LMFDB - Embedded newform 5445.2.a.by.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.74043072.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 9x^{4} + 21x^{2} - 12 $ x^6 - 9x^4 + 21x^2 - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.38098$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.38098 q^{2} +3.66908 q^{4} +1.00000 q^{5} -2.24200 q^{7} -3.97405 q^{8} +O(q^{10})$	$q-2.38098 q^{2} +3.66908 q^{4} +1.00000 q^{5} -2.24200 q^{7} -3.97405 q^{8} -2.38098 q^{10} +5.33816 q^{14} +2.12398 q^{16} -0.509947 q^{17} -2.24200 q^{19} +3.66908 q^{20} +4.45490 q^{23} +1.00000 q^{25} -8.22607 q^{28} +3.46410 q^{29} -7.24797 q^{31} +2.89093 q^{32} +1.21417 q^{34} -2.24200 q^{35} +10.7931 q^{37} +5.33816 q^{38} -3.97405 q^{40} -7.94810 q^{41} -6.05983 q^{43} -10.6071 q^{46} +12.2214 q^{47} -1.97345 q^{49} -2.38098 q^{50} -3.79306 q^{53} +8.90981 q^{56} -8.24797 q^{58} -4.67632 q^{59} -7.79188 q^{61} +17.2573 q^{62} -11.1312 q^{64} -8.79306 q^{67} -1.87104 q^{68} +5.33816 q^{70} +16.6763 q^{71} +12.7858 q^{73} -25.6981 q^{74} -8.22607 q^{76} -6.49402 q^{79} +2.12398 q^{80} +18.9243 q^{82} +12.9880 q^{83} -0.509947 q^{85} +14.4283 q^{86} +0.180384 q^{89} +16.3454 q^{92} -29.0990 q^{94} -2.24200 q^{95} -10.6127 q^{97} +4.69874 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{4} + 6 q^{5}+O(q^{10}) $ 6 * q + 6 * q^4 + 6 * q^5	$ 6 q + 6 q^{4} + 6 q^{5} - 6 q^{16} + 6 q^{20} + 24 q^{23} + 6 q^{25} - 6 q^{31} - 6 q^{34} + 30 q^{37} + 12 q^{47} + 12 q^{49} + 12 q^{53} + 48 q^{56} - 12 q^{58} + 36 q^{59} - 18 q^{67} + 36 q^{71} - 6 q^{80} + 12 q^{82} + 60 q^{86} + 12 q^{89} + 18 q^{92} - 18 q^{97}+O(q^{100}) $ 6 * q + 6 * q^4 + 6 * q^5 - 6 * q^16 + 6 * q^20 + 24 * q^23 + 6 * q^25 - 6 * q^31 - 6 * q^34 + 30 * q^37 + 12 * q^47 + 12 * q^49 + 12 * q^53 + 48 * q^56 - 12 * q^58 + 36 * q^59 - 18 * q^67 + 36 * q^71 - 6 * q^80 + 12 * q^82 + 60 * q^86 + 12 * q^89 + 18 * q^92 - 18 * q^97

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 $

$2$	−2.38098	−1.68361	−0.841805	−	0.539782i	$-0.818506\pi$
$2$	−2.38098	−1.68361	−0.841805	+	0.539782i	$0.818506\pi$
$3$	0	0
$3$	0	0
$4$	3.66908	1.83454
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.24200	−0.847396	−0.423698	−	0.905804i	$-0.639268\pi$
$7$	−2.24200	−0.847396	−0.423698	+	0.905804i	$0.639268\pi$
$8$	−3.97405	−1.40504
$9$	0	0
$10$	−2.38098	−0.752933
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	5.33816	1.42668
$15$	0	0
$16$	2.12398	0.530996
$17$	−0.509947	−0.123680	−0.0618402	−	0.998086i	$-0.519697\pi$
$17$	−0.509947	−0.123680	−0.0618402	+	0.998086i	$0.519697\pi$
$18$	0	0
$19$	−2.24200	−0.514350	−0.257175	−	0.966365i	$-0.582792\pi$
$19$	−2.24200	−0.514350	−0.257175	+	0.966365i	$0.582792\pi$
$20$	3.66908	0.820431
$21$	0	0
$22$	0	0
$23$	4.45490	0.928912	0.464456	−	0.885596i	$-0.346250\pi$
$23$	4.45490	0.928912	0.464456	+	0.885596i	$0.346250\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−8.22607	−1.55458
$29$	3.46410	0.643268	0.321634	−	0.946864i	$-0.395768\pi$
$29$	3.46410	0.643268	0.321634	+	0.946864i	$0.395768\pi$
$30$	0	0
$31$	−7.24797	−1.30177	−0.650887	−	0.759175i	$-0.725603\pi$
$31$	−7.24797	−1.30177	−0.650887	+	0.759175i	$0.725603\pi$
$32$	2.89093	0.511049
$33$	0	0
$34$	1.21417	0.208229
$35$	−2.24200	−0.378967
$36$	0	0
$37$	10.7931	1.77437	0.887184	−	0.461415i	$-0.152658\pi$
$37$	10.7931	1.77437	0.887184	+	0.461415i	$0.152658\pi$
$38$	5.33816	0.865964
$39$	0	0
$40$	−3.97405	−0.628352
$41$	−7.94810	−1.24128	−0.620642	−	0.784094i	$-0.713128\pi$
$41$	−7.94810	−1.24128	−0.620642	+	0.784094i	$0.713128\pi$
$42$	0	0
$43$	−6.05983	−0.924115	−0.462058	−	0.886850i	$-0.652889\pi$
$43$	−6.05983	−0.924115	−0.462058	+	0.886850i	$0.652889\pi$
$44$	0	0
$45$	0	0
$46$	−10.6071	−1.56392
$47$	12.2214	1.78268	0.891338	−	0.453339i	$-0.149767\pi$
$47$	12.2214	1.78268	0.891338	+	0.453339i	$0.149767\pi$
$48$	0	0
$49$	−1.97345	−0.281921
$50$	−2.38098	−0.336722
$51$	0	0
$52$	0	0
$53$	−3.79306	−0.521017	−0.260509	−	0.965472i	$-0.583890\pi$
$53$	−3.79306	−0.521017	−0.260509	+	0.965472i	$0.583890\pi$
$54$	0	0
$55$	0	0
$56$	8.90981	1.19062
$57$	0	0
$58$	−8.24797	−1.08301
$59$	−4.67632	−0.608805	−0.304402	−	0.952544i	$-0.598457\pi$
$59$	−4.67632	−0.608805	−0.304402	+	0.952544i	$0.598457\pi$
$60$	0	0
$61$	−7.79188	−0.997648	−0.498824	−	0.866703i	$-0.666235\pi$
$61$	−7.79188	−0.997648	−0.498824	+	0.866703i	$0.666235\pi$
$62$	17.2573	2.19168
$63$	0	0
$64$	−11.1312	−1.39140
$65$	0	0
$66$	0	0
$67$	−8.79306	−1.07424	−0.537122	−	0.843505i	$-0.680488\pi$
$67$	−8.79306	−1.07424	−0.537122	+	0.843505i	$0.680488\pi$
$68$	−1.87104	−0.226896
$69$	0	0
$70$	5.33816	0.638032
$71$	16.6763	1.97911	0.989557	−	0.144140i	$-0.0460415\pi$
$71$	16.6763	1.97911	0.989557	+	0.144140i	$0.0460415\pi$
$72$	0	0
$73$	12.7858	1.49647	0.748234	−	0.663435i	$-0.230902\pi$
$73$	12.7858	1.49647	0.748234	+	0.663435i	$0.230902\pi$
$74$	−25.6981	−2.98734
$75$	0	0
$76$	−8.22607	−0.943595
$77$	0	0
$78$	0	0
$79$	−6.49402	−0.730634	−0.365317	−	0.930883i	$-0.619039\pi$
$79$	−6.49402	−0.730634	−0.365317	+	0.930883i	$0.619039\pi$
$80$	2.12398	0.237469
$81$	0	0
$82$	18.9243	2.08984
$83$	12.9880	1.42562	0.712811	−	0.701356i	$-0.247422\pi$
$83$	12.9880	1.42562	0.712811	+	0.701356i	$0.247422\pi$
$84$	0	0
$85$	−0.509947	−0.0553115
$86$	14.4283	1.55585
$87$	0	0
$88$	0	0
$89$	0.180384	0.0191206	0.00956031	−	0.999954i	$-0.496957\pi$
$89$	0.180384	0.0191206	0.00956031	+	0.999954i	$0.496957\pi$
$90$	0	0
$91$	0	0
$92$	16.3454	1.70413
$93$	0	0
$94$	−29.0990	−3.00133
$95$	−2.24200	−0.230024
$96$	0	0
$97$	−10.6127	−1.07755	−0.538777	−	0.842448i	$-0.681114\pi$
$97$	−10.6127	−1.07755	−0.538777	+	0.842448i	$0.681114\pi$
$98$	4.69874	0.474644
$99$	0	0
$100$	3.66908	0.366908
$101$	−12.4321	−1.23704	−0.618520	−	0.785769i	$-0.712267\pi$
$101$	−12.4321	−1.23704	−0.618520	+	0.785769i	$0.712267\pi$
$102$	0	0
$103$	−2.06364	−0.203336	−0.101668	−	0.994818i	$-0.532418\pi$
$103$	−2.06364	−0.203336	−0.101668	+	0.994818i	$0.532418\pi$
$104$	0	0
$105$	0	0
$106$	9.03122	0.877189
$107$	6.29181	0.608252	0.304126	−	0.952632i	$-0.401636\pi$
$107$	6.29181	0.608252	0.304126	+	0.952632i	$0.401636\pi$
$108$	0	0
$109$	0.666164	0.0638069	0.0319035	−	0.999491i	$-0.489843\pi$
$109$	0.666164	0.0638069	0.0319035	+	0.999491i	$0.489843\pi$
$110$	0	0
$111$	0	0
$112$	−4.76197	−0.449963
$113$	−6.70287	−0.630553	−0.315277	−	0.949000i	$-0.602097\pi$
$113$	−6.70287	−0.630553	−0.315277	+	0.949000i	$0.602097\pi$
$114$	0	0
$115$	4.45490	0.415422
$116$	12.7101	1.18010
$117$	0	0
$118$	11.1342	1.02499
$119$	1.14330	0.104806
$120$	0	0
$121$	0	0
$122$	18.5523	1.67965
$123$	0	0
$124$	−26.5934	−2.38815
$125$	1.00000	0.0894427
$126$	0	0
$127$	12.0784	1.07178	0.535891	−	0.844287i	$-0.319976\pi$
$127$	12.0784	1.07178	0.535891	+	0.844287i	$0.319976\pi$
$128$	20.7214	1.83153
$129$	0	0
$130$	0	0
$131$	3.61562	0.315898	0.157949	−	0.987447i	$-0.449512\pi$
$131$	3.61562	0.315898	0.157949	+	0.987447i	$0.449512\pi$
$132$	0	0
$133$	5.02655	0.435858
$134$	20.9361	1.80861
$135$	0	0
$136$	2.02655	0.173776
$137$	−6.88325	−0.588076	−0.294038	−	0.955794i	$-0.594999\pi$
$137$	−6.88325	−0.588076	−0.294038	+	0.955794i	$0.594999\pi$
$138$	0	0
$139$	20.7042	1.75610	0.878052	−	0.478566i	$-0.158843\pi$
$139$	20.7042	1.75610	0.878052	+	0.478566i	$0.158843\pi$
$140$	−8.22607	−0.695230
$141$	0	0
$142$	−39.7060	−3.33206
$143$	0	0
$144$	0	0
$145$	3.46410	0.287678
$146$	−30.4428	−2.51947
$147$	0	0
$148$	39.6006	3.25515
$149$	−7.63566	−0.625538	−0.312769	−	0.949829i	$-0.601257\pi$
$149$	−7.63566	−0.625538	−0.312769	+	0.949829i	$0.601257\pi$
$150$	0	0
$151$	6.29181	0.512020	0.256010	−	0.966674i	$-0.417592\pi$
$151$	6.29181	0.512020	0.256010	+	0.966674i	$0.417592\pi$
$152$	8.90981	0.722681
$153$	0	0
$154$	0	0
$155$	−7.24797	−0.582171
$156$	0	0
$157$	24.5596	1.96007	0.980034	−	0.198832i	$-0.0637149\pi$
$157$	24.5596	1.96007	0.980034	+	0.198832i	$0.0637149\pi$
$158$	15.4621	1.23010
$159$	0	0
$160$	2.89093	0.228548
$161$	−9.98789	−0.787156
$162$	0	0
$163$	−10.6127	−0.831249	−0.415625	−	0.909536i	$-0.636437\pi$
$163$	−10.6127	−0.831249	−0.415625	+	0.909536i	$0.636437\pi$
$164$	−29.1622	−2.27719
$165$	0	0
$166$	−30.9243	−2.40019
$167$	6.14029	0.475150	0.237575	−	0.971369i	$-0.423648\pi$
$167$	6.14029	0.475150	0.237575	+	0.971369i	$0.423648\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	1.21417	0.0931230
$171$	0	0
$172$	−22.2340	−1.69533
$173$	16.6037	1.26235	0.631176	−	0.775639i	$-0.282572\pi$
$173$	16.6037	1.26235	0.631176	+	0.775639i	$0.282572\pi$
$174$	0	0
$175$	−2.24200	−0.169479
$176$	0	0
$177$	0	0
$178$	−0.429490	−0.0321917
$179$	19.4057	1.45045	0.725227	−	0.688510i	$-0.241735\pi$
$179$	19.4057	1.45045	0.725227	+	0.688510i	$0.241735\pi$
$180$	0	0
$181$	10.5596	0.784887	0.392443	−	0.919776i	$-0.371630\pi$
$181$	10.5596	0.784887	0.392443	+	0.919776i	$0.371630\pi$
$182$	0	0
$183$	0	0
$184$	−17.7040	−1.30516
$185$	10.7931	0.793522
$186$	0	0
$187$	0	0
$188$	44.8413	3.27039
$189$	0	0
$190$	5.33816	0.387271
$191$	8.90981	0.644691	0.322346	−	0.946622i	$-0.395529\pi$
$191$	8.90981	0.644691	0.322346	+	0.946622i	$0.395529\pi$
$192$	0	0
$193$	−17.6742	−1.27222	−0.636110	−	0.771599i	$-0.719457\pi$
$193$	−17.6742	−1.27222	−0.636110	+	0.771599i	$0.719457\pi$
$194$	25.2686	1.81418
$195$	0	0
$196$	−7.24073	−0.517195
$197$	−13.8564	−0.987228	−0.493614	−	0.869681i	$-0.664324\pi$
$197$	−13.8564	−0.987228	−0.493614	+	0.869681i	$0.664324\pi$
$198$	0	0
$199$	14.1047	0.999853	0.499927	−	0.866068i	$-0.333360\pi$
$199$	14.1047	0.999853	0.499927	+	0.866068i	$0.333360\pi$
$200$	−3.97405	−0.281008
$201$	0	0
$202$	29.6006	2.08269
$203$	−7.76651	−0.545102
$204$	0	0
$205$	−7.94810	−0.555119
$206$	4.91349	0.342339
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	9.08974	0.625764	0.312882	−	0.949792i	$-0.398706\pi$
$211$	9.08974	0.625764	0.312882	+	0.949792i	$0.398706\pi$
$212$	−13.9170	−0.955827
$213$	0	0
$214$	−14.9807	−1.02406
$215$	−6.05983	−0.413277
$216$	0	0
$217$	16.2499	1.10312
$218$	−1.58612	−0.107426
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.70287	0.381892	0.190946	−	0.981600i	$-0.438844\pi$
$223$	5.70287	0.381892	0.190946	+	0.981600i	$0.438844\pi$
$224$	−6.48146	−0.433061
$225$	0	0
$226$	15.9594	1.06160
$227$	7.16018	0.475238	0.237619	−	0.971358i	$-0.423633\pi$
$227$	7.16018	0.475238	0.237619	+	0.971358i	$0.423633\pi$
$228$	0	0
$229$	−14.7029	−0.971593	−0.485797	−	0.874072i	$-0.661470\pi$
$229$	−14.7029	−0.971593	−0.485797	+	0.874072i	$0.661470\pi$
$230$	−10.6071	−0.699408
$231$	0	0
$232$	−13.7665	−0.903816
$233$	14.3664	0.941171	0.470586	−	0.882354i	$-0.344043\pi$
$233$	14.3664	0.941171	0.470586	+	0.882354i	$0.344043\pi$
$234$	0	0
$235$	12.2214	0.797237
$236$	−17.1578	−1.11688
$237$	0	0
$238$	−2.72218	−0.176453
$239$	1.33233	0.0861811	0.0430906	−	0.999071i	$-0.486280\pi$
$239$	1.33233	0.0861811	0.0430906	+	0.999071i	$0.486280\pi$
$240$	0	0
$241$	−6.56978	−0.423197	−0.211598	−	0.977357i	$-0.567867\pi$
$241$	−6.56978	−0.423197	−0.211598	+	0.977357i	$0.567867\pi$
$242$	0	0
$243$	0	0
$244$	−28.5890	−1.83022
$245$	−1.97345	−0.126079
$246$	0	0
$247$	0	0
$248$	28.8038	1.82904
$249$	0	0
$250$	−2.38098	−0.150587
$251$	6.18038	0.390102	0.195051	−	0.980793i	$-0.437513\pi$
$251$	6.18038	0.390102	0.195051	+	0.980793i	$0.437513\pi$
$252$	0	0
$253$	0	0
$254$	−28.7584	−1.80446
$255$	0	0
$256$	−27.0748	−1.69218
$257$	−2.46938	−0.154036	−0.0770178	−	0.997030i	$-0.524540\pi$
$257$	−2.46938	−0.154036	−0.0770178	+	0.997030i	$0.524540\pi$
$258$	0	0
$259$	−24.1980	−1.50359
$260$	0	0
$261$	0	0
$262$	−8.60873	−0.531849
$263$	−22.0365	−1.35883	−0.679414	−	0.733755i	$-0.737766\pi$
$263$	−22.0365	−1.35883	−0.679414	+	0.733755i	$0.737766\pi$
$264$	0	0
$265$	−3.79306	−0.233006
$266$	−11.9681	−0.733814
$267$	0	0
$268$	−32.2624	−1.97074
$269$	−13.5861	−0.828361	−0.414180	−	0.910195i	$-0.635932\pi$
$269$	−13.5861	−0.828361	−0.414180	+	0.910195i	$0.635932\pi$
$270$	0	0
$271$	21.7240	1.31964	0.659821	−	0.751423i	$-0.270632\pi$
$271$	21.7240	1.31964	0.659821	+	0.751423i	$0.270632\pi$
$272$	−1.08312	−0.0656737
$273$	0	0
$274$	16.3889	0.990090
$275$	0	0
$276$	0	0
$277$	13.8057	0.829505	0.414753	−	0.909934i	$-0.363868\pi$
$277$	13.8057	0.829505	0.414753	+	0.909934i	$0.363868\pi$
$278$	−49.2962	−2.95659
$279$	0	0
$280$	8.90981	0.532463
$281$	28.4203	1.69541	0.847706	−	0.530467i	$-0.177983\pi$
$281$	28.4203	1.69541	0.847706	+	0.530467i	$0.177983\pi$
$282$	0	0
$283$	−9.17020	−0.545112	−0.272556	−	0.962140i	$-0.587869\pi$
$283$	−9.17020	−0.545112	−0.272556	+	0.962140i	$0.587869\pi$
$284$	61.1867	3.63076
$285$	0	0
$286$	0	0
$287$	17.8196	1.05186
$288$	0	0
$289$	−16.7400	−0.984703
$290$	−8.24797	−0.484337
$291$	0	0
$292$	46.9122	2.74533
$293$	−19.0019	−1.11010	−0.555051	−	0.831817i	$-0.687301\pi$
$293$	−19.0019	−1.11010	−0.555051	+	0.831817i	$0.687301\pi$
$294$	0	0
$295$	−4.67632	−0.272266
$296$	−42.8922	−2.49306
$297$	0	0
$298$	18.1804	1.05316
$299$	0	0
$300$	0	0
$301$	13.5861	0.783091
$302$	−14.9807	−0.862041
$303$	0	0
$304$	−4.76197	−0.273117
$305$	−7.79188	−0.446162
$306$	0	0
$307$	−4.68621	−0.267456	−0.133728	−	0.991018i	$-0.542695\pi$
$307$	−4.68621	−0.267456	−0.133728	+	0.991018i	$0.542695\pi$
$308$	0	0
$309$	0	0
$310$	17.2573	0.980148
$311$	−19.4057	−1.10040	−0.550199	−	0.835033i	$-0.685448\pi$
$311$	−19.4057	−1.10040	−0.550199	+	0.835033i	$0.685448\pi$
$312$	0	0
$313$	−17.1722	−0.970633	−0.485316	−	0.874339i	$-0.661296\pi$
$313$	−17.1722	−0.970633	−0.485316	+	0.874339i	$0.661296\pi$
$314$	−58.4759	−3.29999
$315$	0	0
$316$	−23.8271	−1.34038
$317$	−19.1457	−1.07533	−0.537665	−	0.843159i	$-0.680693\pi$
$317$	−19.1457	−1.07533	−0.537665	+	0.843159i	$0.680693\pi$
$318$	0	0
$319$	0	0
$320$	−11.1312	−0.622254
$321$	0	0
$322$	23.7810	1.32526
$323$	1.14330	0.0636149
$324$	0	0
$325$	0	0
$326$	25.2686	1.39950
$327$	0	0
$328$	31.5861	1.74405
$329$	−27.4004	−1.51063
$330$	0	0
$331$	19.5104	1.07239	0.536194	−	0.844094i	$-0.319861\pi$
$331$	19.5104	1.07239	0.536194	+	0.844094i	$0.319861\pi$
$332$	47.6541	2.61536
$333$	0	0
$334$	−14.6199	−0.799966
$335$	−8.79306	−0.480416
$336$	0	0
$337$	−12.0784	−0.657950	−0.328975	−	0.944339i	$-0.606703\pi$
$337$	−12.0784	−0.657950	−0.328975	+	0.944339i	$0.606703\pi$
$338$	30.9528	1.68361
$339$	0	0
$340$	−1.87104	−0.101471
$341$	0	0
$342$	0	0
$343$	20.1184	1.08629
$344$	24.0821	1.29842
$345$	0	0
$346$	−39.5330	−2.12531
$347$	8.73601	0.468974	0.234487	−	0.972119i	$-0.424659\pi$
$347$	8.73601	0.468974	0.234487	+	0.972119i	$0.424659\pi$
$348$	0	0
$349$	−8.10431	−0.433814	−0.216907	−	0.976192i	$-0.569597\pi$
$349$	−8.10431	−0.433814	−0.216907	+	0.976192i	$0.569597\pi$
$350$	5.33816	0.285337
$351$	0	0
$352$	0	0
$353$	−1.24402	−0.0662126	−0.0331063	−	0.999452i	$-0.510540\pi$
$353$	−1.24402	−0.0662126	−0.0331063	+	0.999452i	$0.510540\pi$
$354$	0	0
$355$	16.6763	0.885087
$356$	0.661842	0.0350775
$357$	0	0
$358$	−46.2047	−2.44200
$359$	−19.9162	−1.05114	−0.525569	−	0.850751i	$-0.676148\pi$
$359$	−19.9162	−1.05114	−0.525569	+	0.850751i	$0.676148\pi$
$360$	0	0
$361$	−13.9734	−0.735445
$362$	−25.1422	−1.32144
$363$	0	0
$364$	0	0
$365$	12.7858	0.669241
$366$	0	0
$367$	14.1804	0.740210	0.370105	−	0.928990i	$-0.379322\pi$
$367$	14.1804	0.740210	0.370105	+	0.928990i	$0.379322\pi$
$368$	9.46214	0.493248
$369$	0	0
$370$	−25.6981	−1.33598
$371$	8.50404	0.441508
$372$	0	0
$373$	34.2779	1.77484	0.887421	−	0.460960i	$-0.152495\pi$
$373$	34.2779	1.77484	0.887421	+	0.460960i	$0.152495\pi$
$374$	0	0
$375$	0	0
$376$	−48.5685	−2.50473
$377$	0	0
$378$	0	0
$379$	−26.7173	−1.37238	−0.686189	−	0.727423i	$-0.740718\pi$
$379$	−26.7173	−1.37238	−0.686189	+	0.727423i	$0.740718\pi$
$380$	−8.22607	−0.421988
$381$	0	0
$382$	−21.2141	−1.08541
$383$	0.360767	0.0184343	0.00921717	−	0.999958i	$-0.497066\pi$
$383$	0.360767	0.0184343	0.00921717	+	0.999958i	$0.497066\pi$
$384$	0	0
$385$	0	0
$386$	42.0821	2.14192
$387$	0	0
$388$	−38.9388	−1.97682
$389$	19.9469	1.01135	0.505674	−	0.862725i	$-0.331244\pi$
$389$	19.9469	1.01135	0.505674	+	0.862725i	$0.331244\pi$
$390$	0	0
$391$	−2.27176	−0.114888
$392$	7.84257	0.396110
$393$	0	0
$394$	32.9919	1.66211
$395$	−6.49402	−0.326749
$396$	0	0
$397$	18.1457	0.910706	0.455353	−	0.890311i	$-0.349513\pi$
$397$	18.1457	0.910706	0.455353	+	0.890311i	$0.349513\pi$
$398$	−33.5830	−1.68336
$399$	0	0
$400$	2.12398	0.106199
$401$	−1.94689	−0.0972231	−0.0486116	−	0.998818i	$-0.515480\pi$
$401$	−1.94689	−0.0972231	−0.0486116	+	0.998818i	$0.515480\pi$
$402$	0	0
$403$	0	0
$404$	−45.6143	−2.26940
$405$	0	0
$406$	18.4919	0.917739
$407$	0	0
$408$	0	0
$409$	35.3438	1.74764	0.873819	−	0.486252i	$-0.161636\pi$
$409$	35.3438	1.74764	0.873819	+	0.486252i	$0.161636\pi$
$410$	18.9243	0.934604
$411$	0	0
$412$	−7.57165	−0.373028
$413$	10.4843	0.515898
$414$	0	0
$415$	12.9880	0.637557
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−28.7584	−1.40494	−0.702469	−	0.711714i	$-0.747919\pi$
$419$	−28.7584	−1.40494	−0.702469	+	0.711714i	$0.747919\pi$
$420$	0	0
$421$	22.3421	1.08889	0.544444	−	0.838797i	$-0.316741\pi$
$421$	22.3421	1.08889	0.544444	+	0.838797i	$0.316741\pi$
$422$	−21.6425	−1.05354
$423$	0	0
$424$	15.0738	0.732049
$425$	−0.509947	−0.0247361
$426$	0	0
$427$	17.4694	0.845402
$428$	23.0851	1.11586
$429$	0	0
$430$	14.4283	0.695797
$431$	38.2566	1.84276	0.921379	−	0.388666i	$-0.127064\pi$
$431$	38.2566	1.84276	0.921379	+	0.388666i	$0.127064\pi$
$432$	0	0
$433$	−16.6127	−0.798354	−0.399177	−	0.916874i	$-0.630704\pi$
$433$	−16.6127	−0.798354	−0.399177	+	0.916874i	$0.630704\pi$
$434$	−38.6908	−1.85722
$435$	0	0
$436$	2.44421	0.117056
$437$	−9.98789	−0.477785
$438$	0	0
$439$	23.9660	1.14384	0.571918	−	0.820310i	$-0.306199\pi$
$439$	23.9660	1.14384	0.571918	+	0.820310i	$0.306199\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.2254	−0.628356	−0.314178	−	0.949364i	$-0.601729\pi$
$443$	−13.2254	−0.628356	−0.314178	+	0.949364i	$0.601729\pi$
$444$	0	0
$445$	0.180384	0.00855100
$446$	−13.5784	−0.642958
$447$	0	0
$448$	24.9562	1.17907
$449$	28.7584	1.35719	0.678596	−	0.734512i	$-0.262589\pi$
$449$	28.7584	1.35719	0.678596	+	0.734512i	$0.262589\pi$
$450$	0	0
$451$	0	0
$452$	−24.5934	−1.15677
$453$	0	0
$454$	−17.0483	−0.800115
$455$	0	0
$456$	0	0
$457$	26.9960	1.26282	0.631409	−	0.775450i	$-0.282477\pi$
$457$	26.9960	1.26282	0.631409	+	0.775450i	$0.282477\pi$
$458$	35.0073	1.63578
$459$	0	0
$460$	16.3454	0.762108
$461$	31.1675	1.45162	0.725808	−	0.687897i	$-0.241466\pi$
$461$	31.1675	1.45162	0.725808	+	0.687897i	$0.241466\pi$
$462$	0	0
$463$	1.73756	0.0807512	0.0403756	−	0.999185i	$-0.487145\pi$
$463$	1.73756	0.0807512	0.0403756	+	0.999185i	$0.487145\pi$
$464$	7.35769	0.341572
$465$	0	0
$466$	−34.2060	−1.58456
$467$	29.0781	1.34557	0.672787	−	0.739836i	$-0.265097\pi$
$467$	29.0781	1.34557	0.672787	+	0.739836i	$0.265097\pi$
$468$	0	0
$469$	19.7140	0.910309
$470$	−29.0990	−1.34224
$471$	0	0
$472$	18.5839	0.855394
$473$	0	0
$474$	0	0
$475$	−2.24200	−0.102870
$476$	4.19486	0.192271
$477$	0	0
$478$	−3.17225	−0.145095
$479$	−19.7553	−0.902644	−0.451322	−	0.892361i	$-0.649047\pi$
$479$	−19.7553	−0.902644	−0.451322	+	0.892361i	$0.649047\pi$
$480$	0	0
$481$	0	0
$482$	15.6425	0.712497
$483$	0	0
$484$	0	0
$485$	−10.6127	−0.481897
$486$	0	0
$487$	40.0821	1.81629	0.908146	−	0.418654i	$-0.137498\pi$
$487$	40.0821	1.81629	0.908146	+	0.418654i	$0.137498\pi$
$488$	30.9653	1.40173
$489$	0	0
$490$	4.69874	0.212267
$491$	12.5836	0.567891	0.283945	−	0.958840i	$-0.408357\pi$
$491$	12.5836	0.567891	0.283945	+	0.958840i	$0.408357\pi$
$492$	0	0
$493$	−1.76651	−0.0795595
$494$	0	0
$495$	0	0
$496$	−15.3946	−0.691236
$497$	−37.3883	−1.67709
$498$	0	0
$499$	−10.7400	−0.480786	−0.240393	−	0.970676i	$-0.577276\pi$
$499$	−10.7400	−0.480786	−0.240393	+	0.970676i	$0.577276\pi$
$500$	3.66908	0.164086
$501$	0	0
$502$	−14.7154	−0.656780
$503$	23.4514	1.04565	0.522823	−	0.852441i	$-0.324879\pi$
$503$	23.4514	1.04565	0.522823	+	0.852441i	$0.324879\pi$
$504$	0	0
$505$	−12.4321	−0.553221
$506$	0	0
$507$	0	0
$508$	44.3165	1.96623
$509$	44.8115	1.98623	0.993117	−	0.117126i	$-0.0373683\pi$
$509$	44.8115	1.98623	0.993117	+	0.117126i	$0.0373683\pi$
$510$	0	0
$511$	−28.6658	−1.26810
$512$	23.0219	1.01743
$513$	0	0
$514$	5.87955	0.259336
$515$	−2.06364	−0.0909347
$516$	0	0
$517$	0	0
$518$	57.6151	2.53146
$519$	0	0
$520$	0	0
$521$	40.7584	1.78566	0.892828	−	0.450397i	$-0.148718\pi$
$521$	40.7584	1.78566	0.892828	+	0.450397i	$0.148718\pi$
$522$	0	0
$523$	13.4933	0.590020	0.295010	−	0.955494i	$-0.404677\pi$
$523$	13.4933	0.590020	0.295010	+	0.955494i	$0.404677\pi$
$524$	13.2660	0.579528
$525$	0	0
$526$	52.4685	2.28773
$527$	3.69608	0.161004
$528$	0	0
$529$	−3.15383	−0.137123
$530$	9.03122	0.392291
$531$	0	0
$532$	18.4428	0.799598
$533$	0	0
$534$	0	0
$535$	6.29181	0.272019
$536$	34.9441	1.50935
$537$	0	0
$538$	32.3483	1.39464
$539$	0	0
$540$	0	0
$541$	−8.66495	−0.372535	−0.186268	−	0.982499i	$-0.559639\pi$
$541$	−8.66495	−0.372535	−0.186268	+	0.982499i	$0.559639\pi$
$542$	−51.7246	−2.22176
$543$	0	0
$544$	−1.47422	−0.0632067
$545$	0.666164	0.0285353
$546$	0	0
$547$	9.37241	0.400735	0.200368	−	0.979721i	$-0.435786\pi$
$547$	9.37241	0.400735	0.200368	+	0.979721i	$0.435786\pi$
$548$	−25.2552	−1.07885
$549$	0	0
$550$	0	0
$551$	−7.76651	−0.330864
$552$	0	0
$553$	14.5596	0.619136
$554$	−32.8712	−1.39656
$555$	0	0
$556$	75.9652	3.22164
$557$	−39.7865	−1.68581	−0.842904	−	0.538065i	$-0.819156\pi$
$557$	−39.7865	−1.68581	−0.842904	+	0.538065i	$0.819156\pi$
$558$	0	0
$559$	0	0
$560$	−4.76197	−0.201230
$561$	0	0
$562$	−67.6682	−2.85441
$563$	−8.50404	−0.358402	−0.179201	−	0.983812i	$-0.557351\pi$
$563$	−8.50404	−0.358402	−0.179201	+	0.983812i	$0.557351\pi$
$564$	0	0
$565$	−6.70287	−0.281992
$566$	21.8341	0.917755
$567$	0	0
$568$	−66.2725	−2.78073
$569$	−46.0438	−1.93026	−0.965129	−	0.261776i	$-0.915692\pi$
$569$	−46.0438	−1.93026	−0.965129	+	0.261776i	$0.915692\pi$
$570$	0	0
$571$	0.829214	0.0347015	0.0173508	−	0.999849i	$-0.494477\pi$
$571$	0.829214	0.0347015	0.0173508	+	0.999849i	$0.494477\pi$
$572$	0	0
$573$	0	0
$574$	−42.4282	−1.77092
$575$	4.45490	0.185782
$576$	0	0
$577$	−14.6127	−0.608334	−0.304167	−	0.952619i	$-0.598378\pi$
$577$	−14.6127	−0.608334	−0.304167	+	0.952619i	$0.598378\pi$
$578$	39.8575	1.65786
$579$	0	0
$580$	12.7101	0.527757
$581$	−29.1191	−1.20807
$582$	0	0
$583$	0	0
$584$	−50.8115	−2.10259
$585$	0	0
$586$	45.2431	1.86898
$587$	−3.13122	−0.129239	−0.0646196	−	0.997910i	$-0.520583\pi$
$587$	−3.13122	−0.129239	−0.0646196	+	0.997910i	$0.520583\pi$
$588$	0	0
$589$	16.2499	0.669566
$590$	11.1342	0.458389
$591$	0	0
$592$	22.9243	0.942182
$593$	21.0877	0.865966	0.432983	−	0.901402i	$-0.357461\pi$
$593$	21.0877	0.865966	0.432983	+	0.901402i	$0.357461\pi$
$594$	0	0
$595$	1.14330	0.0468707
$596$	−28.0159	−1.14757
$597$	0	0
$598$	0	0
$599$	38.3897	1.56856	0.784281	−	0.620406i	$-0.213032\pi$
$599$	38.3897	1.56856	0.784281	+	0.620406i	$0.213032\pi$
$600$	0	0
$601$	10.0386	0.409482	0.204741	−	0.978816i	$-0.434365\pi$
$601$	10.0386	0.409482	0.204741	+	0.978816i	$0.434365\pi$
$602$	−32.3483	−1.31842
$603$	0	0
$604$	23.0851	0.939321
$605$	0	0
$606$	0	0
$607$	28.7233	1.16584	0.582922	−	0.812528i	$-0.301909\pi$
$607$	28.7233	1.16584	0.582922	+	0.812528i	$0.301909\pi$
$608$	−6.48146	−0.262858
$609$	0	0
$610$	18.5523	0.751162
$611$	0	0
$612$	0	0
$613$	31.5306	1.27351	0.636755	−	0.771066i	$-0.280276\pi$
$613$	31.5306	1.27351	0.636755	+	0.771066i	$0.280276\pi$
$614$	11.1578	0.450291
$615$	0	0
$616$	0	0
$617$	9.17225	0.369261	0.184630	−	0.982808i	$-0.440891\pi$
$617$	9.17225	0.369261	0.184630	+	0.982808i	$0.440891\pi$
$618$	0	0
$619$	32.5249	1.30729	0.653643	−	0.756803i	$-0.273240\pi$
$619$	32.5249	1.30729	0.653643	+	0.756803i	$0.273240\pi$
$620$	−26.5934	−1.06802
$621$	0	0
$622$	46.2047	1.85264
$623$	−0.404420	−0.0162027
$624$	0	0
$625$	1.00000	0.0400000
$626$	40.8868	1.63417
$627$	0	0
$628$	90.1110	3.59582
$629$	−5.50389	−0.219454
$630$	0	0
$631$	−13.8606	−0.551784	−0.275892	−	0.961189i	$-0.588973\pi$
$631$	−13.8606	−0.551784	−0.275892	+	0.961189i	$0.588973\pi$
$632$	25.8075	1.02657
$633$	0	0
$634$	45.5856	1.81043
$635$	12.0784	0.479315
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	20.7214	0.819085
$641$	37.9469	1.49881	0.749406	−	0.662111i	$-0.230339\pi$
$641$	37.9469	1.49881	0.749406	+	0.662111i	$0.230339\pi$
$642$	0	0
$643$	14.5306	0.573032	0.286516	−	0.958075i	$-0.407503\pi$
$643$	14.5306	0.573032	0.286516	+	0.958075i	$0.407503\pi$
$644$	−36.6463	−1.44407
$645$	0	0
$646$	−2.72218	−0.107103
$647$	7.98792	0.314038	0.157019	−	0.987596i	$-0.449812\pi$
$647$	7.98792	0.314038	0.157019	+	0.987596i	$0.449812\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−38.9388	−1.52496
$653$	47.1191	1.84391	0.921957	−	0.387292i	$-0.126589\pi$
$653$	47.1191	1.84391	0.921957	+	0.387292i	$0.126589\pi$
$654$	0	0
$655$	3.61562	0.141274
$656$	−16.8816	−0.659117
$657$	0	0
$658$	65.2398	2.54331
$659$	−6.76729	−0.263616	−0.131808	−	0.991275i	$-0.542078\pi$
$659$	−6.76729	−0.263616	−0.131808	+	0.991275i	$0.542078\pi$
$660$	0	0
$661$	−32.6127	−1.26849	−0.634243	−	0.773134i	$-0.718688\pi$
$661$	−32.6127	−1.26849	−0.634243	+	0.773134i	$0.718688\pi$
$662$	−46.4539	−1.80548
$663$	0	0
$664$	−51.6151	−2.00305
$665$	5.02655	0.194921
$666$	0	0
$667$	15.4322	0.597539
$668$	22.5292	0.871681
$669$	0	0
$670$	20.9361	0.808833
$671$	0	0
$672$	0	0
$673$	11.4629	0.441862	0.220931	−	0.975289i	$-0.429090\pi$
$673$	11.4629	0.441862	0.220931	+	0.975289i	$0.429090\pi$
$674$	28.7584	1.10773
$675$	0	0
$676$	−47.6980	−1.83454
$677$	36.6808	1.40976	0.704879	−	0.709328i	$-0.251001\pi$
$677$	36.6808	1.40976	0.704879	+	0.709328i	$0.251001\pi$
$678$	0	0
$679$	23.7936	0.913115
$680$	2.02655	0.0777148
$681$	0	0
$682$	0	0
$683$	38.8115	1.48508	0.742540	−	0.669802i	$-0.233621\pi$
$683$	38.8115	1.48508	0.742540	+	0.669802i	$0.233621\pi$
$684$	0	0
$685$	−6.88325	−0.262996
$686$	−47.9017	−1.82889
$687$	0	0
$688$	−12.8710	−0.490701
$689$	0	0
$690$	0	0
$691$	−27.8712	−1.06027	−0.530135	−	0.847913i	$-0.677859\pi$
$691$	−27.8712	−1.06027	−0.530135	+	0.847913i	$0.677859\pi$
$692$	60.9201	2.31584
$693$	0	0
$694$	−20.8003	−0.789569
$695$	20.7042	0.785353
$696$	0	0
$697$	4.05311	0.153522
$698$	19.2962	0.730373
$699$	0	0
$700$	−8.22607	−0.310916
$701$	−6.92820	−0.261675	−0.130837	−	0.991404i	$-0.541767\pi$
$701$	−6.92820	−0.261675	−0.130837	+	0.991404i	$0.541767\pi$
$702$	0	0
$703$	−24.1980	−0.912646
$704$	0	0
$705$	0	0
$706$	2.96199	0.111476
$707$	27.8727	1.04826
$708$	0	0
$709$	−12.5756	−0.472286	−0.236143	−	0.971718i	$-0.575883\pi$
$709$	−12.5756	−0.472286	−0.236143	+	0.971718i	$0.575883\pi$
$710$	−39.7060	−1.49014
$711$	0	0
$712$	−0.716853	−0.0268652
$713$	−32.2890	−1.20923
$714$	0	0
$715$	0	0
$716$	71.2012	2.66091
$717$	0	0
$718$	47.4202	1.76971
$719$	−28.3155	−1.05599	−0.527996	−	0.849247i	$-0.677056\pi$
$719$	−28.3155	−1.05599	−0.527996	+	0.849247i	$0.677056\pi$
$720$	0	0
$721$	4.62667	0.172306
$722$	33.2705	1.23820
$723$	0	0
$724$	38.7439	1.43991
$725$	3.46410	0.128654
$726$	0	0
$727$	21.4588	0.795865	0.397932	−	0.917415i	$-0.369728\pi$
$727$	21.4588	0.795865	0.397932	+	0.917415i	$0.369728\pi$
$728$	0	0
$729$	0	0
$730$	−30.4428	−1.12674
$731$	3.09019	0.114295
$732$	0	0
$733$	26.2791	0.970641	0.485320	−	0.874336i	$-0.338703\pi$
$733$	26.2791	0.970641	0.485320	+	0.874336i	$0.338703\pi$
$734$	−33.7632	−1.24622
$735$	0	0
$736$	12.8788	0.474719
$737$	0	0
$738$	0	0
$739$	29.4699	1.08407	0.542035	−	0.840356i	$-0.317654\pi$
$739$	29.4699	1.08407	0.542035	+	0.840356i	$0.317654\pi$
$740$	39.6006	1.45575
$741$	0	0
$742$	−20.2480	−0.743326
$743$	25.4912	0.935181	0.467591	−	0.883945i	$-0.345122\pi$
$743$	25.4912	0.935181	0.467591	+	0.883945i	$0.345122\pi$
$744$	0	0
$745$	−7.63566	−0.279749
$746$	−81.6151	−2.98814
$747$	0	0
$748$	0	0
$749$	−14.1062	−0.515430
$750$	0	0
$751$	−40.6006	−1.48154	−0.740768	−	0.671760i	$-0.765538\pi$
$751$	−40.6006	−1.48154	−0.740768	+	0.671760i	$0.765538\pi$
$752$	25.9581	0.946594
$753$	0	0
$754$	0	0
$755$	6.29181	0.228982
$756$	0	0
$757$	24.3792	0.886077	0.443038	−	0.896503i	$-0.353901\pi$
$757$	24.3792	0.886077	0.443038	+	0.896503i	$0.353901\pi$
$758$	63.6135	2.31055
$759$	0	0
$760$	8.90981	0.323193
$761$	27.6114	1.00091	0.500457	−	0.865761i	$-0.333165\pi$
$761$	27.6114	1.00091	0.500457	+	0.865761i	$0.333165\pi$
$762$	0	0
$763$	−1.49354	−0.0540697
$764$	32.6908	1.18271
$765$	0	0
$766$	−0.858981	−0.0310362
$767$	0	0
$768$	0	0
$769$	18.0233	0.649936	0.324968	−	0.945725i	$-0.394646\pi$
$769$	18.0233	0.649936	0.324968	+	0.945725i	$0.394646\pi$
$770$	0	0
$771$	0	0
$772$	−64.8482	−2.33394
$773$	−20.7318	−0.745672	−0.372836	−	0.927897i	$-0.621615\pi$
$773$	−20.7318	−0.745672	−0.372836	+	0.927897i	$0.621615\pi$
$774$	0	0
$775$	−7.24797	−0.260355
$776$	42.1753	1.51401
$777$	0	0
$778$	−47.4932	−1.70271
$779$	17.8196	0.638454
$780$	0	0
$781$	0	0
$782$	5.40903	0.193427
$783$	0	0
$784$	−4.19157	−0.149699
$785$	24.5596	0.876569
$786$	0	0
$787$	−19.9162	−0.709937	−0.354969	−	0.934878i	$-0.615508\pi$
$787$	−19.9162	−0.709937	−0.354969	+	0.934878i	$0.615508\pi$
$788$	−50.8403	−1.81111
$789$	0	0
$790$	15.4621	0.550118
$791$	15.0278	0.534328
$792$	0	0
$793$	0	0
$794$	−43.2046	−1.53327
$795$	0	0
$796$	51.7511	1.83427
$797$	−4.05311	−0.143568	−0.0717842	−	0.997420i	$-0.522869\pi$
$797$	−4.05311	−0.143568	−0.0717842	+	0.997420i	$0.522869\pi$
$798$	0	0
$799$	−6.23227	−0.220482
$800$	2.89093	0.102210
$801$	0	0
$802$	4.63552	0.163686
$803$	0	0
$804$	0	0
$805$	−9.98789	−0.352027
$806$	0	0
$807$	0	0
$808$	49.4057	1.73809
$809$	22.1169	0.777590	0.388795	−	0.921324i	$-0.372891\pi$
$809$	22.1169	0.777590	0.388795	+	0.921324i	$0.372891\pi$
$810$	0	0
$811$	−14.1986	−0.498581	−0.249290	−	0.968429i	$-0.580197\pi$
$811$	−14.1986	−0.498581	−0.249290	+	0.968429i	$0.580197\pi$
$812$	−28.4959	−1.00001
$813$	0	0
$814$	0	0
$815$	−10.6127	−0.371746
$816$	0	0
$817$	13.5861	0.475318
$818$	−84.1529	−2.94234
$819$	0	0
$820$	−29.1622	−1.01839
$821$	38.0037	1.32634	0.663170	−	0.748469i	$-0.269211\pi$
$821$	38.0037	1.32634	0.663170	+	0.748469i	$0.269211\pi$
$822$	0	0
$823$	1.02655	0.0357834	0.0178917	−	0.999840i	$-0.494305\pi$
$823$	1.02655	0.0357834	0.0178917	+	0.999840i	$0.494305\pi$
$824$	8.20100	0.285695
$825$	0	0
$826$	−24.9629	−0.868571
$827$	−20.2193	−0.703093	−0.351547	−	0.936170i	$-0.614344\pi$
$827$	−20.2193	−0.703093	−0.351547	+	0.936170i	$0.614344\pi$
$828$	0	0
$829$	−25.2624	−0.877401	−0.438700	−	0.898633i	$-0.644561\pi$
$829$	−25.2624	−0.877401	−0.438700	+	0.898633i	$0.644561\pi$
$830$	−30.9243	−1.07340
$831$	0	0
$832$	0	0
$833$	1.00635	0.0348681
$834$	0	0
$835$	6.14029	0.212493
$836$	0	0
$837$	0	0
$838$	68.4732	2.36537
$839$	20.5490	0.709432	0.354716	−	0.934974i	$-0.384578\pi$
$839$	20.5490	0.709432	0.354716	+	0.934974i	$0.384578\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	−53.1962	−1.83326
$843$	0	0
$844$	33.3510	1.14799
$845$	−13.0000	−0.447214
$846$	0	0
$847$	0	0
$848$	−8.05640	−0.276658
$849$	0	0
$850$	1.21417	0.0416459
$851$	48.0821	1.64823
$852$	0	0
$853$	−9.01868	−0.308794	−0.154397	−	0.988009i	$-0.549343\pi$
$853$	−9.01868	−0.308794	−0.154397	+	0.988009i	$0.549343\pi$
$854$	−41.5943	−1.42333
$855$	0	0
$856$	−25.0039	−0.854617
$857$	−21.6896	−0.740902	−0.370451	−	0.928852i	$-0.620797\pi$
$857$	−21.6896	−0.740902	−0.370451	+	0.928852i	$0.620797\pi$
$858$	0	0
$859$	1.38732	0.0473348	0.0236674	−	0.999720i	$-0.492466\pi$
$859$	1.38732	0.0473348	0.0236674	+	0.999720i	$0.492466\pi$
$860$	−22.2340	−0.758173
$861$	0	0
$862$	−91.0884	−3.10248
$863$	43.7665	1.48983	0.744915	−	0.667160i	$-0.232490\pi$
$863$	43.7665	1.48983	0.744915	+	0.667160i	$0.232490\pi$
$864$	0	0
$865$	16.6037	0.564541
$866$	39.5545	1.34412
$867$	0	0
$868$	59.6223	2.02371
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−2.64737	−0.0896512
$873$	0	0
$874$	23.7810	0.804404
$875$	−2.24200	−0.0757934
$876$	0	0
$877$	24.6024	0.830765	0.415383	−	0.909647i	$-0.363648\pi$
$877$	24.6024	0.830765	0.415383	+	0.909647i	$0.363648\pi$
$878$	−57.0627	−1.92577
$879$	0	0
$880$	0	0
$881$	−23.8196	−0.802503	−0.401252	−	0.915968i	$-0.631425\pi$
$881$	−23.8196	−0.802503	−0.401252	+	0.915968i	$0.631425\pi$
$882$	0	0
$883$	−4.92034	−0.165583	−0.0827913	−	0.996567i	$-0.526383\pi$
$883$	−4.92034	−0.165583	−0.0827913	+	0.996567i	$0.526383\pi$
$884$	0	0
$885$	0	0
$886$	31.4893	1.05791
$887$	−45.1848	−1.51716	−0.758579	−	0.651581i	$-0.774106\pi$
$887$	−45.1848	−1.51716	−0.758579	+	0.651581i	$0.774106\pi$
$888$	0	0
$889$	−27.0797	−0.908223
$890$	−0.429490	−0.0143965
$891$	0	0
$892$	20.9243	0.700597
$893$	−27.4004	−0.916919
$894$	0	0
$895$	19.4057	0.648662
$896$	−46.4573	−1.55203
$897$	0	0
$898$	−68.4732	−2.28498
$899$	−25.1077	−0.837388
$900$	0	0
$901$	1.93426	0.0644396
$902$	0	0
$903$	0	0
$904$	26.6375	0.885951
$905$	10.5596	0.351012
$906$	0	0
$907$	−50.1457	−1.66506	−0.832530	−	0.553980i	$-0.813109\pi$
$907$	−50.1457	−1.66506	−0.832530	+	0.553980i	$0.813109\pi$
$908$	26.2713	0.871843
$909$	0	0
$910$	0	0
$911$	12.0821	0.400296	0.200148	−	0.979766i	$-0.435858\pi$
$911$	12.0821	0.400296	0.200148	+	0.979766i	$0.435858\pi$
$912$	0	0
$913$	0	0
$914$	−64.2769	−2.12609
$915$	0	0
$916$	−53.9460	−1.78243
$917$	−8.10622	−0.267691
$918$	0	0
$919$	29.5504	0.974777	0.487389	−	0.873185i	$-0.337949\pi$
$919$	29.5504	0.974777	0.487389	+	0.873185i	$0.337949\pi$
$920$	−17.7040	−0.583684
$921$	0	0
$922$	−74.2093	−2.44395
$923$	0	0
$924$	0	0
$925$	10.7931	0.354874
$926$	−4.13710	−0.135953
$927$	0	0
$928$	10.0145	0.328741
$929$	−29.2995	−0.961286	−0.480643	−	0.876916i	$-0.659597\pi$
$929$	−29.2995	−0.961286	−0.480643	+	0.876916i	$0.659597\pi$
$930$	0	0
$931$	4.42446	0.145006
$932$	52.7113	1.72662
$933$	0	0
$934$	−69.2345	−2.26542
$935$	0	0
$936$	0	0
$937$	−40.4986	−1.32303	−0.661516	−	0.749931i	$-0.730087\pi$
$937$	−40.4986	−1.32303	−0.661516	+	0.749931i	$0.730087\pi$
$938$	−46.9388	−1.53260
$939$	0	0
$940$	44.8413	1.46256
$941$	−11.0078	−0.358843	−0.179422	−	0.983772i	$-0.557423\pi$
$941$	−11.0078	−0.358843	−0.179422	+	0.983772i	$0.557423\pi$
$942$	0	0
$943$	−35.4080	−1.15304
$944$	−9.93242	−0.323273
$945$	0	0
$946$	0	0
$947$	10.0941	0.328015	0.164008	−	0.986459i	$-0.447558\pi$
$947$	10.0941	0.328015	0.164008	+	0.986459i	$0.447558\pi$
$948$	0	0
$949$	0	0
$950$	5.33816	0.173193
$951$	0	0
$952$	−4.54353	−0.147257
$953$	7.23124	0.234243	0.117121	−	0.993118i	$-0.462633\pi$
$953$	7.23124	0.234243	0.117121	+	0.993118i	$0.462633\pi$
$954$	0	0
$955$	8.90981	0.288315
$956$	4.88842	0.158103
$957$	0	0
$958$	47.0371	1.51970
$959$	15.4322	0.498333
$960$	0	0
$961$	21.5330	0.694613
$962$	0	0
$963$	0	0
$964$	−24.1050	−0.776371
$965$	−17.6742	−0.568954
$966$	0	0
$967$	33.1660	1.06655	0.533274	−	0.845943i	$-0.320962\pi$
$967$	33.1660	1.06655	0.533274	+	0.845943i	$0.320962\pi$
$968$	0	0
$969$	0	0
$970$	25.2686	0.811326
$971$	−5.45885	−0.175183	−0.0875914	−	0.996156i	$-0.527917\pi$
$971$	−5.45885	−0.175183	−0.0875914	+	0.996156i	$0.527917\pi$
$972$	0	0
$973$	−46.4187	−1.48811
$974$	−95.4347	−3.05792
$975$	0	0
$976$	−16.5498	−0.529747
$977$	49.5885	1.58648	0.793239	−	0.608911i	$-0.208393\pi$
$977$	49.5885	1.58648	0.793239	+	0.608911i	$0.208393\pi$
$978$	0	0
$979$	0	0
$980$	−7.24073	−0.231297
$981$	0	0
$982$	−29.9614	−0.956106
$983$	−14.1683	−0.451899	−0.225949	−	0.974139i	$-0.572548\pi$
$983$	−14.1683	−0.451899	−0.225949	+	0.974139i	$0.572548\pi$
$984$	0	0
$985$	−13.8564	−0.441502
$986$	4.20603	0.133947
$987$	0	0
$988$	0	0
$989$	−26.9960	−0.858422
$990$	0	0
$991$	−26.2214	−0.832951	−0.416475	−	0.909147i	$-0.636735\pi$
$991$	−26.2214	−0.832951	−0.416475	+	0.909147i	$0.636735\pi$
$992$	−20.9534	−0.665270
$993$	0	0
$994$	89.0208	2.82357
$995$	14.1047	0.447148
$996$	0	0
$997$	15.9469	0.505043	0.252521	−	0.967591i	$-0.418740\pi$
$997$	15.9469	0.505043	0.252521	+	0.967591i	$0.418740\pi$
$998$	25.5716	0.809456
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.by.1.1	yes	6
3.2	odd	2		5445.2.a.bw.1.6	yes	6
11.10	odd	2	inner	5445.2.a.by.1.6	yes	6
33.32	even	2		5445.2.a.bw.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5445.2.a.bw.1.1	✓	6	33.32	even	2
5445.2.a.bw.1.6	yes	6	3.2	odd	2
5445.2.a.by.1.1	yes	6	1.1	even	1	trivial
5445.2.a.by.1.6	yes	6	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q-2.38098 q^{2} +3.66908 q^{4} +1.00000 q^{5} -2.24200 q^{7} -3.97405 q^{8} +O(q^{10})\)	\(q-2.38098 q^{2} +3.66908 q^{4} +1.00000 q^{5} -2.24200 q^{7} -3.97405 q^{8} -2.38098 q^{10} +5.33816 q^{14} +2.12398 q^{16} -0.509947 q^{17} -2.24200 q^{19} +3.66908 q^{20} +4.45490 q^{23} +1.00000 q^{25} -8.22607 q^{28} +3.46410 q^{29} -7.24797 q^{31} +2.89093 q^{32} +1.21417 q^{34} -2.24200 q^{35} +10.7931 q^{37} +5.33816 q^{38} -3.97405 q^{40} -7.94810 q^{41} -6.05983 q^{43} -10.6071 q^{46} +12.2214 q^{47} -1.97345 q^{49} -2.38098 q^{50} -3.79306 q^{53} +8.90981 q^{56} -8.24797 q^{58} -4.67632 q^{59} -7.79188 q^{61} +17.2573 q^{62} -11.1312 q^{64} -8.79306 q^{67} -1.87104 q^{68} +5.33816 q^{70} +16.6763 q^{71} +12.7858 q^{73} -25.6981 q^{74} -8.22607 q^{76} -6.49402 q^{79} +2.12398 q^{80} +18.9243 q^{82} +12.9880 q^{83} -0.509947 q^{85} +14.4283 q^{86} +0.180384 q^{89} +16.3454 q^{92} -29.0990 q^{94} -2.24200 q^{95} -10.6127 q^{97} +4.69874 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} + 6 q^{5}+O(q^{10}) \) 6 * q + 6 * q^4 + 6 * q^5	\( 6 q + 6 q^{4} + 6 q^{5} - 6 q^{16} + 6 q^{20} + 24 q^{23} + 6 q^{25} - 6 q^{31} - 6 q^{34} + 30 q^{37} + 12 q^{47} + 12 q^{49} + 12 q^{53} + 48 q^{56} - 12 q^{58} + 36 q^{59} - 18 q^{67} + 36 q^{71} - 6 q^{80} + 12 q^{82} + 60 q^{86} + 12 q^{89} + 18 q^{92} - 18 q^{97}+O(q^{100}) \) 6 * q + 6 * q^4 + 6 * q^5 - 6 * q^16 + 6 * q^20 + 24 * q^23 + 6 * q^25 - 6 * q^31 - 6 * q^34 + 30 * q^37 + 12 * q^47 + 12 * q^49 + 12 * q^53 + 48 * q^56 - 12 * q^58 + 36 * q^59 - 18 * q^67 + 36 * q^71 - 6 * q^80 + 12 * q^82 + 60 * q^86 + 12 * q^89 + 18 * q^92 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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