LMFDB - Embedded newform 1856.4.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,4,Mod(1,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1856.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1856.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:	$109.507544971$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.4481.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 17x - 8 $ x^3 - 17*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.86388$ of defining polynomial
Character	$\chi$	$=$	1856.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.72775 q^{3} -10.8591 q^{5} -35.3146 q^{7} +49.1737 q^{9} +O(q^{10})$	$q-8.72775 q^{3} -10.8591 q^{5} -35.3146 q^{7} +49.1737 q^{9} -15.0096 q^{11} -63.2256 q^{13} +94.7754 q^{15} +89.5964 q^{17} -150.367 q^{19} +308.217 q^{21} -91.3993 q^{23} -7.08037 q^{25} -193.526 q^{27} +29.0000 q^{29} +4.11646 q^{31} +131.000 q^{33} +383.484 q^{35} +59.1187 q^{37} +551.817 q^{39} -346.113 q^{41} +483.285 q^{43} -533.981 q^{45} -75.7902 q^{47} +904.120 q^{49} -781.975 q^{51} +191.660 q^{53} +162.990 q^{55} +1312.36 q^{57} +774.911 q^{59} +277.199 q^{61} -1736.55 q^{63} +686.572 q^{65} -8.54382 q^{67} +797.710 q^{69} +72.4644 q^{71} +775.045 q^{73} +61.7957 q^{75} +530.058 q^{77} -694.941 q^{79} +361.361 q^{81} -483.553 q^{83} -972.935 q^{85} -253.105 q^{87} -1026.96 q^{89} +2232.79 q^{91} -35.9274 q^{93} +1632.84 q^{95} +933.115 q^{97} -738.077 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9	$ 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9} - 65 q^{11} - 29 q^{13} - 85 q^{15} + 244 q^{17} - 312 q^{19} + 214 q^{21} - 24 q^{23} + 436 q^{25} - 57 q^{27} + 87 q^{29} + 249 q^{31} + 393 q^{33} + 718 q^{35} - 200 q^{37} + 749 q^{39} - 470 q^{41} - 97 q^{43} - 1562 q^{45} + 377 q^{47} + 383 q^{49} - 612 q^{51} - 1007 q^{53} - 1399 q^{55} + 1128 q^{57} + 364 q^{59} + 524 q^{61} - 2244 q^{63} + 301 q^{65} + 820 q^{67} + 824 q^{69} - 782 q^{71} + 1620 q^{73} + 1356 q^{75} - 158 q^{77} + 427 q^{79} + 227 q^{81} - 1520 q^{83} + 68 q^{85} - 87 q^{87} - 2474 q^{89} + 2002 q^{91} + 1807 q^{93} + 568 q^{95} - 170 q^{97} + 1362 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9 - 65 * q^11 - 29 * q^13 - 85 * q^15 + 244 * q^17 - 312 * q^19 + 214 * q^21 - 24 * q^23 + 436 * q^25 - 57 * q^27 + 87 * q^29 + 249 * q^31 + 393 * q^33 + 718 * q^35 - 200 * q^37 + 749 * q^39 - 470 * q^41 - 97 * q^43 - 1562 * q^45 + 377 * q^47 + 383 * q^49 - 612 * q^51 - 1007 * q^53 - 1399 * q^55 + 1128 * q^57 + 364 * q^59 + 524 * q^61 - 2244 * q^63 + 301 * q^65 + 820 * q^67 + 824 * q^69 - 782 * q^71 + 1620 * q^73 + 1356 * q^75 - 158 * q^77 + 427 * q^79 + 227 * q^81 - 1520 * q^83 + 68 * q^85 - 87 * q^87 - 2474 * q^89 + 2002 * q^91 + 1807 * q^93 + 568 * q^95 - 170 * q^97 + 1362 * 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
$3$	−8.72775	−1.67966	−0.839828	−	0.542852i	$-0.817344\pi$
$3$	−8.72775	−1.67966	−0.839828	+	0.542852i	$0.817344\pi$
$4$	0	0
$5$	−10.8591	−0.971266	−0.485633	−	0.874163i	$-0.661411\pi$
$5$	−10.8591	−0.971266	−0.485633	+	0.874163i	$0.661411\pi$
$6$	0	0
$7$	−35.3146	−1.90681	−0.953404	−	0.301696i	$-0.902447\pi$
$7$	−35.3146	−1.90681	−0.953404	+	0.301696i	$0.902447\pi$
$8$	0	0
$9$	49.1737	1.82125
$10$	0	0
$11$	−15.0096	−0.411415	−0.205707	−	0.978614i	$-0.565950\pi$
$11$	−15.0096	−0.411415	−0.205707	+	0.978614i	$0.565950\pi$
$12$	0	0
$13$	−63.2256	−1.34889	−0.674447	−	0.738323i	$-0.735618\pi$
$13$	−63.2256	−1.34889	−0.674447	+	0.738323i	$0.735618\pi$
$14$	0	0
$15$	94.7754	1.63139
$16$	0	0
$17$	89.5964	1.27825	0.639127	−	0.769101i	$-0.279296\pi$
$17$	89.5964	1.27825	0.639127	+	0.769101i	$0.279296\pi$
$18$	0	0
$19$	−150.367	−1.81560	−0.907801	−	0.419401i	$-0.862240\pi$
$19$	−150.367	−1.81560	−0.907801	+	0.419401i	$0.862240\pi$
$20$	0	0
$21$	308.217	3.20278
$22$	0	0
$23$	−91.3993	−0.828611	−0.414306	−	0.910138i	$-0.635976\pi$
$23$	−91.3993	−0.828611	−0.414306	+	0.910138i	$0.635976\pi$
$24$	0	0
$25$	−7.08037	−0.0566429
$26$	0	0
$27$	−193.526	−1.37941
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	4.11646	0.0238496	0.0119248	−	0.999929i	$-0.496204\pi$
$31$	4.11646	0.0238496	0.0119248	+	0.999929i	$0.496204\pi$
$32$	0	0
$33$	131.000	0.691036
$34$	0	0
$35$	383.484	1.85202
$36$	0	0
$37$	59.1187	0.262677	0.131339	−	0.991338i	$-0.458072\pi$
$37$	59.1187	0.262677	0.131339	+	0.991338i	$0.458072\pi$
$38$	0	0
$39$	551.817	2.26568
$40$	0	0
$41$	−346.113	−1.31839	−0.659193	−	0.751974i	$-0.729102\pi$
$41$	−346.113	−1.31839	−0.659193	+	0.751974i	$0.729102\pi$
$42$	0	0
$43$	483.285	1.71396	0.856979	−	0.515351i	$-0.172338\pi$
$43$	483.285	1.71396	0.856979	+	0.515351i	$0.172338\pi$
$44$	0	0
$45$	−533.981	−1.76891
$46$	0	0
$47$	−75.7902	−0.235216	−0.117608	−	0.993060i	$-0.537523\pi$
$47$	−75.7902	−0.235216	−0.117608	+	0.993060i	$0.537523\pi$
$48$	0	0
$49$	904.120	2.63592
$50$	0	0
$51$	−781.975	−2.14703
$52$	0	0
$53$	191.660	0.496726	0.248363	−	0.968667i	$-0.420107\pi$
$53$	191.660	0.496726	0.248363	+	0.968667i	$0.420107\pi$
$54$	0	0
$55$	162.990	0.399593
$56$	0	0
$57$	1312.36	3.04959
$58$	0	0
$59$	774.911	1.70991	0.854956	−	0.518700i	$-0.173584\pi$
$59$	774.911	1.70991	0.854956	+	0.518700i	$0.173584\pi$
$60$	0	0
$61$	277.199	0.581832	0.290916	−	0.956749i	$-0.406040\pi$
$61$	277.199	0.581832	0.290916	+	0.956749i	$0.406040\pi$
$62$	0	0
$63$	−1736.55	−3.47277
$64$	0	0
$65$	686.572	1.31013
$66$	0	0
$67$	−8.54382	−0.0155790	−0.00778950	−	0.999970i	$-0.502480\pi$
$67$	−8.54382	−0.0155790	−0.00778950	+	0.999970i	$0.502480\pi$
$68$	0	0
$69$	797.710	1.39178
$70$	0	0
$71$	72.4644	0.121126	0.0605629	−	0.998164i	$-0.480710\pi$
$71$	72.4644	0.121126	0.0605629	+	0.998164i	$0.480710\pi$
$72$	0	0
$73$	775.045	1.24263	0.621316	−	0.783560i	$-0.286598\pi$
$73$	775.045	1.24263	0.621316	+	0.783560i	$0.286598\pi$
$74$	0	0
$75$	61.7957	0.0951407
$76$	0	0
$77$	530.058	0.784489
$78$	0	0
$79$	−694.941	−0.989708	−0.494854	−	0.868976i	$-0.664778\pi$
$79$	−694.941	−0.989708	−0.494854	+	0.868976i	$0.664778\pi$
$80$	0	0
$81$	361.361	0.495693
$82$	0	0
$83$	−483.553	−0.639480	−0.319740	−	0.947505i	$-0.603595\pi$
$83$	−483.553	−0.639480	−0.319740	+	0.947505i	$0.603595\pi$
$84$	0	0
$85$	−972.935	−1.24152
$86$	0	0
$87$	−253.105	−0.311904
$88$	0	0
$89$	−1026.96	−1.22312	−0.611560	−	0.791198i	$-0.709458\pi$
$89$	−1026.96	−1.22312	−0.611560	+	0.791198i	$0.709458\pi$
$90$	0	0
$91$	2232.79	2.57208
$92$	0	0
$93$	−35.9274	−0.0400591
$94$	0	0
$95$	1632.84	1.76343
$96$	0	0
$97$	933.115	0.976737	0.488369	−	0.872637i	$-0.337592\pi$
$97$	933.115	0.976737	0.488369	+	0.872637i	$0.337592\pi$
$98$	0	0
$99$	−738.077	−0.749288
$100$	0	0
$101$	−386.547	−0.380821	−0.190410	−	0.981705i	$-0.560982\pi$
$101$	−386.547	−0.380821	−0.190410	+	0.981705i	$0.560982\pi$
$102$	0	0
$103$	−79.7090	−0.0762520	−0.0381260	−	0.999273i	$-0.512139\pi$
$103$	−79.7090	−0.0762520	−0.0381260	+	0.999273i	$0.512139\pi$
$104$	0	0
$105$	−3346.95	−3.11075
$106$	0	0
$107$	−584.929	−0.528479	−0.264239	−	0.964457i	$-0.585121\pi$
$107$	−584.929	−0.528479	−0.264239	+	0.964457i	$0.585121\pi$
$108$	0	0
$109$	1077.96	0.947243	0.473621	−	0.880729i	$-0.342947\pi$
$109$	1077.96	0.947243	0.473621	+	0.880729i	$0.342947\pi$
$110$	0	0
$111$	−515.974	−0.441208
$112$	0	0
$113$	724.967	0.603532	0.301766	−	0.953382i	$-0.402424\pi$
$113$	724.967	0.603532	0.301766	+	0.953382i	$0.402424\pi$
$114$	0	0
$115$	992.512	0.804802
$116$	0	0
$117$	−3109.03	−2.45667
$118$	0	0
$119$	−3164.06	−2.43739
$120$	0	0
$121$	−1105.71	−0.830738
$122$	0	0
$123$	3020.79	2.21443
$124$	0	0
$125$	1434.27	1.02628
$126$	0	0
$127$	−141.196	−0.0986542	−0.0493271	−	0.998783i	$-0.515708\pi$
$127$	−141.196	−0.0986542	−0.0493271	+	0.998783i	$0.515708\pi$
$128$	0	0
$129$	−4217.99	−2.87886
$130$	0	0
$131$	−355.950	−0.237401	−0.118700	−	0.992930i	$-0.537873\pi$
$131$	−355.950	−0.237401	−0.118700	+	0.992930i	$0.537873\pi$
$132$	0	0
$133$	5310.13	3.46201
$134$	0	0
$135$	2101.52	1.33978
$136$	0	0
$137$	1699.52	1.05986	0.529928	−	0.848043i	$-0.322219\pi$
$137$	1699.52	1.05986	0.529928	+	0.848043i	$0.322219\pi$
$138$	0	0
$139$	−689.933	−0.421003	−0.210501	−	0.977594i	$-0.567510\pi$
$139$	−689.933	−0.421003	−0.210501	+	0.977594i	$0.567510\pi$
$140$	0	0
$141$	661.478	0.395082
$142$	0	0
$143$	948.990	0.554955
$144$	0	0
$145$	−314.913	−0.180360
$146$	0	0
$147$	−7890.94	−4.42744
$148$	0	0
$149$	−1970.49	−1.08341	−0.541706	−	0.840568i	$-0.682222\pi$
$149$	−1970.49	−1.08341	−0.541706	+	0.840568i	$0.682222\pi$
$150$	0	0
$151$	1438.10	0.775039	0.387520	−	0.921861i	$-0.373332\pi$
$151$	1438.10	0.775039	0.387520	+	0.921861i	$0.373332\pi$
$152$	0	0
$153$	4405.78	2.32802
$154$	0	0
$155$	−44.7009	−0.0231643
$156$	0	0
$157$	1320.69	0.671353	0.335676	−	0.941977i	$-0.391035\pi$
$157$	1320.69	0.671353	0.335676	+	0.941977i	$0.391035\pi$
$158$	0	0
$159$	−1672.76	−0.834329
$160$	0	0
$161$	3227.73	1.58000
$162$	0	0
$163$	3812.83	1.83217	0.916086	−	0.400982i	$-0.131331\pi$
$163$	3812.83	1.83217	0.916086	+	0.400982i	$0.131331\pi$
$164$	0	0
$165$	−1422.54	−0.671179
$166$	0	0
$167$	940.119	0.435620	0.217810	−	0.975991i	$-0.430109\pi$
$167$	940.119	0.435620	0.217810	+	0.975991i	$0.430109\pi$
$168$	0	0
$169$	1800.48	0.819516
$170$	0	0
$171$	−7394.07	−3.30666
$172$	0	0
$173$	−2691.33	−1.18276	−0.591381	−	0.806392i	$-0.701417\pi$
$173$	−2691.33	−1.18276	−0.591381	+	0.806392i	$0.701417\pi$
$174$	0	0
$175$	250.040	0.108007
$176$	0	0
$177$	−6763.23	−2.87207
$178$	0	0
$179$	−3915.79	−1.63508	−0.817541	−	0.575870i	$-0.804664\pi$
$179$	−3915.79	−1.63508	−0.817541	+	0.575870i	$0.804664\pi$
$180$	0	0
$181$	−3607.97	−1.48165	−0.740824	−	0.671699i	$-0.765565\pi$
$181$	−3607.97	−1.48165	−0.740824	+	0.671699i	$0.765565\pi$
$182$	0	0
$183$	−2419.33	−0.977278
$184$	0	0
$185$	−641.975	−0.255129
$186$	0	0
$187$	−1344.81	−0.525893
$188$	0	0
$189$	6834.30	2.63028
$190$	0	0
$191$	−3158.69	−1.19662	−0.598311	−	0.801264i	$-0.704161\pi$
$191$	−3158.69	−1.19662	−0.598311	+	0.801264i	$0.704161\pi$
$192$	0	0
$193$	2845.74	1.06135	0.530675	−	0.847575i	$-0.321938\pi$
$193$	2845.74	1.06135	0.530675	+	0.847575i	$0.321938\pi$
$194$	0	0
$195$	−5992.23	−2.20058
$196$	0	0
$197$	−456.198	−0.164989	−0.0824943	−	0.996592i	$-0.526289\pi$
$197$	−456.198	−0.164989	−0.0824943	+	0.996592i	$0.526289\pi$
$198$	0	0
$199$	−2123.49	−0.756434	−0.378217	−	0.925717i	$-0.623463\pi$
$199$	−2123.49	−0.756434	−0.378217	+	0.925717i	$0.623463\pi$
$200$	0	0
$201$	74.5683	0.0261674
$202$	0	0
$203$	−1024.12	−0.354085
$204$	0	0
$205$	3758.47	1.28050
$206$	0	0
$207$	−4494.44	−1.50911
$208$	0	0
$209$	2256.94	0.746966
$210$	0	0
$211$	2684.61	0.875905	0.437952	−	0.898998i	$-0.355704\pi$
$211$	2684.61	0.875905	0.437952	+	0.898998i	$0.355704\pi$
$212$	0	0
$213$	−632.451	−0.203450
$214$	0	0
$215$	−5248.03	−1.66471
$216$	0	0
$217$	−145.371	−0.0454766
$218$	0	0
$219$	−6764.40	−2.08719
$220$	0	0
$221$	−5664.79	−1.72423
$222$	0	0
$223$	−4442.57	−1.33407	−0.667033	−	0.745028i	$-0.732436\pi$
$223$	−4442.57	−1.33407	−0.667033	+	0.745028i	$0.732436\pi$
$224$	0	0
$225$	−348.168	−0.103161
$226$	0	0
$227$	−394.373	−0.115310	−0.0576551	−	0.998337i	$-0.518362\pi$
$227$	−394.373	−0.115310	−0.0576551	+	0.998337i	$0.518362\pi$
$228$	0	0
$229$	−1426.84	−0.411740	−0.205870	−	0.978579i	$-0.566002\pi$
$229$	−1426.84	−0.411740	−0.205870	+	0.978579i	$0.566002\pi$
$230$	0	0
$231$	−4626.21	−1.31767
$232$	0	0
$233$	−483.144	−0.135845	−0.0679223	−	0.997691i	$-0.521637\pi$
$233$	−483.144	−0.135845	−0.0679223	+	0.997691i	$0.521637\pi$
$234$	0	0
$235$	823.012	0.228457
$236$	0	0
$237$	6065.27	1.66237
$238$	0	0
$239$	665.454	0.180103	0.0900516	−	0.995937i	$-0.471297\pi$
$239$	665.454	0.180103	0.0900516	+	0.995937i	$0.471297\pi$
$240$	0	0
$241$	722.879	0.193215	0.0966073	−	0.995323i	$-0.469201\pi$
$241$	722.879	0.193215	0.0966073	+	0.995323i	$0.469201\pi$
$242$	0	0
$243$	2071.34	0.546818
$244$	0	0
$245$	−9817.91	−2.56018
$246$	0	0
$247$	9507.01	2.44906
$248$	0	0
$249$	4220.33	1.07411
$250$	0	0
$251$	4416.87	1.11072	0.555359	−	0.831610i	$-0.312581\pi$
$251$	4416.87	1.11072	0.555359	+	0.831610i	$0.312581\pi$
$252$	0	0
$253$	1371.87	0.340903
$254$	0	0
$255$	8491.53	2.08534
$256$	0	0
$257$	−1069.36	−0.259551	−0.129776	−	0.991543i	$-0.541426\pi$
$257$	−1069.36	−0.259551	−0.129776	+	0.991543i	$0.541426\pi$
$258$	0	0
$259$	−2087.75	−0.500875
$260$	0	0
$261$	1426.04	0.338197
$262$	0	0
$263$	4059.01	0.951671	0.475835	−	0.879534i	$-0.342146\pi$
$263$	4059.01	0.951671	0.475835	+	0.879534i	$0.342146\pi$
$264$	0	0
$265$	−2081.25	−0.482453
$266$	0	0
$267$	8963.06	2.05442
$268$	0	0
$269$	6169.01	1.39826	0.699128	−	0.714996i	$-0.253572\pi$
$269$	6169.01	1.39826	0.699128	+	0.714996i	$0.253572\pi$
$270$	0	0
$271$	475.127	0.106501	0.0532507	−	0.998581i	$-0.483042\pi$
$271$	475.127	0.106501	0.0532507	+	0.998581i	$0.483042\pi$
$272$	0	0
$273$	−19487.2	−4.32022
$274$	0	0
$275$	106.273	0.0233037
$276$	0	0
$277$	3182.94	0.690414	0.345207	−	0.938527i	$-0.387809\pi$
$277$	3182.94	0.690414	0.345207	+	0.938527i	$0.387809\pi$
$278$	0	0
$279$	202.421	0.0434360
$280$	0	0
$281$	−6286.93	−1.33469	−0.667343	−	0.744751i	$-0.732568\pi$
$281$	−6286.93	−1.33469	−0.667343	+	0.744751i	$0.732568\pi$
$282$	0	0
$283$	−5170.76	−1.08611	−0.543056	−	0.839697i	$-0.682733\pi$
$283$	−5170.76	−1.08611	−0.543056	+	0.839697i	$0.682733\pi$
$284$	0	0
$285$	−14251.0	−2.96196
$286$	0	0
$287$	12222.8	2.51391
$288$	0	0
$289$	3114.52	0.633934
$290$	0	0
$291$	−8144.00	−1.64058
$292$	0	0
$293$	4967.68	0.990495	0.495248	−	0.868752i	$-0.335077\pi$
$293$	4967.68	0.990495	0.495248	+	0.868752i	$0.335077\pi$
$294$	0	0
$295$	−8414.82	−1.66078
$296$	0	0
$297$	2904.75	0.567511
$298$	0	0
$299$	5778.77	1.11771
$300$	0	0
$301$	−17067.0	−3.26819
$302$	0	0
$303$	3373.69	0.639648
$304$	0	0
$305$	−3010.13	−0.565114
$306$	0	0
$307$	−3533.68	−0.656931	−0.328465	−	0.944516i	$-0.606531\pi$
$307$	−3533.68	−0.656931	−0.328465	+	0.944516i	$0.606531\pi$
$308$	0	0
$309$	695.680	0.128077
$310$	0	0
$311$	−2787.67	−0.508278	−0.254139	−	0.967168i	$-0.581792\pi$
$311$	−2787.67	−0.508278	−0.254139	+	0.967168i	$0.581792\pi$
$312$	0	0
$313$	−4644.65	−0.838758	−0.419379	−	0.907811i	$-0.637752\pi$
$313$	−4644.65	−0.838758	−0.419379	+	0.907811i	$0.637752\pi$
$314$	0	0
$315$	18857.3	3.37298
$316$	0	0
$317$	83.0639	0.0147171	0.00735857	−	0.999973i	$-0.497658\pi$
$317$	83.0639	0.0147171	0.00735857	+	0.999973i	$0.497658\pi$
$318$	0	0
$319$	−435.278	−0.0763978
$320$	0	0
$321$	5105.11	0.887663
$322$	0	0
$323$	−13472.3	−2.32080
$324$	0	0
$325$	447.660	0.0764053
$326$	0	0
$327$	−9408.13	−1.59104
$328$	0	0
$329$	2676.50	0.448511
$330$	0	0
$331$	8782.18	1.45834	0.729172	−	0.684330i	$-0.239905\pi$
$331$	8782.18	1.45834	0.729172	+	0.684330i	$0.239905\pi$
$332$	0	0
$333$	2907.08	0.478400
$334$	0	0
$335$	92.7780	0.0151314
$336$	0	0
$337$	−2006.52	−0.324339	−0.162169	−	0.986763i	$-0.551849\pi$
$337$	−2006.52	−0.324339	−0.162169	+	0.986763i	$0.551849\pi$
$338$	0	0
$339$	−6327.33	−1.01373
$340$	0	0
$341$	−61.7863	−0.00981208
$342$	0	0
$343$	−19815.7	−3.11938
$344$	0	0
$345$	−8662.40	−1.35179
$346$	0	0
$347$	−6333.56	−0.979836	−0.489918	−	0.871769i	$-0.662973\pi$
$347$	−6333.56	−0.979836	−0.489918	+	0.871769i	$0.662973\pi$
$348$	0	0
$349$	5109.62	0.783701	0.391851	−	0.920029i	$-0.371835\pi$
$349$	5109.62	0.783701	0.391851	+	0.920029i	$0.371835\pi$
$350$	0	0
$351$	12235.8	1.86068
$352$	0	0
$353$	−3692.69	−0.556776	−0.278388	−	0.960469i	$-0.589800\pi$
$353$	−3692.69	−0.556776	−0.278388	+	0.960469i	$0.589800\pi$
$354$	0	0
$355$	−786.896	−0.117645
$356$	0	0
$357$	27615.1	4.09397
$358$	0	0
$359$	10348.9	1.52144	0.760719	−	0.649082i	$-0.224847\pi$
$359$	10348.9	1.52144	0.760719	+	0.649082i	$0.224847\pi$
$360$	0	0
$361$	15751.1	2.29641
$362$	0	0
$363$	9650.38	1.39535
$364$	0	0
$365$	−8416.27	−1.20693
$366$	0	0
$367$	−3939.11	−0.560272	−0.280136	−	0.959960i	$-0.590380\pi$
$367$	−3939.11	−0.560272	−0.280136	+	0.959960i	$0.590380\pi$
$368$	0	0
$369$	−17019.7	−2.40110
$370$	0	0
$371$	−6768.38	−0.947161
$372$	0	0
$373$	−2464.89	−0.342164	−0.171082	−	0.985257i	$-0.554726\pi$
$373$	−2464.89	−0.342164	−0.171082	+	0.985257i	$0.554726\pi$
$374$	0	0
$375$	−12518.0	−1.72380
$376$	0	0
$377$	−1833.54	−0.250483
$378$	0	0
$379$	−11748.7	−1.59232	−0.796161	−	0.605085i	$-0.793139\pi$
$379$	−11748.7	−1.59232	−0.796161	+	0.605085i	$0.793139\pi$
$380$	0	0
$381$	1232.32	0.165705
$382$	0	0
$383$	13496.5	1.80063	0.900313	−	0.435243i	$-0.143338\pi$
$383$	13496.5	1.80063	0.900313	+	0.435243i	$0.143338\pi$
$384$	0	0
$385$	−5755.94	−0.761947
$386$	0	0
$387$	23764.9	3.12154
$388$	0	0
$389$	−560.330	−0.0730330	−0.0365165	−	0.999333i	$-0.511626\pi$
$389$	−560.330	−0.0730330	−0.0365165	+	0.999333i	$0.511626\pi$
$390$	0	0
$391$	−8189.05	−1.05918
$392$	0	0
$393$	3106.65	0.398752
$394$	0	0
$395$	7546.42	0.961269
$396$	0	0
$397$	−11464.1	−1.44928	−0.724642	−	0.689125i	$-0.757995\pi$
$397$	−11464.1	−1.44928	−0.724642	+	0.689125i	$0.757995\pi$
$398$	0	0
$399$	−46345.5	−5.81498
$400$	0	0
$401$	6943.53	0.864697	0.432348	−	0.901707i	$-0.357685\pi$
$401$	6943.53	0.864697	0.432348	+	0.901707i	$0.357685\pi$
$402$	0	0
$403$	−260.265	−0.0321706
$404$	0	0
$405$	−3924.04	−0.481450
$406$	0	0
$407$	−887.348	−0.108069
$408$	0	0
$409$	−950.076	−0.114861	−0.0574306	−	0.998350i	$-0.518291\pi$
$409$	−950.076	−0.114861	−0.0574306	+	0.998350i	$0.518291\pi$
$410$	0	0
$411$	−14833.0	−1.78019
$412$	0	0
$413$	−27365.7	−3.26047
$414$	0	0
$415$	5250.94	0.621105
$416$	0	0
$417$	6021.57	0.707140
$418$	0	0
$419$	−2429.81	−0.283303	−0.141651	−	0.989917i	$-0.545241\pi$
$419$	−2429.81	−0.283303	−0.141651	+	0.989917i	$0.545241\pi$
$420$	0	0
$421$	−5879.81	−0.680676	−0.340338	−	0.940303i	$-0.610541\pi$
$421$	−5879.81	−0.680676	−0.340338	+	0.940303i	$0.610541\pi$
$422$	0	0
$423$	−3726.88	−0.428386
$424$	0	0
$425$	−634.375	−0.0724041
$426$	0	0
$427$	−9789.19	−1.10944
$428$	0	0
$429$	−8282.55	−0.932134
$430$	0	0
$431$	15127.1	1.69060	0.845300	−	0.534292i	$-0.179422\pi$
$431$	15127.1	1.69060	0.845300	+	0.534292i	$0.179422\pi$
$432$	0	0
$433$	−4474.46	−0.496602	−0.248301	−	0.968683i	$-0.579872\pi$
$433$	−4474.46	−0.496602	−0.248301	+	0.968683i	$0.579872\pi$
$434$	0	0
$435$	2748.49	0.302942
$436$	0	0
$437$	13743.4	1.50443
$438$	0	0
$439$	−8016.32	−0.871522	−0.435761	−	0.900062i	$-0.643521\pi$
$439$	−8016.32	−0.871522	−0.435761	+	0.900062i	$0.643521\pi$
$440$	0	0
$441$	44458.9	4.80066
$442$	0	0
$443$	14409.7	1.54543	0.772714	−	0.634755i	$-0.218899\pi$
$443$	14409.7	1.54543	0.772714	+	0.634755i	$0.218899\pi$
$444$	0	0
$445$	11151.9	1.18797
$446$	0	0
$447$	17197.9	1.81976
$448$	0	0
$449$	4952.41	0.520532	0.260266	−	0.965537i	$-0.416190\pi$
$449$	4952.41	0.520532	0.260266	+	0.965537i	$0.416190\pi$
$450$	0	0
$451$	5195.02	0.542403
$452$	0	0
$453$	−12551.4	−1.30180
$454$	0	0
$455$	−24246.0	−2.49818
$456$	0	0
$457$	15852.6	1.62265	0.811326	−	0.584594i	$-0.198746\pi$
$457$	15852.6	1.62265	0.811326	+	0.584594i	$0.198746\pi$
$458$	0	0
$459$	−17339.3	−1.76324
$460$	0	0
$461$	−4825.97	−0.487566	−0.243783	−	0.969830i	$-0.578388\pi$
$461$	−4825.97	−0.487566	−0.243783	+	0.969830i	$0.578388\pi$
$462$	0	0
$463$	−2052.70	−0.206042	−0.103021	−	0.994679i	$-0.532851\pi$
$463$	−2052.70	−0.206042	−0.103021	+	0.994679i	$0.532851\pi$
$464$	0	0
$465$	390.139	0.0389081
$466$	0	0
$467$	−6460.13	−0.640127	−0.320063	−	0.947396i	$-0.603704\pi$
$467$	−6460.13	−0.640127	−0.320063	+	0.947396i	$0.603704\pi$
$468$	0	0
$469$	301.721	0.0297062
$470$	0	0
$471$	−11526.6	−1.12764
$472$	0	0
$473$	−7253.91	−0.705148
$474$	0	0
$475$	1064.65	0.102841
$476$	0	0
$477$	9424.60	0.904660
$478$	0	0
$479$	6211.92	0.592547	0.296273	−	0.955103i	$-0.404256\pi$
$479$	6211.92	0.592547	0.296273	+	0.955103i	$0.404256\pi$
$480$	0	0
$481$	−3737.82	−0.354324
$482$	0	0
$483$	−28170.8	−2.65386
$484$	0	0
$485$	−10132.8	−0.948671
$486$	0	0
$487$	14172.0	1.31867	0.659336	−	0.751848i	$-0.270837\pi$
$487$	14172.0	1.31867	0.659336	+	0.751848i	$0.270837\pi$
$488$	0	0
$489$	−33277.4	−3.07742
$490$	0	0
$491$	−4875.68	−0.448140	−0.224070	−	0.974573i	$-0.571934\pi$
$491$	−4875.68	−0.448140	−0.224070	+	0.974573i	$0.571934\pi$
$492$	0	0
$493$	2598.30	0.237366
$494$	0	0
$495$	8014.83	0.727758
$496$	0	0
$497$	−2559.05	−0.230964
$498$	0	0
$499$	−5020.87	−0.450431	−0.225215	−	0.974309i	$-0.572309\pi$
$499$	−5020.87	−0.450431	−0.225215	+	0.974309i	$0.572309\pi$
$500$	0	0
$501$	−8205.13	−0.731693
$502$	0	0
$503$	10619.4	0.941339	0.470669	−	0.882310i	$-0.344012\pi$
$503$	10619.4	0.941339	0.470669	+	0.882310i	$0.344012\pi$
$504$	0	0
$505$	4197.55	0.369878
$506$	0	0
$507$	−15714.1	−1.37651
$508$	0	0
$509$	−3280.05	−0.285630	−0.142815	−	0.989749i	$-0.545615\pi$
$509$	−3280.05	−0.285630	−0.142815	+	0.989749i	$0.545615\pi$
$510$	0	0
$511$	−27370.4	−2.36946
$512$	0	0
$513$	29099.9	2.50447
$514$	0	0
$515$	865.566	0.0740610
$516$	0	0
$517$	1137.58	0.0967712
$518$	0	0
$519$	23489.3	1.98664
$520$	0	0
$521$	−21762.9	−1.83004	−0.915020	−	0.403408i	$-0.867825\pi$
$521$	−21762.9	−1.83004	−0.915020	+	0.403408i	$0.867825\pi$
$522$	0	0
$523$	20790.2	1.73822	0.869112	−	0.494615i	$-0.164691\pi$
$523$	20790.2	1.73822	0.869112	+	0.494615i	$0.164691\pi$
$524$	0	0
$525$	−2182.29	−0.181415
$526$	0	0
$527$	368.820	0.0304859
$528$	0	0
$529$	−3813.17	−0.313403
$530$	0	0
$531$	38105.2	3.11417
$532$	0	0
$533$	21883.2	1.77836
$534$	0	0
$535$	6351.79	0.513293
$536$	0	0
$537$	34176.0	2.74638
$538$	0	0
$539$	−13570.5	−1.08446
$540$	0	0
$541$	23195.8	1.84338	0.921688	−	0.387933i	$-0.126811\pi$
$541$	23195.8	1.84338	0.921688	+	0.387933i	$0.126811\pi$
$542$	0	0
$543$	31489.5	2.48866
$544$	0	0
$545$	−11705.6	−0.920024
$546$	0	0
$547$	−3748.76	−0.293026	−0.146513	−	0.989209i	$-0.546805\pi$
$547$	−3748.76	−0.293026	−0.146513	+	0.989209i	$0.546805\pi$
$548$	0	0
$549$	13630.9	1.05966
$550$	0	0
$551$	−4360.63	−0.337149
$552$	0	0
$553$	24541.5	1.88718
$554$	0	0
$555$	5603.00	0.428530
$556$	0	0
$557$	9464.95	0.720005	0.360003	−	0.932951i	$-0.382776\pi$
$557$	9464.95	0.720005	0.360003	+	0.932951i	$0.382776\pi$
$558$	0	0
$559$	−30556.0	−2.31195
$560$	0	0
$561$	11737.1	0.883319
$562$	0	0
$563$	759.432	0.0568495	0.0284247	−	0.999596i	$-0.490951\pi$
$563$	759.432	0.0568495	0.0284247	+	0.999596i	$0.490951\pi$
$564$	0	0
$565$	−7872.47	−0.586190
$566$	0	0
$567$	−12761.3	−0.945192
$568$	0	0
$569$	6609.15	0.486942	0.243471	−	0.969908i	$-0.421714\pi$
$569$	6609.15	0.486942	0.243471	+	0.969908i	$0.421714\pi$
$570$	0	0
$571$	−6972.10	−0.510986	−0.255493	−	0.966811i	$-0.582238\pi$
$571$	−6972.10	−0.510986	−0.255493	+	0.966811i	$0.582238\pi$
$572$	0	0
$573$	27568.3	2.00992
$574$	0	0
$575$	647.140	0.0469350
$576$	0	0
$577$	1569.50	0.113239	0.0566196	−	0.998396i	$-0.481968\pi$
$577$	1569.50	0.113239	0.0566196	+	0.998396i	$0.481968\pi$
$578$	0	0
$579$	−24836.9	−1.78271
$580$	0	0
$581$	17076.5	1.21937
$582$	0	0
$583$	−2876.73	−0.204360
$584$	0	0
$585$	33761.3	2.38608
$586$	0	0
$587$	25741.0	1.80996	0.904979	−	0.425457i	$-0.139887\pi$
$587$	25741.0	1.80996	0.904979	+	0.425457i	$0.139887\pi$
$588$	0	0
$589$	−618.977	−0.0433014
$590$	0	0
$591$	3981.58	0.277124
$592$	0	0
$593$	11222.8	0.777177	0.388588	−	0.921411i	$-0.372963\pi$
$593$	11222.8	0.777177	0.388588	+	0.921411i	$0.372963\pi$
$594$	0	0
$595$	34358.8	2.36735
$596$	0	0
$597$	18533.3	1.27055
$598$	0	0
$599$	25777.2	1.75831	0.879156	−	0.476533i	$-0.158107\pi$
$599$	25777.2	1.75831	0.879156	+	0.476533i	$0.158107\pi$
$600$	0	0
$601$	−11855.1	−0.804624	−0.402312	−	0.915503i	$-0.631793\pi$
$601$	−11855.1	−0.804624	−0.402312	+	0.915503i	$0.631793\pi$
$602$	0	0
$603$	−420.131	−0.0283732
$604$	0	0
$605$	12007.0	0.806867
$606$	0	0
$607$	7293.09	0.487673	0.243837	−	0.969816i	$-0.421594\pi$
$607$	7293.09	0.487673	0.243837	+	0.969816i	$0.421594\pi$
$608$	0	0
$609$	8938.29	0.594742
$610$	0	0
$611$	4791.88	0.317281
$612$	0	0
$613$	28047.6	1.84801	0.924006	−	0.382377i	$-0.124895\pi$
$613$	28047.6	1.84801	0.924006	+	0.382377i	$0.124895\pi$
$614$	0	0
$615$	−32803.0	−2.15080
$616$	0	0
$617$	−6987.42	−0.455920	−0.227960	−	0.973670i	$-0.573206\pi$
$617$	−6987.42	−0.455920	−0.227960	+	0.973670i	$0.573206\pi$
$618$	0	0
$619$	−6670.44	−0.433131	−0.216565	−	0.976268i	$-0.569485\pi$
$619$	−6670.44	−0.433131	−0.216565	+	0.976268i	$0.569485\pi$
$620$	0	0
$621$	17688.2	1.14300
$622$	0	0
$623$	36266.7	2.33226
$624$	0	0
$625$	−14689.8	−0.940149
$626$	0	0
$627$	−19698.0	−1.25465
$628$	0	0
$629$	5296.83	0.335768
$630$	0	0
$631$	28063.1	1.77048	0.885240	−	0.465135i	$-0.153994\pi$
$631$	28063.1	1.77048	0.885240	+	0.465135i	$0.153994\pi$
$632$	0	0
$633$	−23430.6	−1.47122
$634$	0	0
$635$	1533.26	0.0958195
$636$	0	0
$637$	−57163.5	−3.55558
$638$	0	0
$639$	3563.34	0.220600
$640$	0	0
$641$	−11554.7	−0.711989	−0.355995	−	0.934488i	$-0.615858\pi$
$641$	−11554.7	−0.711989	−0.355995	+	0.934488i	$0.615858\pi$
$642$	0	0
$643$	6197.60	0.380108	0.190054	−	0.981774i	$-0.439134\pi$
$643$	6197.60	0.380108	0.190054	+	0.981774i	$0.439134\pi$
$644$	0	0
$645$	45803.5	2.79614
$646$	0	0
$647$	4932.29	0.299704	0.149852	−	0.988708i	$-0.452120\pi$
$647$	4932.29	0.299704	0.149852	+	0.988708i	$0.452120\pi$
$648$	0	0
$649$	−11631.1	−0.703483
$650$	0	0
$651$	1268.76	0.0763851
$652$	0	0
$653$	−22761.0	−1.36402	−0.682012	−	0.731341i	$-0.738895\pi$
$653$	−22761.0	−1.36402	−0.682012	+	0.731341i	$0.738895\pi$
$654$	0	0
$655$	3865.29	0.230579
$656$	0	0
$657$	38111.8	2.26314
$658$	0	0
$659$	26751.4	1.58131	0.790656	−	0.612260i	$-0.209739\pi$
$659$	26751.4	1.58131	0.790656	+	0.612260i	$0.209739\pi$
$660$	0	0
$661$	2558.77	0.150567	0.0752833	−	0.997162i	$-0.476014\pi$
$661$	2558.77	0.150567	0.0752833	+	0.997162i	$0.476014\pi$
$662$	0	0
$663$	49440.9	2.89611
$664$	0	0
$665$	−57663.1	−3.36253
$666$	0	0
$667$	−2650.58	−0.153869
$668$	0	0
$669$	38773.7	2.24077
$670$	0	0
$671$	−4160.65	−0.239374
$672$	0	0
$673$	12155.6	0.696231	0.348116	−	0.937452i	$-0.386822\pi$
$673$	12155.6	0.696231	0.348116	+	0.937452i	$0.386822\pi$
$674$	0	0
$675$	1370.24	0.0781340
$676$	0	0
$677$	−28508.5	−1.61842	−0.809212	−	0.587517i	$-0.800105\pi$
$677$	−28508.5	−1.61842	−0.809212	+	0.587517i	$0.800105\pi$
$678$	0	0
$679$	−32952.6	−1.86245
$680$	0	0
$681$	3441.99	0.193682
$682$	0	0
$683$	−25318.0	−1.41840	−0.709199	−	0.705008i	$-0.750943\pi$
$683$	−25318.0	−1.41840	−0.709199	+	0.705008i	$0.750943\pi$
$684$	0	0
$685$	−18455.3	−1.02940
$686$	0	0
$687$	12453.1	0.691582
$688$	0	0
$689$	−12117.8	−0.670030
$690$	0	0
$691$	4704.87	0.259018	0.129509	−	0.991578i	$-0.458660\pi$
$691$	4704.87	0.259018	0.129509	+	0.991578i	$0.458660\pi$
$692$	0	0
$693$	26064.9	1.42875
$694$	0	0
$695$	7492.04	0.408905
$696$	0	0
$697$	−31010.5	−1.68523
$698$	0	0
$699$	4216.76	0.228172
$700$	0	0
$701$	−1673.98	−0.0901930	−0.0450965	−	0.998983i	$-0.514360\pi$
$701$	−1673.98	−0.0901930	−0.0450965	+	0.998983i	$0.514360\pi$
$702$	0	0
$703$	−8889.48	−0.476917
$704$	0	0
$705$	−7183.05	−0.383729
$706$	0	0
$707$	13650.8	0.726152
$708$	0	0
$709$	11258.9	0.596386	0.298193	−	0.954506i	$-0.403616\pi$
$709$	11258.9	0.596386	0.298193	+	0.954506i	$0.403616\pi$
$710$	0	0
$711$	−34172.8	−1.80250
$712$	0	0
$713$	−376.241	−0.0197621
$714$	0	0
$715$	−10305.2	−0.539009
$716$	0	0
$717$	−5807.92	−0.302511
$718$	0	0
$719$	−15759.7	−0.817440	−0.408720	−	0.912660i	$-0.634025\pi$
$719$	−15759.7	−0.817440	−0.408720	+	0.912660i	$0.634025\pi$
$720$	0	0
$721$	2814.89	0.145398
$722$	0	0
$723$	−6309.11	−0.324534
$724$	0	0
$725$	−205.331	−0.0105183
$726$	0	0
$727$	−16230.9	−0.828022	−0.414011	−	0.910272i	$-0.635873\pi$
$727$	−16230.9	−0.828022	−0.414011	+	0.910272i	$0.635873\pi$
$728$	0	0
$729$	−27834.9	−1.41416
$730$	0	0
$731$	43300.6	2.19088
$732$	0	0
$733$	−31438.2	−1.58417	−0.792086	−	0.610410i	$-0.791005\pi$
$733$	−31438.2	−1.58417	−0.792086	+	0.610410i	$0.791005\pi$
$734$	0	0
$735$	85688.3	4.30022
$736$	0	0
$737$	128.239	0.00640943
$738$	0	0
$739$	−16449.5	−0.818817	−0.409408	−	0.912351i	$-0.634265\pi$
$739$	−16449.5	−0.818817	−0.409408	+	0.912351i	$0.634265\pi$
$740$	0	0
$741$	−82974.9	−4.11357
$742$	0	0
$743$	33179.2	1.63826	0.819129	−	0.573610i	$-0.194457\pi$
$743$	33179.2	1.63826	0.819129	+	0.573610i	$0.194457\pi$
$744$	0	0
$745$	21397.7	1.05228
$746$	0	0
$747$	−23778.1	−1.16465
$748$	0	0
$749$	20656.5	1.00771
$750$	0	0
$751$	3581.31	0.174013	0.0870065	−	0.996208i	$-0.472270\pi$
$751$	3581.31	0.174013	0.0870065	+	0.996208i	$0.472270\pi$
$752$	0	0
$753$	−38549.4	−1.86563
$754$	0	0
$755$	−15616.4	−0.752769
$756$	0	0
$757$	1050.27	0.0504265	0.0252132	−	0.999682i	$-0.491974\pi$
$757$	1050.27	0.0504265	0.0252132	+	0.999682i	$0.491974\pi$
$758$	0	0
$759$	−11973.3	−0.572600
$760$	0	0
$761$	19296.7	0.919192	0.459596	−	0.888128i	$-0.347994\pi$
$761$	19296.7	0.919192	0.459596	+	0.888128i	$0.347994\pi$
$762$	0	0
$763$	−38067.6	−1.80621
$764$	0	0
$765$	−47842.8	−2.26112
$766$	0	0
$767$	−48994.2	−2.30649
$768$	0	0
$769$	17635.2	0.826973	0.413486	−	0.910510i	$-0.364311\pi$
$769$	17635.2	0.826973	0.413486	+	0.910510i	$0.364311\pi$
$770$	0	0
$771$	9333.08	0.435957
$772$	0	0
$773$	6482.52	0.301630	0.150815	−	0.988562i	$-0.451810\pi$
$773$	6482.52	0.301630	0.150815	+	0.988562i	$0.451810\pi$
$774$	0	0
$775$	−29.1460	−0.00135091
$776$	0	0
$777$	18221.4	0.841299
$778$	0	0
$779$	52043.8	2.39366
$780$	0	0
$781$	−1087.66	−0.0498330
$782$	0	0
$783$	−5612.26	−0.256151
$784$	0	0
$785$	−14341.5	−0.652062
$786$	0	0
$787$	−6515.85	−0.295127	−0.147563	−	0.989053i	$-0.547143\pi$
$787$	−6515.85	−0.295127	−0.147563	+	0.989053i	$0.547143\pi$
$788$	0	0
$789$	−35426.0	−1.59848
$790$	0	0
$791$	−25601.9	−1.15082
$792$	0	0
$793$	−17526.1	−0.784830
$794$	0	0
$795$	18164.6	0.810355
$796$	0	0
$797$	31806.7	1.41362	0.706809	−	0.707405i	$-0.250134\pi$
$797$	31806.7	1.41362	0.706809	+	0.707405i	$0.250134\pi$
$798$	0	0
$799$	−6790.53	−0.300666
$800$	0	0
$801$	−50499.4	−2.22760
$802$	0	0
$803$	−11633.1	−0.511237
$804$	0	0
$805$	−35050.1	−1.53460
$806$	0	0
$807$	−53841.6	−2.34859
$808$	0	0
$809$	−36594.8	−1.59036	−0.795181	−	0.606372i	$-0.792624\pi$
$809$	−36594.8	−1.59036	−0.795181	+	0.606372i	$0.792624\pi$
$810$	0	0
$811$	13763.1	0.595917	0.297958	−	0.954579i	$-0.403694\pi$
$811$	13763.1	0.595917	0.297958	+	0.954579i	$0.403694\pi$
$812$	0	0
$813$	−4146.79	−0.178886
$814$	0	0
$815$	−41403.8	−1.77953
$816$	0	0
$817$	−72669.8	−3.11187
$818$	0	0
$819$	109794.	4.68440
$820$	0	0
$821$	−5524.35	−0.234837	−0.117419	−	0.993083i	$-0.537462\pi$
$821$	−5524.35	−0.234837	−0.117419	+	0.993083i	$0.537462\pi$
$822$	0	0
$823$	19038.1	0.806352	0.403176	−	0.915122i	$-0.367906\pi$
$823$	19038.1	0.806352	0.403176	+	0.915122i	$0.367906\pi$
$824$	0	0
$825$	−927.528	−0.0391423
$826$	0	0
$827$	9157.55	0.385053	0.192527	−	0.981292i	$-0.438332\pi$
$827$	9157.55	0.385053	0.192527	+	0.981292i	$0.438332\pi$
$828$	0	0
$829$	4033.20	0.168973	0.0844867	−	0.996425i	$-0.473075\pi$
$829$	4033.20	0.168973	0.0844867	+	0.996425i	$0.473075\pi$
$830$	0	0
$831$	−27779.9	−1.15966
$832$	0	0
$833$	81005.9	3.36937
$834$	0	0
$835$	−10208.8	−0.423103
$836$	0	0
$837$	−796.643	−0.0328984
$838$	0	0
$839$	26260.8	1.08060	0.540301	−	0.841472i	$-0.318311\pi$
$839$	26260.8	1.08060	0.540301	+	0.841472i	$0.318311\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	54870.7	2.24181
$844$	0	0
$845$	−19551.5	−0.795968
$846$	0	0
$847$	39047.8	1.58406
$848$	0	0
$849$	45129.1	1.82429
$850$	0	0
$851$	−5403.41	−0.217657
$852$	0	0
$853$	−5194.92	−0.208524	−0.104262	−	0.994550i	$-0.533248\pi$
$853$	−5194.92	−0.208524	−0.104262	+	0.994550i	$0.533248\pi$
$854$	0	0
$855$	80292.8	3.21165
$856$	0	0
$857$	−22863.1	−0.911307	−0.455654	−	0.890157i	$-0.650594\pi$
$857$	−22863.1	−0.911307	−0.455654	+	0.890157i	$0.650594\pi$
$858$	0	0
$859$	19015.3	0.755288	0.377644	−	0.925951i	$-0.376734\pi$
$859$	19015.3	0.755288	0.377644	+	0.925951i	$0.376734\pi$
$860$	0	0
$861$	−106678.	−4.22250
$862$	0	0
$863$	33368.4	1.31619	0.658096	−	0.752934i	$-0.271362\pi$
$863$	33368.4	1.31619	0.658096	+	0.752934i	$0.271362\pi$
$864$	0	0
$865$	29225.4	1.14878
$866$	0	0
$867$	−27182.8	−1.06479
$868$	0	0
$869$	10430.8	0.407180
$870$	0	0
$871$	540.188	0.0210144
$872$	0	0
$873$	45884.7	1.77888
$874$	0	0
$875$	−50650.7	−1.95692
$876$	0	0
$877$	33775.9	1.30049	0.650246	−	0.759723i	$-0.274666\pi$
$877$	33775.9	1.30049	0.650246	+	0.759723i	$0.274666\pi$
$878$	0	0
$879$	−43356.7	−1.66369
$880$	0	0
$881$	−417.141	−0.0159521	−0.00797606	−	0.999968i	$-0.502539\pi$
$881$	−417.141	−0.0159521	−0.00797606	+	0.999968i	$0.502539\pi$
$882$	0	0
$883$	41417.9	1.57851	0.789254	−	0.614067i	$-0.210468\pi$
$883$	41417.9	1.57851	0.789254	+	0.614067i	$0.210468\pi$
$884$	0	0
$885$	73442.5	2.78954
$886$	0	0
$887$	40342.4	1.52713	0.763565	−	0.645731i	$-0.223447\pi$
$887$	40342.4	1.52713	0.763565	+	0.645731i	$0.223447\pi$
$888$	0	0
$889$	4986.27	0.188115
$890$	0	0
$891$	−5423.87	−0.203936
$892$	0	0
$893$	11396.3	0.427058
$894$	0	0
$895$	42521.9	1.58810
$896$	0	0
$897$	−50435.7	−1.87737
$898$	0	0
$899$	119.377	0.00442876
$900$	0	0
$901$	17172.0	0.634942
$902$	0	0
$903$	148957.	5.48944
$904$	0	0
$905$	39179.2	1.43907
$906$	0	0
$907$	−30573.6	−1.11927	−0.559636	−	0.828739i	$-0.689059\pi$
$907$	−30573.6	−1.11927	−0.559636	+	0.828739i	$0.689059\pi$
$908$	0	0
$909$	−19007.9	−0.693568
$910$	0	0
$911$	15352.6	0.558346	0.279173	−	0.960241i	$-0.409940\pi$
$911$	15352.6	0.558346	0.279173	+	0.960241i	$0.409940\pi$
$912$	0	0
$913$	7257.93	0.263091
$914$	0	0
$915$	26271.7	0.949197
$916$	0	0
$917$	12570.2	0.452678
$918$	0	0
$919$	8524.83	0.305994	0.152997	−	0.988227i	$-0.451108\pi$
$919$	8524.83	0.305994	0.152997	+	0.988227i	$0.451108\pi$
$920$	0	0
$921$	30841.1	1.10342
$922$	0	0
$923$	−4581.60	−0.163386
$924$	0	0
$925$	−418.582	−0.0148788
$926$	0	0
$927$	−3919.58	−0.138874
$928$	0	0
$929$	−20169.6	−0.712319	−0.356159	−	0.934425i	$-0.615914\pi$
$929$	−20169.6	−0.712319	−0.356159	+	0.934425i	$0.615914\pi$
$930$	0	0
$931$	−135949.	−4.78578
$932$	0	0
$933$	24330.1	0.853733
$934$	0	0
$935$	14603.4	0.510782
$936$	0	0
$937$	17975.4	0.626713	0.313356	−	0.949636i	$-0.398547\pi$
$937$	17975.4	0.626713	0.313356	+	0.949636i	$0.398547\pi$
$938$	0	0
$939$	40537.4	1.40883
$940$	0	0
$941$	−41436.9	−1.43550	−0.717750	−	0.696301i	$-0.754828\pi$
$941$	−41436.9	−1.43550	−0.717750	+	0.696301i	$0.754828\pi$
$942$	0	0
$943$	31634.5	1.09243
$944$	0	0
$945$	−74214.2	−2.55470
$946$	0	0
$947$	−44431.8	−1.52465	−0.762323	−	0.647197i	$-0.775941\pi$
$947$	−44431.8	−1.52465	−0.762323	+	0.647197i	$0.775941\pi$
$948$	0	0
$949$	−49002.7	−1.67618
$950$	0	0
$951$	−724.961	−0.0247197
$952$	0	0
$953$	−35372.8	−1.20235	−0.601174	−	0.799118i	$-0.705300\pi$
$953$	−35372.8	−1.20235	−0.601174	+	0.799118i	$0.705300\pi$
$954$	0	0
$955$	34300.5	1.16224
$956$	0	0
$957$	3799.00	0.128322
$958$	0	0
$959$	−60018.0	−2.02094
$960$	0	0
$961$	−29774.1	−0.999431
$962$	0	0
$963$	−28763.1	−0.962490
$964$	0	0
$965$	−30902.1	−1.03085
$966$	0	0
$967$	−37084.4	−1.23325	−0.616626	−	0.787257i	$-0.711501\pi$
$967$	−37084.4	−1.23325	−0.616626	+	0.787257i	$0.711501\pi$
$968$	0	0
$969$	117583.	3.89815
$970$	0	0
$971$	21317.9	0.704557	0.352279	−	0.935895i	$-0.385407\pi$
$971$	21317.9	0.704557	0.352279	+	0.935895i	$0.385407\pi$
$972$	0	0
$973$	24364.7	0.802771
$974$	0	0
$975$	−3907.07	−0.128335
$976$	0	0
$977$	17115.5	0.560464	0.280232	−	0.959932i	$-0.409589\pi$
$977$	17115.5	0.560464	0.280232	+	0.959932i	$0.409589\pi$
$978$	0	0
$979$	15414.3	0.503210
$980$	0	0
$981$	53007.0	1.72516
$982$	0	0
$983$	−2593.88	−0.0841625	−0.0420813	−	0.999114i	$-0.513399\pi$
$983$	−2593.88	−0.0841625	−0.0420813	+	0.999114i	$0.513399\pi$
$984$	0	0
$985$	4953.89	0.160248
$986$	0	0
$987$	−23359.8	−0.753345
$988$	0	0
$989$	−44171.9	−1.42021
$990$	0	0
$991$	−13864.0	−0.444404	−0.222202	−	0.975001i	$-0.571324\pi$
$991$	−13864.0	−0.444404	−0.222202	+	0.975001i	$0.571324\pi$
$992$	0	0
$993$	−76648.7	−2.44952
$994$	0	0
$995$	23059.2	0.734698
$996$	0	0
$997$	−51189.0	−1.62605	−0.813026	−	0.582228i	$-0.802181\pi$
$997$	−51189.0	−1.62605	−0.813026	+	0.582228i	$0.802181\pi$
$998$	0	0
$999$	−11441.0	−0.362340

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.4.a.q.1.1		3
4.3	odd	2		1856.4.a.t.1.3		3
8.3	odd	2		464.4.a.h.1.1		3
8.5	even	2		232.4.a.a.1.3	✓	3
24.5	odd	2		2088.4.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.a.1.3	✓	3	8.5	even	2
464.4.a.h.1.1		3	8.3	odd	2
1856.4.a.q.1.1		3	1.1	even	1	trivial
1856.4.a.t.1.3		3	4.3	odd	2
2088.4.a.a.1.2		3	24.5	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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1856.a (trivial)

\(f(q)\)	\(=\)	\(q-8.72775 q^{3} -10.8591 q^{5} -35.3146 q^{7} +49.1737 q^{9} +O(q^{10})\)	\(q-8.72775 q^{3} -10.8591 q^{5} -35.3146 q^{7} +49.1737 q^{9} -15.0096 q^{11} -63.2256 q^{13} +94.7754 q^{15} +89.5964 q^{17} -150.367 q^{19} +308.217 q^{21} -91.3993 q^{23} -7.08037 q^{25} -193.526 q^{27} +29.0000 q^{29} +4.11646 q^{31} +131.000 q^{33} +383.484 q^{35} +59.1187 q^{37} +551.817 q^{39} -346.113 q^{41} +483.285 q^{43} -533.981 q^{45} -75.7902 q^{47} +904.120 q^{49} -781.975 q^{51} +191.660 q^{53} +162.990 q^{55} +1312.36 q^{57} +774.911 q^{59} +277.199 q^{61} -1736.55 q^{63} +686.572 q^{65} -8.54382 q^{67} +797.710 q^{69} +72.4644 q^{71} +775.045 q^{73} +61.7957 q^{75} +530.058 q^{77} -694.941 q^{79} +361.361 q^{81} -483.553 q^{83} -972.935 q^{85} -253.105 q^{87} -1026.96 q^{89} +2232.79 q^{91} -35.9274 q^{93} +1632.84 q^{95} +933.115 q^{97} -738.077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9	\( 3 q - 3 q^{3} - 11 q^{5} - 38 q^{7} + 58 q^{9} - 65 q^{11} - 29 q^{13} - 85 q^{15} + 244 q^{17} - 312 q^{19} + 214 q^{21} - 24 q^{23} + 436 q^{25} - 57 q^{27} + 87 q^{29} + 249 q^{31} + 393 q^{33} + 718 q^{35} - 200 q^{37} + 749 q^{39} - 470 q^{41} - 97 q^{43} - 1562 q^{45} + 377 q^{47} + 383 q^{49} - 612 q^{51} - 1007 q^{53} - 1399 q^{55} + 1128 q^{57} + 364 q^{59} + 524 q^{61} - 2244 q^{63} + 301 q^{65} + 820 q^{67} + 824 q^{69} - 782 q^{71} + 1620 q^{73} + 1356 q^{75} - 158 q^{77} + 427 q^{79} + 227 q^{81} - 1520 q^{83} + 68 q^{85} - 87 q^{87} - 2474 q^{89} + 2002 q^{91} + 1807 q^{93} + 568 q^{95} - 170 q^{97} + 1362 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 11 * q^5 - 38 * q^7 + 58 * q^9 - 65 * q^11 - 29 * q^13 - 85 * q^15 + 244 * q^17 - 312 * q^19 + 214 * q^21 - 24 * q^23 + 436 * q^25 - 57 * q^27 + 87 * q^29 + 249 * q^31 + 393 * q^33 + 718 * q^35 - 200 * q^37 + 749 * q^39 - 470 * q^41 - 97 * q^43 - 1562 * q^45 + 377 * q^47 + 383 * q^49 - 612 * q^51 - 1007 * q^53 - 1399 * q^55 + 1128 * q^57 + 364 * q^59 + 524 * q^61 - 2244 * q^63 + 301 * q^65 + 820 * q^67 + 824 * q^69 - 782 * q^71 + 1620 * q^73 + 1356 * q^75 - 158 * q^77 + 427 * q^79 + 227 * q^81 - 1520 * q^83 + 68 * q^85 - 87 * q^87 - 2474 * q^89 + 2002 * q^91 + 1807 * q^93 + 568 * q^95 - 170 * q^97 + 1362 * q^99

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