LMFDB - Embedded newform 7744.2.a.ds.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7744,2,Mod(1,7744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7744, 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("7744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7744 = 2^{6} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7744.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:	$61.8361513253$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48718$ of defining polynomial
Character	$\chi$	$=$	7744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.919131 q^{3} +4.02435 q^{5} -3.72325 q^{7} -2.15520 q^{9} +1.04998 q^{13} -3.69890 q^{15} +0.944761 q^{17} -2.38197 q^{19} +3.42216 q^{21} +1.73830 q^{23} +11.1954 q^{25} +4.73830 q^{27} -2.74888 q^{29} -4.78828 q^{31} -14.9837 q^{35} -2.11501 q^{37} -0.965069 q^{39} +9.12957 q^{41} -0.431946 q^{43} -8.67327 q^{45} -6.32545 q^{47} +6.86261 q^{49} -0.868359 q^{51} -0.316146 q^{53} +2.18934 q^{57} +8.09017 q^{59} -13.6326 q^{61} +8.02435 q^{63} +4.22549 q^{65} +5.68178 q^{67} -1.59773 q^{69} +12.8687 q^{71} +3.52132 q^{73} -10.2900 q^{75} -8.83698 q^{79} +2.11048 q^{81} -14.6667 q^{83} +3.80205 q^{85} +2.52658 q^{87} +2.43195 q^{89} -3.90934 q^{91} +4.40106 q^{93} -9.58586 q^{95} +1.48193 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - q^{5} + q^{7} + 6 q^{9} + q^{13} - 16 q^{15} + 12 q^{17} - 14 q^{19} - q^{21} + 2 q^{23} + 11 q^{25} + 14 q^{27} - 9 q^{29} - 11 q^{31} - 18 q^{35} - 13 q^{37} - 18 q^{39} + 8 q^{41}+ \cdots + 4 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 - q^5 + q^7 + 6 * q^9 + q^13 - 16 * q^15 + 12 * q^17 - 14 * q^19 - q^21 + 2 * q^23 + 11 * q^25 + 14 * q^27 - 9 * q^29 - 11 * q^31 - 18 * q^35 - 13 * q^37 - 18 * q^39 + 8 * q^41 - 3 * q^43 - 22 * q^45 - 7 * q^47 - q^49 - 14 * q^51 - 11 * q^53 - 7 * q^57 + 10 * q^59 - 17 * q^61 + 15 * q^63 + 5 * q^65 - 5 * q^67 + 4 * q^69 + 5 * q^71 + 6 * q^73 + 7 * q^75 + 7 * q^79 - 8 * q^81 - 20 * q^83 - 13 * q^85 - 3 * q^87 + 11 * q^89 + 6 * q^91 + 10 * q^93 + 6 * q^95 + 4 * q^97

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
$3$	−0.919131	−0.530660	−0.265330	−	0.964158i	$-0.585481\pi$
$3$	−0.919131	−0.530660	−0.265330	+	0.964158i	$0.585481\pi$
$4$	0	0
$5$	4.02435	1.79974	0.899872	−	0.436154i	$-0.143660\pi$
$5$	4.02435	1.79974	0.899872	+	0.436154i	$0.143660\pi$
$6$	0	0
$7$	−3.72325	−1.40726	−0.703629	−	0.710568i	$-0.748438\pi$
$7$	−3.72325	−1.40726	−0.703629	+	0.710568i	$0.748438\pi$
$8$	0	0
$9$	−2.15520	−0.718400
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.04998	0.291212	0.145606	−	0.989343i	$-0.453487\pi$
$13$	1.04998	0.291212	0.145606	+	0.989343i	$0.453487\pi$
$14$	0	0
$15$	−3.69890	−0.955053
$16$	0	0
$17$	0.944761	0.229138	0.114569	−	0.993415i	$-0.463451\pi$
$17$	0.944761	0.229138	0.114569	+	0.993415i	$0.463451\pi$
$18$	0	0
$19$	−2.38197	−0.546460	−0.273230	−	0.961949i	$-0.588092\pi$
$19$	−2.38197	−0.546460	−0.273230	+	0.961949i	$0.588092\pi$
$20$	0	0
$21$	3.42216	0.746776
$22$	0	0
$23$	1.73830	0.362461	0.181230	−	0.983441i	$-0.441992\pi$
$23$	1.73830	0.362461	0.181230	+	0.983441i	$0.441992\pi$
$24$	0	0
$25$	11.1954	2.23908
$26$	0	0
$27$	4.73830	0.911887
$28$	0	0
$29$	−2.74888	−0.510455	−0.255227	−	0.966881i	$-0.582150\pi$
$29$	−2.74888	−0.510455	−0.255227	+	0.966881i	$0.582150\pi$
$30$	0	0
$31$	−4.78828	−0.860001	−0.430000	−	0.902829i	$-0.641487\pi$
$31$	−4.78828	−0.860001	−0.430000	+	0.902829i	$0.641487\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−14.9837	−2.53270
$36$	0	0
$37$	−2.11501	−0.347705	−0.173853	−	0.984772i	$-0.555622\pi$
$37$	−2.11501	−0.347705	−0.173853	+	0.984772i	$0.555622\pi$
$38$	0	0
$39$	−0.965069	−0.154535
$40$	0	0
$41$	9.12957	1.42580	0.712900	−	0.701266i	$-0.247382\pi$
$41$	9.12957	1.42580	0.712900	+	0.701266i	$0.247382\pi$
$42$	0	0
$43$	−0.431946	−0.0658711	−0.0329356	−	0.999457i	$-0.510486\pi$
$43$	−0.431946	−0.0658711	−0.0329356	+	0.999457i	$0.510486\pi$
$44$	0	0
$45$	−8.67327	−1.29294
$46$	0	0
$47$	−6.32545	−0.922661	−0.461331	−	0.887228i	$-0.652628\pi$
$47$	−6.32545	−0.922661	−0.461331	+	0.887228i	$0.652628\pi$
$48$	0	0
$49$	6.86261	0.980373
$50$	0	0
$51$	−0.868359	−0.121595
$52$	0	0
$53$	−0.316146	−0.0434259	−0.0217130	−	0.999764i	$-0.506912\pi$
$53$	−0.316146	−0.0434259	−0.0217130	+	0.999764i	$0.506912\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.18934	0.289985
$58$	0	0
$59$	8.09017	1.05325	0.526625	−	0.850098i	$-0.323457\pi$
$59$	8.09017	1.05325	0.526625	+	0.850098i	$0.323457\pi$
$60$	0	0
$61$	−13.6326	−1.74547	−0.872737	−	0.488190i	$-0.837657\pi$
$61$	−13.6326	−1.74547	−0.872737	+	0.488190i	$0.837657\pi$
$62$	0	0
$63$	8.02435	1.01097
$64$	0	0
$65$	4.22549	0.524107
$66$	0	0
$67$	5.68178	0.694140	0.347070	−	0.937839i	$-0.387177\pi$
$67$	5.68178	0.694140	0.347070	+	0.937839i	$0.387177\pi$
$68$	0	0
$69$	−1.59773	−0.192344
$70$	0	0
$71$	12.8687	1.52723	0.763615	−	0.645672i	$-0.223423\pi$
$71$	12.8687	1.52723	0.763615	+	0.645672i	$0.223423\pi$
$72$	0	0
$73$	3.52132	0.412140	0.206070	−	0.978537i	$-0.433933\pi$
$73$	3.52132	0.412140	0.206070	+	0.978537i	$0.433933\pi$
$74$	0	0
$75$	−10.2900	−1.18819
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.83698	−0.994238	−0.497119	−	0.867682i	$-0.665609\pi$
$79$	−8.83698	−0.994238	−0.497119	+	0.867682i	$0.665609\pi$
$80$	0	0
$81$	2.11048	0.234498
$82$	0	0
$83$	−14.6667	−1.60988	−0.804942	−	0.593354i	$-0.797803\pi$
$83$	−14.6667	−1.60988	−0.804942	+	0.593354i	$0.797803\pi$
$84$	0	0
$85$	3.80205	0.412390
$86$	0	0
$87$	2.52658	0.270878
$88$	0	0
$89$	2.43195	0.257786	0.128893	−	0.991659i	$-0.458858\pi$
$89$	2.43195	0.257786	0.128893	+	0.991659i	$0.458858\pi$
$90$	0	0
$91$	−3.90934	−0.409810
$92$	0	0
$93$	4.40106	0.456368
$94$	0	0
$95$	−9.58586	−0.983489
$96$	0	0
$97$	1.48193	0.150467	0.0752334	−	0.997166i	$-0.476030\pi$
$97$	1.48193	0.150467	0.0752334	+	0.997166i	$0.476030\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.34783	−0.432625	−0.216312	−	0.976324i	$-0.569403\pi$
$101$	−4.34783	−0.432625	−0.216312	+	0.976324i	$0.569403\pi$
$102$	0	0
$103$	−11.0894	−1.09267	−0.546334	−	0.837567i	$-0.683977\pi$
$103$	−11.0894	−1.09267	−0.546334	+	0.837567i	$0.683977\pi$
$104$	0	0
$105$	13.7720	1.34400
$106$	0	0
$107$	−9.55301	−0.923524	−0.461762	−	0.887004i	$-0.652783\pi$
$107$	−9.55301	−0.923524	−0.461762	+	0.887004i	$0.652783\pi$
$108$	0	0
$109$	4.14866	0.397369	0.198685	−	0.980063i	$-0.436333\pi$
$109$	4.14866	0.397369	0.198685	+	0.980063i	$0.436333\pi$
$110$	0	0
$111$	1.94397	0.184513
$112$	0	0
$113$	−9.29456	−0.874358	−0.437179	−	0.899374i	$-0.644022\pi$
$113$	−9.29456	−0.874358	−0.437179	+	0.899374i	$0.644022\pi$
$114$	0	0
$115$	6.99553	0.652337
$116$	0	0
$117$	−2.26292	−0.209207
$118$	0	0
$119$	−3.51758	−0.322456
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−8.39127	−0.756615
$124$	0	0
$125$	24.9324	2.23002
$126$	0	0
$127$	7.95607	0.705987	0.352994	−	0.935626i	$-0.385164\pi$
$127$	7.95607	0.705987	0.352994	+	0.935626i	$0.385164\pi$
$128$	0	0
$129$	0.397015	0.0349552
$130$	0	0
$131$	−12.2670	−1.07177	−0.535885	−	0.844291i	$-0.680022\pi$
$131$	−12.2670	−1.07177	−0.535885	+	0.844291i	$0.680022\pi$
$132$	0	0
$133$	8.86866	0.769010
$134$	0	0
$135$	19.0686	1.64116
$136$	0	0
$137$	−2.51807	−0.215134	−0.107567	−	0.994198i	$-0.534306\pi$
$137$	−2.51807	−0.215134	−0.107567	+	0.994198i	$0.534306\pi$
$138$	0	0
$139$	−14.0954	−1.19556	−0.597779	−	0.801661i	$-0.703950\pi$
$139$	−14.0954	−1.19556	−0.597779	+	0.801661i	$0.703950\pi$
$140$	0	0
$141$	5.81391	0.489620
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−11.0625	−0.918688
$146$	0	0
$147$	−6.30764	−0.520245
$148$	0	0
$149$	10.6752	0.874550	0.437275	−	0.899328i	$-0.355944\pi$
$149$	10.6752	0.874550	0.437275	+	0.899328i	$0.355944\pi$
$150$	0	0
$151$	−20.2441	−1.64744	−0.823720	−	0.566996i	$-0.808105\pi$
$151$	−20.2441	−1.64744	−0.823720	+	0.566996i	$0.808105\pi$
$152$	0	0
$153$	−2.03615	−0.164613
$154$	0	0
$155$	−19.2697	−1.54778
$156$	0	0
$157$	−19.4103	−1.54911	−0.774555	−	0.632507i	$-0.782026\pi$
$157$	−19.4103	−1.54911	−0.774555	+	0.632507i	$0.782026\pi$
$158$	0	0
$159$	0.290579	0.0230444
$160$	0	0
$161$	−6.47214	−0.510076
$162$	0	0
$163$	18.8739	1.47832	0.739160	−	0.673530i	$-0.235223\pi$
$163$	18.8739	1.47832	0.739160	+	0.673530i	$0.235223\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.7642	−1.37464	−0.687319	−	0.726356i	$-0.741212\pi$
$167$	−17.7642	−1.37464	−0.687319	+	0.726356i	$0.741212\pi$
$168$	0	0
$169$	−11.8975	−0.915196
$170$	0	0
$171$	5.13361	0.392577
$172$	0	0
$173$	−6.35554	−0.483203	−0.241602	−	0.970376i	$-0.577673\pi$
$173$	−6.35554	−0.483203	−0.241602	+	0.970376i	$0.577673\pi$
$174$	0	0
$175$	−41.6833	−3.15096
$176$	0	0
$177$	−7.43592	−0.558918
$178$	0	0
$179$	−1.40760	−0.105209	−0.0526044	−	0.998615i	$-0.516752\pi$
$179$	−1.40760	−0.105209	−0.0526044	+	0.998615i	$0.516752\pi$
$180$	0	0
$181$	−17.3408	−1.28893	−0.644466	−	0.764633i	$-0.722920\pi$
$181$	−17.3408	−1.28893	−0.644466	+	0.764633i	$0.722920\pi$
$182$	0	0
$183$	12.5301	0.926254
$184$	0	0
$185$	−8.51153	−0.625780
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−17.6419	−1.28326
$190$	0	0
$191$	6.98942	0.505737	0.252868	−	0.967501i	$-0.418626\pi$
$191$	6.98942	0.505737	0.252868	+	0.967501i	$0.418626\pi$
$192$	0	0
$193$	−23.3392	−1.67999	−0.839997	−	0.542592i	$-0.817443\pi$
$193$	−23.3392	−1.67999	−0.839997	+	0.542592i	$0.817443\pi$
$194$	0	0
$195$	−3.88377	−0.278123
$196$	0	0
$197$	−10.8142	−0.770481	−0.385240	−	0.922816i	$-0.625881\pi$
$197$	−10.8142	−0.770481	−0.385240	+	0.922816i	$0.625881\pi$
$198$	0	0
$199$	−9.07433	−0.643262	−0.321631	−	0.946865i	$-0.604231\pi$
$199$	−9.07433	−0.643262	−0.321631	+	0.946865i	$0.604231\pi$
$200$	0	0
$201$	−5.22230	−0.368353
$202$	0	0
$203$	10.2348	0.718341
$204$	0	0
$205$	36.7406	2.56607
$206$	0	0
$207$	−3.74639	−0.260392
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2.38643	0.164289	0.0821444	−	0.996620i	$-0.473823\pi$
$211$	2.38643	0.164289	0.0821444	+	0.996620i	$0.473823\pi$
$212$	0	0
$213$	−11.8280	−0.810440
$214$	0	0
$215$	−1.73830	−0.118551
$216$	0	0
$217$	17.8280	1.21024
$218$	0	0
$219$	−3.23656	−0.218706
$220$	0	0
$221$	0.991980	0.0667278
$222$	0	0
$223$	15.3802	1.02993	0.514967	−	0.857210i	$-0.327804\pi$
$223$	15.3802	1.02993	0.514967	+	0.857210i	$0.327804\pi$
$224$	0	0
$225$	−24.1283	−1.60855
$226$	0	0
$227$	12.5411	0.832385	0.416192	−	0.909277i	$-0.363364\pi$
$227$	12.5411	0.832385	0.416192	+	0.909277i	$0.363364\pi$
$228$	0	0
$229$	−13.3408	−0.881585	−0.440792	−	0.897609i	$-0.645302\pi$
$229$	−13.3408	−0.881585	−0.440792	+	0.897609i	$0.645302\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.4794	1.53818	0.769092	−	0.639138i	$-0.220709\pi$
$233$	23.4794	1.53818	0.769092	+	0.639138i	$0.220709\pi$
$234$	0	0
$235$	−25.4558	−1.66055
$236$	0	0
$237$	8.12234	0.527603
$238$	0	0
$239$	3.73377	0.241518	0.120759	−	0.992682i	$-0.461467\pi$
$239$	3.73377	0.241518	0.120759	+	0.992682i	$0.461467\pi$
$240$	0	0
$241$	−15.9621	−1.02821	−0.514104	−	0.857728i	$-0.671875\pi$
$241$	−15.9621	−1.02821	−0.514104	+	0.857728i	$0.671875\pi$
$242$	0	0
$243$	−16.1547	−1.03633
$244$	0	0
$245$	27.6175	1.76442
$246$	0	0
$247$	−2.50102	−0.159136
$248$	0	0
$249$	13.4806	0.854301
$250$	0	0
$251$	−11.0817	−0.699468	−0.349734	−	0.936849i	$-0.613728\pi$
$251$	−11.0817	−0.699468	−0.349734	+	0.936849i	$0.613728\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.49458	−0.218839
$256$	0	0
$257$	14.9848	0.934729	0.467365	−	0.884065i	$-0.345204\pi$
$257$	14.9848	0.934729	0.467365	+	0.884065i	$0.345204\pi$
$258$	0	0
$259$	7.87471	0.489311
$260$	0	0
$261$	5.92439	0.366711
$262$	0	0
$263$	8.96129	0.552577	0.276288	−	0.961075i	$-0.410896\pi$
$263$	8.96129	0.552577	0.276288	+	0.961075i	$0.410896\pi$
$264$	0	0
$265$	−1.27228	−0.0781555
$266$	0	0
$267$	−2.23528	−0.136797
$268$	0	0
$269$	−1.04998	−0.0640184	−0.0320092	−	0.999488i	$-0.510191\pi$
$269$	−1.04998	−0.0640184	−0.0320092	+	0.999488i	$0.510191\pi$
$270$	0	0
$271$	18.1653	1.10346	0.551731	−	0.834022i	$-0.313967\pi$
$271$	18.1653	1.10346	0.551731	+	0.834022i	$0.313967\pi$
$272$	0	0
$273$	3.59320	0.217470
$274$	0	0
$275$	0	0
$276$	0	0
$277$	19.9430	1.19826	0.599129	−	0.800652i	$-0.295514\pi$
$277$	19.9430	1.19826	0.599129	+	0.800652i	$0.295514\pi$
$278$	0	0
$279$	10.3197	0.617824
$280$	0	0
$281$	21.0072	1.25319	0.626593	−	0.779347i	$-0.284449\pi$
$281$	21.0072	1.25319	0.626593	+	0.779347i	$0.284449\pi$
$282$	0	0
$283$	−6.54640	−0.389143	−0.194572	−	0.980888i	$-0.562332\pi$
$283$	−6.54640	−0.389143	−0.194572	+	0.980888i	$0.562332\pi$
$284$	0	0
$285$	8.81066	0.521899
$286$	0	0
$287$	−33.9917	−2.00647
$288$	0	0
$289$	−16.1074	−0.947496
$290$	0	0
$291$	−1.36208	−0.0798468
$292$	0	0
$293$	−16.0231	−0.936081	−0.468041	−	0.883707i	$-0.655040\pi$
$293$	−16.0231	−0.936081	−0.468041	+	0.883707i	$0.655040\pi$
$294$	0	0
$295$	32.5577	1.89558
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.82518	0.105553
$300$	0	0
$301$	1.60824	0.0926976
$302$	0	0
$303$	3.99622	0.229577
$304$	0	0
$305$	−54.8623	−3.14141
$306$	0	0
$307$	−10.8016	−0.616478	−0.308239	−	0.951309i	$-0.599740\pi$
$307$	−10.8016	−0.616478	−0.308239	+	0.951309i	$0.599740\pi$
$308$	0	0
$309$	10.1926	0.579836
$310$	0	0
$311$	−3.18934	−0.180851	−0.0904254	−	0.995903i	$-0.528823\pi$
$311$	−3.18934	−0.180851	−0.0904254	+	0.995903i	$0.528823\pi$
$312$	0	0
$313$	15.0552	0.850972	0.425486	−	0.904965i	$-0.360103\pi$
$313$	15.0552	0.850972	0.425486	+	0.904965i	$0.360103\pi$
$314$	0	0
$315$	32.2928	1.81949
$316$	0	0
$317$	−13.2499	−0.744189	−0.372094	−	0.928195i	$-0.621360\pi$
$317$	−13.2499	−0.744189	−0.372094	+	0.928195i	$0.621360\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	8.78046	0.490078
$322$	0	0
$323$	−2.25039	−0.125215
$324$	0	0
$325$	11.7549	0.652046
$326$	0	0
$327$	−3.81316	−0.210868
$328$	0	0
$329$	23.5512	1.29842
$330$	0	0
$331$	−33.5540	−1.84429	−0.922147	−	0.386839i	$-0.873567\pi$
$331$	−33.5540	−1.84429	−0.922147	+	0.386839i	$0.873567\pi$
$332$	0	0
$333$	4.55826	0.249791
$334$	0	0
$335$	22.8655	1.24927
$336$	0	0
$337$	−6.90536	−0.376159	−0.188080	−	0.982154i	$-0.560226\pi$
$337$	−6.90536	−0.376159	−0.188080	+	0.982154i	$0.560226\pi$
$338$	0	0
$339$	8.54291	0.463987
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0.511534	0.0276203
$344$	0	0
$345$	−6.42981	−0.346169
$346$	0	0
$347$	20.6991	1.11119	0.555593	−	0.831454i	$-0.312491\pi$
$347$	20.6991	1.11119	0.555593	+	0.831454i	$0.312491\pi$
$348$	0	0
$349$	−11.0567	−0.591853	−0.295926	−	0.955211i	$-0.595628\pi$
$349$	−11.0567	−0.591853	−0.295926	+	0.955211i	$0.595628\pi$
$350$	0	0
$351$	4.97512	0.265552
$352$	0	0
$353$	13.5005	0.718557	0.359279	−	0.933230i	$-0.383023\pi$
$353$	13.5005	0.718557	0.359279	+	0.933230i	$0.383023\pi$
$354$	0	0
$355$	51.7880	2.74862
$356$	0	0
$357$	3.23312	0.171115
$358$	0	0
$359$	33.4263	1.76417	0.882087	−	0.471086i	$-0.156138\pi$
$359$	33.4263	1.76417	0.882087	+	0.471086i	$0.156138\pi$
$360$	0	0
$361$	−13.3262	−0.701381
$362$	0	0
$363$	0	0
$364$	0	0
$365$	14.1710	0.741746
$366$	0	0
$367$	−20.0938	−1.04889	−0.524445	−	0.851444i	$-0.675727\pi$
$367$	−20.0938	−1.04889	−0.524445	+	0.851444i	$0.675727\pi$
$368$	0	0
$369$	−19.6760	−1.02429
$370$	0	0
$371$	1.17709	0.0611115
$372$	0	0
$373$	−26.4379	−1.36890	−0.684451	−	0.729059i	$-0.739958\pi$
$373$	−26.4379	−1.36890	−0.684451	+	0.729059i	$0.739958\pi$
$374$	0	0
$375$	−22.9161	−1.18338
$376$	0	0
$377$	−2.88627	−0.148651
$378$	0	0
$379$	20.2580	1.04058	0.520291	−	0.853989i	$-0.325824\pi$
$379$	20.2580	1.04058	0.520291	+	0.853989i	$0.325824\pi$
$380$	0	0
$381$	−7.31267	−0.374639
$382$	0	0
$383$	4.60953	0.235536	0.117768	−	0.993041i	$-0.462426\pi$
$383$	4.60953	0.235536	0.117768	+	0.993041i	$0.462426\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.930929	0.0473218
$388$	0	0
$389$	−30.8267	−1.56298	−0.781488	−	0.623920i	$-0.785539\pi$
$389$	−30.8267	−1.56298	−0.781488	+	0.623920i	$0.785539\pi$
$390$	0	0
$391$	1.64228	0.0830537
$392$	0	0
$393$	11.2749	0.568745
$394$	0	0
$395$	−35.5631	−1.78937
$396$	0	0
$397$	−15.9382	−0.799916	−0.399958	−	0.916533i	$-0.630975\pi$
$397$	−15.9382	−0.799916	−0.399958	+	0.916533i	$0.630975\pi$
$398$	0	0
$399$	−8.15146	−0.408083
$400$	0	0
$401$	23.0680	1.15196	0.575981	−	0.817463i	$-0.304620\pi$
$401$	23.0680	1.15196	0.575981	+	0.817463i	$0.304620\pi$
$402$	0	0
$403$	−5.02760	−0.250443
$404$	0	0
$405$	8.49330	0.422035
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−14.5347	−0.718693	−0.359347	−	0.933204i	$-0.617000\pi$
$409$	−14.5347	−0.718693	−0.359347	+	0.933204i	$0.617000\pi$
$410$	0	0
$411$	2.31444	0.114163
$412$	0	0
$413$	−30.1217	−1.48219
$414$	0	0
$415$	−59.0241	−2.89738
$416$	0	0
$417$	12.9555	0.634436
$418$	0	0
$419$	7.34710	0.358929	0.179465	−	0.983764i	$-0.442563\pi$
$419$	7.34710	0.358929	0.179465	+	0.983764i	$0.442563\pi$
$420$	0	0
$421$	−1.76801	−0.0861677	−0.0430839	−	0.999071i	$-0.513718\pi$
$421$	−1.76801	−0.0861677	−0.0430839	+	0.999071i	$0.513718\pi$
$422$	0	0
$423$	13.6326	0.662839
$424$	0	0
$425$	10.5770	0.513058
$426$	0	0
$427$	50.7576	2.45633
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.2860	−0.784471	−0.392236	−	0.919865i	$-0.628298\pi$
$431$	−16.2860	−0.784471	−0.392236	+	0.919865i	$0.628298\pi$
$432$	0	0
$433$	−2.24935	−0.108097	−0.0540483	−	0.998538i	$-0.517213\pi$
$433$	−2.24935	−0.108097	−0.0540483	+	0.998538i	$0.517213\pi$
$434$	0	0
$435$	10.1679	0.487511
$436$	0	0
$437$	−4.14058	−0.198071
$438$	0	0
$439$	−8.85337	−0.422549	−0.211274	−	0.977427i	$-0.567761\pi$
$439$	−8.85337	−0.422549	−0.211274	+	0.977427i	$0.567761\pi$
$440$	0	0
$441$	−14.7903	−0.704300
$442$	0	0
$443$	15.9649	0.758517	0.379259	−	0.925291i	$-0.376179\pi$
$443$	15.9649	0.758517	0.379259	+	0.925291i	$0.376179\pi$
$444$	0	0
$445$	9.78700	0.463948
$446$	0	0
$447$	−9.81194	−0.464089
$448$	0	0
$449$	23.7996	1.12317	0.561586	−	0.827418i	$-0.310191\pi$
$449$	23.7996	1.12317	0.561586	+	0.827418i	$0.310191\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	18.6070	0.874231
$454$	0	0
$455$	−15.7326	−0.737553
$456$	0	0
$457$	−29.6427	−1.38663	−0.693313	−	0.720636i	$-0.743850\pi$
$457$	−29.6427	−1.38663	−0.693313	+	0.720636i	$0.743850\pi$
$458$	0	0
$459$	4.47656	0.208948
$460$	0	0
$461$	9.00605	0.419454	0.209727	−	0.977760i	$-0.432743\pi$
$461$	9.00605	0.419454	0.209727	+	0.977760i	$0.432743\pi$
$462$	0	0
$463$	−16.0634	−0.746528	−0.373264	−	0.927725i	$-0.621761\pi$
$463$	−16.0634	−0.746528	−0.373264	+	0.927725i	$0.621761\pi$
$464$	0	0
$465$	17.7114	0.821346
$466$	0	0
$467$	−4.07463	−0.188551	−0.0942757	−	0.995546i	$-0.530054\pi$
$467$	−4.07463	−0.188551	−0.0942757	+	0.995546i	$0.530054\pi$
$468$	0	0
$469$	−21.1547	−0.976834
$470$	0	0
$471$	17.8406	0.822051
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−26.6670	−1.22357
$476$	0	0
$477$	0.681356	0.0311972
$478$	0	0
$479$	0.349410	0.0159649	0.00798247	−	0.999968i	$-0.497459\pi$
$479$	0.349410	0.0159649	0.00798247	+	0.999968i	$0.497459\pi$
$480$	0	0
$481$	−2.22072	−0.101256
$482$	0	0
$483$	5.94874	0.270677
$484$	0	0
$485$	5.96379	0.270802
$486$	0	0
$487$	−39.1502	−1.77406	−0.887032	−	0.461708i	$-0.847237\pi$
$487$	−39.1502	−1.77406	−0.887032	+	0.461708i	$0.847237\pi$
$488$	0	0
$489$	−17.3476	−0.784486
$490$	0	0
$491$	8.78651	0.396530	0.198265	−	0.980148i	$-0.436469\pi$
$491$	8.78651	0.396530	0.198265	+	0.980148i	$0.436469\pi$
$492$	0	0
$493$	−2.59704	−0.116965
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−47.9133	−2.14920
$498$	0	0
$499$	−25.7478	−1.15263	−0.576315	−	0.817228i	$-0.695510\pi$
$499$	−25.7478	−1.15263	−0.576315	+	0.817228i	$0.695510\pi$
$500$	0	0
$501$	16.3277	0.729466
$502$	0	0
$503$	14.8659	0.662836	0.331418	−	0.943484i	$-0.392473\pi$
$503$	14.8659	0.662836	0.331418	+	0.943484i	$0.392473\pi$
$504$	0	0
$505$	−17.4972	−0.778614
$506$	0	0
$507$	10.9354	0.485658
$508$	0	0
$509$	−3.44295	−0.152606	−0.0763031	−	0.997085i	$-0.524312\pi$
$509$	−3.44295	−0.152606	−0.0763031	+	0.997085i	$0.524312\pi$
$510$	0	0
$511$	−13.1108	−0.579987
$512$	0	0
$513$	−11.2865	−0.498310
$514$	0	0
$515$	−44.6275	−1.96652
$516$	0	0
$517$	0	0
$518$	0	0
$519$	5.84158	0.256417
$520$	0	0
$521$	40.2208	1.76211	0.881053	−	0.473017i	$-0.156835\pi$
$521$	40.2208	1.76211	0.881053	+	0.473017i	$0.156835\pi$
$522$	0	0
$523$	34.4269	1.50538	0.752691	−	0.658374i	$-0.228755\pi$
$523$	34.4269	1.50538	0.752691	+	0.658374i	$0.228755\pi$
$524$	0	0
$525$	38.3124	1.67209
$526$	0	0
$527$	−4.52378	−0.197059
$528$	0	0
$529$	−19.9783	−0.868622
$530$	0	0
$531$	−17.4359	−0.756655
$532$	0	0
$533$	9.58586	0.415210
$534$	0	0
$535$	−38.4446	−1.66211
$536$	0	0
$537$	1.29377	0.0558301
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−18.9239	−0.813600	−0.406800	−	0.913517i	$-0.633355\pi$
$541$	−18.9239	−0.813600	−0.406800	+	0.913517i	$0.633355\pi$
$542$	0	0
$543$	15.9385	0.683985
$544$	0	0
$545$	16.6957	0.715163
$546$	0	0
$547$	−32.8385	−1.40407	−0.702036	−	0.712142i	$-0.747725\pi$
$547$	−32.8385	−1.40407	−0.702036	+	0.712142i	$0.747725\pi$
$548$	0	0
$549$	29.3809	1.25395
$550$	0	0
$551$	6.54775	0.278943
$552$	0	0
$553$	32.9023	1.39915
$554$	0	0
$555$	7.82321	0.332077
$556$	0	0
$557$	−32.0612	−1.35848	−0.679238	−	0.733918i	$-0.737690\pi$
$557$	−32.0612	−1.35848	−0.679238	+	0.733918i	$0.737690\pi$
$558$	0	0
$559$	−0.453535	−0.0191825
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.25289	−0.0528029	−0.0264014	−	0.999651i	$-0.508405\pi$
$563$	−1.25289	−0.0528029	−0.0264014	+	0.999651i	$0.508405\pi$
$564$	0	0
$565$	−37.4045	−1.57362
$566$	0	0
$567$	−7.85784	−0.329998
$568$	0	0
$569$	−5.16194	−0.216400	−0.108200	−	0.994129i	$-0.534509\pi$
$569$	−5.16194	−0.216400	−0.108200	+	0.994129i	$0.534509\pi$
$570$	0	0
$571$	−1.46767	−0.0614200	−0.0307100	−	0.999528i	$-0.509777\pi$
$571$	−1.46767	−0.0614200	−0.0307100	+	0.999528i	$0.509777\pi$
$572$	0	0
$573$	−6.42419	−0.268374
$574$	0	0
$575$	19.4610	0.811578
$576$	0	0
$577$	−19.2013	−0.799362	−0.399681	−	0.916654i	$-0.630879\pi$
$577$	−19.2013	−0.799362	−0.399681	+	0.916654i	$0.630879\pi$
$578$	0	0
$579$	21.4518	0.891506
$580$	0	0
$581$	54.6080	2.26552
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−9.10676	−0.376518
$586$	0	0
$587$	−5.71149	−0.235739	−0.117869	−	0.993029i	$-0.537606\pi$
$587$	−5.71149	−0.235739	−0.117869	+	0.993029i	$0.537606\pi$
$588$	0	0
$589$	11.4055	0.469956
$590$	0	0
$591$	9.93968	0.408864
$592$	0	0
$593$	−36.3301	−1.49190	−0.745949	−	0.666003i	$-0.768004\pi$
$593$	−36.3301	−1.49190	−0.745949	+	0.666003i	$0.768004\pi$
$594$	0	0
$595$	−14.1560	−0.580339
$596$	0	0
$597$	8.34050	0.341354
$598$	0	0
$599$	41.2464	1.68528	0.842640	−	0.538477i	$-0.181000\pi$
$599$	41.2464	1.68528	0.842640	+	0.538477i	$0.181000\pi$
$600$	0	0
$601$	−21.4570	−0.875249	−0.437624	−	0.899158i	$-0.644180\pi$
$601$	−21.4570	−0.875249	−0.437624	+	0.899158i	$0.644180\pi$
$602$	0	0
$603$	−12.2454	−0.498670
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.72447	0.151172	0.0755858	−	0.997139i	$-0.475917\pi$
$607$	3.72447	0.151172	0.0755858	+	0.997139i	$0.475917\pi$
$608$	0	0
$609$	−9.40711	−0.381195
$610$	0	0
$611$	−6.64159	−0.268690
$612$	0	0
$613$	−14.2457	−0.575377	−0.287689	−	0.957724i	$-0.592887\pi$
$613$	−14.2457	−0.575377	−0.287689	+	0.957724i	$0.592887\pi$
$614$	0	0
$615$	−33.7694	−1.36171
$616$	0	0
$617$	33.2403	1.33821	0.669103	−	0.743170i	$-0.266678\pi$
$617$	33.2403	1.33821	0.669103	+	0.743170i	$0.266678\pi$
$618$	0	0
$619$	−15.6642	−0.629596	−0.314798	−	0.949159i	$-0.601937\pi$
$619$	−15.6642	−0.629596	−0.314798	+	0.949159i	$0.601937\pi$
$620$	0	0
$621$	8.23660	0.330523
$622$	0	0
$623$	−9.05475	−0.362771
$624$	0	0
$625$	44.3598	1.77439
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.99818	−0.0796726
$630$	0	0
$631$	−7.63381	−0.303897	−0.151949	−	0.988388i	$-0.548555\pi$
$631$	−7.63381	−0.303897	−0.151949	+	0.988388i	$0.548555\pi$
$632$	0	0
$633$	−2.19344	−0.0871816
$634$	0	0
$635$	32.0180	1.27060
$636$	0	0
$637$	7.20560	0.285496
$638$	0	0
$639$	−27.7345	−1.09716
$640$	0	0
$641$	−18.9809	−0.749701	−0.374850	−	0.927085i	$-0.622306\pi$
$641$	−18.9809	−0.749701	−0.374850	+	0.927085i	$0.622306\pi$
$642$	0	0
$643$	16.4749	0.649707	0.324853	−	0.945764i	$-0.394685\pi$
$643$	16.4749	0.649707	0.324853	+	0.945764i	$0.394685\pi$
$644$	0	0
$645$	1.59773	0.0629104
$646$	0	0
$647$	−39.8210	−1.56553	−0.782763	−	0.622320i	$-0.786190\pi$
$647$	−39.8210	−1.56553	−0.782763	+	0.622320i	$0.786190\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−16.3862	−0.642228
$652$	0	0
$653$	24.6544	0.964803	0.482401	−	0.875950i	$-0.339765\pi$
$653$	24.6544	0.964803	0.482401	+	0.875950i	$0.339765\pi$
$654$	0	0
$655$	−49.3665	−1.92891
$656$	0	0
$657$	−7.58915	−0.296081
$658$	0	0
$659$	11.3599	0.442518	0.221259	−	0.975215i	$-0.428983\pi$
$659$	11.3599	0.442518	0.221259	+	0.975215i	$0.428983\pi$
$660$	0	0
$661$	5.05889	0.196768	0.0983841	−	0.995149i	$-0.468633\pi$
$661$	5.05889	0.196768	0.0983841	+	0.995149i	$0.468633\pi$
$662$	0	0
$663$	−0.911760	−0.0354098
$664$	0	0
$665$	35.6906	1.38402
$666$	0	0
$667$	−4.77839	−0.185020
$668$	0	0
$669$	−14.1364	−0.546545
$670$	0	0
$671$	0	0
$672$	0	0
$673$	40.3560	1.55561	0.777806	−	0.628505i	$-0.216333\pi$
$673$	40.3560	1.55561	0.777806	+	0.628505i	$0.216333\pi$
$674$	0	0
$675$	53.0471	2.04178
$676$	0	0
$677$	31.4789	1.20983	0.604917	−	0.796289i	$-0.293206\pi$
$677$	31.4789	1.20983	0.604917	+	0.796289i	$0.293206\pi$
$678$	0	0
$679$	−5.51758	−0.211745
$680$	0	0
$681$	−11.5269	−0.441714
$682$	0	0
$683$	40.0582	1.53279	0.766393	−	0.642372i	$-0.222050\pi$
$683$	40.0582	1.53279	0.766393	+	0.642372i	$0.222050\pi$
$684$	0	0
$685$	−10.1336	−0.387185
$686$	0	0
$687$	12.2619	0.467822
$688$	0	0
$689$	−0.331946	−0.0126462
$690$	0	0
$691$	−12.9739	−0.493550	−0.246775	−	0.969073i	$-0.579371\pi$
$691$	−12.9739	−0.493550	−0.246775	+	0.969073i	$0.579371\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−56.7249	−2.15170
$696$	0	0
$697$	8.62526	0.326705
$698$	0	0
$699$	−21.5806	−0.816253
$700$	0	0
$701$	13.1844	0.497969	0.248984	−	0.968508i	$-0.419903\pi$
$701$	13.1844	0.497969	0.248984	+	0.968508i	$0.419903\pi$
$702$	0	0
$703$	5.03788	0.190007
$704$	0	0
$705$	23.3972	0.881190
$706$	0	0
$707$	16.1881	0.608815
$708$	0	0
$709$	−16.3437	−0.613800	−0.306900	−	0.951742i	$-0.599292\pi$
$709$	−16.3437	−0.613800	−0.306900	+	0.951742i	$0.599292\pi$
$710$	0	0
$711$	19.0454	0.714260
$712$	0	0
$713$	−8.32348	−0.311717
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.43182	−0.128164
$718$	0	0
$719$	5.60213	0.208924	0.104462	−	0.994529i	$-0.466688\pi$
$719$	5.60213	0.208924	0.104462	+	0.994529i	$0.466688\pi$
$720$	0	0
$721$	41.2886	1.53767
$722$	0	0
$723$	14.6712	0.545629
$724$	0	0
$725$	−30.7748	−1.14295
$726$	0	0
$727$	29.1822	1.08231	0.541155	−	0.840923i	$-0.317987\pi$
$727$	29.1822	1.08231	0.541155	+	0.840923i	$0.317987\pi$
$728$	0	0
$729$	8.51686	0.315439
$730$	0	0
$731$	−0.408086	−0.0150936
$732$	0	0
$733$	24.7911	0.915680	0.457840	−	0.889035i	$-0.348623\pi$
$733$	24.7911	0.915680	0.457840	+	0.889035i	$0.348623\pi$
$734$	0	0
$735$	−25.3841	−0.936308
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−26.7504	−0.984028	−0.492014	−	0.870587i	$-0.663739\pi$
$739$	−26.7504	−0.984028	−0.492014	+	0.870587i	$0.663739\pi$
$740$	0	0
$741$	2.29876	0.0844471
$742$	0	0
$743$	36.7239	1.34727	0.673635	−	0.739064i	$-0.264732\pi$
$743$	36.7239	1.34727	0.673635	+	0.739064i	$0.264732\pi$
$744$	0	0
$745$	42.9609	1.57397
$746$	0	0
$747$	31.6097	1.15654
$748$	0	0
$749$	35.5683	1.29964
$750$	0	0
$751$	22.8122	0.832431	0.416215	−	0.909266i	$-0.363356\pi$
$751$	22.8122	0.832431	0.416215	+	0.909266i	$0.363356\pi$
$752$	0	0
$753$	10.1855	0.371180
$754$	0	0
$755$	−81.4693	−2.96497
$756$	0	0
$757$	26.7174	0.971062	0.485531	−	0.874219i	$-0.338626\pi$
$757$	26.7174	0.971062	0.485531	+	0.874219i	$0.338626\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−21.7222	−0.787428	−0.393714	−	0.919233i	$-0.628810\pi$
$761$	−21.7222	−0.787428	−0.393714	+	0.919233i	$0.628810\pi$
$762$	0	0
$763$	−15.4465	−0.559201
$764$	0	0
$765$	−8.19417	−0.296261
$766$	0	0
$767$	8.49452	0.306719
$768$	0	0
$769$	53.7688	1.93895	0.969476	−	0.245185i	$-0.0788488\pi$
$769$	53.7688	1.93895	0.969476	+	0.245185i	$0.0788488\pi$
$770$	0	0
$771$	−13.7730	−0.496024
$772$	0	0
$773$	−9.73219	−0.350042	−0.175021	−	0.984565i	$-0.555999\pi$
$773$	−9.73219	−0.350042	−0.175021	+	0.984565i	$0.555999\pi$
$774$	0	0
$775$	−53.6067	−1.92561
$776$	0	0
$777$	−7.23789	−0.259658
$778$	0	0
$779$	−21.7463	−0.779143
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−13.0250	−0.465477
$784$	0	0
$785$	−78.1138	−2.78800
$786$	0	0
$787$	−38.2155	−1.36223	−0.681117	−	0.732174i	$-0.738506\pi$
$787$	−38.2155	−1.36223	−0.681117	+	0.732174i	$0.738506\pi$
$788$	0	0
$789$	−8.23660	−0.293231
$790$	0	0
$791$	34.6060	1.23045
$792$	0	0
$793$	−14.3139	−0.508303
$794$	0	0
$795$	1.16939	0.0414741
$796$	0	0
$797$	−21.7617	−0.770838	−0.385419	−	0.922742i	$-0.625943\pi$
$797$	−21.7617	−0.770838	−0.385419	+	0.922742i	$0.625943\pi$
$798$	0	0
$799$	−5.97604	−0.211417
$800$	0	0
$801$	−5.24133	−0.185193
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−26.0461	−0.918006
$806$	0	0
$807$	0.965069	0.0339720
$808$	0	0
$809$	13.6681	0.480544	0.240272	−	0.970706i	$-0.422763\pi$
$809$	13.6681	0.480544	0.240272	+	0.970706i	$0.422763\pi$
$810$	0	0
$811$	−35.4756	−1.24572	−0.622858	−	0.782335i	$-0.714029\pi$
$811$	−35.4756	−1.24572	−0.622858	+	0.782335i	$0.714029\pi$
$812$	0	0
$813$	−16.6963	−0.585564
$814$	0	0
$815$	75.9553	2.66060
$816$	0	0
$817$	1.02888	0.0359960
$818$	0	0
$819$	8.42541	0.294408
$820$	0	0
$821$	−14.9658	−0.522311	−0.261155	−	0.965297i	$-0.584103\pi$
$821$	−14.9658	−0.522311	−0.261155	+	0.965297i	$0.584103\pi$
$822$	0	0
$823$	−12.2152	−0.425795	−0.212898	−	0.977075i	$-0.568290\pi$
$823$	−12.2152	−0.425795	−0.212898	+	0.977075i	$0.568290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.9035	−0.483472	−0.241736	−	0.970342i	$-0.577717\pi$
$827$	−13.9035	−0.483472	−0.241736	+	0.970342i	$0.577717\pi$
$828$	0	0
$829$	12.3466	0.428815	0.214408	−	0.976744i	$-0.431218\pi$
$829$	12.3466	0.428815	0.214408	+	0.976744i	$0.431218\pi$
$830$	0	0
$831$	−18.3302	−0.635868
$832$	0	0
$833$	6.48353	0.224641
$834$	0	0
$835$	−71.4895	−2.47400
$836$	0	0
$837$	−22.6883	−0.784223
$838$	0	0
$839$	−20.3651	−0.703081	−0.351540	−	0.936173i	$-0.614342\pi$
$839$	−20.3651	−0.703081	−0.351540	+	0.936173i	$0.614342\pi$
$840$	0	0
$841$	−21.4436	−0.739436
$842$	0	0
$843$	−19.3084	−0.665016
$844$	0	0
$845$	−47.8799	−1.64712
$846$	0	0
$847$	0	0
$848$	0	0
$849$	6.01700	0.206503
$850$	0	0
$851$	−3.67652	−0.126030
$852$	0	0
$853$	−52.2362	−1.78853	−0.894266	−	0.447536i	$-0.852302\pi$
$853$	−52.2362	−1.78853	−0.894266	+	0.447536i	$0.852302\pi$
$854$	0	0
$855$	20.6594	0.706538
$856$	0	0
$857$	38.4691	1.31408	0.657041	−	0.753855i	$-0.271808\pi$
$857$	38.4691	1.31408	0.657041	+	0.753855i	$0.271808\pi$
$858$	0	0
$859$	33.3955	1.13944	0.569720	−	0.821839i	$-0.307052\pi$
$859$	33.3955	1.13944	0.569720	+	0.821839i	$0.307052\pi$
$860$	0	0
$861$	31.2428	1.06475
$862$	0	0
$863$	15.1147	0.514511	0.257255	−	0.966343i	$-0.417182\pi$
$863$	15.1147	0.514511	0.257255	+	0.966343i	$0.417182\pi$
$864$	0	0
$865$	−25.5769	−0.869642
$866$	0	0
$867$	14.8048	0.502798
$868$	0	0
$869$	0	0
$870$	0	0
$871$	5.96576	0.202142
$872$	0	0
$873$	−3.19384	−0.108095
$874$	0	0
$875$	−92.8297	−3.13822
$876$	0	0
$877$	15.9971	0.540182	0.270091	−	0.962835i	$-0.412946\pi$
$877$	15.9971	0.540182	0.270091	+	0.962835i	$0.412946\pi$
$878$	0	0
$879$	14.7274	0.496741
$880$	0	0
$881$	−43.0908	−1.45177	−0.725883	−	0.687818i	$-0.758569\pi$
$881$	−43.0908	−1.45177	−0.725883	+	0.687818i	$0.758569\pi$
$882$	0	0
$883$	−44.9734	−1.51347	−0.756737	−	0.653720i	$-0.773208\pi$
$883$	−44.9734	−1.51347	−0.756737	+	0.653720i	$0.773208\pi$
$884$	0	0
$885$	−29.9248	−1.00591
$886$	0	0
$887$	−45.3094	−1.52134	−0.760670	−	0.649139i	$-0.775129\pi$
$887$	−45.3094	−1.52134	−0.760670	+	0.649139i	$0.775129\pi$
$888$	0	0
$889$	−29.6225	−0.993505
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.0670	0.504198
$894$	0	0
$895$	−5.66466	−0.189349
$896$	0	0
$897$	−1.67758	−0.0560128
$898$	0	0
$899$	13.1624	0.438992
$900$	0	0
$901$	−0.298682	−0.00995054
$902$	0	0
$903$	−1.47819	−0.0491910
$904$	0	0
$905$	−69.7854	−2.31975
$906$	0	0
$907$	−21.2120	−0.704332	−0.352166	−	0.935938i	$-0.614555\pi$
$907$	−21.2120	−0.704332	−0.352166	+	0.935938i	$0.614555\pi$
$908$	0	0
$909$	9.37043	0.310798
$910$	0	0
$911$	19.1313	0.633850	0.316925	−	0.948451i	$-0.397350\pi$
$911$	19.1313	0.633850	0.316925	+	0.948451i	$0.397350\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	50.4256	1.66702
$916$	0	0
$917$	45.6730	1.50825
$918$	0	0
$919$	41.1068	1.35599	0.677993	−	0.735068i	$-0.262850\pi$
$919$	41.1068	1.35599	0.677993	+	0.735068i	$0.262850\pi$
$920$	0	0
$921$	9.92805	0.327140
$922$	0	0
$923$	13.5118	0.444748
$924$	0	0
$925$	−23.6783	−0.778539
$926$	0	0
$927$	23.8998	0.784973
$928$	0	0
$929$	−47.4720	−1.55750	−0.778752	−	0.627331i	$-0.784147\pi$
$929$	−47.4720	−1.55750	−0.778752	+	0.627331i	$0.784147\pi$
$930$	0	0
$931$	−16.3465	−0.535735
$932$	0	0
$933$	2.93142	0.0959703
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.4615	0.897126	0.448563	−	0.893751i	$-0.351936\pi$
$937$	27.4615	0.897126	0.448563	+	0.893751i	$0.351936\pi$
$938$	0	0
$939$	−13.8377	−0.451577
$940$	0	0
$941$	−16.1641	−0.526934	−0.263467	−	0.964668i	$-0.584866\pi$
$941$	−16.1641	−0.526934	−0.263467	+	0.964668i	$0.584866\pi$
$942$	0	0
$943$	15.8699	0.516796
$944$	0	0
$945$	−70.9971	−2.30954
$946$	0	0
$947$	−28.3252	−0.920445	−0.460223	−	0.887804i	$-0.652230\pi$
$947$	−28.3252	−0.920445	−0.460223	+	0.887804i	$0.652230\pi$
$948$	0	0
$949$	3.69732	0.120020
$950$	0	0
$951$	12.1784	0.394911
$952$	0	0
$953$	53.3846	1.72930	0.864648	−	0.502379i	$-0.167542\pi$
$953$	53.3846	1.72930	0.864648	+	0.502379i	$0.167542\pi$
$954$	0	0
$955$	28.1279	0.910196
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.37543	0.302748
$960$	0	0
$961$	−8.07236	−0.260399
$962$	0	0
$963$	20.5886	0.663459
$964$	0	0
$965$	−93.9252	−3.02356
$966$	0	0
$967$	−45.0985	−1.45027	−0.725136	−	0.688606i	$-0.758223\pi$
$967$	−45.0985	−1.45027	−0.725136	+	0.688606i	$0.758223\pi$
$968$	0	0
$969$	2.06840	0.0664466
$970$	0	0
$971$	−47.5369	−1.52553	−0.762766	−	0.646674i	$-0.776159\pi$
$971$	−47.5369	−1.52553	−0.762766	+	0.646674i	$0.776159\pi$
$972$	0	0
$973$	52.4808	1.68246
$974$	0	0
$975$	−10.8043	−0.346015
$976$	0	0
$977$	−5.31562	−0.170062	−0.0850308	−	0.996378i	$-0.527099\pi$
$977$	−5.31562	−0.170062	−0.0850308	+	0.996378i	$0.527099\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−8.94118	−0.285470
$982$	0	0
$983$	−3.55660	−0.113438	−0.0567189	−	0.998390i	$-0.518064\pi$
$983$	−3.55660	−0.113438	−0.0567189	+	0.998390i	$0.518064\pi$
$984$	0	0
$985$	−43.5202	−1.38667
$986$	0	0
$987$	−21.6467	−0.689021
$988$	0	0
$989$	−0.750852	−0.0238757
$990$	0	0
$991$	38.1592	1.21217	0.606083	−	0.795401i	$-0.292740\pi$
$991$	38.1592	1.21217	0.606083	+	0.795401i	$0.292740\pi$
$992$	0	0
$993$	30.8405	0.978694
$994$	0	0
$995$	−36.5183	−1.15771
$996$	0	0
$997$	29.1688	0.923785	0.461892	−	0.886936i	$-0.347171\pi$
$997$	29.1688	0.923785	0.461892	+	0.886936i	$0.347171\pi$
$998$	0	0
$999$	−10.0215	−0.317068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.ds.1.2		4
4.3	odd	2		7744.2.a.dh.1.3		4
8.3	odd	2		968.2.a.m.1.2		4
8.5	even	2		1936.2.a.bc.1.3		4
11.7	odd	10		704.2.m.i.577.2		8
11.8	odd	10		704.2.m.i.449.2		8
11.10	odd	2		7744.2.a.dr.1.2		4
24.11	even	2		8712.2.a.cd.1.4		4
44.7	even	10		704.2.m.l.577.1		8
44.19	even	10		704.2.m.l.449.1		8
44.43	even	2		7744.2.a.di.1.3		4
88.3	odd	10		968.2.i.p.9.2		8
88.19	even	10		88.2.i.b.9.2	✓	8
88.21	odd	2		1936.2.a.bb.1.3		4
88.27	odd	10		968.2.i.t.729.1		8
88.29	odd	10		176.2.m.d.49.1		8
88.35	even	10		968.2.i.s.81.1		8
88.43	even	2		968.2.a.n.1.2		4
88.51	even	10		88.2.i.b.49.2	yes	8
88.59	odd	10		968.2.i.p.753.2		8
88.75	odd	10		968.2.i.t.81.1		8
88.83	even	10		968.2.i.s.729.1		8
88.85	odd	10		176.2.m.d.97.1		8
264.107	odd	10		792.2.r.g.361.2		8
264.131	odd	2		8712.2.a.ce.1.4		4
264.227	odd	10		792.2.r.g.577.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.i.b.9.2	✓	8	88.19	even	10
88.2.i.b.49.2	yes	8	88.51	even	10
176.2.m.d.49.1		8	88.29	odd	10
176.2.m.d.97.1		8	88.85	odd	10
704.2.m.i.449.2		8	11.8	odd	10
704.2.m.i.577.2		8	11.7	odd	10
704.2.m.l.449.1		8	44.19	even	10
704.2.m.l.577.1		8	44.7	even	10
792.2.r.g.361.2		8	264.107	odd	10
792.2.r.g.577.2		8	264.227	odd	10
968.2.a.m.1.2		4	8.3	odd	2
968.2.a.n.1.2		4	88.43	even	2
968.2.i.p.9.2		8	88.3	odd	10
968.2.i.p.753.2		8	88.59	odd	10
968.2.i.s.81.1		8	88.35	even	10
968.2.i.s.729.1		8	88.83	even	10
968.2.i.t.81.1		8	88.75	odd	10
968.2.i.t.729.1		8	88.27	odd	10
1936.2.a.bb.1.3		4	88.21	odd	2
1936.2.a.bc.1.3		4	8.5	even	2
7744.2.a.dh.1.3		4	4.3	odd	2
7744.2.a.di.1.3		4	44.43	even	2
7744.2.a.dr.1.2		4	11.10	odd	2
7744.2.a.ds.1.2		4	1.1	even	1	trivial
8712.2.a.cd.1.4		4	24.11	even	2
8712.2.a.ce.1.4		4	264.131	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:	\( N \)	\(=\)	\( 7744 = 2^{6} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7744.a (trivial)

\(f(q)\)	\(=\)	\(q-0.919131 q^{3} +4.02435 q^{5} -3.72325 q^{7} -2.15520 q^{9} +1.04998 q^{13} -3.69890 q^{15} +0.944761 q^{17} -2.38197 q^{19} +3.42216 q^{21} +1.73830 q^{23} +11.1954 q^{25} +4.73830 q^{27} -2.74888 q^{29} -4.78828 q^{31} -14.9837 q^{35} -2.11501 q^{37} -0.965069 q^{39} +9.12957 q^{41} -0.431946 q^{43} -8.67327 q^{45} -6.32545 q^{47} +6.86261 q^{49} -0.868359 q^{51} -0.316146 q^{53} +2.18934 q^{57} +8.09017 q^{59} -13.6326 q^{61} +8.02435 q^{63} +4.22549 q^{65} +5.68178 q^{67} -1.59773 q^{69} +12.8687 q^{71} +3.52132 q^{73} -10.2900 q^{75} -8.83698 q^{79} +2.11048 q^{81} -14.6667 q^{83} +3.80205 q^{85} +2.52658 q^{87} +2.43195 q^{89} -3.90934 q^{91} +4.40106 q^{93} -9.58586 q^{95} +1.48193 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - q^{5} + q^{7} + 6 q^{9} + q^{13} - 16 q^{15} + 12 q^{17} - 14 q^{19} - q^{21} + 2 q^{23} + 11 q^{25} + 14 q^{27} - 9 q^{29} - 11 q^{31} - 18 q^{35} - 13 q^{37} - 18 q^{39} + 8 q^{41}+ \cdots + 4 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 - q^5 + q^7 + 6 * q^9 + q^13 - 16 * q^15 + 12 * q^17 - 14 * q^19 - q^21 + 2 * q^23 + 11 * q^25 + 14 * q^27 - 9 * q^29 - 11 * q^31 - 18 * q^35 - 13 * q^37 - 18 * q^39 + 8 * q^41 - 3 * q^43 - 22 * q^45 - 7 * q^47 - q^49 - 14 * q^51 - 11 * q^53 - 7 * q^57 + 10 * q^59 - 17 * q^61 + 15 * q^63 + 5 * q^65 - 5 * q^67 + 4 * q^69 + 5 * q^71 + 6 * q^73 + 7 * q^75 + 7 * q^79 - 8 * q^81 - 20 * q^83 - 13 * q^85 - 3 * q^87 + 11 * q^89 + 6 * q^91 + 10 * q^93 + 6 * q^95 + 4 * q^97

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