LMFDB - Embedded newform 7448.2.a.bo.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 13x^{4} + 50x^{3} - 53x^{2} - 25x + 28 $ x^7 - x^6 - 14x^5 + 13x^4 + 50x^3 - 53x^2 - 25*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.13946$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.13946 q^{3} -3.27400 q^{5} -1.70164 q^{9} +O(q^{10})$	$q-1.13946 q^{3} -3.27400 q^{5} -1.70164 q^{9} -1.97279 q^{11} -0.682878 q^{13} +3.73058 q^{15} -1.78449 q^{17} -1.00000 q^{19} +1.76178 q^{23} +5.71907 q^{25} +5.35731 q^{27} +3.74137 q^{29} -5.29276 q^{31} +2.24791 q^{33} +9.86746 q^{37} +0.778109 q^{39} +5.13101 q^{41} +8.66516 q^{43} +5.57117 q^{45} +1.79063 q^{47} +2.03335 q^{51} -0.579980 q^{53} +6.45892 q^{55} +1.13946 q^{57} +3.54556 q^{59} -5.79063 q^{61} +2.23574 q^{65} -8.48295 q^{67} -2.00747 q^{69} +1.95738 q^{71} -2.77976 q^{73} -6.51663 q^{75} -0.161751 q^{79} -0.999497 q^{81} +8.45213 q^{83} +5.84241 q^{85} -4.26312 q^{87} +6.62966 q^{89} +6.03087 q^{93} +3.27400 q^{95} -8.22000 q^{97} +3.35698 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{3} - q^{5} + 8 q^{9}+O(q^{10}) $ 7 * q - q^3 - q^5 + 8 * q^9	$ 7 q - q^{3} - q^{5} + 8 q^{9} + 3 q^{11} - 6 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{19} - 2 q^{23} + 8 q^{25} + 2 q^{27} + 3 q^{29} + 6 q^{31} - 21 q^{33} - 7 q^{37} - 2 q^{39} - 9 q^{41} + q^{43} - 24 q^{45} - 15 q^{47} + 6 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 9 q^{59} - 13 q^{61} - 6 q^{65} - 2 q^{67} + 20 q^{69} - q^{71} - 42 q^{73} - 40 q^{75} - 3 q^{79} - q^{81} + 12 q^{83} - 16 q^{85} + 2 q^{87} - 41 q^{89} - 2 q^{93} + q^{95} - 5 q^{97} + 9 q^{99}+O(q^{100}) $ 7 * q - q^3 - q^5 + 8 * q^9 + 3 * q^11 - 6 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^19 - 2 * q^23 + 8 * q^25 + 2 * q^27 + 3 * q^29 + 6 * q^31 - 21 * q^33 - 7 * q^37 - 2 * q^39 - 9 * q^41 + q^43 - 24 * q^45 - 15 * q^47 + 6 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 9 * q^59 - 13 * q^61 - 6 * q^65 - 2 * q^67 + 20 * q^69 - q^71 - 42 * q^73 - 40 * q^75 - 3 * q^79 - q^81 + 12 * q^83 - 16 * q^85 + 2 * q^87 - 41 * q^89 - 2 * q^93 + q^95 - 5 * q^97 + 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.13946	−0.657865	−0.328933	−	0.944353i	$-0.606689\pi$
$3$	−1.13946	−0.657865	−0.328933	+	0.944353i	$0.606689\pi$
$4$	0	0
$5$	−3.27400	−1.46418	−0.732088	−	0.681210i	$-0.761454\pi$
$5$	−3.27400	−1.46418	−0.732088	+	0.681210i	$0.761454\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.70164	−0.567214
$10$	0	0
$11$	−1.97279	−0.594819	−0.297410	−	0.954750i	$-0.596123\pi$
$11$	−1.97279	−0.594819	−0.297410	+	0.954750i	$0.596123\pi$
$12$	0	0
$13$	−0.682878	−0.189396	−0.0946981	−	0.995506i	$-0.530189\pi$
$13$	−0.682878	−0.189396	−0.0946981	+	0.995506i	$0.530189\pi$
$14$	0	0
$15$	3.73058	0.963231
$16$	0	0
$17$	−1.78449	−0.432802	−0.216401	−	0.976305i	$-0.569432\pi$
$17$	−1.78449	−0.432802	−0.216401	+	0.976305i	$0.569432\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.76178	0.367357	0.183678	−	0.982986i	$-0.441200\pi$
$23$	1.76178	0.367357	0.183678	+	0.982986i	$0.441200\pi$
$24$	0	0
$25$	5.71907	1.14381
$26$	0	0
$27$	5.35731	1.03102
$28$	0	0
$29$	3.74137	0.694754	0.347377	−	0.937726i	$-0.387072\pi$
$29$	3.74137	0.694754	0.347377	+	0.937726i	$0.387072\pi$
$30$	0	0
$31$	−5.29276	−0.950608	−0.475304	−	0.879822i	$-0.657662\pi$
$31$	−5.29276	−0.950608	−0.475304	+	0.879822i	$0.657662\pi$
$32$	0	0
$33$	2.24791	0.391311
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.86746	1.62220	0.811101	−	0.584907i	$-0.198869\pi$
$37$	9.86746	1.62220	0.811101	+	0.584907i	$0.198869\pi$
$38$	0	0
$39$	0.778109	0.124597
$40$	0	0
$41$	5.13101	0.801329	0.400665	−	0.916225i	$-0.368779\pi$
$41$	5.13101	0.801329	0.400665	+	0.916225i	$0.368779\pi$
$42$	0	0
$43$	8.66516	1.32142	0.660712	−	0.750639i	$-0.270254\pi$
$43$	8.66516	1.32142	0.660712	+	0.750639i	$0.270254\pi$
$44$	0	0
$45$	5.57117	0.830501
$46$	0	0
$47$	1.79063	0.261190	0.130595	−	0.991436i	$-0.458311\pi$
$47$	1.79063	0.261190	0.130595	+	0.991436i	$0.458311\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.03335	0.284725
$52$	0	0
$53$	−0.579980	−0.0796664	−0.0398332	−	0.999206i	$-0.512683\pi$
$53$	−0.579980	−0.0796664	−0.0398332	+	0.999206i	$0.512683\pi$
$54$	0	0
$55$	6.45892	0.870921
$56$	0	0
$57$	1.13946	0.150925
$58$	0	0
$59$	3.54556	0.461593	0.230797	−	0.973002i	$-0.425867\pi$
$59$	3.54556	0.461593	0.230797	+	0.973002i	$0.425867\pi$
$60$	0	0
$61$	−5.79063	−0.741414	−0.370707	−	0.928750i	$-0.620885\pi$
$61$	−5.79063	−0.741414	−0.370707	+	0.928750i	$0.620885\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.23574	0.277310
$66$	0	0
$67$	−8.48295	−1.03636	−0.518178	−	0.855273i	$-0.673390\pi$
$67$	−8.48295	−1.03636	−0.518178	+	0.855273i	$0.673390\pi$
$68$	0	0
$69$	−2.00747	−0.241671
$70$	0	0
$71$	1.95738	0.232298	0.116149	−	0.993232i	$-0.462945\pi$
$71$	1.95738	0.232298	0.116149	+	0.993232i	$0.462945\pi$
$72$	0	0
$73$	−2.77976	−0.325347	−0.162673	−	0.986680i	$-0.552012\pi$
$73$	−2.77976	−0.325347	−0.162673	+	0.986680i	$0.552012\pi$
$74$	0	0
$75$	−6.51663	−0.752475
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−0.161751	−0.0181984	−0.00909919	−	0.999959i	$-0.502896\pi$
$79$	−0.161751	−0.0181984	−0.00909919	+	0.999959i	$0.502896\pi$
$80$	0	0
$81$	−0.999497	−0.111055
$82$	0	0
$83$	8.45213	0.927742	0.463871	−	0.885903i	$-0.346460\pi$
$83$	8.45213	0.927742	0.463871	+	0.885903i	$0.346460\pi$
$84$	0	0
$85$	5.84241	0.633699
$86$	0	0
$87$	−4.26312	−0.457054
$88$	0	0
$89$	6.62966	0.702742	0.351371	−	0.936236i	$-0.385716\pi$
$89$	6.62966	0.702742	0.351371	+	0.936236i	$0.385716\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.03087	0.625372
$94$	0	0
$95$	3.27400	0.335905
$96$	0	0
$97$	−8.22000	−0.834614	−0.417307	−	0.908766i	$-0.637026\pi$
$97$	−8.22000	−0.834614	−0.417307	+	0.908766i	$0.637026\pi$
$98$	0	0
$99$	3.35698	0.337390
$100$	0	0
$101$	−6.19909	−0.616833	−0.308416	−	0.951251i	$-0.599799\pi$
$101$	−6.19909	−0.616833	−0.308416	+	0.951251i	$0.599799\pi$
$102$	0	0
$103$	−16.2031	−1.59654	−0.798271	−	0.602298i	$-0.794252\pi$
$103$	−16.2031	−1.59654	−0.798271	+	0.602298i	$0.794252\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.23341	−0.505932	−0.252966	−	0.967475i	$-0.581406\pi$
$107$	−5.23341	−0.505932	−0.252966	+	0.967475i	$0.581406\pi$
$108$	0	0
$109$	9.42735	0.902977	0.451488	−	0.892277i	$-0.350893\pi$
$109$	9.42735	0.902977	0.451488	+	0.892277i	$0.350893\pi$
$110$	0	0
$111$	−11.2435	−1.06719
$112$	0	0
$113$	−0.579705	−0.0545341	−0.0272671	−	0.999628i	$-0.508680\pi$
$113$	−0.579705	−0.0545341	−0.0272671	+	0.999628i	$0.508680\pi$
$114$	0	0
$115$	−5.76807	−0.537875
$116$	0	0
$117$	1.16201	0.107428
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.10809	−0.646190
$122$	0	0
$123$	−5.84656	−0.527167
$124$	0	0
$125$	−2.35424	−0.210569
$126$	0	0
$127$	13.3396	1.18370	0.591849	−	0.806049i	$-0.298398\pi$
$127$	13.3396	1.18370	0.591849	+	0.806049i	$0.298398\pi$
$128$	0	0
$129$	−9.87356	−0.869319
$130$	0	0
$131$	−2.43655	−0.212882	−0.106441	−	0.994319i	$-0.533946\pi$
$131$	−2.43655	−0.212882	−0.106441	+	0.994319i	$0.533946\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−17.5398	−1.50959
$136$	0	0
$137$	16.8197	1.43700	0.718500	−	0.695527i	$-0.244829\pi$
$137$	16.8197	1.43700	0.718500	+	0.695527i	$0.244829\pi$
$138$	0	0
$139$	−5.88129	−0.498844	−0.249422	−	0.968395i	$-0.580241\pi$
$139$	−5.88129	−0.498844	−0.249422	+	0.968395i	$0.580241\pi$
$140$	0	0
$141$	−2.04034	−0.171828
$142$	0	0
$143$	1.34718	0.112657
$144$	0	0
$145$	−12.2492	−1.01724
$146$	0	0
$147$	0	0
$148$	0	0
$149$	18.2145	1.49219	0.746096	−	0.665838i	$-0.231926\pi$
$149$	18.2145	1.49219	0.746096	+	0.665838i	$0.231926\pi$
$150$	0	0
$151$	−11.1520	−0.907541	−0.453770	−	0.891119i	$-0.649921\pi$
$151$	−11.1520	−0.907541	−0.453770	+	0.891119i	$0.649921\pi$
$152$	0	0
$153$	3.03656	0.245491
$154$	0	0
$155$	17.3285	1.39186
$156$	0	0
$157$	−1.30364	−0.104042	−0.0520208	−	0.998646i	$-0.516566\pi$
$157$	−1.30364	−0.104042	−0.0520208	+	0.998646i	$0.516566\pi$
$158$	0	0
$159$	0.660862	0.0524097
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−8.56937	−0.671205	−0.335603	−	0.942004i	$-0.608940\pi$
$163$	−8.56937	−0.671205	−0.335603	+	0.942004i	$0.608940\pi$
$164$	0	0
$165$	−7.35966	−0.572948
$166$	0	0
$167$	12.4994	0.967230	0.483615	−	0.875281i	$-0.339324\pi$
$167$	12.4994	0.967230	0.483615	+	0.875281i	$0.339324\pi$
$168$	0	0
$169$	−12.5337	−0.964129
$170$	0	0
$171$	1.70164	0.130128
$172$	0	0
$173$	−6.86098	−0.521630	−0.260815	−	0.965389i	$-0.583991\pi$
$173$	−6.86098	−0.521630	−0.260815	+	0.965389i	$0.583991\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.04001	−0.303666
$178$	0	0
$179$	14.8751	1.11182	0.555908	−	0.831244i	$-0.312371\pi$
$179$	14.8751	1.11182	0.555908	+	0.831244i	$0.312371\pi$
$180$	0	0
$181$	−2.71293	−0.201651	−0.100825	−	0.994904i	$-0.532148\pi$
$181$	−2.71293	−0.201651	−0.100825	+	0.994904i	$0.532148\pi$
$182$	0	0
$183$	6.59816	0.487750
$184$	0	0
$185$	−32.3061	−2.37519
$186$	0	0
$187$	3.52043	0.257439
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.0918	1.30908	0.654540	−	0.756028i	$-0.272862\pi$
$191$	18.0918	1.30908	0.654540	+	0.756028i	$0.272862\pi$
$192$	0	0
$193$	10.1820	0.732918	0.366459	−	0.930434i	$-0.380570\pi$
$193$	10.1820	0.732918	0.366459	+	0.930434i	$0.380570\pi$
$194$	0	0
$195$	−2.54753	−0.182432
$196$	0	0
$197$	−12.4262	−0.885333	−0.442666	−	0.896686i	$-0.645967\pi$
$197$	−12.4262	−0.885333	−0.442666	+	0.896686i	$0.645967\pi$
$198$	0	0
$199$	−1.12519	−0.0797628	−0.0398814	−	0.999204i	$-0.512698\pi$
$199$	−1.12519	−0.0797628	−0.0398814	+	0.999204i	$0.512698\pi$
$200$	0	0
$201$	9.66594	0.681783
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−16.7989	−1.17329
$206$	0	0
$207$	−2.99792	−0.208370
$208$	0	0
$209$	1.97279	0.136461
$210$	0	0
$211$	5.91712	0.407351	0.203676	−	0.979038i	$-0.434711\pi$
$211$	5.91712	0.407351	0.203676	+	0.979038i	$0.434711\pi$
$212$	0	0
$213$	−2.23035	−0.152821
$214$	0	0
$215$	−28.3697	−1.93480
$216$	0	0
$217$	0	0
$218$	0	0
$219$	3.16742	0.214034
$220$	0	0
$221$	1.21859	0.0819711
$222$	0	0
$223$	6.40167	0.428687	0.214344	−	0.976758i	$-0.431239\pi$
$223$	6.40167	0.428687	0.214344	+	0.976758i	$0.431239\pi$
$224$	0	0
$225$	−9.73180	−0.648787
$226$	0	0
$227$	12.3309	0.818430	0.409215	−	0.912438i	$-0.365803\pi$
$227$	12.3309	0.818430	0.409215	+	0.912438i	$0.365803\pi$
$228$	0	0
$229$	−12.7784	−0.844419	−0.422209	−	0.906498i	$-0.638745\pi$
$229$	−12.7784	−0.844419	−0.422209	+	0.906498i	$0.638745\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.41267	−0.223571	−0.111786	−	0.993732i	$-0.535657\pi$
$233$	−3.41267	−0.223571	−0.111786	+	0.993732i	$0.535657\pi$
$234$	0	0
$235$	−5.86251	−0.382428
$236$	0	0
$237$	0.184308	0.0119721
$238$	0	0
$239$	−18.2380	−1.17972	−0.589861	−	0.807505i	$-0.700817\pi$
$239$	−18.2380	−1.17972	−0.589861	+	0.807505i	$0.700817\pi$
$240$	0	0
$241$	−14.3514	−0.924454	−0.462227	−	0.886762i	$-0.652950\pi$
$241$	−14.3514	−0.924454	−0.462227	+	0.886762i	$0.652950\pi$
$242$	0	0
$243$	−14.9331	−0.957956
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.682878	0.0434505
$248$	0	0
$249$	−9.63083	−0.610329
$250$	0	0
$251$	13.1945	0.832832	0.416416	−	0.909174i	$-0.363286\pi$
$251$	13.1945	0.832832	0.416416	+	0.909174i	$0.363286\pi$
$252$	0	0
$253$	−3.47563	−0.218511
$254$	0	0
$255$	−6.65717	−0.416888
$256$	0	0
$257$	−18.7406	−1.16901	−0.584504	−	0.811391i	$-0.698711\pi$
$257$	−18.7406	−1.16901	−0.584504	+	0.811391i	$0.698711\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.36646	−0.394074
$262$	0	0
$263$	−2.63500	−0.162481	−0.0812404	−	0.996695i	$-0.525888\pi$
$263$	−2.63500	−0.162481	−0.0812404	+	0.996695i	$0.525888\pi$
$264$	0	0
$265$	1.89885	0.116646
$266$	0	0
$267$	−7.55420	−0.462310
$268$	0	0
$269$	−29.8782	−1.82171	−0.910853	−	0.412731i	$-0.864575\pi$
$269$	−29.8782	−1.82171	−0.910853	+	0.412731i	$0.864575\pi$
$270$	0	0
$271$	−13.6224	−0.827504	−0.413752	−	0.910390i	$-0.635782\pi$
$271$	−13.6224	−0.827504	−0.413752	+	0.910390i	$0.635782\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.2825	−0.680363
$276$	0	0
$277$	28.6716	1.72271	0.861355	−	0.508004i	$-0.169617\pi$
$277$	28.6716	1.72271	0.861355	+	0.508004i	$0.169617\pi$
$278$	0	0
$279$	9.00638	0.539198
$280$	0	0
$281$	−21.9244	−1.30790	−0.653951	−	0.756537i	$-0.726890\pi$
$281$	−21.9244	−1.30790	−0.653951	+	0.756537i	$0.726890\pi$
$282$	0	0
$283$	25.5286	1.51752	0.758758	−	0.651373i	$-0.225806\pi$
$283$	25.5286	1.51752	0.758758	+	0.651373i	$0.225806\pi$
$284$	0	0
$285$	−3.73058	−0.220980
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−13.8156	−0.812682
$290$	0	0
$291$	9.36632	0.549064
$292$	0	0
$293$	−4.76582	−0.278422	−0.139211	−	0.990263i	$-0.544457\pi$
$293$	−4.76582	−0.278422	−0.139211	+	0.990263i	$0.544457\pi$
$294$	0	0
$295$	−11.6082	−0.675854
$296$	0	0
$297$	−10.5689	−0.613268
$298$	0	0
$299$	−1.20308	−0.0695760
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7.06359	0.405793
$304$	0	0
$305$	18.9585	1.08556
$306$	0	0
$307$	21.5708	1.23111	0.615554	−	0.788095i	$-0.288932\pi$
$307$	21.5708	1.23111	0.615554	+	0.788095i	$0.288932\pi$
$308$	0	0
$309$	18.4628	1.05031
$310$	0	0
$311$	4.47756	0.253899	0.126950	−	0.991909i	$-0.459481\pi$
$311$	4.47756	0.253899	0.126950	+	0.991909i	$0.459481\pi$
$312$	0	0
$313$	−21.1779	−1.19705	−0.598524	−	0.801105i	$-0.704246\pi$
$313$	−21.1779	−1.19705	−0.598524	+	0.801105i	$0.704246\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.13801	−0.232414	−0.116207	−	0.993225i	$-0.537074\pi$
$317$	−4.13801	−0.232414	−0.116207	+	0.993225i	$0.537074\pi$
$318$	0	0
$319$	−7.38094	−0.413253
$320$	0	0
$321$	5.96323	0.332835
$322$	0	0
$323$	1.78449	0.0992916
$324$	0	0
$325$	−3.90543	−0.216634
$326$	0	0
$327$	−10.7421	−0.594037
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−9.64167	−0.529954	−0.264977	−	0.964255i	$-0.585364\pi$
$331$	−9.64167	−0.529954	−0.264977	+	0.964255i	$0.585364\pi$
$332$	0	0
$333$	−16.7909	−0.920134
$334$	0	0
$335$	27.7732	1.51741
$336$	0	0
$337$	−26.1546	−1.42473	−0.712365	−	0.701809i	$-0.752376\pi$
$337$	−26.1546	−1.42473	−0.712365	+	0.701809i	$0.752376\pi$
$338$	0	0
$339$	0.660549	0.0358761
$340$	0	0
$341$	10.4415	0.565440
$342$	0	0
$343$	0	0
$344$	0	0
$345$	6.57246	0.353849
$346$	0	0
$347$	8.08063	0.433791	0.216896	−	0.976195i	$-0.430407\pi$
$347$	8.08063	0.433791	0.216896	+	0.976195i	$0.430407\pi$
$348$	0	0
$349$	−21.4682	−1.14917	−0.574584	−	0.818445i	$-0.694836\pi$
$349$	−21.4682	−1.14917	−0.574584	+	0.818445i	$0.694836\pi$
$350$	0	0
$351$	−3.65839	−0.195270
$352$	0	0
$353$	23.1131	1.23019	0.615094	−	0.788454i	$-0.289118\pi$
$353$	23.1131	1.23019	0.615094	+	0.788454i	$0.289118\pi$
$354$	0	0
$355$	−6.40846	−0.340126
$356$	0	0
$357$	0	0
$358$	0	0
$359$	12.9970	0.685953	0.342977	−	0.939344i	$-0.388565\pi$
$359$	12.9970	0.685953	0.342977	+	0.939344i	$0.388565\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	8.09935	0.425106
$364$	0	0
$365$	9.10094	0.476365
$366$	0	0
$367$	−21.5571	−1.12527	−0.562637	−	0.826704i	$-0.690213\pi$
$367$	−21.5571	−1.12527	−0.562637	+	0.826704i	$0.690213\pi$
$368$	0	0
$369$	−8.73114	−0.454525
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4.63758	0.240125	0.120062	−	0.992766i	$-0.461691\pi$
$373$	4.63758	0.240125	0.120062	+	0.992766i	$0.461691\pi$
$374$	0	0
$375$	2.68255	0.138526
$376$	0	0
$377$	−2.55490	−0.131584
$378$	0	0
$379$	−25.7660	−1.32351	−0.661755	−	0.749721i	$-0.730188\pi$
$379$	−25.7660	−1.32351	−0.661755	+	0.749721i	$0.730188\pi$
$380$	0	0
$381$	−15.1999	−0.778713
$382$	0	0
$383$	−7.14949	−0.365322	−0.182661	−	0.983176i	$-0.558471\pi$
$383$	−7.14949	−0.365322	−0.182661	+	0.983176i	$0.558471\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−14.7450	−0.749530
$388$	0	0
$389$	−14.6967	−0.745152	−0.372576	−	0.928002i	$-0.621525\pi$
$389$	−14.6967	−0.745152	−0.372576	+	0.928002i	$0.621525\pi$
$390$	0	0
$391$	−3.14388	−0.158993
$392$	0	0
$393$	2.77634	0.140048
$394$	0	0
$395$	0.529572	0.0266456
$396$	0	0
$397$	35.5637	1.78489	0.892446	−	0.451154i	$-0.148988\pi$
$397$	35.5637	1.78489	0.892446	+	0.451154i	$0.148988\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−19.5840	−0.977978	−0.488989	−	0.872290i	$-0.662634\pi$
$401$	−19.5840	−0.977978	−0.488989	+	0.872290i	$0.662634\pi$
$402$	0	0
$403$	3.61431	0.180042
$404$	0	0
$405$	3.27235	0.162604
$406$	0	0
$407$	−19.4665	−0.964917
$408$	0	0
$409$	−23.3882	−1.15647	−0.578236	−	0.815870i	$-0.696259\pi$
$409$	−23.3882	−1.15647	−0.578236	+	0.815870i	$0.696259\pi$
$410$	0	0
$411$	−19.1653	−0.945352
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−27.6723	−1.35838
$416$	0	0
$417$	6.70146	0.328172
$418$	0	0
$419$	14.8069	0.723366	0.361683	−	0.932301i	$-0.382202\pi$
$419$	14.8069	0.723366	0.361683	+	0.932301i	$0.382202\pi$
$420$	0	0
$421$	11.0471	0.538404	0.269202	−	0.963084i	$-0.413240\pi$
$421$	11.0471	0.538404	0.269202	+	0.963084i	$0.413240\pi$
$422$	0	0
$423$	−3.04700	−0.148150
$424$	0	0
$425$	−10.2056	−0.495045
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.53505	−0.0741128
$430$	0	0
$431$	−1.33807	−0.0644526	−0.0322263	−	0.999481i	$-0.510260\pi$
$431$	−1.33807	−0.0644526	−0.0322263	+	0.999481i	$0.510260\pi$
$432$	0	0
$433$	12.5315	0.602226	0.301113	−	0.953588i	$-0.402642\pi$
$433$	12.5315	0.602226	0.301113	+	0.953588i	$0.402642\pi$
$434$	0	0
$435$	13.9575	0.669209
$436$	0	0
$437$	−1.76178	−0.0842774
$438$	0	0
$439$	17.9604	0.857205	0.428602	−	0.903493i	$-0.359006\pi$
$439$	17.9604	0.857205	0.428602	+	0.903493i	$0.359006\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	21.8285	1.03710	0.518552	−	0.855046i	$-0.326471\pi$
$443$	21.8285	1.03710	0.518552	+	0.855046i	$0.326471\pi$
$444$	0	0
$445$	−21.7055	−1.02894
$446$	0	0
$447$	−20.7547	−0.981661
$448$	0	0
$449$	−39.4566	−1.86207	−0.931035	−	0.364929i	$-0.881093\pi$
$449$	−39.4566	−1.86207	−0.931035	+	0.364929i	$0.881093\pi$
$450$	0	0
$451$	−10.1224	−0.476646
$452$	0	0
$453$	12.7073	0.597039
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33.2909	−1.55728	−0.778642	−	0.627469i	$-0.784091\pi$
$457$	−33.2909	−1.55728	−0.778642	+	0.627469i	$0.784091\pi$
$458$	0	0
$459$	−9.56006	−0.446225
$460$	0	0
$461$	25.3922	1.18263	0.591315	−	0.806440i	$-0.298609\pi$
$461$	25.3922	1.18263	0.591315	+	0.806440i	$0.298609\pi$
$462$	0	0
$463$	−15.4985	−0.720278	−0.360139	−	0.932899i	$-0.617271\pi$
$463$	−15.4985	−0.720278	−0.360139	+	0.932899i	$0.617271\pi$
$464$	0	0
$465$	−19.7451	−0.915655
$466$	0	0
$467$	5.47078	0.253157	0.126579	−	0.991957i	$-0.459600\pi$
$467$	5.47078	0.253157	0.126579	+	0.991957i	$0.459600\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	1.48544	0.0684453
$472$	0	0
$473$	−17.0946	−0.786009
$474$	0	0
$475$	−5.71907	−0.262409
$476$	0	0
$477$	0.986918	0.0451879
$478$	0	0
$479$	3.21537	0.146914	0.0734569	−	0.997298i	$-0.476597\pi$
$479$	3.21537	0.146914	0.0734569	+	0.997298i	$0.476597\pi$
$480$	0	0
$481$	−6.73827	−0.307239
$482$	0	0
$483$	0	0
$484$	0	0
$485$	26.9123	1.22202
$486$	0	0
$487$	−40.4712	−1.83393	−0.916963	−	0.398971i	$-0.869367\pi$
$487$	−40.4712	−1.83393	−0.916963	+	0.398971i	$0.869367\pi$
$488$	0	0
$489$	9.76442	0.441562
$490$	0	0
$491$	−7.30252	−0.329558	−0.164779	−	0.986330i	$-0.552691\pi$
$491$	−7.30252	−0.329558	−0.164779	+	0.986330i	$0.552691\pi$
$492$	0	0
$493$	−6.67642	−0.300691
$494$	0	0
$495$	−10.9908	−0.493998
$496$	0	0
$497$	0	0
$498$	0	0
$499$	24.6871	1.10514	0.552572	−	0.833465i	$-0.313646\pi$
$499$	24.6871	1.10514	0.552572	+	0.833465i	$0.313646\pi$
$500$	0	0
$501$	−14.2425	−0.636307
$502$	0	0
$503$	10.1955	0.454596	0.227298	−	0.973825i	$-0.427011\pi$
$503$	10.1955	0.454596	0.227298	+	0.973825i	$0.427011\pi$
$504$	0	0
$505$	20.2958	0.903152
$506$	0	0
$507$	14.2816	0.634267
$508$	0	0
$509$	−12.8405	−0.569145	−0.284572	−	0.958655i	$-0.591852\pi$
$509$	−12.8405	−0.569145	−0.284572	+	0.958655i	$0.591852\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.35731	−0.236531
$514$	0	0
$515$	53.0490	2.33762
$516$	0	0
$517$	−3.53253	−0.155361
$518$	0	0
$519$	7.81778	0.343162
$520$	0	0
$521$	−10.6515	−0.466650	−0.233325	−	0.972399i	$-0.574961\pi$
$521$	−10.6515	−0.466650	−0.233325	+	0.972399i	$0.574961\pi$
$522$	0	0
$523$	7.85199	0.343344	0.171672	−	0.985154i	$-0.445083\pi$
$523$	7.85199	0.343344	0.171672	+	0.985154i	$0.445083\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.44487	0.411425
$528$	0	0
$529$	−19.8961	−0.865049
$530$	0	0
$531$	−6.03328	−0.261822
$532$	0	0
$533$	−3.50385	−0.151769
$534$	0	0
$535$	17.1342	0.740775
$536$	0	0
$537$	−16.9495	−0.731424
$538$	0	0
$539$	0	0
$540$	0	0
$541$	9.99100	0.429547	0.214773	−	0.976664i	$-0.431099\pi$
$541$	9.99100	0.429547	0.214773	+	0.976664i	$0.431099\pi$
$542$	0	0
$543$	3.09127	0.132659
$544$	0	0
$545$	−30.8651	−1.32212
$546$	0	0
$547$	−4.00912	−0.171418	−0.0857089	−	0.996320i	$-0.527315\pi$
$547$	−4.00912	−0.171418	−0.0857089	+	0.996320i	$0.527315\pi$
$548$	0	0
$549$	9.85357	0.420540
$550$	0	0
$551$	−3.74137	−0.159388
$552$	0	0
$553$	0	0
$554$	0	0
$555$	36.8113	1.56255
$556$	0	0
$557$	−19.5727	−0.829324	−0.414662	−	0.909976i	$-0.636100\pi$
$557$	−19.5727	−0.829324	−0.414662	+	0.909976i	$0.636100\pi$
$558$	0	0
$559$	−5.91725	−0.250273
$560$	0	0
$561$	−4.01137	−0.169360
$562$	0	0
$563$	−35.5624	−1.49878	−0.749388	−	0.662131i	$-0.769652\pi$
$563$	−35.5624	−1.49878	−0.749388	+	0.662131i	$0.769652\pi$
$564$	0	0
$565$	1.89796	0.0798476
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16.0750	0.673899	0.336950	−	0.941523i	$-0.390605\pi$
$569$	16.0750	0.673899	0.336950	+	0.941523i	$0.390605\pi$
$570$	0	0
$571$	10.2549	0.429156	0.214578	−	0.976707i	$-0.431162\pi$
$571$	10.2549	0.429156	0.214578	+	0.976707i	$0.431162\pi$
$572$	0	0
$573$	−20.6148	−0.861198
$574$	0	0
$575$	10.0757	0.420188
$576$	0	0
$577$	−19.5901	−0.815546	−0.407773	−	0.913083i	$-0.633694\pi$
$577$	−19.5901	−0.815546	−0.407773	+	0.913083i	$0.633694\pi$
$578$	0	0
$579$	−11.6020	−0.482161
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.14418	0.0473871
$584$	0	0
$585$	−3.80443	−0.157294
$586$	0	0
$587$	−20.4591	−0.844436	−0.422218	−	0.906494i	$-0.638748\pi$
$587$	−20.4591	−0.844436	−0.422218	+	0.906494i	$0.638748\pi$
$588$	0	0
$589$	5.29276	0.218084
$590$	0	0
$591$	14.1591	0.582429
$592$	0	0
$593$	−1.71138	−0.0702782	−0.0351391	−	0.999382i	$-0.511187\pi$
$593$	−1.71138	−0.0702782	−0.0351391	+	0.999382i	$0.511187\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.28211	0.0524732
$598$	0	0
$599$	−6.91714	−0.282627	−0.141313	−	0.989965i	$-0.545132\pi$
$599$	−6.91714	−0.282627	−0.141313	+	0.989965i	$0.545132\pi$
$600$	0	0
$601$	45.0866	1.83912	0.919560	−	0.392950i	$-0.128545\pi$
$601$	45.0866	1.83912	0.919560	+	0.392950i	$0.128545\pi$
$602$	0	0
$603$	14.4349	0.587836
$604$	0	0
$605$	23.2719	0.946136
$606$	0	0
$607$	20.3086	0.824300	0.412150	−	0.911116i	$-0.364778\pi$
$607$	20.3086	0.824300	0.412150	+	0.911116i	$0.364778\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.22278	−0.0494683
$612$	0	0
$613$	−7.52107	−0.303773	−0.151887	−	0.988398i	$-0.548535\pi$
$613$	−7.52107	−0.303773	−0.151887	+	0.988398i	$0.548535\pi$
$614$	0	0
$615$	19.1416	0.771865
$616$	0	0
$617$	1.04667	0.0421372	0.0210686	−	0.999778i	$-0.493293\pi$
$617$	1.04667	0.0421372	0.0210686	+	0.999778i	$0.493293\pi$
$618$	0	0
$619$	11.1371	0.447638	0.223819	−	0.974631i	$-0.428147\pi$
$619$	11.1371	0.447638	0.223819	+	0.974631i	$0.428147\pi$
$620$	0	0
$621$	9.43841	0.378750
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−20.8876	−0.835503
$626$	0	0
$627$	−2.24791	−0.0897729
$628$	0	0
$629$	−17.6084	−0.702092
$630$	0	0
$631$	−29.0078	−1.15478	−0.577391	−	0.816468i	$-0.695929\pi$
$631$	−29.0078	−1.15478	−0.577391	+	0.816468i	$0.695929\pi$
$632$	0	0
$633$	−6.74230	−0.267982
$634$	0	0
$635$	−43.6738	−1.73314
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.33076	−0.131763
$640$	0	0
$641$	−5.86183	−0.231528	−0.115764	−	0.993277i	$-0.536932\pi$
$641$	−5.86183	−0.231528	−0.115764	+	0.993277i	$0.536932\pi$
$642$	0	0
$643$	17.3986	0.686134	0.343067	−	0.939311i	$-0.388534\pi$
$643$	17.3986	0.686134	0.343067	+	0.939311i	$0.388534\pi$
$644$	0	0
$645$	32.3260	1.27284
$646$	0	0
$647$	20.5154	0.806545	0.403273	−	0.915080i	$-0.367873\pi$
$647$	20.5154	0.806545	0.403273	+	0.915080i	$0.367873\pi$
$648$	0	0
$649$	−6.99466	−0.274564
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.43765	0.173659	0.0868293	−	0.996223i	$-0.472327\pi$
$653$	4.43765	0.173659	0.0868293	+	0.996223i	$0.472327\pi$
$654$	0	0
$655$	7.97726	0.311698
$656$	0	0
$657$	4.73016	0.184541
$658$	0	0
$659$	15.3319	0.597245	0.298622	−	0.954371i	$-0.403473\pi$
$659$	15.3319	0.597245	0.298622	+	0.954371i	$0.403473\pi$
$660$	0	0
$661$	−49.2968	−1.91742	−0.958712	−	0.284379i	$-0.908212\pi$
$661$	−49.2968	−1.91742	−0.958712	+	0.284379i	$0.908212\pi$
$662$	0	0
$663$	−1.38853	−0.0539259
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.59147	0.255223
$668$	0	0
$669$	−7.29442	−0.282018
$670$	0	0
$671$	11.4237	0.441007
$672$	0	0
$673$	10.9839	0.423399	0.211699	−	0.977335i	$-0.432100\pi$
$673$	10.9839	0.423399	0.211699	+	0.977335i	$0.432100\pi$
$674$	0	0
$675$	30.6388	1.17929
$676$	0	0
$677$	−37.7842	−1.45216	−0.726082	−	0.687608i	$-0.758661\pi$
$677$	−37.7842	−1.45216	−0.726082	+	0.687608i	$0.758661\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14.0505	−0.538416
$682$	0	0
$683$	16.9169	0.647307	0.323654	−	0.946176i	$-0.395089\pi$
$683$	16.9169	0.647307	0.323654	+	0.946176i	$0.395089\pi$
$684$	0	0
$685$	−55.0676	−2.10402
$686$	0	0
$687$	14.5604	0.555514
$688$	0	0
$689$	0.396056	0.0150885
$690$	0	0
$691$	−15.1700	−0.577093	−0.288547	−	0.957466i	$-0.593172\pi$
$691$	−15.1700	−0.577093	−0.288547	+	0.957466i	$0.593172\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	19.2553	0.730396
$696$	0	0
$697$	−9.15623	−0.346817
$698$	0	0
$699$	3.88859	0.147080
$700$	0	0
$701$	−7.49053	−0.282914	−0.141457	−	0.989944i	$-0.545179\pi$
$701$	−7.49053	−0.282914	−0.141457	+	0.989944i	$0.545179\pi$
$702$	0	0
$703$	−9.86746	−0.372158
$704$	0	0
$705$	6.68007	0.251586
$706$	0	0
$707$	0	0
$708$	0	0
$709$	5.43385	0.204072	0.102036	−	0.994781i	$-0.467464\pi$
$709$	5.43385	0.204072	0.102036	+	0.994781i	$0.467464\pi$
$710$	0	0
$711$	0.275242	0.0103224
$712$	0	0
$713$	−9.32469	−0.349212
$714$	0	0
$715$	−4.41065	−0.164949
$716$	0	0
$717$	20.7814	0.776097
$718$	0	0
$719$	52.5305	1.95906	0.979529	−	0.201302i	$-0.0645172\pi$
$719$	52.5305	1.95906	0.979529	+	0.201302i	$0.0645172\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	16.3528	0.608166
$724$	0	0
$725$	21.3971	0.794669
$726$	0	0
$727$	7.70787	0.285869	0.142935	−	0.989732i	$-0.454346\pi$
$727$	7.70787	0.285869	0.142935	+	0.989732i	$0.454346\pi$
$728$	0	0
$729$	20.0140	0.741261
$730$	0	0
$731$	−15.4629	−0.571915
$732$	0	0
$733$	46.5655	1.71994	0.859968	−	0.510349i	$-0.170484\pi$
$733$	46.5655	1.71994	0.859968	+	0.510349i	$0.170484\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.7351	0.616445
$738$	0	0
$739$	−13.9901	−0.514635	−0.257318	−	0.966327i	$-0.582839\pi$
$739$	−13.9901	−0.514635	−0.257318	+	0.966327i	$0.582839\pi$
$740$	0	0
$741$	−0.778109	−0.0285846
$742$	0	0
$743$	2.63020	0.0964926	0.0482463	−	0.998835i	$-0.484637\pi$
$743$	2.63020	0.0964926	0.0482463	+	0.998835i	$0.484637\pi$
$744$	0	0
$745$	−59.6344	−2.18483
$746$	0	0
$747$	−14.3825	−0.526228
$748$	0	0
$749$	0	0
$750$	0	0
$751$	32.1461	1.17303	0.586514	−	0.809939i	$-0.300500\pi$
$751$	32.1461	1.17303	0.586514	+	0.809939i	$0.300500\pi$
$752$	0	0
$753$	−15.0346	−0.547891
$754$	0	0
$755$	36.5118	1.32880
$756$	0	0
$757$	51.8753	1.88544	0.942719	−	0.333587i	$-0.108259\pi$
$757$	51.8753	1.88544	0.942719	+	0.333587i	$0.108259\pi$
$758$	0	0
$759$	3.96032	0.143751
$760$	0	0
$761$	−23.2826	−0.843994	−0.421997	−	0.906597i	$-0.638671\pi$
$761$	−23.2826	−0.843994	−0.421997	+	0.906597i	$0.638671\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−9.94169	−0.359442
$766$	0	0
$767$	−2.42119	−0.0874240
$768$	0	0
$769$	−14.0009	−0.504886	−0.252443	−	0.967612i	$-0.581234\pi$
$769$	−14.0009	−0.504886	−0.252443	+	0.967612i	$0.581234\pi$
$770$	0	0
$771$	21.3541	0.769050
$772$	0	0
$773$	4.11389	0.147966	0.0739831	−	0.997259i	$-0.476429\pi$
$773$	4.11389	0.147966	0.0739831	+	0.997259i	$0.476429\pi$
$774$	0	0
$775$	−30.2697	−1.08732
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.13101	−0.183838
$780$	0	0
$781$	−3.86151	−0.138176
$782$	0	0
$783$	20.0437	0.716302
$784$	0	0
$785$	4.26811	0.152335
$786$	0	0
$787$	−17.1488	−0.611289	−0.305645	−	0.952146i	$-0.598872\pi$
$787$	−17.1488	−0.611289	−0.305645	+	0.952146i	$0.598872\pi$
$788$	0	0
$789$	3.00246	0.106890
$790$	0	0
$791$	0	0
$792$	0	0
$793$	3.95429	0.140421
$794$	0	0
$795$	−2.16366	−0.0767371
$796$	0	0
$797$	−4.33905	−0.153697	−0.0768486	−	0.997043i	$-0.524486\pi$
$797$	−4.33905	−0.153697	−0.0768486	+	0.997043i	$0.524486\pi$
$798$	0	0
$799$	−3.19535	−0.113043
$800$	0	0
$801$	−11.2813	−0.398605
$802$	0	0
$803$	5.48389	0.193522
$804$	0	0
$805$	0	0
$806$	0	0
$807$	34.0449	1.19844
$808$	0	0
$809$	23.9695	0.842723	0.421362	−	0.906893i	$-0.361552\pi$
$809$	23.9695	0.842723	0.421362	+	0.906893i	$0.361552\pi$
$810$	0	0
$811$	15.8866	0.557853	0.278927	−	0.960312i	$-0.410021\pi$
$811$	15.8866	0.557853	0.278927	+	0.960312i	$0.410021\pi$
$812$	0	0
$813$	15.5222	0.544386
$814$	0	0
$815$	28.0561	0.982763
$816$	0	0
$817$	−8.66516	−0.303156
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.8816	0.693871	0.346936	−	0.937889i	$-0.387222\pi$
$821$	19.8816	0.693871	0.346936	+	0.937889i	$0.387222\pi$
$822$	0	0
$823$	−46.8908	−1.63451	−0.817256	−	0.576275i	$-0.804506\pi$
$823$	−46.8908	−1.63451	−0.817256	+	0.576275i	$0.804506\pi$
$824$	0	0
$825$	12.8560	0.447587
$826$	0	0
$827$	−54.2638	−1.88694	−0.943469	−	0.331461i	$-0.892458\pi$
$827$	−54.2638	−1.88694	−0.943469	+	0.331461i	$0.892458\pi$
$828$	0	0
$829$	−29.4027	−1.02120	−0.510599	−	0.859819i	$-0.670576\pi$
$829$	−29.4027	−1.02120	−0.510599	+	0.859819i	$0.670576\pi$
$830$	0	0
$831$	−32.6700	−1.13331
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−40.9229	−1.41620
$836$	0	0
$837$	−28.3550	−0.980091
$838$	0	0
$839$	39.3747	1.35937	0.679683	−	0.733506i	$-0.262117\pi$
$839$	39.3747	1.35937	0.679683	+	0.733506i	$0.262117\pi$
$840$	0	0
$841$	−15.0022	−0.517317
$842$	0	0
$843$	24.9819	0.860423
$844$	0	0
$845$	41.0353	1.41166
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−29.0887	−0.998321
$850$	0	0
$851$	17.3843	0.595926
$852$	0	0
$853$	−53.7402	−1.84003	−0.920015	−	0.391882i	$-0.871824\pi$
$853$	−53.7402	−1.84003	−0.920015	+	0.391882i	$0.871824\pi$
$854$	0	0
$855$	−5.57117	−0.190530
$856$	0	0
$857$	25.4926	0.870809	0.435405	−	0.900235i	$-0.356605\pi$
$857$	25.4926	0.870809	0.435405	+	0.900235i	$0.356605\pi$
$858$	0	0
$859$	−27.9383	−0.953243	−0.476622	−	0.879109i	$-0.658139\pi$
$859$	−27.9383	−0.953243	−0.476622	+	0.879109i	$0.658139\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−7.68162	−0.261485	−0.130743	−	0.991416i	$-0.541736\pi$
$863$	−7.68162	−0.261485	−0.130743	+	0.991416i	$0.541736\pi$
$864$	0	0
$865$	22.4628	0.763759
$866$	0	0
$867$	15.7423	0.534635
$868$	0	0
$869$	0.319101	0.0108247
$870$	0	0
$871$	5.79282	0.196282
$872$	0	0
$873$	13.9875	0.473405
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.1507	0.444067	0.222034	−	0.975039i	$-0.428731\pi$
$877$	13.1507	0.444067	0.222034	+	0.975039i	$0.428731\pi$
$878$	0	0
$879$	5.43044	0.183164
$880$	0	0
$881$	−42.6715	−1.43764	−0.718819	−	0.695197i	$-0.755317\pi$
$881$	−42.6715	−1.43764	−0.718819	+	0.695197i	$0.755317\pi$
$882$	0	0
$883$	0.993670	0.0334396	0.0167198	−	0.999860i	$-0.494678\pi$
$883$	0.993670	0.0334396	0.0167198	+	0.999860i	$0.494678\pi$
$884$	0	0
$885$	13.2270	0.444621
$886$	0	0
$887$	−30.4879	−1.02368	−0.511842	−	0.859079i	$-0.671037\pi$
$887$	−30.4879	−1.02368	−0.511842	+	0.859079i	$0.671037\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.97180	0.0660578
$892$	0	0
$893$	−1.79063	−0.0599210
$894$	0	0
$895$	−48.7010	−1.62789
$896$	0	0
$897$	1.37086	0.0457716
$898$	0	0
$899$	−19.8022	−0.660439
$900$	0	0
$901$	1.03497	0.0344798
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8.88214	0.295252
$906$	0	0
$907$	−47.5186	−1.57783	−0.788915	−	0.614502i	$-0.789357\pi$
$907$	−47.5186	−1.57783	−0.788915	+	0.614502i	$0.789357\pi$
$908$	0	0
$909$	10.5486	0.349876
$910$	0	0
$911$	19.3352	0.640603	0.320302	−	0.947316i	$-0.396216\pi$
$911$	19.3352	0.640603	0.320302	+	0.947316i	$0.396216\pi$
$912$	0	0
$913$	−16.6743	−0.551839
$914$	0	0
$915$	−21.6024	−0.714153
$916$	0	0
$917$	0	0
$918$	0	0
$919$	36.1132	1.19126	0.595632	−	0.803257i	$-0.296902\pi$
$919$	36.1132	1.19126	0.595632	+	0.803257i	$0.296902\pi$
$920$	0	0
$921$	−24.5789	−0.809903
$922$	0	0
$923$	−1.33665	−0.0439964
$924$	0	0
$925$	56.4327	1.85550
$926$	0	0
$927$	27.5719	0.905580
$928$	0	0
$929$	28.2153	0.925714	0.462857	−	0.886433i	$-0.346824\pi$
$929$	28.2153	0.925714	0.462857	+	0.886433i	$0.346824\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5.10198	−0.167031
$934$	0	0
$935$	−11.5259	−0.376936
$936$	0	0
$937$	−15.2550	−0.498358	−0.249179	−	0.968458i	$-0.580161\pi$
$937$	−15.2550	−0.498358	−0.249179	+	0.968458i	$0.580161\pi$
$938$	0	0
$939$	24.1313	0.787496
$940$	0	0
$941$	31.0140	1.01103	0.505514	−	0.862819i	$-0.331303\pi$
$941$	31.0140	1.01103	0.505514	+	0.862819i	$0.331303\pi$
$942$	0	0
$943$	9.03972	0.294374
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.96098	−0.226202	−0.113101	−	0.993584i	$-0.536078\pi$
$947$	−6.96098	−0.226202	−0.113101	+	0.993584i	$0.536078\pi$
$948$	0	0
$949$	1.89824	0.0616194
$950$	0	0
$951$	4.71508	0.152897
$952$	0	0
$953$	−25.4996	−0.826014	−0.413007	−	0.910728i	$-0.635521\pi$
$953$	−25.4996	−0.826014	−0.413007	+	0.910728i	$0.635521\pi$
$954$	0	0
$955$	−59.2327	−1.91672
$956$	0	0
$957$	8.41025	0.271865
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.98667	−0.0963442
$962$	0	0
$963$	8.90538	0.286972
$964$	0	0
$965$	−33.3359	−1.07312
$966$	0	0
$967$	1.87600	0.0603280	0.0301640	−	0.999545i	$-0.490397\pi$
$967$	1.87600	0.0603280	0.0301640	+	0.999545i	$0.490397\pi$
$968$	0	0
$969$	−2.03335	−0.0653205
$970$	0	0
$971$	59.5065	1.90965	0.954827	−	0.297162i	$-0.0960402\pi$
$971$	59.5065	1.90965	0.954827	+	0.297162i	$0.0960402\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.45006	0.142516
$976$	0	0
$977$	−27.5507	−0.881426	−0.440713	−	0.897648i	$-0.645274\pi$
$977$	−27.5507	−0.881426	−0.440713	+	0.897648i	$0.645274\pi$
$978$	0	0
$979$	−13.0789	−0.418005
$980$	0	0
$981$	−16.0420	−0.512181
$982$	0	0
$983$	28.1136	0.896683	0.448342	−	0.893862i	$-0.352015\pi$
$983$	28.1136	0.896683	0.448342	+	0.893862i	$0.352015\pi$
$984$	0	0
$985$	40.6835	1.29628
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.2661	0.485434
$990$	0	0
$991$	50.3916	1.60074	0.800371	−	0.599505i	$-0.204636\pi$
$991$	50.3916	1.60074	0.800371	+	0.599505i	$0.204636\pi$
$992$	0	0
$993$	10.9863	0.348638
$994$	0	0
$995$	3.68388	0.116787
$996$	0	0
$997$	36.3612	1.15157	0.575785	−	0.817601i	$-0.304697\pi$
$997$	36.3612	1.15157	0.575785	+	0.817601i	$0.304697\pi$
$998$	0	0
$999$	52.8631	1.67251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bo.1.3	✓	7
7.6	odd	2		7448.2.a.bp.1.5	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bo.1.3	✓	7	1.1	even	1	trivial
7448.2.a.bp.1.5	yes	7	7.6	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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q-1.13946 q^{3} -3.27400 q^{5} -1.70164 q^{9} +O(q^{10})\)	\(q-1.13946 q^{3} -3.27400 q^{5} -1.70164 q^{9} -1.97279 q^{11} -0.682878 q^{13} +3.73058 q^{15} -1.78449 q^{17} -1.00000 q^{19} +1.76178 q^{23} +5.71907 q^{25} +5.35731 q^{27} +3.74137 q^{29} -5.29276 q^{31} +2.24791 q^{33} +9.86746 q^{37} +0.778109 q^{39} +5.13101 q^{41} +8.66516 q^{43} +5.57117 q^{45} +1.79063 q^{47} +2.03335 q^{51} -0.579980 q^{53} +6.45892 q^{55} +1.13946 q^{57} +3.54556 q^{59} -5.79063 q^{61} +2.23574 q^{65} -8.48295 q^{67} -2.00747 q^{69} +1.95738 q^{71} -2.77976 q^{73} -6.51663 q^{75} -0.161751 q^{79} -0.999497 q^{81} +8.45213 q^{83} +5.84241 q^{85} -4.26312 q^{87} +6.62966 q^{89} +6.03087 q^{93} +3.27400 q^{95} -8.22000 q^{97} +3.35698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} - q^{5} + 8 q^{9}+O(q^{10}) \) 7 * q - q^3 - q^5 + 8 * q^9	\( 7 q - q^{3} - q^{5} + 8 q^{9} + 3 q^{11} - 6 q^{13} - 4 q^{15} - 10 q^{17} - 7 q^{19} - 2 q^{23} + 8 q^{25} + 2 q^{27} + 3 q^{29} + 6 q^{31} - 21 q^{33} - 7 q^{37} - 2 q^{39} - 9 q^{41} + q^{43} - 24 q^{45} - 15 q^{47} + 6 q^{51} + 5 q^{53} + 6 q^{55} + q^{57} + 9 q^{59} - 13 q^{61} - 6 q^{65} - 2 q^{67} + 20 q^{69} - q^{71} - 42 q^{73} - 40 q^{75} - 3 q^{79} - q^{81} + 12 q^{83} - 16 q^{85} + 2 q^{87} - 41 q^{89} - 2 q^{93} + q^{95} - 5 q^{97} + 9 q^{99}+O(q^{100}) \) 7 * q - q^3 - q^5 + 8 * q^9 + 3 * q^11 - 6 * q^13 - 4 * q^15 - 10 * q^17 - 7 * q^19 - 2 * q^23 + 8 * q^25 + 2 * q^27 + 3 * q^29 + 6 * q^31 - 21 * q^33 - 7 * q^37 - 2 * q^39 - 9 * q^41 + q^43 - 24 * q^45 - 15 * q^47 + 6 * q^51 + 5 * q^53 + 6 * q^55 + q^57 + 9 * q^59 - 13 * q^61 - 6 * q^65 - 2 * q^67 + 20 * q^69 - q^71 - 42 * q^73 - 40 * q^75 - 3 * q^79 - q^81 + 12 * q^83 - 16 * q^85 + 2 * q^87 - 41 * q^89 - 2 * q^93 + q^95 - 5 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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