LMFDB - Embedded newform 6896.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6896,2,Mod(1,6896)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6896, 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("6896.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6896 = 2^{4} \cdot 431 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6896.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:	$55.0648372339$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 431)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.35567$ of defining polynomial
Character	$\chi$	$=$	6896.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35567 q^{3} +0.355674 q^{5} -2.19353 q^{7} +2.54920 q^{9} -1.31313 q^{11} -0.219819 q^{13} +0.837853 q^{15} -4.72333 q^{17} +3.76902 q^{19} -5.16724 q^{21} +0.150986 q^{23} -4.87350 q^{25} -1.06193 q^{27} -1.01939 q^{29} +4.64433 q^{31} -3.09331 q^{33} -0.780181 q^{35} -1.24723 q^{37} -0.517822 q^{39} +0.961212 q^{41} +2.70626 q^{43} +0.906685 q^{45} +0.791342 q^{47} -2.18844 q^{49} -11.1266 q^{51} -4.41446 q^{53} -0.467048 q^{55} +8.87858 q^{57} +4.46097 q^{59} -3.10155 q^{61} -5.59174 q^{63} -0.0781839 q^{65} -9.44584 q^{67} +0.355674 q^{69} -2.60980 q^{71} -10.8065 q^{73} -11.4804 q^{75} +2.88039 q^{77} +4.27546 q^{79} -10.1492 q^{81} -5.53916 q^{83} -1.67997 q^{85} -2.40136 q^{87} -14.6037 q^{89} +0.482178 q^{91} +10.9405 q^{93} +1.34054 q^{95} -9.16806 q^{97} -3.34744 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 5 q^{5} - 2 q^{7} - 3 q^{9} - q^{11} - 5 q^{13} + 3 q^{15} + 2 q^{17} + 6 q^{19} - 3 q^{21} - 4 q^{23} - 7 q^{25} - 3 q^{27} - 5 q^{29} + 25 q^{31} - 4 q^{33} + q^{35} - q^{37} + 4 q^{39}+ \cdots - 11 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 5 * q^5 - 2 * q^7 - 3 * q^9 - q^11 - 5 * q^13 + 3 * q^15 + 2 * q^17 + 6 * q^19 - 3 * q^21 - 4 * q^23 - 7 * q^25 - 3 * q^27 - 5 * q^29 + 25 * q^31 - 4 * q^33 + q^35 - q^37 + 4 * q^39 + 2 * q^41 + 16 * q^43 + 12 * q^45 + 4 * q^47 - 20 * q^49 + 3 * q^51 + 4 * q^53 - 2 * q^55 + 5 * q^57 - 5 * q^59 - 7 * q^63 + 14 * q^65 - 9 * q^67 - 5 * q^69 - 10 * q^71 - 50 * q^73 - 24 * q^75 + 9 * q^77 - 8 * q^79 - 8 * q^81 + 15 * q^83 - q^85 - 15 * q^87 - 35 * q^89 + 8 * q^91 + 12 * q^93 - 7 * q^95 - 6 * q^97 - 11 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.35567	1.36005	0.680025	−	0.733189i	$-0.261969\pi$
$3$	2.35567	1.36005	0.680025	+	0.733189i	$0.261969\pi$
$4$	0	0
$5$	0.355674	0.159062	0.0795312	−	0.996832i	$-0.474658\pi$
$5$	0.355674	0.159062	0.0795312	+	0.996832i	$0.474658\pi$
$6$	0	0
$7$	−2.19353	−0.829075	−0.414538	−	0.910032i	$-0.636057\pi$
$7$	−2.19353	−0.829075	−0.414538	+	0.910032i	$0.636057\pi$
$8$	0	0
$9$	2.54920	0.849734
$10$	0	0
$11$	−1.31313	−0.395925	−0.197962	−	0.980210i	$-0.563432\pi$
$11$	−1.31313	−0.395925	−0.197962	+	0.980210i	$0.563432\pi$
$12$	0	0
$13$	−0.219819	−0.0609668	−0.0304834	−	0.999535i	$-0.509705\pi$
$13$	−0.219819	−0.0609668	−0.0304834	+	0.999535i	$0.509705\pi$
$14$	0	0
$15$	0.837853	0.216333
$16$	0	0
$17$	−4.72333	−1.14558	−0.572788	−	0.819703i	$-0.694138\pi$
$17$	−4.72333	−1.14558	−0.572788	+	0.819703i	$0.694138\pi$
$18$	0	0
$19$	3.76902	0.864673	0.432336	−	0.901712i	$-0.357689\pi$
$19$	3.76902	0.864673	0.432336	+	0.901712i	$0.357689\pi$
$20$	0	0
$21$	−5.16724	−1.12758
$22$	0	0
$23$	0.150986	0.0314828	0.0157414	−	0.999876i	$-0.494989\pi$
$23$	0.150986	0.0314828	0.0157414	+	0.999876i	$0.494989\pi$
$24$	0	0
$25$	−4.87350	−0.974699
$26$	0	0
$27$	−1.06193	−0.204369
$28$	0	0
$29$	−1.01939	−0.189297	−0.0946483	−	0.995511i	$-0.530173\pi$
$29$	−1.01939	−0.189297	−0.0946483	+	0.995511i	$0.530173\pi$
$30$	0	0
$31$	4.64433	0.834146	0.417073	−	0.908873i	$-0.363056\pi$
$31$	4.64433	0.834146	0.417073	+	0.908873i	$0.363056\pi$
$32$	0	0
$33$	−3.09331	−0.538477
$34$	0	0
$35$	−0.780181	−0.131875
$36$	0	0
$37$	−1.24723	−0.205043	−0.102522	−	0.994731i	$-0.532691\pi$
$37$	−1.24723	−0.205043	−0.102522	+	0.994731i	$0.532691\pi$
$38$	0	0
$39$	−0.517822	−0.0829178
$40$	0	0
$41$	0.961212	0.150116	0.0750581	−	0.997179i	$-0.476086\pi$
$41$	0.961212	0.150116	0.0750581	+	0.997179i	$0.476086\pi$
$42$	0	0
$43$	2.70626	0.412701	0.206350	−	0.978478i	$-0.433841\pi$
$43$	2.70626	0.412701	0.206350	+	0.978478i	$0.433841\pi$
$44$	0	0
$45$	0.906685	0.135161
$46$	0	0
$47$	0.791342	0.115429	0.0577146	−	0.998333i	$-0.481619\pi$
$47$	0.791342	0.115429	0.0577146	+	0.998333i	$0.481619\pi$
$48$	0	0
$49$	−2.18844	−0.312634
$50$	0	0
$51$	−11.1266	−1.55804
$52$	0	0
$53$	−4.41446	−0.606373	−0.303187	−	0.952931i	$-0.598051\pi$
$53$	−4.41446	−0.606373	−0.303187	+	0.952931i	$0.598051\pi$
$54$	0	0
$55$	−0.467048	−0.0629767
$56$	0	0
$57$	8.87858	1.17600
$58$	0	0
$59$	4.46097	0.580769	0.290385	−	0.956910i	$-0.406217\pi$
$59$	4.46097	0.580769	0.290385	+	0.956910i	$0.406217\pi$
$60$	0	0
$61$	−3.10155	−0.397112	−0.198556	−	0.980089i	$-0.563625\pi$
$61$	−3.10155	−0.397112	−0.198556	+	0.980089i	$0.563625\pi$
$62$	0	0
$63$	−5.59174	−0.704493
$64$	0	0
$65$	−0.0781839	−0.00969752
$66$	0	0
$67$	−9.44584	−1.15399	−0.576997	−	0.816746i	$-0.695775\pi$
$67$	−9.44584	−1.15399	−0.576997	+	0.816746i	$0.695775\pi$
$68$	0	0
$69$	0.355674	0.0428182
$70$	0	0
$71$	−2.60980	−0.309726	−0.154863	−	0.987936i	$-0.549494\pi$
$71$	−2.60980	−0.309726	−0.154863	+	0.987936i	$0.549494\pi$
$72$	0	0
$73$	−10.8065	−1.26480	−0.632401	−	0.774641i	$-0.717930\pi$
$73$	−10.8065	−1.26480	−0.632401	+	0.774641i	$0.717930\pi$
$74$	0	0
$75$	−11.4804	−1.32564
$76$	0	0
$77$	2.88039	0.328251
$78$	0	0
$79$	4.27546	0.481027	0.240514	−	0.970646i	$-0.422684\pi$
$79$	4.27546	0.481027	0.240514	+	0.970646i	$0.422684\pi$
$80$	0	0
$81$	−10.1492	−1.12769
$82$	0	0
$83$	−5.53916	−0.608002	−0.304001	−	0.952672i	$-0.598323\pi$
$83$	−5.53916	−0.608002	−0.304001	+	0.952672i	$0.598323\pi$
$84$	0	0
$85$	−1.67997	−0.182218
$86$	0	0
$87$	−2.40136	−0.257453
$88$	0	0
$89$	−14.6037	−1.54799	−0.773996	−	0.633190i	$-0.781745\pi$
$89$	−14.6037	−1.54799	−0.773996	+	0.633190i	$0.781745\pi$
$90$	0	0
$91$	0.482178	0.0505460
$92$	0	0
$93$	10.9405	1.13448
$94$	0	0
$95$	1.34054	0.137537
$96$	0	0
$97$	−9.16806	−0.930875	−0.465438	−	0.885081i	$-0.654103\pi$
$97$	−9.16806	−0.930875	−0.465438	+	0.885081i	$0.654103\pi$
$98$	0	0
$99$	−3.34744	−0.336431
$100$	0	0
$101$	5.98194	0.595225	0.297613	−	0.954687i	$-0.403810\pi$
$101$	5.98194	0.595225	0.297613	+	0.954687i	$0.403810\pi$
$102$	0	0
$103$	10.5400	1.03854	0.519268	−	0.854612i	$-0.326205\pi$
$103$	10.5400	1.03854	0.519268	+	0.854612i	$0.326205\pi$
$104$	0	0
$105$	−1.83785	−0.179356
$106$	0	0
$107$	−10.8015	−1.04422	−0.522111	−	0.852877i	$-0.674855\pi$
$107$	−10.8015	−1.04422	−0.522111	+	0.852877i	$0.674855\pi$
$108$	0	0
$109$	−19.3739	−1.85568	−0.927840	−	0.372979i	$-0.878336\pi$
$109$	−19.3739	−1.85568	−0.927840	+	0.372979i	$0.878336\pi$
$110$	0	0
$111$	−2.93807	−0.278869
$112$	0	0
$113$	11.7803	1.10820	0.554099	−	0.832451i	$-0.313063\pi$
$113$	11.7803	1.10820	0.554099	+	0.832451i	$0.313063\pi$
$114$	0	0
$115$	0.0537019	0.00500773
$116$	0	0
$117$	−0.560362	−0.0518055
$118$	0	0
$119$	10.3608	0.949770
$120$	0	0
$121$	−9.27568	−0.843244
$122$	0	0
$123$	2.26430	0.204165
$124$	0	0
$125$	−3.51175	−0.314100
$126$	0	0
$127$	−11.0823	−0.983394	−0.491697	−	0.870766i	$-0.663623\pi$
$127$	−11.0823	−0.983394	−0.491697	+	0.870766i	$0.663623\pi$
$128$	0	0
$129$	6.37507	0.561293
$130$	0	0
$131$	−9.41103	−0.822245	−0.411123	−	0.911580i	$-0.634863\pi$
$131$	−9.41103	−0.822245	−0.411123	+	0.911580i	$0.634863\pi$
$132$	0	0
$133$	−8.26745	−0.716879
$134$	0	0
$135$	−0.377703	−0.0325075
$136$	0	0
$137$	14.4355	1.23331	0.616654	−	0.787234i	$-0.288488\pi$
$137$	14.4355	1.23331	0.616654	+	0.787234i	$0.288488\pi$
$138$	0	0
$139$	6.29486	0.533923	0.266961	−	0.963707i	$-0.413980\pi$
$139$	6.29486	0.533923	0.266961	+	0.963707i	$0.413980\pi$
$140$	0	0
$141$	1.86414	0.156989
$142$	0	0
$143$	0.288651	0.0241382
$144$	0	0
$145$	−0.362572	−0.0301100
$146$	0	0
$147$	−5.15525	−0.425198
$148$	0	0
$149$	1.27486	0.104440	0.0522201	−	0.998636i	$-0.483370\pi$
$149$	1.27486	0.104440	0.0522201	+	0.998636i	$0.483370\pi$
$150$	0	0
$151$	13.7654	1.12021	0.560106	−	0.828421i	$-0.310760\pi$
$151$	13.7654	1.12021	0.560106	+	0.828421i	$0.310760\pi$
$152$	0	0
$153$	−12.0407	−0.973435
$154$	0	0
$155$	1.65187	0.132681
$156$	0	0
$157$	0.224691	0.0179323	0.00896613	−	0.999960i	$-0.497146\pi$
$157$	0.224691	0.0179323	0.00896613	+	0.999960i	$0.497146\pi$
$158$	0	0
$159$	−10.3990	−0.824698
$160$	0	0
$161$	−0.331192	−0.0261016
$162$	0	0
$163$	−10.7653	−0.843201	−0.421600	−	0.906782i	$-0.638531\pi$
$163$	−10.7653	−0.843201	−0.421600	+	0.906782i	$0.638531\pi$
$164$	0	0
$165$	−1.10021	−0.0856514
$166$	0	0
$167$	−5.99953	−0.464257	−0.232129	−	0.972685i	$-0.574569\pi$
$167$	−5.99953	−0.464257	−0.232129	+	0.972685i	$0.574569\pi$
$168$	0	0
$169$	−12.9517	−0.996283
$170$	0	0
$171$	9.60799	0.734741
$172$	0	0
$173$	−3.24856	−0.246984	−0.123492	−	0.992346i	$-0.539409\pi$
$173$	−3.24856	−0.246984	−0.123492	+	0.992346i	$0.539409\pi$
$174$	0	0
$175$	10.6901	0.808099
$176$	0	0
$177$	10.5086	0.789875
$178$	0	0
$179$	4.62709	0.345845	0.172923	−	0.984935i	$-0.444679\pi$
$179$	4.62709	0.345845	0.172923	+	0.984935i	$0.444679\pi$
$180$	0	0
$181$	−9.89238	−0.735295	−0.367647	−	0.929965i	$-0.619837\pi$
$181$	−9.89238	−0.735295	−0.367647	+	0.929965i	$0.619837\pi$
$182$	0	0
$183$	−7.30624	−0.540092
$184$	0	0
$185$	−0.443607	−0.0326147
$186$	0	0
$187$	6.20237	0.453562
$188$	0	0
$189$	2.32938	0.169438
$190$	0	0
$191$	−8.84323	−0.639874	−0.319937	−	0.947439i	$-0.603662\pi$
$191$	−8.84323	−0.639874	−0.319937	+	0.947439i	$0.603662\pi$
$192$	0	0
$193$	9.17608	0.660508	0.330254	−	0.943892i	$-0.392866\pi$
$193$	9.17608	0.660508	0.330254	+	0.943892i	$0.392866\pi$
$194$	0	0
$195$	−0.184176	−0.0131891
$196$	0	0
$197$	−0.248564	−0.0177094	−0.00885472	−	0.999961i	$-0.502819\pi$
$197$	−0.248564	−0.0177094	−0.00885472	+	0.999961i	$0.502819\pi$
$198$	0	0
$199$	−5.15802	−0.365642	−0.182821	−	0.983146i	$-0.558523\pi$
$199$	−5.15802	−0.365642	−0.182821	+	0.983146i	$0.558523\pi$
$200$	0	0
$201$	−22.2513	−1.56949
$202$	0	0
$203$	2.23607	0.156941
$204$	0	0
$205$	0.341879	0.0238778
$206$	0	0
$207$	0.384894	0.0267520
$208$	0	0
$209$	−4.94923	−0.342345
$210$	0	0
$211$	−17.5140	−1.20571	−0.602856	−	0.797850i	$-0.705971\pi$
$211$	−17.5140	−1.20571	−0.602856	+	0.797850i	$0.705971\pi$
$212$	0	0
$213$	−6.14784	−0.421243
$214$	0	0
$215$	0.962547	0.0656452
$216$	0	0
$217$	−10.1875	−0.691569
$218$	0	0
$219$	−25.4565	−1.72019
$220$	0	0
$221$	1.03828	0.0698421
$222$	0	0
$223$	18.8858	1.26469	0.632344	−	0.774687i	$-0.282093\pi$
$223$	18.8858	1.26469	0.632344	+	0.774687i	$0.282093\pi$
$224$	0	0
$225$	−12.4235	−0.828235
$226$	0	0
$227$	−4.29688	−0.285194	−0.142597	−	0.989781i	$-0.545545\pi$
$227$	−4.29688	−0.285194	−0.142597	+	0.989781i	$0.545545\pi$
$228$	0	0
$229$	5.94217	0.392670	0.196335	−	0.980537i	$-0.437096\pi$
$229$	5.94217	0.392670	0.196335	+	0.980537i	$0.437096\pi$
$230$	0	0
$231$	6.78527	0.446438
$232$	0	0
$233$	2.48037	0.162494	0.0812472	−	0.996694i	$-0.474110\pi$
$233$	2.48037	0.162494	0.0812472	+	0.996694i	$0.474110\pi$
$234$	0	0
$235$	0.281460	0.0183604
$236$	0	0
$237$	10.0716	0.654221
$238$	0	0
$239$	20.0791	1.29881	0.649406	−	0.760442i	$-0.275018\pi$
$239$	20.0791	1.29881	0.649406	+	0.760442i	$0.275018\pi$
$240$	0	0
$241$	17.7876	1.14580	0.572899	−	0.819626i	$-0.305819\pi$
$241$	17.7876	1.14580	0.572899	+	0.819626i	$0.305819\pi$
$242$	0	0
$243$	−20.7223	−1.32934
$244$	0	0
$245$	−0.778371	−0.0497283
$246$	0	0
$247$	−0.828502	−0.0527163
$248$	0	0
$249$	−13.0485	−0.826912
$250$	0	0
$251$	7.46524	0.471202	0.235601	−	0.971850i	$-0.424294\pi$
$251$	7.46524	0.471202	0.235601	+	0.971850i	$0.424294\pi$
$252$	0	0
$253$	−0.198265	−0.0124648
$254$	0	0
$255$	−3.95746	−0.247826
$256$	0	0
$257$	−3.16460	−0.197402	−0.0987012	−	0.995117i	$-0.531469\pi$
$257$	−3.16460	−0.197402	−0.0987012	+	0.995117i	$0.531469\pi$
$258$	0	0
$259$	2.73583	0.169996
$260$	0	0
$261$	−2.59864	−0.160852
$262$	0	0
$263$	7.20962	0.444564	0.222282	−	0.974982i	$-0.428649\pi$
$263$	7.20962	0.444564	0.222282	+	0.974982i	$0.428649\pi$
$264$	0	0
$265$	−1.57011	−0.0964512
$266$	0	0
$267$	−34.4016	−2.10535
$268$	0	0
$269$	1.81673	0.110768	0.0553840	−	0.998465i	$-0.482362\pi$
$269$	1.81673	0.110768	0.0553840	+	0.998465i	$0.482362\pi$
$270$	0	0
$271$	−15.4315	−0.937399	−0.468700	−	0.883358i	$-0.655277\pi$
$271$	−15.4315	−0.937399	−0.468700	+	0.883358i	$0.655277\pi$
$272$	0	0
$273$	1.13586	0.0687451
$274$	0	0
$275$	6.39955	0.385907
$276$	0	0
$277$	−3.68852	−0.221621	−0.110811	−	0.993842i	$-0.535345\pi$
$277$	−3.68852	−0.221621	−0.110811	+	0.993842i	$0.535345\pi$
$278$	0	0
$279$	11.8393	0.708802
$280$	0	0
$281$	−28.8380	−1.72033	−0.860165	−	0.510016i	$-0.829640\pi$
$281$	−28.8380	−1.72033	−0.860165	+	0.510016i	$0.829640\pi$
$282$	0	0
$283$	19.5214	1.16043	0.580214	−	0.814464i	$-0.302969\pi$
$283$	19.5214	1.16043	0.580214	+	0.814464i	$0.302969\pi$
$284$	0	0
$285$	3.15788	0.187057
$286$	0	0
$287$	−2.10845	−0.124458
$288$	0	0
$289$	5.30989	0.312346
$290$	0	0
$291$	−21.5970	−1.26604
$292$	0	0
$293$	19.2030	1.12185	0.560926	−	0.827866i	$-0.310445\pi$
$293$	19.2030	1.12185	0.560926	+	0.827866i	$0.310445\pi$
$294$	0	0
$295$	1.58665	0.0923786
$296$	0	0
$297$	1.39446	0.0809149
$298$	0	0
$299$	−0.0331896	−0.00191940
$300$	0	0
$301$	−5.93626	−0.342160
$302$	0	0
$303$	14.0915	0.809536
$304$	0	0
$305$	−1.10314	−0.0631657
$306$	0	0
$307$	−8.44718	−0.482106	−0.241053	−	0.970512i	$-0.577493\pi$
$307$	−8.44718	−0.482106	−0.241053	+	0.970512i	$0.577493\pi$
$308$	0	0
$309$	24.8288	1.41246
$310$	0	0
$311$	−24.4490	−1.38637	−0.693187	−	0.720758i	$-0.743794\pi$
$311$	−24.4490	−1.38637	−0.693187	+	0.720758i	$0.743794\pi$
$312$	0	0
$313$	−4.86660	−0.275076	−0.137538	−	0.990496i	$-0.543919\pi$
$313$	−4.86660	−0.275076	−0.137538	+	0.990496i	$0.543919\pi$
$314$	0	0
$315$	−1.98884	−0.112058
$316$	0	0
$317$	34.5036	1.93792	0.968959	−	0.247221i	$-0.0795175\pi$
$317$	34.5036	1.93792	0.968959	+	0.247221i	$0.0795175\pi$
$318$	0	0
$319$	1.33860	0.0749472
$320$	0	0
$321$	−25.4449	−1.42019
$322$	0	0
$323$	−17.8023	−0.990549
$324$	0	0
$325$	1.07129	0.0594243
$326$	0	0
$327$	−45.6385	−2.52382
$328$	0	0
$329$	−1.73583	−0.0956994
$330$	0	0
$331$	5.74204	0.315611	0.157805	−	0.987470i	$-0.449558\pi$
$331$	5.74204	0.315611	0.157805	+	0.987470i	$0.449558\pi$
$332$	0	0
$333$	−3.17944	−0.174232
$334$	0	0
$335$	−3.35964	−0.183557
$336$	0	0
$337$	−12.3242	−0.671343	−0.335671	−	0.941979i	$-0.608963\pi$
$337$	−12.3242	−0.671343	−0.335671	+	0.941979i	$0.608963\pi$
$338$	0	0
$339$	27.7506	1.50720
$340$	0	0
$341$	−6.09862	−0.330259
$342$	0	0
$343$	20.1551	1.08827
$344$	0	0
$345$	0.126504	0.00681076
$346$	0	0
$347$	−11.8379	−0.635489	−0.317745	−	0.948176i	$-0.602925\pi$
$347$	−11.8379	−0.635489	−0.317745	+	0.948176i	$0.602925\pi$
$348$	0	0
$349$	0.350720	0.0187736	0.00938680	−	0.999956i	$-0.497012\pi$
$349$	0.350720	0.0187736	0.00938680	+	0.999956i	$0.497012\pi$
$350$	0	0
$351$	0.233433	0.0124597
$352$	0	0
$353$	7.12371	0.379157	0.189578	−	0.981866i	$-0.439288\pi$
$353$	7.12371	0.379157	0.189578	+	0.981866i	$0.439288\pi$
$354$	0	0
$355$	−0.928239	−0.0492658
$356$	0	0
$357$	24.4066	1.29173
$358$	0	0
$359$	19.6384	1.03647	0.518237	−	0.855237i	$-0.326588\pi$
$359$	19.6384	1.03647	0.518237	+	0.855237i	$0.326588\pi$
$360$	0	0
$361$	−4.79449	−0.252341
$362$	0	0
$363$	−21.8505	−1.14685
$364$	0	0
$365$	−3.84358	−0.201182
$366$	0	0
$367$	1.92083	0.100267	0.0501333	−	0.998743i	$-0.484035\pi$
$367$	1.92083	0.100267	0.0501333	+	0.998743i	$0.484035\pi$
$368$	0	0
$369$	2.45032	0.127559
$370$	0	0
$371$	9.68325	0.502729
$372$	0	0
$373$	38.1400	1.97481	0.987406	−	0.158206i	$-0.0505710\pi$
$373$	38.1400	1.97481	0.987406	+	0.158206i	$0.0505710\pi$
$374$	0	0
$375$	−8.27254	−0.427192
$376$	0	0
$377$	0.224082	0.0115408
$378$	0	0
$379$	−37.7074	−1.93690	−0.968448	−	0.249215i	$-0.919828\pi$
$379$	−37.7074	−1.93690	−0.968448	+	0.249215i	$0.919828\pi$
$380$	0	0
$381$	−26.1063	−1.33746
$382$	0	0
$383$	−7.78393	−0.397740	−0.198870	−	0.980026i	$-0.563727\pi$
$383$	−7.78393	−0.397740	−0.198870	+	0.980026i	$0.563727\pi$
$384$	0	0
$385$	1.02448	0.0522124
$386$	0	0
$387$	6.89880	0.350686
$388$	0	0
$389$	−15.3669	−0.779132	−0.389566	−	0.920998i	$-0.627375\pi$
$389$	−15.3669	−0.779132	−0.389566	+	0.920998i	$0.627375\pi$
$390$	0	0
$391$	−0.713158	−0.0360660
$392$	0	0
$393$	−22.1693	−1.11829
$394$	0	0
$395$	1.52067	0.0765133
$396$	0	0
$397$	8.41124	0.422148	0.211074	−	0.977470i	$-0.432304\pi$
$397$	8.41124	0.422148	0.211074	+	0.977470i	$0.432304\pi$
$398$	0	0
$399$	−19.4754	−0.974990
$400$	0	0
$401$	35.1299	1.75430	0.877152	−	0.480213i	$-0.159440\pi$
$401$	35.1299	1.75430	0.877152	+	0.480213i	$0.159440\pi$
$402$	0	0
$403$	−1.02091	−0.0508552
$404$	0	0
$405$	−3.60980	−0.179372
$406$	0	0
$407$	1.63778	0.0811816
$408$	0	0
$409$	30.9050	1.52816	0.764078	−	0.645124i	$-0.223194\pi$
$409$	30.9050	1.52816	0.764078	+	0.645124i	$0.223194\pi$
$410$	0	0
$411$	34.0054	1.67736
$412$	0	0
$413$	−9.78527	−0.481502
$414$	0	0
$415$	−1.97014	−0.0967102
$416$	0	0
$417$	14.8286	0.726161
$418$	0	0
$419$	−34.1501	−1.66834	−0.834172	−	0.551505i	$-0.814054\pi$
$419$	−34.1501	−1.66834	−0.834172	+	0.551505i	$0.814054\pi$
$420$	0	0
$421$	−10.1575	−0.495048	−0.247524	−	0.968882i	$-0.579617\pi$
$421$	−10.1575	−0.495048	−0.247524	+	0.968882i	$0.579617\pi$
$422$	0	0
$423$	2.01729	0.0980840
$424$	0	0
$425$	23.0192	1.11659
$426$	0	0
$427$	6.80333	0.329236
$428$	0	0
$429$	0.679969	0.0328292
$430$	0	0
$431$	1.00000	0.0481683
$431$	1.00000	0.0481683
$432$	0	0
$433$	5.06771	0.243539	0.121769	−	0.992558i	$-0.461143\pi$
$433$	5.06771	0.243539	0.121769	+	0.992558i	$0.461143\pi$
$434$	0	0
$435$	−0.854102	−0.0409511
$436$	0	0
$437$	0.569070	0.0272223
$438$	0	0
$439$	21.0411	1.00424	0.502118	−	0.864799i	$-0.332554\pi$
$439$	21.0411	1.00424	0.502118	+	0.864799i	$0.332554\pi$
$440$	0	0
$441$	−5.57877	−0.265656
$442$	0	0
$443$	−16.4981	−0.783846	−0.391923	−	0.919998i	$-0.628190\pi$
$443$	−16.4981	−0.783846	−0.391923	+	0.919998i	$0.628190\pi$
$444$	0	0
$445$	−5.19417	−0.246227
$446$	0	0
$447$	3.00314	0.142044
$448$	0	0
$449$	30.4997	1.43937	0.719684	−	0.694301i	$-0.244286\pi$
$449$	30.4997	1.43937	0.719684	+	0.694301i	$0.244286\pi$
$450$	0	0
$451$	−1.26220	−0.0594347
$452$	0	0
$453$	32.4268	1.52354
$454$	0	0
$455$	0.171498	0.00803997
$456$	0	0
$457$	11.8416	0.553929	0.276964	−	0.960880i	$-0.410672\pi$
$457$	11.8416	0.553929	0.276964	+	0.960880i	$0.410672\pi$
$458$	0	0
$459$	5.01587	0.234121
$460$	0	0
$461$	−11.9837	−0.558138	−0.279069	−	0.960271i	$-0.590026\pi$
$461$	−11.9837	−0.558138	−0.279069	+	0.960271i	$0.590026\pi$
$462$	0	0
$463$	−4.46877	−0.207682	−0.103841	−	0.994594i	$-0.533113\pi$
$463$	−4.46877	−0.207682	−0.103841	+	0.994594i	$0.533113\pi$
$464$	0	0
$465$	3.89126	0.180453
$466$	0	0
$467$	29.9185	1.38446	0.692231	−	0.721676i	$-0.256628\pi$
$467$	29.9185	1.38446	0.692231	+	0.721676i	$0.256628\pi$
$468$	0	0
$469$	20.7197	0.956748
$470$	0	0
$471$	0.529298	0.0243887
$472$	0	0
$473$	−3.55368	−0.163398
$474$	0	0
$475$	−18.3683	−0.842796
$476$	0	0
$477$	−11.2534	−0.515256
$478$	0	0
$479$	33.6205	1.53616	0.768080	−	0.640354i	$-0.221212\pi$
$479$	33.6205	1.53616	0.768080	+	0.640354i	$0.221212\pi$
$480$	0	0
$481$	0.274164	0.0125008
$482$	0	0
$483$	−0.780181	−0.0354995
$484$	0	0
$485$	−3.26084	−0.148067
$486$	0	0
$487$	7.66808	0.347474	0.173737	−	0.984792i	$-0.444416\pi$
$487$	7.66808	0.347474	0.173737	+	0.984792i	$0.444416\pi$
$488$	0	0
$489$	−25.3595	−1.14679
$490$	0	0
$491$	23.7671	1.07259	0.536296	−	0.844030i	$-0.319823\pi$
$491$	23.7671	1.07259	0.536296	+	0.844030i	$0.319823\pi$
$492$	0	0
$493$	4.81494	0.216854
$494$	0	0
$495$	−1.19060	−0.0535134
$496$	0	0
$497$	5.72467	0.256787
$498$	0	0
$499$	−29.6770	−1.32853	−0.664263	−	0.747499i	$-0.731254\pi$
$499$	−29.6770	−1.32853	−0.664263	+	0.747499i	$0.731254\pi$
$500$	0	0
$501$	−14.1329	−0.631413
$502$	0	0
$503$	36.5805	1.63105	0.815523	−	0.578725i	$-0.196450\pi$
$503$	36.5805	1.63105	0.815523	+	0.578725i	$0.196450\pi$
$504$	0	0
$505$	2.12762	0.0946780
$506$	0	0
$507$	−30.5099	−1.35499
$508$	0	0
$509$	−38.2366	−1.69481	−0.847403	−	0.530951i	$-0.821835\pi$
$509$	−38.2366	−1.69481	−0.847403	+	0.530951i	$0.821835\pi$
$510$	0	0
$511$	23.7043	1.04862
$512$	0	0
$513$	−4.00245	−0.176713
$514$	0	0
$515$	3.74880	0.165192
$516$	0	0
$517$	−1.03914	−0.0457012
$518$	0	0
$519$	−7.65256	−0.335910
$520$	0	0
$521$	−31.7558	−1.39125	−0.695624	−	0.718406i	$-0.744872\pi$
$521$	−31.7558	−1.39125	−0.695624	+	0.718406i	$0.744872\pi$
$522$	0	0
$523$	−4.30013	−0.188031	−0.0940157	−	0.995571i	$-0.529970\pi$
$523$	−4.30013	−0.188031	−0.0940157	+	0.995571i	$0.529970\pi$
$524$	0	0
$525$	25.1825	1.09905
$526$	0	0
$527$	−21.9367	−0.955578
$528$	0	0
$529$	−22.9772	−0.999009
$530$	0	0
$531$	11.3719	0.493499
$532$	0	0
$533$	−0.211293	−0.00915210
$534$	0	0
$535$	−3.84182	−0.166096
$536$	0	0
$537$	10.8999	0.470366
$538$	0	0
$539$	2.87371	0.123780
$540$	0	0
$541$	−25.1363	−1.08069	−0.540347	−	0.841443i	$-0.681707\pi$
$541$	−25.1363	−1.08069	−0.540347	+	0.841443i	$0.681707\pi$
$542$	0	0
$543$	−23.3032	−1.00004
$544$	0	0
$545$	−6.89079	−0.295169
$546$	0	0
$547$	12.8712	0.550333	0.275167	−	0.961397i	$-0.411267\pi$
$547$	12.8712	0.550333	0.275167	+	0.961397i	$0.411267\pi$
$548$	0	0
$549$	−7.90647	−0.337440
$550$	0	0
$551$	−3.84212	−0.163680
$552$	0	0
$553$	−9.37835	−0.398808
$554$	0	0
$555$	−1.04499	−0.0443575
$556$	0	0
$557$	−3.93164	−0.166589	−0.0832945	−	0.996525i	$-0.526544\pi$
$557$	−3.93164	−0.166589	−0.0832945	+	0.996525i	$0.526544\pi$
$558$	0	0
$559$	−0.594887	−0.0251610
$560$	0	0
$561$	14.6108	0.616867
$562$	0	0
$563$	17.0326	0.717838	0.358919	−	0.933369i	$-0.383145\pi$
$563$	17.0326	0.717838	0.358919	+	0.933369i	$0.383145\pi$
$564$	0	0
$565$	4.18996	0.176273
$566$	0	0
$567$	22.2625	0.934937
$568$	0	0
$569$	18.0132	0.755154	0.377577	−	0.925978i	$-0.376757\pi$
$569$	18.0132	0.755154	0.377577	+	0.925978i	$0.376757\pi$
$570$	0	0
$571$	30.4440	1.27404	0.637020	−	0.770847i	$-0.280167\pi$
$571$	30.4440	1.27404	0.637020	+	0.770847i	$0.280167\pi$
$572$	0	0
$573$	−20.8318	−0.870260
$574$	0	0
$575$	−0.735831	−0.0306863
$576$	0	0
$577$	34.3629	1.43055	0.715273	−	0.698846i	$-0.246303\pi$
$577$	34.3629	1.43055	0.715273	+	0.698846i	$0.246303\pi$
$578$	0	0
$579$	21.6158	0.898324
$580$	0	0
$581$	12.1503	0.504079
$582$	0	0
$583$	5.79678	0.240078
$584$	0	0
$585$	−0.199306	−0.00824031
$586$	0	0
$587$	35.3347	1.45842	0.729209	−	0.684291i	$-0.239888\pi$
$587$	35.3347	1.45842	0.729209	+	0.684291i	$0.239888\pi$
$588$	0	0
$589$	17.5046	0.721263
$590$	0	0
$591$	−0.585536	−0.0240857
$592$	0	0
$593$	−9.31148	−0.382377	−0.191188	−	0.981553i	$-0.561234\pi$
$593$	−9.31148	−0.382377	−0.191188	+	0.981553i	$0.561234\pi$
$594$	0	0
$595$	3.68506	0.151073
$596$	0	0
$597$	−12.1506	−0.497291
$598$	0	0
$599$	−15.9577	−0.652013	−0.326006	−	0.945368i	$-0.605703\pi$
$599$	−15.9577	−0.652013	−0.326006	+	0.945368i	$0.605703\pi$
$600$	0	0
$601$	−40.6737	−1.65912	−0.829558	−	0.558421i	$-0.811407\pi$
$601$	−40.6737	−1.65912	−0.829558	+	0.558421i	$0.811407\pi$
$602$	0	0
$603$	−24.0794	−0.980587
$604$	0	0
$605$	−3.29912	−0.134128
$606$	0	0
$607$	30.2347	1.22719	0.613594	−	0.789621i	$-0.289723\pi$
$607$	30.2347	1.22719	0.613594	+	0.789621i	$0.289723\pi$
$608$	0	0
$609$	5.26745	0.213448
$610$	0	0
$611$	−0.173952	−0.00703734
$612$	0	0
$613$	13.2408	0.534789	0.267395	−	0.963587i	$-0.413837\pi$
$613$	13.2408	0.534789	0.267395	+	0.963587i	$0.413837\pi$
$614$	0	0
$615$	0.805354	0.0324750
$616$	0	0
$617$	4.31606	0.173758	0.0868790	−	0.996219i	$-0.472311\pi$
$617$	4.31606	0.173758	0.0868790	+	0.996219i	$0.472311\pi$
$618$	0	0
$619$	8.65208	0.347757	0.173878	−	0.984767i	$-0.444370\pi$
$619$	8.65208	0.347757	0.173878	+	0.984767i	$0.444370\pi$
$620$	0	0
$621$	−0.160337	−0.00643412
$622$	0	0
$623$	32.0337	1.28340
$624$	0	0
$625$	23.1184	0.924738
$626$	0	0
$627$	−11.6588	−0.465606
$628$	0	0
$629$	5.89108	0.234893
$630$	0	0
$631$	−4.03074	−0.160461	−0.0802305	−	0.996776i	$-0.525566\pi$
$631$	−4.03074	−0.160461	−0.0802305	+	0.996776i	$0.525566\pi$
$632$	0	0
$633$	−41.2572	−1.63983
$634$	0	0
$635$	−3.94168	−0.156421
$636$	0	0
$637$	0.481060	0.0190603
$638$	0	0
$639$	−6.65291	−0.263185
$640$	0	0
$641$	−42.7002	−1.68656	−0.843279	−	0.537476i	$-0.819378\pi$
$641$	−42.7002	−1.68656	−0.843279	+	0.537476i	$0.819378\pi$
$642$	0	0
$643$	2.51260	0.0990873	0.0495436	−	0.998772i	$-0.484223\pi$
$643$	2.51260	0.0990873	0.0495436	+	0.998772i	$0.484223\pi$
$644$	0	0
$645$	2.26745	0.0892807
$646$	0	0
$647$	5.08862	0.200054	0.100027	−	0.994985i	$-0.468107\pi$
$647$	5.08862	0.200054	0.100027	+	0.994985i	$0.468107\pi$
$648$	0	0
$649$	−5.85786	−0.229941
$650$	0	0
$651$	−23.9983	−0.940568
$652$	0	0
$653$	−43.4653	−1.70093	−0.850465	−	0.526031i	$-0.823679\pi$
$653$	−43.4653	−1.70093	−0.850465	+	0.526031i	$0.823679\pi$
$654$	0	0
$655$	−3.34726	−0.130788
$656$	0	0
$657$	−27.5479	−1.07474
$658$	0	0
$659$	−0.206667	−0.00805059	−0.00402530	−	0.999992i	$-0.501281\pi$
$659$	−0.206667	−0.00805059	−0.00402530	+	0.999992i	$0.501281\pi$
$660$	0	0
$661$	35.9230	1.39724	0.698621	−	0.715492i	$-0.253797\pi$
$661$	35.9230	1.39724	0.698621	+	0.715492i	$0.253797\pi$
$662$	0	0
$663$	2.44584	0.0949887
$664$	0	0
$665$	−2.94052	−0.114028
$666$	0	0
$667$	−0.153914	−0.00595959
$668$	0	0
$669$	44.4889	1.72004
$670$	0	0
$671$	4.07275	0.157227
$672$	0	0
$673$	35.1183	1.35371	0.676855	−	0.736116i	$-0.263342\pi$
$673$	35.1183	1.35371	0.676855	+	0.736116i	$0.263342\pi$
$674$	0	0
$675$	5.17533	0.199199
$676$	0	0
$677$	−5.23428	−0.201170	−0.100585	−	0.994928i	$-0.532071\pi$
$677$	−5.23428	−0.201170	−0.100585	+	0.994928i	$0.532071\pi$
$678$	0	0
$679$	20.1104	0.771766
$680$	0	0
$681$	−10.1221	−0.387878
$682$	0	0
$683$	−20.7013	−0.792113	−0.396057	−	0.918226i	$-0.629622\pi$
$683$	−20.7013	−0.792113	−0.396057	+	0.918226i	$0.629622\pi$
$684$	0	0
$685$	5.13434	0.196173
$686$	0	0
$687$	13.9978	0.534050
$688$	0	0
$689$	0.970382	0.0369686
$690$	0	0
$691$	−14.7800	−0.562258	−0.281129	−	0.959670i	$-0.590709\pi$
$691$	−14.7800	−0.562258	−0.281129	+	0.959670i	$0.590709\pi$
$692$	0	0
$693$	7.34270	0.278926
$694$	0	0
$695$	2.23892	0.0849270
$696$	0	0
$697$	−4.54013	−0.171970
$698$	0	0
$699$	5.84294	0.221000
$700$	0	0
$701$	34.1735	1.29072	0.645358	−	0.763881i	$-0.276708\pi$
$701$	34.1735	1.29072	0.645358	+	0.763881i	$0.276708\pi$
$702$	0	0
$703$	−4.70083	−0.177295
$704$	0	0
$705$	0.663028	0.0249711
$706$	0	0
$707$	−13.1215	−0.493487
$708$	0	0
$709$	8.21388	0.308479	0.154239	−	0.988034i	$-0.450707\pi$
$709$	8.21388	0.308479	0.154239	+	0.988034i	$0.450707\pi$
$710$	0	0
$711$	10.8990	0.408745
$712$	0	0
$713$	0.701229	0.0262612
$714$	0	0
$715$	0.102666	0.00383949
$716$	0	0
$717$	47.2999	1.76645
$718$	0	0
$719$	33.7298	1.25791	0.628954	−	0.777443i	$-0.283483\pi$
$719$	33.7298	1.25791	0.628954	+	0.777443i	$0.283483\pi$
$720$	0	0
$721$	−23.1197	−0.861024
$722$	0	0
$723$	41.9017	1.55834
$724$	0	0
$725$	4.96801	0.184507
$726$	0	0
$727$	21.4570	0.795798	0.397899	−	0.917429i	$-0.369740\pi$
$727$	21.4570	0.795798	0.397899	+	0.917429i	$0.369740\pi$
$728$	0	0
$729$	−18.3676	−0.680281
$730$	0	0
$731$	−12.7826	−0.472781
$732$	0	0
$733$	−25.9785	−0.959539	−0.479770	−	0.877395i	$-0.659280\pi$
$733$	−25.9785	−0.959539	−0.479770	+	0.877395i	$0.659280\pi$
$734$	0	0
$735$	−1.83359	−0.0676330
$736$	0	0
$737$	12.4037	0.456894
$738$	0	0
$739$	−34.0810	−1.25369	−0.626845	−	0.779144i	$-0.715654\pi$
$739$	−34.0810	−1.25369	−0.626845	+	0.779144i	$0.715654\pi$
$740$	0	0
$741$	−1.95168	−0.0716967
$742$	0	0
$743$	−8.04036	−0.294972	−0.147486	−	0.989064i	$-0.547118\pi$
$743$	−8.04036	−0.294972	−0.147486	+	0.989064i	$0.547118\pi$
$744$	0	0
$745$	0.453433	0.0166125
$746$	0	0
$747$	−14.1204	−0.516640
$748$	0	0
$749$	23.6934	0.865739
$750$	0	0
$751$	−36.3502	−1.32644	−0.663220	−	0.748425i	$-0.730810\pi$
$751$	−36.3502	−1.32644	−0.663220	+	0.748425i	$0.730810\pi$
$752$	0	0
$753$	17.5857	0.640857
$754$	0	0
$755$	4.89600	0.178184
$756$	0	0
$757$	−6.55362	−0.238195	−0.119098	−	0.992883i	$-0.538000\pi$
$757$	−6.55362	−0.238195	−0.119098	+	0.992883i	$0.538000\pi$
$758$	0	0
$759$	−0.467048	−0.0169528
$760$	0	0
$761$	−28.9735	−1.05029	−0.525144	−	0.851014i	$-0.675988\pi$
$761$	−28.9735	−1.05029	−0.525144	+	0.851014i	$0.675988\pi$
$762$	0	0
$763$	42.4971	1.53850
$764$	0	0
$765$	−4.28258	−0.154837
$766$	0	0
$767$	−0.980606	−0.0354076
$768$	0	0
$769$	−20.7996	−0.750052	−0.375026	−	0.927014i	$-0.622366\pi$
$769$	−20.7996	−0.750052	−0.375026	+	0.927014i	$0.622366\pi$
$770$	0	0
$771$	−7.45477	−0.268477
$772$	0	0
$773$	34.7699	1.25059	0.625293	−	0.780390i	$-0.284980\pi$
$773$	34.7699	1.25059	0.625293	+	0.780390i	$0.284980\pi$
$774$	0	0
$775$	−22.6341	−0.813041
$776$	0	0
$777$	6.44473	0.231203
$778$	0	0
$779$	3.62283	0.129801
$780$	0	0
$781$	3.42702	0.122628
$782$	0	0
$783$	1.08253	0.0386865
$784$	0	0
$785$	0.0799166	0.00285235
$786$	0	0
$787$	−18.6268	−0.663973	−0.331986	−	0.943284i	$-0.607719\pi$
$787$	−18.6268	−0.663973	−0.331986	+	0.943284i	$0.607719\pi$
$788$	0	0
$789$	16.9835	0.604629
$790$	0	0
$791$	−25.8404	−0.918780
$792$	0	0
$793$	0.681778	0.0242107
$794$	0	0
$795$	−3.69867	−0.131178
$796$	0	0
$797$	−8.55696	−0.303103	−0.151552	−	0.988449i	$-0.548427\pi$
$797$	−8.55696	−0.303103	−0.151552	+	0.988449i	$0.548427\pi$
$798$	0	0
$799$	−3.73777	−0.132233
$800$	0	0
$801$	−37.2278	−1.31538
$802$	0	0
$803$	14.1903	0.500766
$804$	0	0
$805$	−0.117797	−0.00415178
$806$	0	0
$807$	4.27963	0.150650
$808$	0	0
$809$	36.5255	1.28417	0.642084	−	0.766634i	$-0.278070\pi$
$809$	36.5255	1.28417	0.642084	+	0.766634i	$0.278070\pi$
$810$	0	0
$811$	−0.291636	−0.0102407	−0.00512037	−	0.999987i	$-0.501630\pi$
$811$	−0.291636	−0.0102407	−0.00512037	+	0.999987i	$0.501630\pi$
$812$	0	0
$813$	−36.3517	−1.27491
$814$	0	0
$815$	−3.82893	−0.134122
$816$	0	0
$817$	10.2000	0.356851
$818$	0	0
$819$	1.22917	0.0429507
$820$	0	0
$821$	−24.0196	−0.838291	−0.419145	−	0.907919i	$-0.637670\pi$
$821$	−24.0196	−0.838291	−0.419145	+	0.907919i	$0.637670\pi$
$822$	0	0
$823$	37.3060	1.30040	0.650202	−	0.759761i	$-0.274684\pi$
$823$	37.3060	1.30040	0.650202	+	0.759761i	$0.274684\pi$
$824$	0	0
$825$	15.0753	0.524853
$826$	0	0
$827$	−2.15858	−0.0750610	−0.0375305	−	0.999295i	$-0.511949\pi$
$827$	−2.15858	−0.0750610	−0.0375305	+	0.999295i	$0.511949\pi$
$828$	0	0
$829$	39.2679	1.36383	0.681915	−	0.731431i	$-0.261147\pi$
$829$	39.2679	1.36383	0.681915	+	0.731431i	$0.261147\pi$
$830$	0	0
$831$	−8.68894	−0.301416
$832$	0	0
$833$	10.3367	0.358146
$834$	0	0
$835$	−2.13388	−0.0738459
$836$	0	0
$837$	−4.93197	−0.170474
$838$	0	0
$839$	−19.4462	−0.671358	−0.335679	−	0.941976i	$-0.608966\pi$
$839$	−19.4462	−0.671358	−0.335679	+	0.941976i	$0.608966\pi$
$840$	0	0
$841$	−27.9608	−0.964167
$842$	0	0
$843$	−67.9329	−2.33973
$844$	0	0
$845$	−4.60658	−0.158471
$846$	0	0
$847$	20.3465	0.699113
$848$	0	0
$849$	45.9861	1.57824
$850$	0	0
$851$	−0.188314	−0.00645533
$852$	0	0
$853$	−31.5937	−1.08175	−0.540874	−	0.841104i	$-0.681906\pi$
$853$	−31.5937	−1.08175	−0.540874	+	0.841104i	$0.681906\pi$
$854$	0	0
$855$	3.41732	0.116870
$856$	0	0
$857$	−38.8076	−1.32564	−0.662822	−	0.748777i	$-0.730641\pi$
$857$	−38.8076	−1.32564	−0.662822	+	0.748777i	$0.730641\pi$
$858$	0	0
$859$	30.1896	1.03005	0.515027	−	0.857174i	$-0.327782\pi$
$859$	30.1896	1.03005	0.515027	+	0.857174i	$0.327782\pi$
$860$	0	0
$861$	−4.96681	−0.169268
$862$	0	0
$863$	−11.4522	−0.389837	−0.194918	−	0.980819i	$-0.562444\pi$
$863$	−11.4522	−0.389837	−0.194918	+	0.980819i	$0.562444\pi$
$864$	0	0
$865$	−1.15543	−0.0392858
$866$	0	0
$867$	12.5084	0.424807
$868$	0	0
$869$	−5.61425	−0.190451
$870$	0	0
$871$	2.07637	0.0703553
$872$	0	0
$873$	−23.3712	−0.790996
$874$	0	0
$875$	7.70312	0.260413
$876$	0	0
$877$	−37.2682	−1.25846	−0.629229	−	0.777220i	$-0.716629\pi$
$877$	−37.2682	−1.25846	−0.629229	+	0.777220i	$0.716629\pi$
$878$	0	0
$879$	45.2360	1.52577
$880$	0	0
$881$	−7.29814	−0.245880	−0.122940	−	0.992414i	$-0.539232\pi$
$881$	−7.29814	−0.245880	−0.122940	+	0.992414i	$0.539232\pi$
$882$	0	0
$883$	19.7549	0.664805	0.332402	−	0.943138i	$-0.392141\pi$
$883$	19.7549	0.664805	0.332402	+	0.943138i	$0.392141\pi$
$884$	0	0
$885$	3.73764	0.125639
$886$	0	0
$887$	−48.7214	−1.63591	−0.817953	−	0.575285i	$-0.804891\pi$
$887$	−48.7214	−1.63591	−0.817953	+	0.575285i	$0.804891\pi$
$888$	0	0
$889$	24.3093	0.815308
$890$	0	0
$891$	13.3272	0.446479
$892$	0	0
$893$	2.98258	0.0998084
$894$	0	0
$895$	1.64574	0.0550109
$896$	0	0
$897$	−0.0781839	−0.00261048
$898$	0	0
$899$	−4.73440	−0.157901
$900$	0	0
$901$	20.8510	0.694647
$902$	0	0
$903$	−13.9839	−0.465355
$904$	0	0
$905$	−3.51847	−0.116958
$906$	0	0
$907$	−7.89607	−0.262185	−0.131092	−	0.991370i	$-0.541848\pi$
$907$	−7.89607	−0.262185	−0.131092	+	0.991370i	$0.541848\pi$
$908$	0	0
$909$	15.2492	0.505783
$910$	0	0
$911$	−41.9892	−1.39117	−0.695583	−	0.718446i	$-0.744854\pi$
$911$	−41.9892	−1.39117	−0.695583	+	0.718446i	$0.744854\pi$
$912$	0	0
$913$	7.27365	0.240723
$914$	0	0
$915$	−2.59864	−0.0859084
$916$	0	0
$917$	20.6433	0.681703
$918$	0	0
$919$	−47.6809	−1.57285	−0.786424	−	0.617687i	$-0.788070\pi$
$919$	−47.6809	−1.57285	−0.786424	+	0.617687i	$0.788070\pi$
$920$	0	0
$921$	−19.8988	−0.655688
$922$	0	0
$923$	0.573683	0.0188830
$924$	0	0
$925$	6.07837	0.199855
$926$	0	0
$927$	26.8685	0.882479
$928$	0	0
$929$	9.33964	0.306424	0.153212	−	0.988193i	$-0.451038\pi$
$929$	9.33964	0.306424	0.153212	+	0.988193i	$0.451038\pi$
$930$	0	0
$931$	−8.24827	−0.270326
$932$	0	0
$933$	−57.5938	−1.88554
$934$	0	0
$935$	2.20602	0.0721447
$936$	0	0
$937$	34.6393	1.13162	0.565809	−	0.824536i	$-0.308564\pi$
$937$	34.6393	1.13162	0.565809	+	0.824536i	$0.308564\pi$
$938$	0	0
$939$	−11.4641	−0.374117
$940$	0	0
$941$	48.6872	1.58716	0.793579	−	0.608467i	$-0.208215\pi$
$941$	48.6872	1.58716	0.793579	+	0.608467i	$0.208215\pi$
$942$	0	0
$943$	0.145130	0.00472608
$944$	0	0
$945$	0.828502	0.0269512
$946$	0	0
$947$	35.0995	1.14058	0.570290	−	0.821444i	$-0.306831\pi$
$947$	35.0995	1.14058	0.570290	+	0.821444i	$0.306831\pi$
$948$	0	0
$949$	2.37547	0.0771109
$950$	0	0
$951$	81.2794	2.63566
$952$	0	0
$953$	41.5286	1.34524	0.672622	−	0.739986i	$-0.265168\pi$
$953$	41.5286	1.34524	0.672622	+	0.739986i	$0.265168\pi$
$954$	0	0
$955$	−3.14531	−0.101780
$956$	0	0
$957$	3.15331	0.101932
$958$	0	0
$959$	−31.6647	−1.02251
$960$	0	0
$961$	−9.43024	−0.304201
$962$	0	0
$963$	−27.5352	−0.887311
$964$	0	0
$965$	3.26369	0.105062
$966$	0	0
$967$	−25.9307	−0.833875	−0.416937	−	0.908935i	$-0.636897\pi$
$967$	−25.9307	−0.833875	−0.416937	+	0.908935i	$0.636897\pi$
$968$	0	0
$969$	−41.9365	−1.34720
$970$	0	0
$971$	34.0711	1.09339	0.546697	−	0.837331i	$-0.315885\pi$
$971$	34.0711	1.09339	0.546697	+	0.837331i	$0.315885\pi$
$972$	0	0
$973$	−13.8079	−0.442662
$974$	0	0
$975$	2.52360	0.0808199
$976$	0	0
$977$	57.9989	1.85555	0.927775	−	0.373139i	$-0.121719\pi$
$977$	57.9989	1.85555	0.927775	+	0.373139i	$0.121719\pi$
$978$	0	0
$979$	19.1766	0.612888
$980$	0	0
$981$	−49.3879	−1.57683
$982$	0	0
$983$	34.3211	1.09467	0.547337	−	0.836912i	$-0.315641\pi$
$983$	34.3211	1.09467	0.547337	+	0.836912i	$0.315641\pi$
$984$	0	0
$985$	−0.0884078	−0.00281691
$986$	0	0
$987$	−4.08905	−0.130156
$988$	0	0
$989$	0.408608	0.0129930
$990$	0	0
$991$	−49.4157	−1.56974	−0.784870	−	0.619660i	$-0.787270\pi$
$991$	−49.4157	−1.56974	−0.784870	+	0.619660i	$0.787270\pi$
$992$	0	0
$993$	13.5264	0.429246
$994$	0	0
$995$	−1.83457	−0.0581599
$996$	0	0
$997$	−18.9248	−0.599354	−0.299677	−	0.954041i	$-0.596879\pi$
$997$	−18.9248	−0.599354	−0.299677	+	0.954041i	$0.596879\pi$
$998$	0	0
$999$	1.32448	0.0419046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6896.2.a.o.1.4		4
4.3	odd	2		431.2.a.e.1.2	✓	4
12.11	even	2		3879.2.a.m.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
431.2.a.e.1.2	✓	4	4.3	odd	2
3879.2.a.m.1.3		4	12.11	even	2
6896.2.a.o.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6896 = 2^{4} \cdot 431 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6896.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35567 q^{3} +0.355674 q^{5} -2.19353 q^{7} +2.54920 q^{9} -1.31313 q^{11} -0.219819 q^{13} +0.837853 q^{15} -4.72333 q^{17} +3.76902 q^{19} -5.16724 q^{21} +0.150986 q^{23} -4.87350 q^{25} -1.06193 q^{27} -1.01939 q^{29} +4.64433 q^{31} -3.09331 q^{33} -0.780181 q^{35} -1.24723 q^{37} -0.517822 q^{39} +0.961212 q^{41} +2.70626 q^{43} +0.906685 q^{45} +0.791342 q^{47} -2.18844 q^{49} -11.1266 q^{51} -4.41446 q^{53} -0.467048 q^{55} +8.87858 q^{57} +4.46097 q^{59} -3.10155 q^{61} -5.59174 q^{63} -0.0781839 q^{65} -9.44584 q^{67} +0.355674 q^{69} -2.60980 q^{71} -10.8065 q^{73} -11.4804 q^{75} +2.88039 q^{77} +4.27546 q^{79} -10.1492 q^{81} -5.53916 q^{83} -1.67997 q^{85} -2.40136 q^{87} -14.6037 q^{89} +0.482178 q^{91} +10.9405 q^{93} +1.34054 q^{95} -9.16806 q^{97} -3.34744 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 5 q^{5} - 2 q^{7} - 3 q^{9} - q^{11} - 5 q^{13} + 3 q^{15} + 2 q^{17} + 6 q^{19} - 3 q^{21} - 4 q^{23} - 7 q^{25} - 3 q^{27} - 5 q^{29} + 25 q^{31} - 4 q^{33} + q^{35} - q^{37} + 4 q^{39}+ \cdots - 11 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 5 * q^5 - 2 * q^7 - 3 * q^9 - q^11 - 5 * q^13 + 3 * q^15 + 2 * q^17 + 6 * q^19 - 3 * q^21 - 4 * q^23 - 7 * q^25 - 3 * q^27 - 5 * q^29 + 25 * q^31 - 4 * q^33 + q^35 - q^37 + 4 * q^39 + 2 * q^41 + 16 * q^43 + 12 * q^45 + 4 * q^47 - 20 * q^49 + 3 * q^51 + 4 * q^53 - 2 * q^55 + 5 * q^57 - 5 * q^59 - 7 * q^63 + 14 * q^65 - 9 * q^67 - 5 * q^69 - 10 * q^71 - 50 * q^73 - 24 * q^75 + 9 * q^77 - 8 * q^79 - 8 * q^81 + 15 * q^83 - q^85 - 15 * q^87 - 35 * q^89 + 8 * q^91 + 12 * q^93 - 7 * q^95 - 6 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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