$202$	0	0
$203$	−4.29273	−0.301291
$204$	0	0
$205$	−14.2070	−0.992263
$206$	0	0
$207$	8.39312	0.583362
$208$	0	0
$209$	24.1151	1.66807
$210$	0	0
$211$	20.3503	1.40097	0.700485	−	0.713667i	$-0.252967\pi$
$211$	20.3503	1.40097	0.700485	+	0.713667i	$0.252967\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	20.2008	1.37768
$216$	0	0
$217$	−7.66442	−0.520295
$218$	0	0
$219$	8.45065	0.571042
$220$	0	0
$221$	1.62158	0.109079
$222$	0	0
$223$	−0.786230	−0.0526499	−0.0263249	−	0.999653i	$-0.508380\pi$
$223$	−0.786230	−0.0526499	−0.0263249	+	0.999653i	$0.508380\pi$
$224$	0	0
$225$	−16.3288	−1.08859
$226$	0	0
$227$	−10.8536	−0.720381	−0.360191	−	0.932879i	$-0.617288\pi$
$227$	−10.8536	−0.720381	−0.360191	+	0.932879i	$0.617288\pi$
$228$	0	0
$229$	10.1249	0.669075	0.334538	−	0.942382i	$-0.391420\pi$
$229$	10.1249	0.669075	0.334538	+	0.942382i	$0.391420\pi$
$230$	0	0
$231$	−5.37169	−0.353431
$232$	0	0
$233$	−18.7862	−1.23073	−0.615363	−	0.788244i	$-0.710991\pi$
$233$	−18.7862	−1.23073	−0.615363	+	0.788244i	$0.710991\pi$
$234$	0	0
$235$	8.78623	0.573150
$236$	0	0
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−7.02142	−0.454178	−0.227089	−	0.973874i	$-0.572921\pi$
$239$	−7.02142	−0.454178	−0.227089	+	0.973874i	$0.572921\pi$
$240$	0	0
$241$	−28.4078	−1.82991	−0.914954	−	0.403558i	$-0.867773\pi$
$241$	−28.4078	−1.82991	−0.914954	+	0.403558i	$0.867773\pi$
$242$	0	0
$243$	14.8536	0.952861
$244$	0	0
$245$	−3.83221	−0.244831
$246$	0	0
$247$	28.5082	1.81393
$248$	0	0
$249$	5.51492	0.349494
$250$	0	0
$251$	2.51806	0.158938	0.0794692	−	0.996837i	$-0.474677\pi$
$251$	2.51806	0.158938	0.0794692	+	0.996837i	$0.474677\pi$
$252$	0	0
$253$	−23.3288	−1.46667
$254$	0	0
$255$	−1.28600	−0.0805325
$256$	0	0
$257$	−1.41454	−0.0882365	−0.0441182	−	0.999026i	$-0.514048\pi$
$257$	−1.41454	−0.0882365	−0.0441182	+	0.999026i	$0.514048\pi$
$258$	0	0
$259$	−9.66442	−0.600518
$260$	0	0
$261$	−7.23688	−0.447952
$262$	0	0
$263$	−27.9143	−1.72127	−0.860635	−	0.509222i	$-0.829933\pi$
$263$	−27.9143	−1.72127	−0.860635	+	0.509222i	$0.829933\pi$
$264$	0	0
$265$	7.66442	0.470822
$266$	0	0
$267$	−2.96388	−0.181387
$268$	0	0
$269$	18.1249	1.10510	0.552549	−	0.833481i	$-0.313655\pi$
$269$	18.1249	1.10510	0.552549	+	0.833481i	$0.313655\pi$
$270$	0	0
$271$	−24.1151	−1.46489	−0.732443	−	0.680828i	$-0.761620\pi$
$271$	−24.1151	−1.46489	−0.732443	+	0.680828i	$0.761620\pi$
$272$	0	0
$273$	−6.35027	−0.384336
$274$	0	0
$275$	45.3864	2.73690
$276$	0	0
$277$	0.628308	0.0377513	0.0188757	−	0.999822i	$-0.493991\pi$
$277$	0.628308	0.0377513	0.0188757	+	0.999822i	$0.493991\pi$
$278$	0	0
$279$	−12.9210	−0.773562
$280$	0	0
$281$	0.743385	0.0443466	0.0221733	−	0.999754i	$-0.492941\pi$
$281$	0.743385	0.0443466	0.0221733	+	0.999754i	$0.492941\pi$
$282$	0	0
$283$	−0.475212	−0.0282484	−0.0141242	−	0.999900i	$-0.504496\pi$
$283$	−0.475212	−0.0282484	−0.0141242	+	0.999900i	$0.504496\pi$
$284$	0	0
$285$	−22.6086	−1.33922
$286$	0	0
$287$	−3.70727	−0.218833
$288$	0	0
$289$	−16.9143	−0.994959
$290$	0	0
$291$	−16.3356	−0.957608
$292$	0	0
$293$	−1.53948	−0.0899374	−0.0449687	−	0.998988i	$-0.514319\pi$
$293$	−1.53948	−0.0899374	−0.0449687	+	0.998988i	$0.514319\pi$
$294$	0	0
$295$	−38.0638	−2.21616
$296$	0	0
$297$	−25.1709	−1.46057
$298$	0	0
$299$	−27.5787	−1.59492
$300$	0	0
$301$	5.27131	0.303833
$302$	0	0
$303$	2.14950	0.123486
$304$	0	0
$305$	−15.9718	−0.914545
$306$	0	0
$307$	−21.2614	−1.21345	−0.606727	−	0.794910i	$-0.707518\pi$
$307$	−21.2614	−1.21345	−0.606727	+	0.794910i	$0.707518\pi$
$308$	0	0
$309$	8.78623	0.499831
$310$	0	0
$311$	15.3288	0.869219	0.434610	−	0.900619i	$-0.356886\pi$
$311$	15.3288	0.869219	0.434610	+	0.900619i	$0.356886\pi$
$312$	0	0
$313$	3.70727	0.209547	0.104774	−	0.994496i	$-0.466588\pi$
$313$	3.70727	0.209547	0.104774	+	0.994496i	$0.466588\pi$
$314$	0	0
$315$	−6.46052	−0.364009
$316$	0	0
$317$	−22.5855	−1.26853	−0.634263	−	0.773117i	$-0.718696\pi$
$317$	−22.5855	−1.26853	−0.634263	+	0.773117i	$0.718696\pi$
$318$	0	0
$319$	20.1151	1.12623
$320$	0	0
$321$	−2.74338	−0.153121
$322$	0	0
$323$	1.50650	0.0838239
$324$	0	0
$325$	53.6546	2.97622
$326$	0	0
$327$	−1.23688	−0.0683998
$328$	0	0
$329$	2.29273	0.126402
$330$	0	0
$331$	14.6430	0.804852	0.402426	−	0.915452i	$-0.368167\pi$
$331$	14.6430	0.804852	0.402426	+	0.915452i	$0.368167\pi$
$332$	0	0
$333$	−16.2927	−0.892836
$334$	0	0
$335$	42.0722	2.29865
$336$	0	0
$337$	21.4721	1.16966	0.584829	−	0.811156i	$-0.301162\pi$
$337$	21.4721	1.16966	0.584829	+	0.811156i	$0.301162\pi$
$338$	0	0
$339$	10.6283	0.577250
$340$	0	0
$341$	35.9143	1.94487
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	21.8715	1.17752
$346$	0	0
$347$	−28.6002	−1.53534	−0.767668	−	0.640847i	$-0.778583\pi$
$347$	−28.6002	−1.53534	−0.767668	+	0.640847i	$0.778583\pi$
$348$	0	0
$349$	23.8322	1.27571	0.637855	−	0.770157i	$-0.279822\pi$
$349$	23.8322	1.27571	0.637855	+	0.770157i	$0.279822\pi$
$350$	0	0
$351$	−29.7564	−1.58828
$352$	0	0
$353$	35.2860	1.87808	0.939042	−	0.343802i	$-0.111715\pi$
$353$	35.2860	1.87808	0.939042	+	0.343802i	$0.111715\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.335577	−0.0177606
$358$	0	0
$359$	−0.393115	−0.0207478	−0.0103739	−	0.999946i	$-0.503302\pi$
$359$	−0.393115	−0.0207478	−0.0103739	+	0.999946i	$0.503302\pi$
$360$	0	0
$361$	7.48508	0.393951
$362$	0	0
$363$	12.5609	0.659276
$364$	0	0
$365$	−28.2499	−1.47867
$366$	0	0
$367$	2.62831	0.137197	0.0685983	−	0.997644i	$-0.478147\pi$
$367$	2.62831	0.137197	0.0685983	+	0.997644i	$0.478147\pi$
$368$	0	0
$369$	−6.24989	−0.325356
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	14.7862	0.765602	0.382801	−	0.923831i	$-0.374959\pi$
$373$	14.7862	0.765602	0.382801	+	0.923831i	$0.374959\pi$
$374$	0	0
$375$	−20.5855	−1.06303
$376$	0	0
$377$	23.7795	1.22471
$378$	0	0
$379$	−4.10038	−0.210623	−0.105311	−	0.994439i	$-0.533584\pi$
$379$	−4.10038	−0.210623	−0.105311	+	0.994439i	$0.533584\pi$
$380$	0	0
$381$	7.27973	0.372952
$382$	0	0
$383$	11.4637	0.585765	0.292883	−	0.956148i	$-0.405385\pi$
$383$	11.4637	0.585765	0.292883	+	0.956148i	$0.405385\pi$
$384$	0	0
$385$	17.9572	0.915181
$386$	0	0
$387$	8.88661	0.451732
$388$	0	0
$389$	24.9933	1.26721	0.633605	−	0.773657i	$-0.281575\pi$
$389$	24.9933	1.26721	0.633605	+	0.773657i	$0.281575\pi$
$390$	0	0
$391$	−1.45738	−0.0737031
$392$	0	0
$393$	−12.0575	−0.608222
$394$	0	0
$395$	53.4868	2.69121
$396$	0	0
$397$	7.83221	0.393087	0.196544	−	0.980495i	$-0.437028\pi$
$397$	7.83221	0.393087	0.196544	+	0.980495i	$0.437028\pi$
$398$	0	0
$399$	−5.89962	−0.295350
$400$	0	0
$401$	37.3864	1.86699	0.933493	−	0.358594i	$-0.116744\pi$
$401$	37.3864	1.86699	0.933493	+	0.358594i	$0.116744\pi$
$402$	0	0
$403$	42.4569	2.11493
$404$	0	0
$405$	4.21691	0.209540
$406$	0	0
$407$	45.2860	2.24474
$408$	0	0
$409$	−6.24989	−0.309037	−0.154518	−	0.987990i	$-0.549383\pi$
$409$	−6.24989	−0.309037	−0.154518	+	0.987990i	$0.549383\pi$
$410$	0	0
$411$	−13.0361	−0.643024
$412$	0	0
$413$	−9.93260	−0.488751
$414$	0	0
$415$	−18.4360	−0.904986
$416$	0	0
$417$	12.4422	0.609299
$418$	0	0
$419$	−19.4391	−0.949662	−0.474831	−	0.880077i	$-0.657491\pi$
$419$	−19.4391	−0.949662	−0.474831	+	0.880077i	$0.657491\pi$
$420$	0	0
$421$	−35.2860	−1.71973	−0.859867	−	0.510518i	$-0.829454\pi$
$421$	−35.2860	−1.71973	−0.859867	+	0.510518i	$0.829454\pi$
$422$	0	0
$423$	3.86519	0.187932
$424$	0	0
$425$	2.83535	0.137535
$426$	0	0
$427$	−4.16779	−0.201693
$428$	0	0
$429$	29.7564	1.43665
$430$	0	0
$431$	−7.80765	−0.376081	−0.188041	−	0.982161i	$-0.560214\pi$
$431$	−7.80765	−0.376081	−0.188041	+	0.982161i	$0.560214\pi$
$432$	0	0
$433$	17.4637	0.839250	0.419625	−	0.907698i	$-0.362162\pi$
$433$	17.4637	0.839250	0.419625	+	0.907698i	$0.362162\pi$
$434$	0	0
$435$	−18.8585	−0.904194
$436$	0	0
$437$	−25.6216	−1.22565
$438$	0	0
$439$	−32.7862	−1.56480	−0.782401	−	0.622775i	$-0.786005\pi$
$439$	−32.7862	−1.56480	−0.782401	+	0.622775i	$0.786005\pi$
$440$	0	0
$441$	−1.68585	−0.0802784
$442$	0	0
$443$	−34.5082	−1.63953	−0.819767	−	0.572697i	$-0.805897\pi$
$443$	−34.5082	−1.63953	−0.819767	+	0.572697i	$0.805897\pi$
$444$	0	0
$445$	9.90804	0.469686
$446$	0	0
$447$	−3.19404	−0.151073
$448$	0	0
$449$	22.7862	1.07535	0.537674	−	0.843153i	$-0.319303\pi$
$449$	22.7862	1.07535	0.537674	+	0.843153i	$0.319303\pi$
$450$	0	0
$451$	17.3717	0.818001
$452$	0	0
$453$	1.79296	0.0842406
$454$	0	0
$455$	21.2285	0.995206
$456$	0	0
$457$	−8.01469	−0.374912	−0.187456	−	0.982273i	$-0.560024\pi$
$457$	−8.01469	−0.374912	−0.187456	+	0.982273i	$0.560024\pi$
$458$	0	0
$459$	−1.57246	−0.0733962
$460$	0	0
$461$	−31.4966	−1.46694	−0.733472	−	0.679719i	$-0.762102\pi$
$461$	−31.4966	−1.46694	−0.733472	+	0.679719i	$0.762102\pi$
$462$	0	0
$463$	−28.7862	−1.33781	−0.668905	−	0.743348i	$-0.733237\pi$
$463$	−28.7862	−1.33781	−0.668905	+	0.743348i	$0.733237\pi$
$464$	0	0
$465$	−33.6707	−1.56144
$466$	0	0
$467$	16.5609	0.766347	0.383174	−	0.923676i	$-0.374831\pi$
$467$	16.5609	0.766347	0.383174	+	0.923676i	$0.374831\pi$
$468$	0	0
$469$	10.9786	0.506944
$470$	0	0
$471$	16.8635	0.777029
$472$	0	0
$473$	−24.7005	−1.13573
$474$	0	0
$475$	49.8469	2.28713
$476$	0	0
$477$	3.37169	0.154379
$478$	0	0
$479$	−15.6644	−0.715726	−0.357863	−	0.933774i	$-0.616494\pi$
$479$	−15.6644	−0.715726	−0.357863	+	0.933774i	$0.616494\pi$
$480$	0	0
$481$	53.5359	2.44103
$482$	0	0
$483$	5.70727	0.259690
$484$	0	0
$485$	54.6086	2.47965
$486$	0	0
$487$	−31.7220	−1.43746	−0.718730	−	0.695290i	$-0.755276\pi$
$487$	−31.7220	−1.43746	−0.718730	+	0.695290i	$0.755276\pi$
$488$	0	0
$489$	−0.115077	−0.00520396
$490$	0	0
$491$	35.4783	1.60112	0.800558	−	0.599256i	$-0.204537\pi$
$491$	35.4783	1.60112	0.800558	+	0.599256i	$0.204537\pi$
$492$	0	0
$493$	1.25662	0.0565951
$494$	0	0
$495$	30.2730	1.36067
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15.6497	0.700578	0.350289	−	0.936642i	$-0.386083\pi$
$499$	15.6497	0.700578	0.350289	+	0.936642i	$0.386083\pi$
$500$	0	0
$501$	−6.54262	−0.292303
$502$	0	0
$503$	−37.3717	−1.66632	−0.833161	−	0.553031i	$-0.813471\pi$
$503$	−37.3717	−1.66632	−0.833161	+	0.553031i	$0.813471\pi$
$504$	0	0
$505$	−7.18562	−0.319756
$506$	0	0
$507$	20.2744	0.900420
$508$	0	0
$509$	18.5756	0.823349	0.411674	−	0.911331i	$-0.364944\pi$
$509$	18.5756	0.823349	0.411674	+	0.911331i	$0.364944\pi$
$510$	0	0
$511$	−7.37169	−0.326104
$512$	0	0
$513$	−27.6447	−1.22054
$514$	0	0
$515$	−29.3717	−1.29427
$516$	0	0
$517$	−10.7434	−0.472494
$518$	0	0
$519$	14.7518	0.647532
$520$	0	0
$521$	17.5787	0.770138	0.385069	−	0.922888i	$-0.374178\pi$
$521$	17.5787	0.770138	0.385069	+	0.922888i	$0.374178\pi$
$522$	0	0
$523$	14.5181	0.634830	0.317415	−	0.948287i	$-0.397185\pi$
$523$	14.5181	0.634830	0.317415	+	0.948287i	$0.397185\pi$
$524$	0	0
$525$	−11.1035	−0.484597
$526$	0	0
$527$	2.24361	0.0977334
$528$	0	0
$529$	1.78623	0.0776622
$530$	0	0
$531$	−16.7448	−0.726664
$532$	0	0
$533$	20.5363	0.889528
$534$	0	0
$535$	9.17092	0.396494
$536$	0	0
$537$	−30.1579	−1.30141
$538$	0	0
$539$	4.68585	0.201834
$540$	0	0
$541$	12.5426	0.539249	0.269625	−	0.962966i	$-0.413100\pi$
$541$	12.5426	0.539249	0.269625	+	0.962966i	$0.413100\pi$
$542$	0	0
$543$	3.72196	0.159725
$544$	0	0
$545$	4.13481	0.177116
$546$	0	0
$547$	−0.771538	−0.0329886	−0.0164943	−	0.999864i	$-0.505251\pi$
$547$	−0.771538	−0.0329886	−0.0164943	+	0.999864i	$0.505251\pi$
$548$	0	0
$549$	−7.02625	−0.299873
$550$	0	0
$551$	22.0920	0.941149
$552$	0	0
$553$	13.9572	0.593519
$554$	0	0
$555$	−42.4569	−1.80220
$556$	0	0
$557$	−28.7434	−1.21790	−0.608948	−	0.793210i	$-0.708408\pi$
$557$	−28.7434	−1.21790	−0.608948	+	0.793210i	$0.708408\pi$
$558$	0	0
$559$	−29.2003	−1.23504
$560$	0	0
$561$	1.57246	0.0663893
$562$	0	0
$563$	−5.34713	−0.225355	−0.112677	−	0.993632i	$-0.535943\pi$
$563$	−5.34713	−0.225355	−0.112677	+	0.993632i	$0.535943\pi$
$564$	0	0
$565$	−35.5296	−1.49474
$566$	0	0
$567$	1.10038	0.0462118
$568$	0	0
$569$	−31.2285	−1.30917	−0.654583	−	0.755990i	$-0.727156\pi$
$569$	−31.2285	−1.30917	−0.654583	+	0.755990i	$0.727156\pi$
$570$	0	0
$571$	18.7287	0.783771	0.391886	−	0.920014i	$-0.371823\pi$
$571$	18.7287	0.783771	0.391886	+	0.920014i	$0.371823\pi$
$572$	0	0
$573$	19.9143	0.831932
$574$	0	0
$575$	−48.2217	−2.01099
$576$	0	0
$577$	0.542616	0.0225894	0.0112947	−	0.999936i	$-0.496405\pi$
$577$	0.542616	0.0225894	0.0112947	+	0.999936i	$0.496405\pi$
$578$	0	0
$579$	31.4439	1.30676
$580$	0	0
$581$	−4.81079	−0.199585
$582$	0	0
$583$	−9.37169	−0.388136
$584$	0	0
$585$	35.7879	1.47965
$586$	0	0
$587$	7.43910	0.307044	0.153522	−	0.988145i	$-0.450938\pi$
$587$	7.43910	0.307044	0.153522	+	0.988145i	$0.450938\pi$
$588$	0	0
$589$	39.4439	1.62526
$590$	0	0
$591$	−11.4637	−0.471552
$592$	0	0
$593$	−9.91431	−0.407132	−0.203566	−	0.979061i	$-0.565253\pi$
$593$	−9.91431	−0.407132	−0.203566	+	0.979061i	$0.565253\pi$
$594$	0	0
$595$	1.12181	0.0459896
$596$	0	0
$597$	−4.87192	−0.199394
$598$	0	0
$599$	−29.4868	−1.20480	−0.602398	−	0.798196i	$-0.705788\pi$
$599$	−29.4868	−1.20480	−0.602398	+	0.798196i	$0.705788\pi$
$600$	0	0
$601$	−33.2432	−1.35602	−0.678008	−	0.735054i	$-0.737157\pi$
$601$	−33.2432	−1.35602	−0.678008	+	0.735054i	$0.737157\pi$
$602$	0	0
$603$	18.5082	0.753712
$604$	0	0
$605$	−41.9901	−1.70714
$606$	0	0
$607$	18.0722	0.733529	0.366765	−	0.930314i	$-0.380465\pi$
$607$	18.0722	0.733529	0.366765	+	0.930314i	$0.380465\pi$
$608$	0	0
$609$	−4.92104	−0.199411
$610$	0	0
$611$	−12.7005	−0.513809
$612$	0	0
$613$	−37.7795	−1.52590	−0.762950	−	0.646458i	$-0.776250\pi$
$613$	−37.7795	−1.52590	−0.762950	+	0.646458i	$0.776250\pi$
$614$	0	0
$615$	−16.2865	−0.656733
$616$	0	0
$617$	−16.4851	−0.663664	−0.331832	−	0.943338i	$-0.607667\pi$
$617$	−16.4851	−0.663664	−0.331832	+	0.943338i	$0.607667\pi$
$618$	0	0
$619$	22.1825	0.891589	0.445795	−	0.895135i	$-0.352921\pi$
$619$	22.1825	0.891589	0.445795	+	0.895135i	$0.352921\pi$
$620$	0	0
$621$	26.7434	1.07318
$622$	0	0
$623$	2.58546	0.103584
$624$	0	0
$625$	20.3864	0.815455
$626$	0	0
$627$	27.6447	1.10402
$628$	0	0
$629$	2.82908	0.112803
$630$	0	0
$631$	−12.9870	−0.517004	−0.258502	−	0.966011i	$-0.583229\pi$
$631$	−12.9870	−0.517004	−0.258502	+	0.966011i	$0.583229\pi$
$632$	0	0
$633$	23.3288	0.927238
$634$	0	0
$635$	−24.3356	−0.965728
$636$	0	0
$637$	5.53948	0.219482
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.72869	0.265767	0.132884	−	0.991132i	$-0.457576\pi$
$641$	6.72869	0.265767	0.132884	+	0.991132i	$0.457576\pi$
$642$	0	0
$643$	−45.1758	−1.78156	−0.890779	−	0.454437i	$-0.849840\pi$
$643$	−45.1758	−1.78156	−0.890779	+	0.454437i	$0.849840\pi$
$644$	0	0
$645$	23.1575	0.911824
$646$	0	0
$647$	34.4078	1.35271	0.676355	−	0.736576i	$-0.263558\pi$
$647$	34.4078	1.35271	0.676355	+	0.736576i	$0.263558\pi$
$648$	0	0
$649$	46.5426	1.82696
$650$	0	0
$651$	−8.78623	−0.344360
$652$	0	0
$653$	15.0361	0.588409	0.294204	−	0.955743i	$-0.404945\pi$
$653$	15.0361	0.588409	0.294204	+	0.955743i	$0.404945\pi$
$654$	0	0
$655$	40.3074	1.57494
$656$	0	0
$657$	−12.4275	−0.484844
$658$	0	0
$659$	20.8866	0.813627	0.406813	−	0.913511i	$-0.366640\pi$
$659$	20.8866	0.813627	0.406813	+	0.913511i	$0.366640\pi$
$660$	0	0
$661$	−6.79610	−0.264337	−0.132169	−	0.991227i	$-0.542194\pi$
$661$	−6.79610	−0.264337	−0.132169	+	0.991227i	$0.542194\pi$
$662$	0	0
$663$	1.85892	0.0721945
$664$	0	0
$665$	19.7220	0.764785
$666$	0	0
$667$	−21.3717	−0.827515
$668$	0	0
$669$	−0.901307	−0.0348466
$670$	0	0
$671$	19.5296	0.753932
$672$	0	0
$673$	−8.15792	−0.314465	−0.157232	−	0.987562i	$-0.550257\pi$
$673$	−8.15792	−0.314465	−0.157232	+	0.987562i	$0.550257\pi$
$674$	0	0
$675$	−52.0294	−2.00261
$676$	0	0
$677$	5.58860	0.214787	0.107394	−	0.994217i	$-0.465749\pi$
$677$	5.58860	0.214787	0.107394	+	0.994217i	$0.465749\pi$
$678$	0	0
$679$	14.2499	0.546860
$680$	0	0
$681$	−12.4422	−0.476787
$682$	0	0
$683$	−6.30742	−0.241347	−0.120673	−	0.992692i	$-0.538505\pi$
$683$	−6.30742	−0.241347	−0.120673	+	0.992692i	$0.538505\pi$
$684$	0	0
$685$	43.5787	1.66506
$686$	0	0
$687$	11.6069	0.442830
$688$	0	0
$689$	−11.0790	−0.422075
$690$	0	0
$691$	−0.945597	−0.0359722	−0.0179861	−	0.999838i	$-0.505725\pi$
$691$	−0.945597	−0.0359722	−0.0179861	+	0.999838i	$0.505725\pi$
$692$	0	0
$693$	7.89962	0.300082
$694$	0	0
$695$	−41.5934	−1.57773
$696$	0	0
$697$	1.08523	0.0411061
$698$	0	0
$699$	−21.5359	−0.814562
$700$	0	0
$701$	−12.2070	−0.461054	−0.230527	−	0.973066i	$-0.574045\pi$
$701$	−12.2070	−0.461054	−0.230527	+	0.973066i	$0.574045\pi$
$702$	0	0
$703$	49.7367	1.87585
$704$	0	0
$705$	10.0722	0.379342
$706$	0	0
$707$	−1.87506	−0.0705188
$708$	0	0
$709$	−34.2499	−1.28628	−0.643141	−	0.765748i	$-0.722369\pi$
$709$	−34.2499	−1.28628	−0.643141	+	0.765748i	$0.722369\pi$
$710$	0	0
$711$	23.5296	0.882430
$712$	0	0
$713$	−38.1579	−1.42903
$714$	0	0
$715$	−99.4733	−3.72009
$716$	0	0
$717$	−8.04912	−0.300600
$718$	0	0
$719$	−25.1218	−0.936885	−0.468443	−	0.883494i	$-0.655185\pi$
$719$	−25.1218	−0.936885	−0.468443	+	0.883494i	$0.655185\pi$
$720$	0	0
$721$	−7.66442	−0.285438
$722$	0	0
$723$	−32.5657	−1.21113
$724$	0	0
$725$	41.5787	1.54420
$726$	0	0
$727$	31.2797	1.16010	0.580050	−	0.814581i	$-0.303033\pi$
$727$	31.2797	1.16010	0.580050	+	0.814581i	$0.303033\pi$
$728$	0	0
$729$	20.3288	0.752920
$730$	0	0
$731$	−1.54308	−0.0570727
$732$	0	0
$733$	−21.0031	−0.775769	−0.387884	−	0.921708i	$-0.626794\pi$
$733$	−21.0031	−0.775769	−0.387884	+	0.921708i	$0.626794\pi$
$734$	0	0
$735$	−4.39312	−0.162042
$736$	0	0
$737$	−51.4439	−1.89496
$738$	0	0
$739$	37.3864	1.37528	0.687640	−	0.726052i	$-0.258647\pi$
$739$	37.3864	1.37528	0.687640	+	0.726052i	$0.258647\pi$
$740$	0	0
$741$	32.6808	1.20056
$742$	0	0
$743$	19.9227	0.730894	0.365447	−	0.930832i	$-0.380916\pi$
$743$	19.9227	0.730894	0.365447	+	0.930832i	$0.380916\pi$
$744$	0	0
$745$	10.6774	0.391191
$746$	0	0
$747$	−8.11025	−0.296739
$748$	0	0
$749$	2.39312	0.0874425
$750$	0	0
$751$	−44.8929	−1.63816	−0.819082	−	0.573676i	$-0.805517\pi$
$751$	−44.8929	−1.63816	−0.819082	+	0.573676i	$0.805517\pi$
$752$	0	0
$753$	2.88661	0.105194
$754$	0	0
$755$	−5.99373	−0.218134
$756$	0	0
$757$	−9.46365	−0.343962	−0.171981	−	0.985100i	$-0.555017\pi$
$757$	−9.46365	−0.343962	−0.171981	+	0.985100i	$0.555017\pi$
$758$	0	0
$759$	−26.7434	−0.970723
$760$	0	0
$761$	−25.7795	−0.934506	−0.467253	−	0.884124i	$-0.654756\pi$
$761$	−25.7795	−0.934506	−0.467253	+	0.884124i	$0.654756\pi$
$762$	0	0
$763$	1.07896	0.0390610
$764$	0	0
$765$	1.89119	0.0683763
$766$	0	0
$767$	55.0214	1.98671
$768$	0	0
$769$	−19.5065	−0.703422	−0.351711	−	0.936109i	$-0.614400\pi$
$769$	−19.5065	−0.703422	−0.351711	+	0.936109i	$0.614400\pi$
$770$	0	0
$771$	−1.62158	−0.0583997
$772$	0	0
$773$	44.6676	1.60658	0.803290	−	0.595588i	$-0.203081\pi$
$773$	44.6676	1.60658	0.803290	+	0.595588i	$0.203081\pi$
$774$	0	0
$775$	74.2364	2.66665
$776$	0	0
$777$	−11.0790	−0.397456
$778$	0	0
$779$	19.0790	0.683575
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−23.0592	−0.824070
$784$	0	0
$785$	−56.3734	−2.01205
$786$	0	0
$787$	−34.1825	−1.21847	−0.609237	−	0.792988i	$-0.708524\pi$
$787$	−34.1825	−1.21847	−0.609237	+	0.792988i	$0.708524\pi$
$788$	0	0
$789$	−32.0000	−1.13923
$790$	0	0
$791$	−9.27131	−0.329650
$792$	0	0
$793$	23.0874	0.819857
$794$	0	0
$795$	8.78623	0.311615
$796$	0	0
$797$	−24.6184	−0.872030	−0.436015	−	0.899939i	$-0.643611\pi$
$797$	−24.6184	−0.872030	−0.436015	+	0.899939i	$0.643611\pi$
$798$	0	0
$799$	−0.671153	−0.0237437
$800$	0	0
$801$	4.35869	0.154007
$802$	0	0
$803$	34.5426	1.21898
$804$	0	0
$805$	−19.0790	−0.672445
$806$	0	0
$807$	20.7778	0.731414
$808$	0	0
$809$	31.8139	1.11852	0.559259	−	0.828993i	$-0.311086\pi$
$809$	31.8139	1.11852	0.559259	+	0.828993i	$0.311086\pi$
$810$	0	0
$811$	43.9389	1.54290	0.771451	−	0.636289i	$-0.219531\pi$
$811$	43.9389	1.54290	0.771451	+	0.636289i	$0.219531\pi$
$812$	0	0
$813$	−27.6447	−0.969542
$814$	0	0
$815$	0.384694	0.0134752
$816$	0	0
$817$	−27.1281	−0.949091
$818$	0	0
$819$	9.33871	0.326321
$820$	0	0
$821$	−45.1281	−1.57498	−0.787490	−	0.616327i	$-0.788620\pi$
$821$	−45.1281	−1.57498	−0.787490	+	0.616327i	$0.788620\pi$
$822$	0	0
$823$	14.1579	0.493514	0.246757	−	0.969077i	$-0.420635\pi$
$823$	14.1579	0.493514	0.246757	+	0.969077i	$0.420635\pi$
$824$	0	0
$825$	52.0294	1.81143
$826$	0	0
$827$	14.5082	0.504499	0.252250	−	0.967662i	$-0.418830\pi$
$827$	14.5082	0.504499	0.252250	+	0.967662i	$0.418830\pi$
$828$	0	0
$829$	−15.7465	−0.546899	−0.273450	−	0.961886i	$-0.588165\pi$
$829$	−15.7465	−0.546899	−0.273450	+	0.961886i	$0.588165\pi$
$830$	0	0
$831$	0.720270	0.0249859
$832$	0	0
$833$	0.292731	0.0101425
$834$	0	0
$835$	21.8715	0.756893
$836$	0	0
$837$	−41.1709	−1.42308
$838$	0	0
$839$	−7.66442	−0.264605	−0.132303	−	0.991209i	$-0.542237\pi$
$839$	−7.66442	−0.264605	−0.132303	+	0.991209i	$0.542237\pi$
$840$	0	0
$841$	−10.5725	−0.364568
$842$	0	0
$843$	0.852191	0.0293510
$844$	0	0
$845$	−67.7759	−2.33156
$846$	0	0
$847$	−10.9572	−0.376492
$848$	0	0
$849$	−0.544767	−0.0186963
$850$	0	0
$851$	−48.1151	−1.64936
$852$	0	0
$853$	16.9540	0.580495	0.290247	−	0.956952i	$-0.406262\pi$
$853$	16.9540	0.580495	0.290247	+	0.956952i	$0.406262\pi$
$854$	0	0
$855$	33.2482	1.13706
$856$	0	0
$857$	17.0790	0.583406	0.291703	−	0.956509i	$-0.405778\pi$
$857$	17.0790	0.583406	0.291703	+	0.956509i	$0.405778\pi$
$858$	0	0
$859$	10.5181	0.358872	0.179436	−	0.983770i	$-0.442573\pi$
$859$	10.5181	0.358872	0.179436	+	0.983770i	$0.442573\pi$
$860$	0	0
$861$	−4.24989	−0.144836
$862$	0	0
$863$	−36.1151	−1.22937	−0.614686	−	0.788772i	$-0.710717\pi$
$863$	−36.1151	−1.22937	−0.614686	+	0.788772i	$0.710717\pi$
$864$	0	0
$865$	−49.3142	−1.67673
$866$	0	0
$867$	−19.3900	−0.658518
$868$	0	0
$869$	−65.4011	−2.21858
$870$	0	0
$871$	−60.8156	−2.06066
$872$	0	0
$873$	24.0231	0.813059
$874$	0	0
$875$	17.9572	0.607063
$876$	0	0
$877$	−0.177654	−0.00599895	−0.00299947	−	0.999996i	$-0.500955\pi$
$877$	−0.177654	−0.00599895	−0.00299947	+	0.999996i	$0.500955\pi$
$878$	0	0
$879$	−1.76481	−0.0595255
$880$	0	0
$881$	−1.52962	−0.0515340	−0.0257670	−	0.999668i	$-0.508203\pi$
$881$	−1.52962	−0.0515340	−0.0257670	+	0.999668i	$0.508203\pi$
$882$	0	0
$883$	43.7942	1.47379	0.736896	−	0.676006i	$-0.236291\pi$
$883$	43.7942	1.47379	0.736896	+	0.676006i	$0.236291\pi$
$884$	0	0
$885$	−43.6350	−1.46678
$886$	0	0
$887$	−35.6938	−1.19848	−0.599240	−	0.800569i	$-0.704531\pi$
$887$	−35.6938	−1.19848	−0.599240	+	0.800569i	$0.704531\pi$
$888$	0	0
$889$	−6.35027	−0.212981
$890$	0	0
$891$	−5.15623	−0.172740
$892$	0	0
$893$	−11.7992	−0.394846
$894$	0	0
$895$	100.816	3.36989
$896$	0	0
$897$	−31.6153	−1.05560
$898$	0	0
$899$	32.9013	1.09732
$900$	0	0
$901$	−0.585462	−0.0195046
$902$	0	0
$903$	6.04285	0.201093
$904$	0	0
$905$	−12.4422	−0.413594
$906$	0	0
$907$	−33.3373	−1.10695	−0.553473	−	0.832867i	$-0.686698\pi$
$907$	−33.3373	−1.10695	−0.553473	+	0.832867i	$0.686698\pi$
$908$	0	0
$909$	−3.16106	−0.104846
$910$	0	0
$911$	45.5640	1.50960	0.754802	−	0.655953i	$-0.227733\pi$
$911$	45.5640	1.50960	0.754802	+	0.655953i	$0.227733\pi$
$912$	0	0
$913$	22.5426	0.746052
$914$	0	0
$915$	−18.3096	−0.605296
$916$	0	0
$917$	10.5181	0.347337
$918$	0	0
$919$	6.15792	0.203131	0.101566	−	0.994829i	$-0.467615\pi$
$919$	6.15792	0.203131	0.101566	+	0.994829i	$0.467615\pi$
$920$	0	0
$921$	−24.3734	−0.803130
$922$	0	0
$923$	0	0
$924$	0	0
$925$	93.6081	3.07782
$926$	0	0
$927$	−12.9210	−0.424383
$928$	0	0
$929$	−42.9504	−1.40916	−0.704579	−	0.709626i	$-0.748864\pi$
$929$	−42.9504	−1.40916	−0.704579	+	0.709626i	$0.748864\pi$
$930$	0	0
$931$	5.14637	0.168665
$932$	0	0
$933$	17.5725	0.575297
$934$	0	0
$935$	−5.25662	−0.171910
$936$	0	0
$937$	−41.1281	−1.34360	−0.671798	−	0.740735i	$-0.734478\pi$
$937$	−41.1281	−1.34360	−0.671798	+	0.740735i	$0.734478\pi$
$938$	0	0
$939$	4.24989	0.138690
$940$	0	0
$941$	4.61844	0.150557	0.0752785	−	0.997163i	$-0.476015\pi$
$941$	4.61844	0.150557	0.0752785	+	0.997163i	$0.476015\pi$
$942$	0	0
$943$	−18.4569	−0.601040
$944$	0	0
$945$	−20.5855	−0.669645
$946$	0	0
$947$	0.800923	0.0260265	0.0130133	−	0.999915i	$-0.495858\pi$
$947$	0.800923	0.0260265	0.0130133	+	0.999915i	$0.495858\pi$
$948$	0	0
$949$	40.8353	1.32557
$950$	0	0
$951$	−25.8912	−0.839579
$952$	0	0
$953$	13.9143	0.450729	0.225364	−	0.974275i	$-0.427643\pi$
$953$	13.9143	0.450729	0.225364	+	0.974275i	$0.427643\pi$
$954$	0	0
$955$	−66.5720	−2.15422
$956$	0	0
$957$	23.0592	0.745399
$958$	0	0
$959$	11.3717	0.367211
$960$	0	0
$961$	27.7434	0.894948
$962$	0	0
$963$	4.03442	0.130007
$964$	0	0
$965$	−105.115	−3.38376
$966$	0	0
$967$	−27.6363	−0.888723	−0.444361	−	0.895848i	$-0.646569\pi$
$967$	−27.6363	−0.888723	−0.444361	+	0.895848i	$0.646569\pi$
$968$	0	0
$969$	1.72700	0.0554792
$970$	0	0
$971$	−0.810789	−0.0260195	−0.0130097	−	0.999915i	$-0.504141\pi$
$971$	−0.810789	−0.0260195	−0.0130097	+	0.999915i	$0.504141\pi$
$972$	0	0
$973$	−10.8536	−0.347952
$974$	0	0
$975$	61.5077	1.96982
$976$	0	0
$977$	−4.62831	−0.148073	−0.0740363	−	0.997256i	$-0.523588\pi$
$977$	−4.62831	−0.148073	−0.0740363	+	0.997256i	$0.523588\pi$
$978$	0	0
$979$	−12.1151	−0.387200
$980$	0	0
$981$	1.81896	0.0580750
$982$	0	0
$983$	5.42081	0.172897	0.0864485	−	0.996256i	$-0.472448\pi$
$983$	5.42081	0.172897	0.0864485	+	0.996256i	$0.472448\pi$
$984$	0	0
$985$	38.3221	1.22104
$986$	0	0
$987$	2.62831	0.0836600
$988$	0	0
$989$	26.2436	0.834498
$990$	0	0
$991$	−16.2008	−0.514634	−0.257317	−	0.966327i	$-0.582839\pi$
$991$	−16.2008	−0.514634	−0.257317	+	0.966327i	$0.582839\pi$
$992$	0	0
$993$	16.7862	0.532695
$994$	0	0
$995$	16.2865	0.516315
$996$	0	0
$997$	−17.9901	−0.569753	−0.284877	−	0.958564i	$-0.591953\pi$
$997$	−17.9901	−0.569753	−0.284877	+	0.958564i	$0.591953\pi$
$998$	0	0
$999$	−51.9143	−1.64250

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	896.2.a.l.1.2	yes	3
3.2	odd	2		8064.2.a.bu.1.3		3
4.3	odd	2		896.2.a.j.1.2	yes	3
7.6	odd	2		6272.2.a.u.1.2		3
8.3	odd	2		896.2.a.k.1.2	yes	3
8.5	even	2		896.2.a.i.1.2	✓	3
12.11	even	2		8064.2.a.cb.1.3		3
16.3	odd	4		1792.2.b.o.897.3		6
16.5	even	4		1792.2.b.p.897.3		6
16.11	odd	4		1792.2.b.o.897.4		6
16.13	even	4		1792.2.b.p.897.4		6
24.5	odd	2		8064.2.a.ce.1.1		3
24.11	even	2		8064.2.a.ch.1.1		3
28.27	even	2		6272.2.a.w.1.2		3
56.13	odd	2		6272.2.a.x.1.2		3
56.27	even	2		6272.2.a.v.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
896.2.a.i.1.2	✓	3	8.5	even	2
896.2.a.j.1.2	yes	3	4.3	odd	2
896.2.a.k.1.2	yes	3	8.3	odd	2
896.2.a.l.1.2	yes	3	1.1	even	1	trivial
1792.2.b.o.897.3		6	16.3	odd	4
1792.2.b.o.897.4		6	16.11	odd	4
1792.2.b.p.897.3		6	16.5	even	4
1792.2.b.p.897.4		6	16.13	even	4
6272.2.a.u.1.2		3	7.6	odd	2
6272.2.a.v.1.2		3	56.27	even	2
6272.2.a.w.1.2		3	28.27	even	2
6272.2.a.x.1.2		3	56.13	odd	2
8064.2.a.bu.1.3		3	3.2	odd	2
8064.2.a.cb.1.3		3	12.11	even	2
8064.2.a.ce.1.1		3	24.5	odd	2
8064.2.a.ch.1.1		3	24.11	even	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:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.a (trivial)

\(f(q)\)	\(=\)	\(q+1.14637 q^{3} -3.83221 q^{5} -1.00000 q^{7} -1.68585 q^{9} +O(q^{10})\)	\(q+1.14637 q^{3} -3.83221 q^{5} -1.00000 q^{7} -1.68585 q^{9} +4.68585 q^{11} +5.53948 q^{13} -4.39312 q^{15} +0.292731 q^{17} +5.14637 q^{19} -1.14637 q^{21} -4.97858 q^{23} +9.68585 q^{25} -5.37169 q^{27} +4.29273 q^{29} +7.66442 q^{31} +5.37169 q^{33} +3.83221 q^{35} +9.66442 q^{37} +6.35027 q^{39} +3.70727 q^{41} -5.27131 q^{43} +6.46052 q^{45} -2.29273 q^{47} +1.00000 q^{49} +0.335577 q^{51} -2.00000 q^{53} -17.9572 q^{55} +5.89962 q^{57} +9.93260 q^{59} +4.16779 q^{61} +1.68585 q^{63} -21.2285 q^{65} -10.9786 q^{67} -5.70727 q^{69} +7.37169 q^{73} +11.1035 q^{75} -4.68585 q^{77} -13.9572 q^{79} -1.10038 q^{81} +4.81079 q^{83} -1.12181 q^{85} +4.92104 q^{87} -2.58546 q^{89} -5.53948 q^{91} +8.78623 q^{93} -19.7220 q^{95} -14.2499 q^{97} -7.89962 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} + 2 q^{5} - 3 q^{7} + 7 q^{9}+O(q^{10}) \) 3 * q + 2 * q^3 + 2 * q^5 - 3 * q^7 + 7 * q^9	\( 3 q + 2 q^{3} + 2 q^{5} - 3 q^{7} + 7 q^{9} + 2 q^{11} + 6 q^{13} - 4 q^{15} - 2 q^{17} + 14 q^{19} - 2 q^{21} + 17 q^{25} + 8 q^{27} + 10 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} + 2 q^{37} - 20 q^{39} + 14 q^{41} + 2 q^{43} + 30 q^{45} - 4 q^{47} + 3 q^{49} + 28 q^{51} - 6 q^{53} - 24 q^{55} + 24 q^{57} + 10 q^{59} + 26 q^{61} - 7 q^{63} - 16 q^{65} - 18 q^{67} - 20 q^{69} - 2 q^{73} + 2 q^{75} - 2 q^{77} - 12 q^{79} + 3 q^{81} - 14 q^{83} - 12 q^{85} + 36 q^{87} - 2 q^{89} - 6 q^{91} + 8 q^{93} + 4 q^{95} - 10 q^{97} - 30 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 + 2 * q^5 - 3 * q^7 + 7 * q^9 + 2 * q^11 + 6 * q^13 - 4 * q^15 - 2 * q^17 + 14 * q^19 - 2 * q^21 + 17 * q^25 + 8 * q^27 + 10 * q^29 - 4 * q^31 - 8 * q^33 - 2 * q^35 + 2 * q^37 - 20 * q^39 + 14 * q^41 + 2 * q^43 + 30 * q^45 - 4 * q^47 + 3 * q^49 + 28 * q^51 - 6 * q^53 - 24 * q^55 + 24 * q^57 + 10 * q^59 + 26 * q^61 - 7 * q^63 - 16 * q^65 - 18 * q^67 - 20 * q^69 - 2 * q^73 + 2 * q^75 - 2 * q^77 - 12 * q^79 + 3 * q^81 - 14 * q^83 - 12 * q^85 + 36 * q^87 - 2 * q^89 - 6 * q^91 + 8 * q^93 + 4 * q^95 - 10 * q^97 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more