LMFDB - Embedded newform 7744.2.a.dv.1.1

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:	$6$
Coefficient field:	6.6.19898000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5 $ x^6 - x^5 - 10x^4 + 7x^3 + 24x^2 - 15x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 352)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.86564$ 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)

$f(q)$	$=$	$q-1.86564 q^{3} +2.22111 q^{5} -0.973509 q^{7} +0.480595 q^{9} -5.76323 q^{13} -4.14378 q^{15} -3.97580 q^{17} +2.81392 q^{19} +1.81621 q^{21} +1.20615 q^{23} -0.0666695 q^{25} +4.70029 q^{27} -2.08367 q^{29} +6.84856 q^{31} -2.16227 q^{35} +9.41004 q^{37} +10.7521 q^{39} +4.40102 q^{41} +8.84372 q^{43} +1.06745 q^{45} +1.44564 q^{47} -6.05228 q^{49} +7.41739 q^{51} +10.2399 q^{53} -5.24976 q^{57} -3.52945 q^{59} -4.13538 q^{61} -0.467864 q^{63} -12.8008 q^{65} +12.2596 q^{67} -2.25024 q^{69} -9.44515 q^{71} -6.62134 q^{73} +0.124381 q^{75} +7.76432 q^{79} -10.2108 q^{81} -17.2597 q^{83} -8.83068 q^{85} +3.88737 q^{87} -0.470432 q^{89} +5.61056 q^{91} -12.7769 q^{93} +6.25003 q^{95} -9.06025 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} - 2 q^{7} + 7 q^{9} - 4 q^{13} - 8 q^{15} - 9 q^{17} + 5 q^{19} - 12 q^{21} - 6 q^{23} + 4 q^{25} + 26 q^{27} - 10 q^{29} - 12 q^{31} - 26 q^{35} + 12 q^{37} - 2 q^{39} - 17 q^{41} + 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100}) $ 6 * q + 5 * q^3 - 2 * q^7 + 7 * q^9 - 4 * q^13 - 8 * q^15 - 9 * q^17 + 5 * q^19 - 12 * q^21 - 6 * q^23 + 4 * q^25 + 26 * q^27 - 10 * q^29 - 12 * q^31 - 26 * q^35 + 12 * q^37 - 2 * q^39 - 17 * q^41 + 11 * q^43 - 14 * q^45 - 22 * q^47 - 6 * q^51 - 18 * q^53 - 6 * q^57 + 7 * q^59 - 18 * q^61 - 26 * q^63 - 26 * q^65 + 31 * q^67 - 16 * q^69 - 22 * q^71 + q^73 + 11 * q^75 + 28 * q^79 + 34 * q^81 - 13 * q^83 - 2 * q^85 - 38 * q^87 - q^89 + 26 * q^91 - 34 * q^93 - 2 * q^95 - 21 * 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
$3$	−1.86564	−1.07713	−0.538563	−	0.842586i	$-0.681032\pi$
$3$	−1.86564	−1.07713	−0.538563	+	0.842586i	$0.681032\pi$
$4$	0	0
$5$	2.22111	0.993311	0.496655	−	0.867948i	$-0.334561\pi$
$5$	2.22111	0.993311	0.496655	+	0.867948i	$0.334561\pi$
$6$	0	0
$7$	−0.973509	−0.367952	−0.183976	−	0.982931i	$-0.558897\pi$
$7$	−0.973509	−0.367952	−0.183976	+	0.982931i	$0.558897\pi$
$8$	0	0
$9$	0.480595	0.160198
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.76323	−1.59843	−0.799216	−	0.601044i	$-0.794752\pi$
$13$	−5.76323	−1.59843	−0.799216	+	0.601044i	$0.794752\pi$
$14$	0	0
$15$	−4.14378	−1.06992
$16$	0	0
$17$	−3.97580	−0.964273	−0.482136	−	0.876096i	$-0.660139\pi$
$17$	−3.97580	−0.964273	−0.482136	+	0.876096i	$0.660139\pi$
$18$	0	0
$19$	2.81392	0.645558	0.322779	−	0.946474i	$-0.395383\pi$
$19$	2.81392	0.645558	0.322779	+	0.946474i	$0.395383\pi$
$20$	0	0
$21$	1.81621	0.396330
$22$	0	0
$23$	1.20615	0.251500	0.125750	−	0.992062i	$-0.459866\pi$
$23$	1.20615	0.251500	0.125750	+	0.992062i	$0.459866\pi$
$24$	0	0
$25$	−0.0666695	−0.0133339
$26$	0	0
$27$	4.70029	0.904571
$28$	0	0
$29$	−2.08367	−0.386928	−0.193464	−	0.981107i	$-0.561972\pi$
$29$	−2.08367	−0.386928	−0.193464	+	0.981107i	$0.561972\pi$
$30$	0	0
$31$	6.84856	1.23004	0.615019	−	0.788512i	$-0.289148\pi$
$31$	6.84856	1.23004	0.615019	+	0.788512i	$0.289148\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.16227	−0.365490
$36$	0	0
$37$	9.41004	1.54700	0.773501	−	0.633795i	$-0.218504\pi$
$37$	9.41004	1.54700	0.773501	+	0.633795i	$0.218504\pi$
$38$	0	0
$39$	10.7521	1.72171
$40$	0	0
$41$	4.40102	0.687323	0.343662	−	0.939094i	$-0.388333\pi$
$41$	4.40102	0.687323	0.343662	+	0.939094i	$0.388333\pi$
$42$	0	0
$43$	8.84372	1.34865	0.674327	−	0.738432i	$-0.264434\pi$
$43$	8.84372	1.34865	0.674327	+	0.738432i	$0.264434\pi$
$44$	0	0
$45$	1.06745	0.159127
$46$	0	0
$47$	1.44564	0.210869	0.105434	−	0.994426i	$-0.466377\pi$
$47$	1.44564	0.210869	0.105434	+	0.994426i	$0.466377\pi$
$48$	0	0
$49$	−6.05228	−0.864611
$50$	0	0
$51$	7.41739	1.03864
$52$	0	0
$53$	10.2399	1.40656	0.703282	−	0.710911i	$-0.251717\pi$
$53$	10.2399	1.40656	0.703282	+	0.710911i	$0.251717\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.24976	−0.695347
$58$	0	0
$59$	−3.52945	−0.459495	−0.229748	−	0.973250i	$-0.573790\pi$
$59$	−3.52945	−0.459495	−0.229748	+	0.973250i	$0.573790\pi$
$60$	0	0
$61$	−4.13538	−0.529482	−0.264741	−	0.964320i	$-0.585286\pi$
$61$	−4.13538	−0.529482	−0.264741	+	0.964320i	$0.585286\pi$
$62$	0	0
$63$	−0.467864	−0.0589453
$64$	0	0
$65$	−12.8008	−1.58774
$66$	0	0
$67$	12.2596	1.49774	0.748872	−	0.662714i	$-0.230596\pi$
$67$	12.2596	1.49774	0.748872	+	0.662714i	$0.230596\pi$
$68$	0	0
$69$	−2.25024	−0.270897
$70$	0	0
$71$	−9.44515	−1.12093	−0.560467	−	0.828177i	$-0.689378\pi$
$71$	−9.44515	−1.12093	−0.560467	+	0.828177i	$0.689378\pi$
$72$	0	0
$73$	−6.62134	−0.774969	−0.387485	−	0.921876i	$-0.626656\pi$
$73$	−6.62134	−0.774969	−0.387485	+	0.921876i	$0.626656\pi$
$74$	0	0
$75$	0.124381	0.0143623
$76$	0	0
$77$	0	0
$78$	0	0
$79$	7.76432	0.873554	0.436777	−	0.899570i	$-0.356120\pi$
$79$	7.76432	0.873554	0.436777	+	0.899570i	$0.356120\pi$
$80$	0	0
$81$	−10.2108	−1.13453
$82$	0	0
$83$	−17.2597	−1.89450	−0.947251	−	0.320492i	$-0.896152\pi$
$83$	−17.2597	−1.89450	−0.947251	+	0.320492i	$0.896152\pi$
$84$	0	0
$85$	−8.83068	−0.957822
$86$	0	0
$87$	3.88737	0.416770
$88$	0	0
$89$	−0.470432	−0.0498657	−0.0249328	−	0.999689i	$-0.507937\pi$
$89$	−0.470432	−0.0498657	−0.0249328	+	0.999689i	$0.507937\pi$
$90$	0	0
$91$	5.61056	0.588146
$92$	0	0
$93$	−12.7769	−1.32490
$94$	0	0
$95$	6.25003	0.641240
$96$	0	0
$97$	−9.06025	−0.919929	−0.459965	−	0.887937i	$-0.652138\pi$
$97$	−9.06025	−0.919929	−0.459965	+	0.887937i	$0.652138\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.2414	−1.31757	−0.658786	−	0.752331i	$-0.728930\pi$
$101$	−13.2414	−1.31757	−0.658786	+	0.752331i	$0.728930\pi$
$102$	0	0
$103$	−15.2000	−1.49770	−0.748849	−	0.662740i	$-0.769393\pi$
$103$	−15.2000	−1.49770	−0.748849	+	0.662740i	$0.769393\pi$
$104$	0	0
$105$	4.03401	0.393679
$106$	0	0
$107$	20.5113	1.98290	0.991449	−	0.130493i	$-0.0416559\pi$
$107$	20.5113	1.98290	0.991449	+	0.130493i	$0.0416559\pi$
$108$	0	0
$109$	−4.79668	−0.459438	−0.229719	−	0.973257i	$-0.573781\pi$
$109$	−4.79668	−0.459438	−0.229719	+	0.973257i	$0.573781\pi$
$110$	0	0
$111$	−17.5557	−1.66631
$112$	0	0
$113$	6.52271	0.613605	0.306803	−	0.951773i	$-0.400741\pi$
$113$	6.52271	0.613605	0.306803	+	0.951773i	$0.400741\pi$
$114$	0	0
$115$	2.67900	0.249818
$116$	0	0
$117$	−2.76978	−0.256066
$118$	0	0
$119$	3.87047	0.354806
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−8.21069	−0.740333
$124$	0	0
$125$	−11.2536	−1.00656
$126$	0	0
$127$	−13.5342	−1.20096	−0.600482	−	0.799638i	$-0.705025\pi$
$127$	−13.5342	−1.20096	−0.600482	+	0.799638i	$0.705025\pi$
$128$	0	0
$129$	−16.4992	−1.45267
$130$	0	0
$131$	−11.8240	−1.03306	−0.516532	−	0.856268i	$-0.672777\pi$
$131$	−11.8240	−1.03306	−0.516532	+	0.856268i	$0.672777\pi$
$132$	0	0
$133$	−2.73938	−0.237534
$134$	0	0
$135$	10.4399	0.898520
$136$	0	0
$137$	−16.7357	−1.42983	−0.714914	−	0.699212i	$-0.753534\pi$
$137$	−16.7357	−1.42983	−0.714914	+	0.699212i	$0.753534\pi$
$138$	0	0
$139$	11.5502	0.979676	0.489838	−	0.871814i	$-0.337056\pi$
$139$	11.5502	0.979676	0.489838	+	0.871814i	$0.337056\pi$
$140$	0	0
$141$	−2.69705	−0.227132
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−4.62806	−0.384340
$146$	0	0
$147$	11.2913	0.931295
$148$	0	0
$149$	15.5034	1.27009	0.635043	−	0.772476i	$-0.280982\pi$
$149$	15.5034	1.27009	0.635043	+	0.772476i	$0.280982\pi$
$150$	0	0
$151$	−9.59246	−0.780623	−0.390312	−	0.920683i	$-0.627633\pi$
$151$	−9.59246	−0.780623	−0.390312	+	0.920683i	$0.627633\pi$
$152$	0	0
$153$	−1.91075	−0.154475
$154$	0	0
$155$	15.2114	1.22181
$156$	0	0
$157$	3.20127	0.255489	0.127744	−	0.991807i	$-0.459226\pi$
$157$	3.20127	0.255489	0.127744	+	0.991807i	$0.459226\pi$
$158$	0	0
$159$	−19.1040	−1.51505
$160$	0	0
$161$	−1.17420	−0.0925399
$162$	0	0
$163$	23.6441	1.85195	0.925974	−	0.377587i	$-0.123246\pi$
$163$	23.6441	1.85195	0.925974	+	0.377587i	$0.123246\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.84707	−0.607225	−0.303612	−	0.952796i	$-0.598193\pi$
$167$	−7.84707	−0.607225	−0.303612	+	0.952796i	$0.598193\pi$
$168$	0	0
$169$	20.2148	1.55499
$170$	0	0
$171$	1.35236	0.103417
$172$	0	0
$173$	−1.88641	−0.143421	−0.0717107	−	0.997425i	$-0.522846\pi$
$173$	−1.88641	−0.143421	−0.0717107	+	0.997425i	$0.522846\pi$
$174$	0	0
$175$	0.0649034	0.00490623
$176$	0	0
$177$	6.58467	0.494934
$178$	0	0
$179$	−3.06573	−0.229144	−0.114572	−	0.993415i	$-0.536550\pi$
$179$	−3.06573	−0.229144	−0.114572	+	0.993415i	$0.536550\pi$
$180$	0	0
$181$	−6.72316	−0.499729	−0.249864	−	0.968281i	$-0.580386\pi$
$181$	−6.72316	−0.499729	−0.249864	+	0.968281i	$0.580386\pi$
$182$	0	0
$183$	7.71512	0.570318
$184$	0	0
$185$	20.9007	1.53665
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.57577	−0.332839
$190$	0	0
$191$	−1.80547	−0.130639	−0.0653197	−	0.997864i	$-0.520807\pi$
$191$	−1.80547	−0.130639	−0.0653197	+	0.997864i	$0.520807\pi$
$192$	0	0
$193$	−0.810851	−0.0583663	−0.0291832	−	0.999574i	$-0.509291\pi$
$193$	−0.810851	−0.0583663	−0.0291832	+	0.999574i	$0.509291\pi$
$194$	0	0
$195$	23.8816	1.71019
$196$	0	0
$197$	−18.6154	−1.32629	−0.663147	−	0.748489i	$-0.730780\pi$
$197$	−18.6154	−1.32629	−0.663147	+	0.748489i	$0.730780\pi$
$198$	0	0
$199$	−16.8109	−1.19170	−0.595848	−	0.803097i	$-0.703184\pi$
$199$	−16.8109	−1.19170	−0.595848	+	0.803097i	$0.703184\pi$
$200$	0	0
$201$	−22.8719	−1.61326
$202$	0	0
$203$	2.02847	0.142371
$204$	0	0
$205$	9.77514	0.682726
$206$	0	0
$207$	0.579671	0.0402899
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−0.384111	−0.0264433	−0.0132216	−	0.999913i	$-0.504209\pi$
$211$	−0.384111	−0.0264433	−0.0132216	+	0.999913i	$0.504209\pi$
$212$	0	0
$213$	17.6212	1.20739
$214$	0	0
$215$	19.6429	1.33963
$216$	0	0
$217$	−6.66713	−0.452595
$218$	0	0
$219$	12.3530	0.834739
$220$	0	0
$221$	22.9134	1.54132
$222$	0	0
$223$	−23.3143	−1.56124	−0.780622	−	0.625004i	$-0.785098\pi$
$223$	−23.3143	−1.56124	−0.780622	+	0.625004i	$0.785098\pi$
$224$	0	0
$225$	−0.0320410	−0.00213607
$226$	0	0
$227$	2.72537	0.180889	0.0904445	−	0.995901i	$-0.471171\pi$
$227$	2.72537	0.180889	0.0904445	+	0.995901i	$0.471171\pi$
$228$	0	0
$229$	−24.9517	−1.64885	−0.824426	−	0.565969i	$-0.808502\pi$
$229$	−24.9517	−1.64885	−0.824426	+	0.565969i	$0.808502\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.1389	−1.38486	−0.692428	−	0.721487i	$-0.743459\pi$
$233$	−21.1389	−1.38486	−0.692428	+	0.721487i	$0.743459\pi$
$234$	0	0
$235$	3.21094	0.209458
$236$	0	0
$237$	−14.4854	−0.940927
$238$	0	0
$239$	0.208006	0.0134548	0.00672741	−	0.999977i	$-0.497859\pi$
$239$	0.208006	0.0134548	0.00672741	+	0.999977i	$0.497859\pi$
$240$	0	0
$241$	1.25123	0.0805990	0.0402995	−	0.999188i	$-0.487169\pi$
$241$	1.25123	0.0805990	0.0402995	+	0.999188i	$0.487169\pi$
$242$	0	0
$243$	4.94878	0.317465
$244$	0	0
$245$	−13.4428	−0.858828
$246$	0	0
$247$	−16.2173	−1.03188
$248$	0	0
$249$	32.2004	2.04062
$250$	0	0
$251$	−22.3385	−1.40999	−0.704996	−	0.709211i	$-0.749051\pi$
$251$	−22.3385	−1.40999	−0.704996	+	0.709211i	$0.749051\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	16.4748	1.03169
$256$	0	0
$257$	21.6595	1.35108	0.675542	−	0.737321i	$-0.263909\pi$
$257$	21.6595	1.35108	0.675542	+	0.737321i	$0.263909\pi$
$258$	0	0
$259$	−9.16076	−0.569222
$260$	0	0
$261$	−1.00140	−0.0619852
$262$	0	0
$263$	11.3447	0.699545	0.349773	−	0.936835i	$-0.386259\pi$
$263$	11.3447	0.699545	0.349773	+	0.936835i	$0.386259\pi$
$264$	0	0
$265$	22.7440	1.39716
$266$	0	0
$267$	0.877654	0.0537116
$268$	0	0
$269$	−14.3791	−0.876708	−0.438354	−	0.898802i	$-0.644438\pi$
$269$	−14.3791	−0.876708	−0.438354	+	0.898802i	$0.644438\pi$
$270$	0	0
$271$	−11.0116	−0.668908	−0.334454	−	0.942412i	$-0.608552\pi$
$271$	−11.0116	−0.668908	−0.334454	+	0.942412i	$0.608552\pi$
$272$	0	0
$273$	−10.4673	−0.633507
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.90241	0.354641	0.177321	−	0.984153i	$-0.443257\pi$
$277$	5.90241	0.354641	0.177321	+	0.984153i	$0.443257\pi$
$278$	0	0
$279$	3.29139	0.197050
$280$	0	0
$281$	13.1988	0.787376	0.393688	−	0.919244i	$-0.371199\pi$
$281$	13.1988	0.787376	0.393688	+	0.919244i	$0.371199\pi$
$282$	0	0
$283$	−8.74336	−0.519739	−0.259869	−	0.965644i	$-0.583680\pi$
$283$	−8.74336	−0.519739	−0.259869	+	0.965644i	$0.583680\pi$
$284$	0	0
$285$	−11.6603	−0.690696
$286$	0	0
$287$	−4.28443	−0.252902
$288$	0	0
$289$	−1.19303	−0.0701783
$290$	0	0
$291$	16.9031	0.990879
$292$	0	0
$293$	1.51630	0.0885830	0.0442915	−	0.999019i	$-0.485897\pi$
$293$	1.51630	0.0885830	0.0442915	+	0.999019i	$0.485897\pi$
$294$	0	0
$295$	−7.83930	−0.456422
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.95133	−0.402006
$300$	0	0
$301$	−8.60944	−0.496240
$302$	0	0
$303$	24.7037	1.41919
$304$	0	0
$305$	−9.18514	−0.525940
$306$	0	0
$307$	−5.53572	−0.315940	−0.157970	−	0.987444i	$-0.550495\pi$
$307$	−5.53572	−0.315940	−0.157970	+	0.987444i	$0.550495\pi$
$308$	0	0
$309$	28.3576	1.61321
$310$	0	0
$311$	9.16048	0.519443	0.259722	−	0.965684i	$-0.416369\pi$
$311$	9.16048	0.519443	0.259722	+	0.965684i	$0.416369\pi$
$312$	0	0
$313$	16.2744	0.919882	0.459941	−	0.887949i	$-0.347871\pi$
$313$	16.2744	0.919882	0.459941	+	0.887949i	$0.347871\pi$
$314$	0	0
$315$	−1.03918	−0.0585510
$316$	0	0
$317$	3.54644	0.199188	0.0995938	−	0.995028i	$-0.468246\pi$
$317$	3.54644	0.199188	0.0995938	+	0.995028i	$0.468246\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−38.2665	−2.13583
$322$	0	0
$323$	−11.1876	−0.622494
$324$	0	0
$325$	0.384232	0.0213133
$326$	0	0
$327$	8.94885	0.494873
$328$	0	0
$329$	−1.40735	−0.0775896
$330$	0	0
$331$	16.2332	0.892257	0.446128	−	0.894969i	$-0.352802\pi$
$331$	16.2332	0.892257	0.446128	+	0.894969i	$0.352802\pi$
$332$	0	0
$333$	4.52242	0.247827
$334$	0	0
$335$	27.2299	1.48773
$336$	0	0
$337$	−8.46084	−0.460892	−0.230446	−	0.973085i	$-0.574018\pi$
$337$	−8.46084	−0.460892	−0.230446	+	0.973085i	$0.574018\pi$
$338$	0	0
$339$	−12.1690	−0.660929
$340$	0	0
$341$	0	0
$342$	0	0
$343$	12.7065	0.686087
$344$	0	0
$345$	−4.99803	−0.269085
$346$	0	0
$347$	−5.39095	−0.289401	−0.144701	−	0.989475i	$-0.546222\pi$
$347$	−5.39095	−0.289401	−0.144701	+	0.989475i	$0.546222\pi$
$348$	0	0
$349$	−12.9250	−0.691859	−0.345929	−	0.938261i	$-0.612436\pi$
$349$	−12.9250	−0.691859	−0.345929	+	0.938261i	$0.612436\pi$
$350$	0	0
$351$	−27.0889	−1.44590
$352$	0	0
$353$	34.5770	1.84035	0.920173	−	0.391511i	$-0.128047\pi$
$353$	34.5770	1.84035	0.920173	+	0.391511i	$0.128047\pi$
$354$	0	0
$355$	−20.9787	−1.11343
$356$	0	0
$357$	−7.22089	−0.382170
$358$	0	0
$359$	−23.1986	−1.22437	−0.612187	−	0.790713i	$-0.709710\pi$
$359$	−23.1986	−1.22437	−0.612187	+	0.790713i	$0.709710\pi$
$360$	0	0
$361$	−11.0818	−0.583254
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.7067	−0.769785
$366$	0	0
$367$	−15.4707	−0.807566	−0.403783	−	0.914855i	$-0.632305\pi$
$367$	−15.4707	−0.807566	−0.403783	+	0.914855i	$0.632305\pi$
$368$	0	0
$369$	2.11511	0.110108
$370$	0	0
$371$	−9.96868	−0.517548
$372$	0	0
$373$	−30.5787	−1.58331	−0.791653	−	0.610970i	$-0.790779\pi$
$373$	−30.5787	−1.58331	−0.791653	+	0.610970i	$0.790779\pi$
$374$	0	0
$375$	20.9952	1.08419
$376$	0	0
$377$	12.0087	0.618478
$378$	0	0
$379$	−2.71862	−0.139646	−0.0698230	−	0.997559i	$-0.522243\pi$
$379$	−2.71862	−0.139646	−0.0698230	+	0.997559i	$0.522243\pi$
$380$	0	0
$381$	25.2498	1.29359
$382$	0	0
$383$	18.4024	0.940319	0.470159	−	0.882582i	$-0.344196\pi$
$383$	18.4024	0.940319	0.470159	+	0.882582i	$0.344196\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.25025	0.216052
$388$	0	0
$389$	−0.636434	−0.0322685	−0.0161342	−	0.999870i	$-0.505136\pi$
$389$	−0.636434	−0.0322685	−0.0161342	+	0.999870i	$0.505136\pi$
$390$	0	0
$391$	−4.79542	−0.242515
$392$	0	0
$393$	22.0592	1.11274
$394$	0	0
$395$	17.2454	0.867711
$396$	0	0
$397$	19.6411	0.985760	0.492880	−	0.870097i	$-0.335944\pi$
$397$	19.6411	0.985760	0.492880	+	0.870097i	$0.335944\pi$
$398$	0	0
$399$	5.11068	0.255854
$400$	0	0
$401$	−5.75683	−0.287482	−0.143741	−	0.989615i	$-0.545913\pi$
$401$	−5.75683	−0.287482	−0.143741	+	0.989615i	$0.545913\pi$
$402$	0	0
$403$	−39.4698	−1.96613
$404$	0	0
$405$	−22.6793	−1.12695
$406$	0	0
$407$	0	0
$408$	0	0
$409$	23.1324	1.14382	0.571912	−	0.820315i	$-0.306202\pi$
$409$	23.1324	1.14382	0.571912	+	0.820315i	$0.306202\pi$
$410$	0	0
$411$	31.2227	1.54010
$412$	0	0
$413$	3.43595	0.169072
$414$	0	0
$415$	−38.3358	−1.88183
$416$	0	0
$417$	−21.5485	−1.05523
$418$	0	0
$419$	−20.6359	−1.00813	−0.504065	−	0.863666i	$-0.668163\pi$
$419$	−20.6359	−1.00813	−0.504065	+	0.863666i	$0.668163\pi$
$420$	0	0
$421$	−30.1427	−1.46906	−0.734532	−	0.678574i	$-0.762598\pi$
$421$	−30.1427	−1.46906	−0.734532	+	0.678574i	$0.762598\pi$
$422$	0	0
$423$	0.694770	0.0337809
$424$	0	0
$425$	0.265065	0.0128575
$426$	0	0
$427$	4.02583	0.194824
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−32.6765	−1.57397	−0.786985	−	0.616972i	$-0.788359\pi$
$431$	−32.6765	−1.57397	−0.786985	+	0.616972i	$0.788359\pi$
$432$	0	0
$433$	28.0266	1.34687	0.673436	−	0.739245i	$-0.264818\pi$
$433$	28.0266	1.34687	0.673436	+	0.739245i	$0.264818\pi$
$434$	0	0
$435$	8.63428	0.413982
$436$	0	0
$437$	3.39402	0.162358
$438$	0	0
$439$	19.4940	0.930399	0.465200	−	0.885206i	$-0.345983\pi$
$439$	19.4940	0.930399	0.465200	+	0.885206i	$0.345983\pi$
$440$	0	0
$441$	−2.90870	−0.138509
$442$	0	0
$443$	8.91652	0.423637	0.211818	−	0.977309i	$-0.432061\pi$
$443$	8.91652	0.423637	0.211818	+	0.977309i	$0.432061\pi$
$444$	0	0
$445$	−1.04488	−0.0495321
$446$	0	0
$447$	−28.9237	−1.36804
$448$	0	0
$449$	0.730919	0.0344942	0.0172471	−	0.999851i	$-0.494510\pi$
$449$	0.730919	0.0344942	0.0172471	+	0.999851i	$0.494510\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	17.8960	0.840829
$454$	0	0
$455$	12.4617	0.584212
$456$	0	0
$457$	1.63396	0.0764332	0.0382166	−	0.999269i	$-0.487832\pi$
$457$	1.63396	0.0764332	0.0382166	+	0.999269i	$0.487832\pi$
$458$	0	0
$459$	−18.6874	−0.872253
$460$	0	0
$461$	2.38755	0.111199	0.0555996	−	0.998453i	$-0.482293\pi$
$461$	2.38755	0.111199	0.0555996	+	0.998453i	$0.482293\pi$
$462$	0	0
$463$	16.3739	0.760961	0.380480	−	0.924789i	$-0.375759\pi$
$463$	16.3739	0.760961	0.380480	+	0.924789i	$0.375759\pi$
$464$	0	0
$465$	−28.3789	−1.31604
$466$	0	0
$467$	22.0348	1.01965	0.509825	−	0.860278i	$-0.329710\pi$
$467$	22.0348	1.01965	0.509825	+	0.860278i	$0.329710\pi$
$468$	0	0
$469$	−11.9348	−0.551098
$470$	0	0
$471$	−5.97240	−0.275194
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−0.187603	−0.00860781
$476$	0	0
$477$	4.92127	0.225329
$478$	0	0
$479$	−24.6894	−1.12809	−0.564044	−	0.825745i	$-0.690755\pi$
$479$	−24.6894	−1.12809	−0.564044	+	0.825745i	$0.690755\pi$
$480$	0	0
$481$	−54.2323	−2.47278
$482$	0	0
$483$	2.19063	0.0996771
$484$	0	0
$485$	−20.1238	−0.913776
$486$	0	0
$487$	19.5180	0.884446	0.442223	−	0.896905i	$-0.354190\pi$
$487$	19.5180	0.884446	0.442223	+	0.896905i	$0.354190\pi$
$488$	0	0
$489$	−44.1113	−1.99478
$490$	0	0
$491$	14.7670	0.666425	0.333213	−	0.942852i	$-0.391867\pi$
$491$	14.7670	0.666425	0.333213	+	0.942852i	$0.391867\pi$
$492$	0	0
$493$	8.28426	0.373104
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.19494	0.412449
$498$	0	0
$499$	−17.7602	−0.795054	−0.397527	−	0.917590i	$-0.630131\pi$
$499$	−17.7602	−0.795054	−0.397527	+	0.917590i	$0.630131\pi$
$500$	0	0
$501$	14.6398	0.654057
$502$	0	0
$503$	−16.4173	−0.732013	−0.366007	−	0.930612i	$-0.619275\pi$
$503$	−16.4173	−0.732013	−0.366007	+	0.930612i	$0.619275\pi$
$504$	0	0
$505$	−29.4107	−1.30876
$506$	0	0
$507$	−37.7135	−1.67491
$508$	0	0
$509$	13.4609	0.596645	0.298322	−	0.954465i	$-0.403573\pi$
$509$	13.4609	0.596645	0.298322	+	0.954465i	$0.403573\pi$
$510$	0	0
$511$	6.44593	0.285151
$512$	0	0
$513$	13.2263	0.583954
$514$	0	0
$515$	−33.7608	−1.48768
$516$	0	0
$517$	0	0
$518$	0	0
$519$	3.51936	0.154483
$520$	0	0
$521$	−36.4464	−1.59675	−0.798373	−	0.602163i	$-0.794306\pi$
$521$	−36.4464	−1.59675	−0.798373	+	0.602163i	$0.794306\pi$
$522$	0	0
$523$	11.3383	0.495788	0.247894	−	0.968787i	$-0.420261\pi$
$523$	11.3383	0.495788	0.247894	+	0.968787i	$0.420261\pi$
$524$	0	0
$525$	−0.121086	−0.00528463
$526$	0	0
$527$	−27.2285	−1.18609
$528$	0	0
$529$	−21.5452	−0.936748
$530$	0	0
$531$	−1.69624	−0.0736104
$532$	0	0
$533$	−25.3641	−1.09864
$534$	0	0
$535$	45.5578	1.96963
$536$	0	0
$537$	5.71954	0.246816
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−40.8576	−1.75660	−0.878302	−	0.478106i	$-0.841323\pi$
$541$	−40.8576	−1.75660	−0.878302	+	0.478106i	$0.841323\pi$
$542$	0	0
$543$	12.5430	0.538270
$544$	0	0
$545$	−10.6539	−0.456365
$546$	0	0
$547$	−32.7961	−1.40226	−0.701129	−	0.713034i	$-0.747320\pi$
$547$	−32.7961	−1.40226	−0.701129	+	0.713034i	$0.747320\pi$
$548$	0	0
$549$	−1.98744	−0.0848221
$550$	0	0
$551$	−5.86329	−0.249785
$552$	0	0
$553$	−7.55864	−0.321426
$554$	0	0
$555$	−38.9932	−1.65517
$556$	0	0
$557$	7.50056	0.317809	0.158904	−	0.987294i	$-0.449204\pi$
$557$	7.50056	0.317809	0.158904	+	0.987294i	$0.449204\pi$
$558$	0	0
$559$	−50.9684	−2.15573
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11.7203	0.493951	0.246976	−	0.969022i	$-0.420563\pi$
$563$	11.7203	0.493951	0.246976	+	0.969022i	$0.420563\pi$
$564$	0	0
$565$	14.4877	0.609501
$566$	0	0
$567$	9.94032	0.417454
$568$	0	0
$569$	7.42622	0.311323	0.155662	−	0.987810i	$-0.450249\pi$
$569$	7.42622	0.311323	0.155662	+	0.987810i	$0.450249\pi$
$570$	0	0
$571$	−37.7892	−1.58143	−0.790714	−	0.612185i	$-0.790291\pi$
$571$	−37.7892	−1.58143	−0.790714	+	0.612185i	$0.790291\pi$
$572$	0	0
$573$	3.36835	0.140715
$574$	0	0
$575$	−0.0804136	−0.00335348
$576$	0	0
$577$	−21.4699	−0.893802	−0.446901	−	0.894583i	$-0.647472\pi$
$577$	−21.4699	−0.893802	−0.446901	+	0.894583i	$0.647472\pi$
$578$	0	0
$579$	1.51275	0.0628678
$580$	0	0
$581$	16.8025	0.697086
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−6.15199	−0.254353
$586$	0	0
$587$	1.97188	0.0813883	0.0406941	−	0.999172i	$-0.487043\pi$
$587$	1.97188	0.0813883	0.0406941	+	0.999172i	$0.487043\pi$
$588$	0	0
$589$	19.2713	0.794061
$590$	0	0
$591$	34.7296	1.42858
$592$	0	0
$593$	7.57118	0.310911	0.155456	−	0.987843i	$-0.450315\pi$
$593$	7.57118	0.310911	0.155456	+	0.987843i	$0.450315\pi$
$594$	0	0
$595$	8.59675	0.352432
$596$	0	0
$597$	31.3631	1.28361
$598$	0	0
$599$	10.2478	0.418714	0.209357	−	0.977839i	$-0.432863\pi$
$599$	10.2478	0.418714	0.209357	+	0.977839i	$0.432863\pi$
$600$	0	0
$601$	3.45161	0.140794	0.0703970	−	0.997519i	$-0.477573\pi$
$601$	3.45161	0.140794	0.0703970	+	0.997519i	$0.477573\pi$
$602$	0	0
$603$	5.89189	0.239936
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.14665	−0.127719	−0.0638593	−	0.997959i	$-0.520341\pi$
$607$	−3.14665	−0.127719	−0.0638593	+	0.997959i	$0.520341\pi$
$608$	0	0
$609$	−3.78439	−0.153351
$610$	0	0
$611$	−8.33158	−0.337060
$612$	0	0
$613$	−45.8250	−1.85086	−0.925428	−	0.378923i	$-0.876294\pi$
$613$	−45.8250	−1.85086	−0.925428	+	0.378923i	$0.876294\pi$
$614$	0	0
$615$	−18.2369	−0.735381
$616$	0	0
$617$	6.41440	0.258234	0.129117	−	0.991629i	$-0.458786\pi$
$617$	6.41440	0.258234	0.129117	+	0.991629i	$0.458786\pi$
$618$	0	0
$619$	29.7618	1.19623	0.598115	−	0.801411i	$-0.295917\pi$
$619$	29.7618	1.19623	0.598115	+	0.801411i	$0.295917\pi$
$620$	0	0
$621$	5.66927	0.227500
$622$	0	0
$623$	0.457969	0.0183482
$624$	0	0
$625$	−24.6622	−0.986488
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−37.4124	−1.49173
$630$	0	0
$631$	48.7020	1.93879	0.969397	−	0.245498i	$-0.0789514\pi$
$631$	48.7020	1.93879	0.969397	+	0.245498i	$0.0789514\pi$
$632$	0	0
$633$	0.716611	0.0284827
$634$	0	0
$635$	−30.0609	−1.19293
$636$	0	0
$637$	34.8807	1.38202
$638$	0	0
$639$	−4.53929	−0.179572
$640$	0	0
$641$	16.8534	0.665669	0.332835	−	0.942985i	$-0.391995\pi$
$641$	16.8534	0.665669	0.332835	+	0.942985i	$0.391995\pi$
$642$	0	0
$643$	6.56513	0.258903	0.129452	−	0.991586i	$-0.458678\pi$
$643$	6.56513	0.258903	0.129452	+	0.991586i	$0.458678\pi$
$644$	0	0
$645$	−36.6465	−1.44295
$646$	0	0
$647$	3.53053	0.138800	0.0693998	−	0.997589i	$-0.477892\pi$
$647$	3.53053	0.138800	0.0693998	+	0.997589i	$0.477892\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	12.4384	0.487501
$652$	0	0
$653$	6.13537	0.240096	0.120048	−	0.992768i	$-0.461695\pi$
$653$	6.13537	0.240096	0.120048	+	0.992768i	$0.461695\pi$
$654$	0	0
$655$	−26.2623	−1.02615
$656$	0	0
$657$	−3.18218	−0.124149
$658$	0	0
$659$	−27.9511	−1.08882	−0.544410	−	0.838819i	$-0.683246\pi$
$659$	−27.9511	−1.08882	−0.544410	+	0.838819i	$0.683246\pi$
$660$	0	0
$661$	−11.5665	−0.449886	−0.224943	−	0.974372i	$-0.572220\pi$
$661$	−11.5665	−0.449886	−0.224943	+	0.974372i	$0.572220\pi$
$662$	0	0
$663$	−42.7481	−1.66020
$664$	0	0
$665$	−6.08446	−0.235945
$666$	0	0
$667$	−2.51323	−0.0973125
$668$	0	0
$669$	43.4961	1.68165
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−14.1644	−0.545999	−0.273000	−	0.962014i	$-0.588016\pi$
$673$	−14.1644	−0.545999	−0.273000	+	0.962014i	$0.588016\pi$
$674$	0	0
$675$	−0.313366	−0.0120615
$676$	0	0
$677$	−25.3891	−0.975784	−0.487892	−	0.872904i	$-0.662234\pi$
$677$	−25.3891	−0.975784	−0.487892	+	0.872904i	$0.662234\pi$
$678$	0	0
$679$	8.82024	0.338490
$680$	0	0
$681$	−5.08454	−0.194840
$682$	0	0
$683$	−14.6841	−0.561870	−0.280935	−	0.959727i	$-0.590645\pi$
$683$	−14.6841	−0.561870	−0.280935	+	0.959727i	$0.590645\pi$
$684$	0	0
$685$	−37.1719	−1.42026
$686$	0	0
$687$	46.5507	1.77602
$688$	0	0
$689$	−59.0152	−2.24830
$690$	0	0
$691$	31.8808	1.21280	0.606401	−	0.795159i	$-0.292613\pi$
$691$	31.8808	1.21280	0.606401	+	0.795159i	$0.292613\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	25.6543	0.973122
$696$	0	0
$697$	−17.4976	−0.662767
$698$	0	0
$699$	39.4375	1.49166
$700$	0	0
$701$	16.7002	0.630759	0.315380	−	0.948966i	$-0.397868\pi$
$701$	16.7002	0.630759	0.315380	+	0.948966i	$0.397868\pi$
$702$	0	0
$703$	26.4791	0.998680
$704$	0	0
$705$	−5.99044	−0.225613
$706$	0	0
$707$	12.8907	0.484803
$708$	0	0
$709$	26.6741	1.00177	0.500884	−	0.865514i	$-0.333008\pi$
$709$	26.6741	1.00177	0.500884	+	0.865514i	$0.333008\pi$
$710$	0	0
$711$	3.73149	0.139942
$712$	0	0
$713$	8.26041	0.309355
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−0.388064	−0.0144925
$718$	0	0
$719$	19.6224	0.731791	0.365896	−	0.930656i	$-0.380763\pi$
$719$	19.6224	0.731791	0.365896	+	0.930656i	$0.380763\pi$
$720$	0	0
$721$	14.7973	0.551081
$722$	0	0
$723$	−2.33434	−0.0868152
$724$	0	0
$725$	0.138917	0.00515926
$726$	0	0
$727$	−23.9815	−0.889424	−0.444712	−	0.895674i	$-0.646694\pi$
$727$	−23.9815	−0.889424	−0.444712	+	0.895674i	$0.646694\pi$
$728$	0	0
$729$	21.3998	0.792586
$730$	0	0
$731$	−35.1609	−1.30047
$732$	0	0
$733$	15.3420	0.566669	0.283334	−	0.959021i	$-0.408559\pi$
$733$	15.3420	0.566669	0.283334	+	0.959021i	$0.408559\pi$
$734$	0	0
$735$	25.0793	0.925065
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−3.19534	−0.117542	−0.0587712	−	0.998271i	$-0.518718\pi$
$739$	−3.19534	−0.117542	−0.0587712	+	0.998271i	$0.518718\pi$
$740$	0	0
$741$	30.2555	1.11147
$742$	0	0
$743$	−30.9590	−1.13578	−0.567888	−	0.823106i	$-0.692239\pi$
$743$	−30.9590	−1.13578	−0.567888	+	0.823106i	$0.692239\pi$
$744$	0	0
$745$	34.4347	1.26159
$746$	0	0
$747$	−8.29495	−0.303496
$748$	0	0
$749$	−19.9679	−0.729611
$750$	0	0
$751$	−17.9709	−0.655769	−0.327884	−	0.944718i	$-0.606336\pi$
$751$	−17.9709	−0.655769	−0.327884	+	0.944718i	$0.606336\pi$
$752$	0	0
$753$	41.6755	1.51874
$754$	0	0
$755$	−21.3059	−0.775402
$756$	0	0
$757$	−14.6884	−0.533858	−0.266929	−	0.963716i	$-0.586009\pi$
$757$	−14.6884	−0.533858	−0.266929	+	0.963716i	$0.586009\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−13.2646	−0.480841	−0.240420	−	0.970669i	$-0.577285\pi$
$761$	−13.2646	−0.480841	−0.240420	+	0.970669i	$0.577285\pi$
$762$	0	0
$763$	4.66961	0.169051
$764$	0	0
$765$	−4.24398	−0.153442
$766$	0	0
$767$	20.3410	0.734472
$768$	0	0
$769$	−30.0680	−1.08428	−0.542139	−	0.840289i	$-0.682385\pi$
$769$	−30.0680	−1.08428	−0.542139	+	0.840289i	$0.682385\pi$
$770$	0	0
$771$	−40.4088	−1.45529
$772$	0	0
$773$	−10.5402	−0.379104	−0.189552	−	0.981871i	$-0.560703\pi$
$773$	−10.5402	−0.379104	−0.189552	+	0.981871i	$0.560703\pi$
$774$	0	0
$775$	−0.456590	−0.0164012
$776$	0	0
$777$	17.0906	0.613123
$778$	0	0
$779$	12.3841	0.443707
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−9.79386	−0.350004
$784$	0	0
$785$	7.11037	0.253780
$786$	0	0
$787$	37.5551	1.33870	0.669348	−	0.742949i	$-0.266573\pi$
$787$	37.5551	1.33870	0.669348	+	0.742949i	$0.266573\pi$
$788$	0	0
$789$	−21.1651	−0.753498
$790$	0	0
$791$	−6.34992	−0.225777
$792$	0	0
$793$	23.8332	0.846340
$794$	0	0
$795$	−42.4321	−1.50491
$796$	0	0
$797$	−15.7550	−0.558072	−0.279036	−	0.960281i	$-0.590015\pi$
$797$	−15.7550	−0.558072	−0.279036	+	0.960281i	$0.590015\pi$
$798$	0	0
$799$	−5.74759	−0.203335
$800$	0	0
$801$	−0.226087	−0.00798840
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.60803	−0.0919209
$806$	0	0
$807$	26.8261	0.944325
$808$	0	0
$809$	−35.5605	−1.25024	−0.625120	−	0.780529i	$-0.714950\pi$
$809$	−35.5605	−1.25024	−0.625120	+	0.780529i	$0.714950\pi$
$810$	0	0
$811$	7.79005	0.273546	0.136773	−	0.990602i	$-0.456327\pi$
$811$	7.79005	0.273546	0.136773	+	0.990602i	$0.456327\pi$
$812$	0	0
$813$	20.5436	0.720497
$814$	0	0
$815$	52.5161	1.83956
$816$	0	0
$817$	24.8856	0.870635
$818$	0	0
$819$	2.69641	0.0942200
$820$	0	0
$821$	6.03130	0.210494	0.105247	−	0.994446i	$-0.466437\pi$
$821$	6.03130	0.210494	0.105247	+	0.994446i	$0.466437\pi$
$822$	0	0
$823$	−50.0562	−1.74485	−0.872425	−	0.488749i	$-0.837454\pi$
$823$	−50.0562	−1.74485	−0.872425	+	0.488749i	$0.837454\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.3258	0.984985	0.492492	−	0.870317i	$-0.336086\pi$
$827$	28.3258	0.984985	0.492492	+	0.870317i	$0.336086\pi$
$828$	0	0
$829$	5.42617	0.188459	0.0942293	−	0.995551i	$-0.469961\pi$
$829$	5.42617	0.188459	0.0942293	+	0.995551i	$0.469961\pi$
$830$	0	0
$831$	−11.0117	−0.381993
$832$	0	0
$833$	24.0626	0.833721
$834$	0	0
$835$	−17.4292	−0.603163
$836$	0	0
$837$	32.1902	1.11266
$838$	0	0
$839$	40.4123	1.39519	0.697594	−	0.716493i	$-0.254254\pi$
$839$	40.4123	1.39519	0.697594	+	0.716493i	$0.254254\pi$
$840$	0	0
$841$	−24.6583	−0.850287
$842$	0	0
$843$	−24.6242	−0.848102
$844$	0	0
$845$	44.8993	1.54458
$846$	0	0
$847$	0	0
$848$	0	0
$849$	16.3119	0.559824
$850$	0	0
$851$	11.3499	0.389071
$852$	0	0
$853$	24.0406	0.823133	0.411566	−	0.911380i	$-0.364982\pi$
$853$	24.0406	0.823133	0.411566	+	0.911380i	$0.364982\pi$
$854$	0	0
$855$	3.00374	0.102726
$856$	0	0
$857$	42.9442	1.46695	0.733474	−	0.679718i	$-0.237898\pi$
$857$	42.9442	1.46695	0.733474	+	0.679718i	$0.237898\pi$
$858$	0	0
$859$	−12.9486	−0.441800	−0.220900	−	0.975296i	$-0.570899\pi$
$859$	−12.9486	−0.441800	−0.220900	+	0.975296i	$0.570899\pi$
$860$	0	0
$861$	7.99318	0.272407
$862$	0	0
$863$	11.7658	0.400513	0.200256	−	0.979744i	$-0.435822\pi$
$863$	11.7658	0.400513	0.200256	+	0.979744i	$0.435822\pi$
$864$	0	0
$865$	−4.18993	−0.142462
$866$	0	0
$867$	2.22576	0.0755909
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−70.6547	−2.39404
$872$	0	0
$873$	−4.35431	−0.147371
$874$	0	0
$875$	10.9555	0.370364
$876$	0	0
$877$	41.4239	1.39879	0.699393	−	0.714738i	$-0.253454\pi$
$877$	41.4239	1.39879	0.699393	+	0.714738i	$0.253454\pi$
$878$	0	0
$879$	−2.82886	−0.0954150
$880$	0	0
$881$	−44.7091	−1.50629	−0.753143	−	0.657857i	$-0.771463\pi$
$881$	−44.7091	−1.50629	−0.753143	+	0.657857i	$0.771463\pi$
$882$	0	0
$883$	21.9335	0.738119	0.369060	−	0.929406i	$-0.379680\pi$
$883$	21.9335	0.738119	0.369060	+	0.929406i	$0.379680\pi$
$884$	0	0
$885$	14.6253	0.491623
$886$	0	0
$887$	17.4780	0.586855	0.293427	−	0.955981i	$-0.405204\pi$
$887$	17.4780	0.586855	0.293427	+	0.955981i	$0.405204\pi$
$888$	0	0
$889$	13.1756	0.441897
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.06793	0.136128
$894$	0	0
$895$	−6.80933	−0.227611
$896$	0	0
$897$	12.9687	0.433011
$898$	0	0
$899$	−14.2702	−0.475936
$900$	0	0
$901$	−40.7119	−1.35631
$902$	0	0
$903$	16.0621	0.534513
$904$	0	0
$905$	−14.9329	−0.496386
$906$	0	0
$907$	−16.4521	−0.546282	−0.273141	−	0.961974i	$-0.588063\pi$
$907$	−16.4521	−0.546282	−0.273141	+	0.961974i	$0.588063\pi$
$908$	0	0
$909$	−6.36377	−0.211073
$910$	0	0
$911$	−15.1884	−0.503214	−0.251607	−	0.967829i	$-0.580959\pi$
$911$	−15.1884	−0.503214	−0.251607	+	0.967829i	$0.580959\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	17.1361	0.566503
$916$	0	0
$917$	11.5107	0.380118
$918$	0	0
$919$	−41.8476	−1.38042	−0.690212	−	0.723607i	$-0.742483\pi$
$919$	−41.8476	−1.38042	−0.690212	+	0.723607i	$0.742483\pi$
$920$	0	0
$921$	10.3276	0.340307
$922$	0	0
$923$	54.4346	1.79174
$924$	0	0
$925$	−0.627363	−0.0206276
$926$	0	0
$927$	−7.30504	−0.239929
$928$	0	0
$929$	0.313961	0.0103007	0.00515037	−	0.999987i	$-0.498361\pi$
$929$	0.313961	0.0103007	0.00515037	+	0.999987i	$0.498361\pi$
$930$	0	0
$931$	−17.0307	−0.558157
$932$	0	0
$933$	−17.0901	−0.559505
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−10.5890	−0.345927	−0.172963	−	0.984928i	$-0.555334\pi$
$937$	−10.5890	−0.345927	−0.172963	+	0.984928i	$0.555334\pi$
$938$	0	0
$939$	−30.3620	−0.990828
$940$	0	0
$941$	−58.8538	−1.91858	−0.959290	−	0.282422i	$-0.908862\pi$
$941$	−58.8538	−1.91858	−0.959290	+	0.282422i	$0.908862\pi$
$942$	0	0
$943$	5.30830	0.172862
$944$	0	0
$945$	−10.1633	−0.330612
$946$	0	0
$947$	−15.0139	−0.487887	−0.243943	−	0.969789i	$-0.578441\pi$
$947$	−15.0139	−0.487887	−0.243943	+	0.969789i	$0.578441\pi$
$948$	0	0
$949$	38.1603	1.23874
$950$	0	0
$951$	−6.61635	−0.214550
$952$	0	0
$953$	10.7187	0.347213	0.173607	−	0.984815i	$-0.444458\pi$
$953$	10.7187	0.347213	0.173607	+	0.984815i	$0.444458\pi$
$954$	0	0
$955$	−4.01015	−0.129765
$956$	0	0
$957$	0	0
$958$	0	0
$959$	16.2924	0.526108
$960$	0	0
$961$	15.9028	0.512993
$962$	0	0
$963$	9.85761	0.317657
$964$	0	0
$965$	−1.80099	−0.0579759
$966$	0	0
$967$	43.1296	1.38695	0.693477	−	0.720479i	$-0.256078\pi$
$967$	43.1296	1.38695	0.693477	+	0.720479i	$0.256078\pi$
$968$	0	0
$969$	20.8720	0.670504
$970$	0	0
$971$	16.5561	0.531310	0.265655	−	0.964068i	$-0.414412\pi$
$971$	16.5561	0.531310	0.265655	+	0.964068i	$0.414412\pi$
$972$	0	0
$973$	−11.2442	−0.360473
$974$	0	0
$975$	−0.716836	−0.0229571
$976$	0	0
$977$	1.56159	0.0499597	0.0249799	−	0.999688i	$-0.492048\pi$
$977$	1.56159	0.0499597	0.0249799	+	0.999688i	$0.492048\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−2.30526	−0.0736013
$982$	0	0
$983$	−48.6006	−1.55012	−0.775059	−	0.631889i	$-0.782280\pi$
$983$	−48.6006	−1.55012	−0.775059	+	0.631889i	$0.782280\pi$
$984$	0	0
$985$	−41.3469	−1.31742
$986$	0	0
$987$	2.62560	0.0835737
$988$	0	0
$989$	10.6669	0.339187
$990$	0	0
$991$	13.0806	0.415518	0.207759	−	0.978180i	$-0.433383\pi$
$991$	13.0806	0.415518	0.207759	+	0.978180i	$0.433383\pi$
$992$	0	0
$993$	−30.2852	−0.961072
$994$	0	0
$995$	−37.3389	−1.18372
$996$	0	0
$997$	−30.5715	−0.968211	−0.484105	−	0.875010i	$-0.660855\pi$
$997$	−30.5715	−0.968211	−0.484105	+	0.875010i	$0.660855\pi$
$998$	0	0
$999$	44.2299	1.39937

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7744.2.a.dv.1.1		6
4.3	odd	2		7744.2.a.du.1.6		6
8.3	odd	2		3872.2.a.bq.1.1		6
8.5	even	2		3872.2.a.bn.1.6		6
11.3	even	5		704.2.m.n.449.3		12
11.4	even	5		704.2.m.n.577.3		12
11.10	odd	2		7744.2.a.dw.1.1		6
44.3	odd	10		704.2.m.m.449.1		12
44.15	odd	10		704.2.m.m.577.1		12
44.43	even	2		7744.2.a.dt.1.6		6
88.3	odd	10		352.2.m.e.97.3	✓	12
88.21	odd	2		3872.2.a.bo.1.6		6
88.37	even	10		352.2.m.f.225.1	yes	12
88.43	even	2		3872.2.a.bp.1.1		6
88.59	odd	10		352.2.m.e.225.3	yes	12
88.69	even	10		352.2.m.f.97.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
352.2.m.e.97.3	✓	12	88.3	odd	10
352.2.m.e.225.3	yes	12	88.59	odd	10
352.2.m.f.97.1	yes	12	88.69	even	10
352.2.m.f.225.1	yes	12	88.37	even	10
704.2.m.m.449.1		12	44.3	odd	10
704.2.m.m.577.1		12	44.15	odd	10
704.2.m.n.449.3		12	11.3	even	5
704.2.m.n.577.3		12	11.4	even	5
3872.2.a.bn.1.6		6	8.5	even	2
3872.2.a.bo.1.6		6	88.21	odd	2
3872.2.a.bp.1.1		6	88.43	even	2
3872.2.a.bq.1.1		6	8.3	odd	2
7744.2.a.dt.1.6		6	44.43	even	2
7744.2.a.du.1.6		6	4.3	odd	2
7744.2.a.dv.1.1		6	1.1	even	1	trivial
7744.2.a.dw.1.1		6	11.10	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)

\(f(q)\)	\(=\)	\(q-1.86564 q^{3} +2.22111 q^{5} -0.973509 q^{7} +0.480595 q^{9} -5.76323 q^{13} -4.14378 q^{15} -3.97580 q^{17} +2.81392 q^{19} +1.81621 q^{21} +1.20615 q^{23} -0.0666695 q^{25} +4.70029 q^{27} -2.08367 q^{29} +6.84856 q^{31} -2.16227 q^{35} +9.41004 q^{37} +10.7521 q^{39} +4.40102 q^{41} +8.84372 q^{43} +1.06745 q^{45} +1.44564 q^{47} -6.05228 q^{49} +7.41739 q^{51} +10.2399 q^{53} -5.24976 q^{57} -3.52945 q^{59} -4.13538 q^{61} -0.467864 q^{63} -12.8008 q^{65} +12.2596 q^{67} -2.25024 q^{69} -9.44515 q^{71} -6.62134 q^{73} +0.124381 q^{75} +7.76432 q^{79} -10.2108 q^{81} -17.2597 q^{83} -8.83068 q^{85} +3.88737 q^{87} -0.470432 q^{89} +5.61056 q^{91} -12.7769 q^{93} +6.25003 q^{95} -9.06025 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} - 2 q^{7} + 7 q^{9} - 4 q^{13} - 8 q^{15} - 9 q^{17} + 5 q^{19} - 12 q^{21} - 6 q^{23} + 4 q^{25} + 26 q^{27} - 10 q^{29} - 12 q^{31} - 26 q^{35} + 12 q^{37} - 2 q^{39} - 17 q^{41} + 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100}) \) 6 * q + 5 * q^3 - 2 * q^7 + 7 * q^9 - 4 * q^13 - 8 * q^15 - 9 * q^17 + 5 * q^19 - 12 * q^21 - 6 * q^23 + 4 * q^25 + 26 * q^27 - 10 * q^29 - 12 * q^31 - 26 * q^35 + 12 * q^37 - 2 * q^39 - 17 * q^41 + 11 * q^43 - 14 * q^45 - 22 * q^47 - 6 * q^51 - 18 * q^53 - 6 * q^57 + 7 * q^59 - 18 * q^61 - 26 * q^63 - 26 * q^65 + 31 * q^67 - 16 * q^69 - 22 * q^71 + q^73 + 11 * q^75 + 28 * q^79 + 34 * q^81 - 13 * q^83 - 2 * q^85 - 38 * q^87 - q^89 + 26 * q^91 - 34 * q^93 - 2 * q^95 - 21 * q^97

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