LMFDB - Embedded newform 819.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [819,2,Mod(1,819)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(819, 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("819.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 819 = 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	819.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:	$6.53974792554$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.199374400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 10x^{2} - 1 $ x^6 - 8x^4 + 10x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.330837$ of defining polynomial
Character	$\chi$	$=$	819.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69180 q^{2} +5.24581 q^{4} +1.40467 q^{5} -1.00000 q^{7} -8.73709 q^{8} +O(q^{10})$	\(q-2.69180 q^{2} +5.24581 q^{4} +1.40467 q^{5} -1.00000 q^{7} -8.73709 q^{8} -3.78109 q^{10} -6.04528 q^{11} +1.00000 q^{13} +2.69180 q^{14} +13.0269 q^{16} +6.70695 q^{17} +1.53528 q^{19} +7.36863 q^{20} +16.2727 q^{22} -0.742996 q^{23} -3.02690 q^{25} -2.69180 q^{26} -5.24581 q^{28} +6.12660 q^{29} +10.0269 q^{31} -17.5917 q^{32} -18.0538 q^{34} -1.40467 q^{35} -6.49162 q^{37} -4.13268 q^{38} -12.2727 q^{40} +7.53127 q^{41} -1.53528 q^{43} -31.7124 q^{44} +2.00000 q^{46} -5.30229 q^{47} +1.00000 q^{49} +8.14783 q^{50} +5.24581 q^{52} +5.96396 q^{53} -8.49162 q^{55} +8.73709 q^{56} -16.4916 q^{58} +10.1055 q^{59} +10.0000 q^{61} -26.9905 q^{62} +21.2996 q^{64} +1.40467 q^{65} -4.49162 q^{67} +35.1834 q^{68} +3.78109 q^{70} +3.23594 q^{71} +8.02690 q^{73} +17.4742 q^{74} +8.05381 q^{76} +6.04528 q^{77} -10.5185 q^{79} +18.2985 q^{80} -20.2727 q^{82} +8.11162 q^{83} +9.42105 q^{85} +4.13268 q^{86} +52.8181 q^{88} -1.24202 q^{89} -1.00000 q^{91} -3.89762 q^{92} +14.2727 q^{94} +2.15657 q^{95} -0.464716 q^{97} -2.69180 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4} - 6 q^{7}+O(q^{10}) $ 6 * q + 12 * q^4 - 6 * q^7	$ 6 q + 12 q^{4} - 6 q^{7} + 8 q^{10} + 6 q^{13} + 28 q^{16} - 2 q^{19} + 28 q^{22} + 32 q^{25} - 12 q^{28} + 10 q^{31} - 8 q^{34} - 4 q^{40} + 2 q^{43} + 12 q^{46} + 6 q^{49} + 12 q^{52} - 12 q^{55} - 60 q^{58} + 60 q^{61} + 8 q^{64} + 12 q^{67} - 8 q^{70} - 2 q^{73} - 52 q^{76} + 26 q^{79} - 52 q^{82} + 40 q^{85} + 108 q^{88} - 6 q^{91} + 16 q^{94} - 14 q^{97}+O(q^{100}) $ 6 * q + 12 * q^4 - 6 * q^7 + 8 * q^10 + 6 * q^13 + 28 * q^16 - 2 * q^19 + 28 * q^22 + 32 * q^25 - 12 * q^28 + 10 * q^31 - 8 * q^34 - 4 * q^40 + 2 * q^43 + 12 * q^46 + 6 * q^49 + 12 * q^52 - 12 * q^55 - 60 * q^58 + 60 * q^61 + 8 * q^64 + 12 * q^67 - 8 * q^70 - 2 * q^73 - 52 * q^76 + 26 * q^79 - 52 * q^82 + 40 * q^85 + 108 * q^88 - 6 * q^91 + 16 * q^94 - 14 * 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 $

$2$	−2.69180	−1.90339	−0.951697	−	0.307040i	$-0.900661\pi$
$2$	−2.69180	−1.90339	−0.951697	+	0.307040i	$0.900661\pi$
$3$	0	0
$3$	0	0
$4$	5.24581	2.62291
$5$	1.40467	0.628187	0.314094	−	0.949392i	$-0.398299\pi$
$5$	1.40467	0.628187	0.314094	+	0.949392i	$0.398299\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−8.73709	−3.08903
$9$	0	0
$10$	−3.78109	−1.19569
$11$	−6.04528	−1.82272	−0.911360	−	0.411609i	$-0.864967\pi$
$11$	−6.04528	−1.82272	−0.911360	+	0.411609i	$0.864967\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.69180	0.719415
$15$	0	0
$16$	13.0269	3.25673
$17$	6.70695	1.62668	0.813338	−	0.581792i	$-0.197648\pi$
$17$	6.70695	1.62668	0.813338	+	0.581792i	$0.197648\pi$
$18$	0	0
$19$	1.53528	0.352218	0.176109	−	0.984371i	$-0.443649\pi$
$19$	1.53528	0.352218	0.176109	+	0.984371i	$0.443649\pi$
$20$	7.36863	1.64768
$21$	0	0
$22$	16.2727	3.46935
$23$	−0.742996	−0.154925	−0.0774627	−	0.996995i	$-0.524682\pi$
$23$	−0.742996	−0.154925	−0.0774627	+	0.996995i	$0.524682\pi$
$24$	0	0
$25$	−3.02690	−0.605381
$26$	−2.69180	−0.527906
$27$	0	0
$28$	−5.24581	−0.991365
$29$	6.12660	1.13768	0.568841	−	0.822448i	$-0.307392\pi$
$29$	6.12660	1.13768	0.568841	+	0.822448i	$0.307392\pi$
$30$	0	0
$31$	10.0269	1.80089	0.900443	−	0.434975i	$-0.143243\pi$
$31$	10.0269	1.80089	0.900443	+	0.434975i	$0.143243\pi$
$32$	−17.5917	−3.10980
$33$	0	0
$34$	−18.0538	−3.09620
$35$	−1.40467	−0.237432
$36$	0	0
$37$	−6.49162	−1.06722	−0.533608	−	0.845732i	$-0.679164\pi$
$37$	−6.49162	−1.06722	−0.533608	+	0.845732i	$0.679164\pi$
$38$	−4.13268	−0.670410
$39$	0	0
$40$	−12.2727	−1.94049
$41$	7.53127	1.17619	0.588094	−	0.808793i	$-0.299879\pi$
$41$	7.53127	1.17619	0.588094	+	0.808793i	$0.299879\pi$
$42$	0	0
$43$	−1.53528	−0.234129	−0.117064	−	0.993124i	$-0.537348\pi$
$43$	−1.53528	−0.234129	−0.117064	+	0.993124i	$0.537348\pi$
$44$	−31.7124	−4.78082
$45$	0	0
$46$	2.00000	0.294884
$47$	−5.30229	−0.773418	−0.386709	−	0.922202i	$-0.626388\pi$
$47$	−5.30229	−0.773418	−0.386709	+	0.922202i	$0.626388\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	8.14783	1.15228
$51$	0	0
$52$	5.24581	0.727463
$53$	5.96396	0.819213	0.409606	−	0.912262i	$-0.365666\pi$
$53$	5.96396	0.819213	0.409606	+	0.912262i	$0.365666\pi$
$54$	0	0
$55$	−8.49162	−1.14501
$56$	8.73709	1.16754
$57$	0	0
$58$	−16.4916	−2.16546
$59$	10.1055	1.31563	0.657815	−	0.753180i	$-0.271481\pi$
$59$	10.1055	1.31563	0.657815	+	0.753180i	$0.271481\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	−26.9905	−3.42779
$63$	0	0
$64$	21.2996	2.66245
$65$	1.40467	0.174228
$66$	0	0
$67$	−4.49162	−0.548739	−0.274369	−	0.961624i	$-0.588469\pi$
$67$	−4.49162	−0.548739	−0.274369	+	0.961624i	$0.588469\pi$
$68$	35.1834	4.26662
$69$	0	0
$70$	3.78109	0.451927
$71$	3.23594	0.384036	0.192018	−	0.981391i	$-0.438497\pi$
$71$	3.23594	0.384036	0.192018	+	0.981391i	$0.438497\pi$
$72$	0	0
$73$	8.02690	0.939478	0.469739	−	0.882805i	$-0.344348\pi$
$73$	8.02690	0.939478	0.469739	+	0.882805i	$0.344348\pi$
$74$	17.4742	2.03133
$75$	0	0
$76$	8.05381	0.923835
$77$	6.04528	0.688924
$78$	0	0
$79$	−10.5185	−1.18343	−0.591713	−	0.806149i	$-0.701548\pi$
$79$	−10.5185	−1.18343	−0.591713	+	0.806149i	$0.701548\pi$
$80$	18.2985	2.04583
$81$	0	0
$82$	−20.2727	−2.23875
$83$	8.11162	0.890366	0.445183	−	0.895440i	$-0.353139\pi$
$83$	8.11162	0.890366	0.445183	+	0.895440i	$0.353139\pi$
$84$	0	0
$85$	9.42105	1.02186
$86$	4.13268	0.445639
$87$	0	0
$88$	52.8181	5.63043
$89$	−1.24202	−0.131654	−0.0658271	−	0.997831i	$-0.520969\pi$
$89$	−1.24202	−0.131654	−0.0658271	+	0.997831i	$0.520969\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	−3.89762	−0.406355
$93$	0	0
$94$	14.2727	1.47212
$95$	2.15657	0.221259
$96$	0	0
$97$	−0.464716	−0.0471847	−0.0235924	−	0.999722i	$-0.507510\pi$
$97$	−0.464716	−0.0471847	−0.0235924	+	0.999722i	$0.507510\pi$
$98$	−2.69180	−0.271913
$99$	0	0
$100$	−15.8786	−1.58786
$101$	14.6648	1.45921	0.729603	−	0.683871i	$-0.239705\pi$
$101$	14.6648	1.45921	0.729603	+	0.683871i	$0.239705\pi$
$102$	0	0
$103$	−11.5622	−1.13926	−0.569628	−	0.821903i	$-0.692913\pi$
$103$	−11.5622	−1.13926	−0.569628	+	0.821903i	$0.692913\pi$
$104$	−8.73709	−0.856742
$105$	0	0
$106$	−16.0538	−1.55928
$107$	5.22096	0.504729	0.252365	−	0.967632i	$-0.418792\pi$
$107$	5.22096	0.504729	0.252365	+	0.967632i	$0.418792\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	22.8578	2.17940
$111$	0	0
$112$	−13.0269	−1.23093
$113$	−6.12660	−0.576342	−0.288171	−	0.957579i	$-0.593047\pi$
$113$	−6.12660	−0.576342	−0.288171	+	0.957579i	$0.593047\pi$
$114$	0	0
$115$	−1.04366	−0.0973221
$116$	32.1390	2.98403
$117$	0	0
$118$	−27.2021	−2.50416
$119$	−6.70695	−0.614826
$120$	0	0
$121$	25.5454	2.32231
$122$	−26.9180	−2.43705
$123$	0	0
$124$	52.5992	4.72355
$125$	−11.2751	−1.00848
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−22.1510	−1.95789
$129$	0	0
$130$	−3.78109	−0.331624
$131$	−4.29533	−0.375285	−0.187642	−	0.982237i	$-0.560085\pi$
$131$	−4.29533	−0.375285	−0.187642	+	0.982237i	$0.560085\pi$
$132$	0	0
$133$	−1.53528	−0.133126
$134$	12.0906	1.04447
$135$	0	0
$136$	−58.5992	−5.02484
$137$	2.14767	0.183487	0.0917437	−	0.995783i	$-0.470756\pi$
$137$	2.14767	0.183487	0.0917437	+	0.995783i	$0.470756\pi$
$138$	0	0
$139$	21.4749	1.82147	0.910737	−	0.412987i	$-0.135514\pi$
$139$	21.4749	1.82147	0.910737	+	0.412987i	$0.135514\pi$
$140$	−7.36863	−0.622763
$141$	0	0
$142$	−8.71053	−0.730971
$143$	−6.04528	−0.505532
$144$	0	0
$145$	8.60585	0.714677
$146$	−21.6069	−1.78820
$147$	0	0
$148$	−34.0538	−2.79921
$149$	−15.7242	−1.28818	−0.644089	−	0.764950i	$-0.722763\pi$
$149$	−15.7242	−1.28818	−0.644089	+	0.764950i	$0.722763\pi$
$150$	0	0
$151$	11.5622	0.940918	0.470459	−	0.882422i	$-0.344088\pi$
$151$	11.5622	0.940918	0.470459	+	0.882422i	$0.344088\pi$
$152$	−13.4139	−1.08801
$153$	0	0
$154$	−16.2727	−1.31129
$155$	14.0845	1.13129
$156$	0	0
$157$	1.07057	0.0854406	0.0427203	−	0.999087i	$-0.486398\pi$
$157$	1.07057	0.0854406	0.0427203	+	0.999087i	$0.486398\pi$
$158$	28.3138	2.25253
$159$	0	0
$160$	−24.7105	−1.95354
$161$	0.742996	0.0585563
$162$	0	0
$163$	20.9832	1.64353	0.821767	−	0.569823i	$-0.192988\pi$
$163$	20.9832	1.64353	0.821767	+	0.569823i	$0.192988\pi$
$164$	39.5076	3.08503
$165$	0	0
$166$	−21.8349	−1.69472
$167$	−8.27427	−0.640282	−0.320141	−	0.947370i	$-0.603730\pi$
$167$	−8.27427	−0.640282	−0.320141	+	0.947370i	$0.603730\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−25.3596	−1.94499
$171$	0	0
$172$	−8.05381	−0.614097
$173$	−0.235068	−0.0178719	−0.00893594	−	0.999960i	$-0.502844\pi$
$173$	−0.235068	−0.0178719	−0.00893594	+	0.999960i	$0.502844\pi$
$174$	0	0
$175$	3.02690	0.228812
$176$	−78.7513	−5.93610
$177$	0	0
$178$	3.34328	0.250590
$179$	0.580350	0.0433774	0.0216887	−	0.999765i	$-0.493096\pi$
$179$	0.580350	0.0433774	0.0216887	+	0.999765i	$0.493096\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	2.69180	0.199530
$183$	0	0
$184$	6.49162	0.478569
$185$	−9.11858	−0.670411
$186$	0	0
$187$	−40.5454	−2.96498
$188$	−27.8148	−2.02860
$189$	0	0
$190$	−5.80505	−0.421143
$191$	−20.2835	−1.46766	−0.733832	−	0.679331i	$-0.762270\pi$
$191$	−20.2835	−1.46766	−0.733832	+	0.679331i	$0.762270\pi$
$192$	0	0
$193$	−21.5622	−1.55208	−0.776040	−	0.630684i	$-0.782775\pi$
$193$	−21.5622	−1.55208	−0.776040	+	0.630684i	$0.782775\pi$
$194$	1.25092	0.0898111
$195$	0	0
$196$	5.24581	0.374701
$197$	10.1055	0.719990	0.359995	−	0.932954i	$-0.382778\pi$
$197$	10.1055	0.719990	0.359995	+	0.932954i	$0.382778\pi$
$198$	0	0
$199$	0.983241	0.0697001	0.0348501	−	0.999393i	$-0.488905\pi$
$199$	0.983241	0.0697001	0.0348501	+	0.999393i	$0.488905\pi$
$200$	26.4463	1.87004
$201$	0	0
$202$	−39.4749	−2.77744
$203$	−6.12660	−0.430003
$204$	0	0
$205$	10.5789	0.738866
$206$	31.1231	2.16845
$207$	0	0
$208$	13.0269	0.903253
$209$	−9.28122	−0.641996
$210$	0	0
$211$	−2.46472	−0.169678	−0.0848390	−	0.996395i	$-0.527038\pi$
$211$	−2.46472	−0.169678	−0.0848390	+	0.996395i	$0.527038\pi$
$212$	31.2858	2.14872
$213$	0	0
$214$	−14.0538	−0.960699
$215$	−2.15657	−0.147077
$216$	0	0
$217$	−10.0269	−0.680671
$218$	−26.9180	−1.82312
$219$	0	0
$220$	−44.5454	−3.00325
$221$	6.70695	0.451159
$222$	0	0
$223$	2.46472	0.165050	0.0825248	−	0.996589i	$-0.473702\pi$
$223$	2.46472	0.165050	0.0825248	+	0.996589i	$0.473702\pi$
$224$	17.5917	1.17540
$225$	0	0
$226$	16.4916	1.09701
$227$	15.7242	1.04365	0.521827	−	0.853052i	$-0.325251\pi$
$227$	15.7242	1.04365	0.521827	+	0.853052i	$0.325251\pi$
$228$	0	0
$229$	−13.5622	−0.896215	−0.448107	−	0.893980i	$-0.647902\pi$
$229$	−13.5622	−0.896215	−0.448107	+	0.893980i	$0.647902\pi$
$230$	2.80934	0.185242
$231$	0	0
$232$	−53.5287	−3.51433
$233$	−23.2031	−1.52008	−0.760042	−	0.649874i	$-0.774821\pi$
$233$	−23.2031	−1.52008	−0.760042	+	0.649874i	$0.774821\pi$
$234$	0	0
$235$	−7.44796	−0.485851
$236$	53.0118	3.45077
$237$	0	0
$238$	18.0538	1.17025
$239$	16.6499	1.07699	0.538495	−	0.842629i	$-0.318993\pi$
$239$	16.6499	1.07699	0.538495	+	0.842629i	$0.318993\pi$
$240$	0	0
$241$	−24.0807	−1.55118	−0.775588	−	0.631240i	$-0.782546\pi$
$241$	−24.0807	−1.55118	−0.775588	+	0.631240i	$0.782546\pi$
$242$	−68.7633	−4.42027
$243$	0	0
$244$	52.4581	3.35829
$245$	1.40467	0.0897410
$246$	0	0
$247$	1.53528	0.0976878
$248$	−87.6059	−5.56298
$249$	0	0
$250$	30.3505	1.91953
$251$	5.45603	0.344382	0.172191	−	0.985064i	$-0.444915\pi$
$251$	5.45603	0.344382	0.172191	+	0.985064i	$0.444915\pi$
$252$	0	0
$253$	4.49162	0.282386
$254$	−10.7672	−0.675595
$255$	0	0
$256$	17.0269	1.06418
$257$	3.89762	0.243127	0.121563	−	0.992584i	$-0.461209\pi$
$257$	3.89762	0.243127	0.121563	+	0.992584i	$0.461209\pi$
$258$	0	0
$259$	6.49162	0.403370
$260$	7.36863	0.456983
$261$	0	0
$262$	11.5622	0.714314
$263$	4.87568	0.300647	0.150324	−	0.988637i	$-0.451968\pi$
$263$	4.87568	0.300647	0.150324	+	0.988637i	$0.451968\pi$
$264$	0	0
$265$	8.37739	0.514619
$266$	4.13268	0.253391
$267$	0	0
$268$	−23.5622	−1.43929
$269$	−8.03030	−0.489616	−0.244808	−	0.969572i	$-0.578725\pi$
$269$	−8.03030	−0.489616	−0.244808	+	0.969572i	$0.578725\pi$
$270$	0	0
$271$	32.5454	1.97699	0.988497	−	0.151240i	$-0.0483267\pi$
$271$	32.5454	1.97699	0.988497	+	0.151240i	$0.0483267\pi$
$272$	87.3709	5.29764
$273$	0	0
$274$	−5.78109	−0.349249
$275$	18.2985	1.10344
$276$	0	0
$277$	11.5353	0.693088	0.346544	−	0.938034i	$-0.387355\pi$
$277$	11.5353	0.693088	0.346544	+	0.938034i	$0.387355\pi$
$278$	−57.8061	−3.46698
$279$	0	0
$280$	12.2727	0.733435
$281$	−4.48687	−0.267664	−0.133832	−	0.991004i	$-0.542728\pi$
$281$	−4.48687	−0.267664	−0.133832	+	0.991004i	$0.542728\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	16.9751	1.00729
$285$	0	0
$286$	16.2727	0.962226
$287$	−7.53127	−0.444557
$288$	0	0
$289$	27.9832	1.64607
$290$	−23.1653	−1.36031
$291$	0	0
$292$	42.1076	2.46416
$293$	−22.9391	−1.34012	−0.670058	−	0.742308i	$-0.733731\pi$
$293$	−22.9391	−1.34012	−0.670058	+	0.742308i	$0.733731\pi$
$294$	0	0
$295$	14.1949	0.826462
$296$	56.7178	3.29666
$297$	0	0
$298$	42.3265	2.45191
$299$	−0.742996	−0.0429686
$300$	0	0
$301$	1.53528	0.0884923
$302$	−31.1231	−1.79094
$303$	0	0
$304$	20.0000	1.14708
$305$	14.0467	0.804311
$306$	0	0
$307$	−12.6059	−0.719454	−0.359727	−	0.933058i	$-0.617130\pi$
$307$	−12.6059	−0.719454	−0.359727	+	0.933058i	$0.617130\pi$
$308$	31.7124	1.80698
$309$	0	0
$310$	−37.9127	−2.15330
$311$	−18.7251	−1.06180	−0.530901	−	0.847434i	$-0.678147\pi$
$311$	−18.7251	−1.06180	−0.530901	+	0.847434i	$0.678147\pi$
$312$	0	0
$313$	30.9832	1.75128	0.875638	−	0.482968i	$-0.160441\pi$
$313$	30.9832	1.75128	0.875638	+	0.482968i	$0.160441\pi$
$314$	−2.88176	−0.162627
$315$	0	0
$316$	−55.1782	−3.10402
$317$	−5.11965	−0.287548	−0.143774	−	0.989611i	$-0.545924\pi$
$317$	−5.11965	−0.287548	−0.143774	+	0.989611i	$0.545924\pi$
$318$	0	0
$319$	−37.0371	−2.07368
$320$	29.9189	1.67252
$321$	0	0
$322$	−2.00000	−0.111456
$323$	10.2971	0.572945
$324$	0	0
$325$	−3.02690	−0.167902
$326$	−56.4828	−3.12829
$327$	0	0
$328$	−65.8014	−3.63327
$329$	5.30229	0.292324
$330$	0	0
$331$	−16.0538	−0.882397	−0.441199	−	0.897410i	$-0.645447\pi$
$331$	−16.0538	−0.882397	−0.441199	+	0.897410i	$0.645447\pi$
$332$	42.5520	2.33535
$333$	0	0
$334$	22.2727	1.21871
$335$	−6.30924	−0.344711
$336$	0	0
$337$	2.11423	0.115170	0.0575848	−	0.998341i	$-0.481660\pi$
$337$	2.11423	0.115170	0.0575848	+	0.998341i	$0.481660\pi$
$338$	−2.69180	−0.146415
$339$	0	0
$340$	49.4211	2.68023
$341$	−60.6155	−3.28251
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	13.4139	0.723230
$345$	0	0
$346$	0.632757	0.0340172
$347$	−9.51629	−0.510861	−0.255431	−	0.966827i	$-0.582217\pi$
$347$	−9.51629	−0.510861	−0.255431	+	0.966827i	$0.582217\pi$
$348$	0	0
$349$	30.4312	1.62894	0.814472	−	0.580202i	$-0.197027\pi$
$349$	30.4312	1.62894	0.814472	+	0.580202i	$0.197027\pi$
$350$	−8.14783	−0.435520
$351$	0	0
$352$	106.347	5.66830
$353$	33.8890	1.80373	0.901864	−	0.432021i	$-0.142199\pi$
$353$	33.8890	1.80373	0.901864	+	0.432021i	$0.142199\pi$
$354$	0	0
$355$	4.54543	0.241246
$356$	−6.51542	−0.345316
$357$	0	0
$358$	−1.56219	−0.0825642
$359$	−9.70783	−0.512360	−0.256180	−	0.966629i	$-0.582464\pi$
$359$	−9.70783	−0.512360	−0.256180	+	0.966629i	$0.582464\pi$
$360$	0	0
$361$	−16.6429	−0.875942
$362$	−26.9180	−1.41478
$363$	0	0
$364$	−5.24581	−0.274955
$365$	11.2751	0.590168
$366$	0	0
$367$	−20.4916	−1.06965	−0.534827	−	0.844962i	$-0.679623\pi$
$367$	−20.4916	−1.06965	−0.534827	+	0.844962i	$0.679623\pi$
$368$	−9.67894	−0.504550
$369$	0	0
$370$	24.5454	1.27606
$371$	−5.96396	−0.309633
$372$	0	0
$373$	−1.56219	−0.0808871	−0.0404435	−	0.999182i	$-0.512877\pi$
$373$	−1.56219	−0.0808871	−0.0404435	+	0.999182i	$0.512877\pi$
$374$	109.140	5.64351
$375$	0	0
$376$	46.3265	2.38911
$377$	6.12660	0.315536
$378$	0	0
$379$	−28.5992	−1.46904	−0.734522	−	0.678585i	$-0.762594\pi$
$379$	−28.5992	−1.46904	−0.734522	+	0.678585i	$0.762594\pi$
$380$	11.3129	0.580341
$381$	0	0
$382$	54.5992	2.79354
$383$	2.31031	0.118051	0.0590257	−	0.998256i	$-0.481201\pi$
$383$	2.31031	0.118051	0.0590257	+	0.998256i	$0.481201\pi$
$384$	0	0
$385$	8.49162	0.432773
$386$	58.0412	2.95422
$387$	0	0
$388$	−2.43781	−0.123761
$389$	21.5344	1.09184	0.545920	−	0.837838i	$-0.316180\pi$
$389$	21.5344	1.09184	0.545920	+	0.837838i	$0.316180\pi$
$390$	0	0
$391$	−4.98324	−0.252013
$392$	−8.73709	−0.441289
$393$	0	0
$394$	−27.2021	−1.37042
$395$	−14.7750	−0.743413
$396$	0	0
$397$	−13.0101	−0.652960	−0.326480	−	0.945204i	$-0.605863\pi$
$397$	−13.0101	−0.652960	−0.326480	+	0.945204i	$0.605863\pi$
$398$	−2.64669	−0.132667
$399$	0	0
$400$	−39.4312	−1.97156
$401$	−5.11965	−0.255663	−0.127832	−	0.991796i	$-0.540802\pi$
$401$	−5.11965	−0.255663	−0.127832	+	0.991796i	$0.540802\pi$
$402$	0	0
$403$	10.0269	0.499476
$404$	76.9289	3.82736
$405$	0	0
$406$	16.4916	0.818465
$407$	39.2437	1.94524
$408$	0	0
$409$	16.5185	0.816788	0.408394	−	0.912806i	$-0.366089\pi$
$409$	16.5185	0.816788	0.408394	+	0.912806i	$0.366089\pi$
$410$	−28.4765	−1.40635
$411$	0	0
$412$	−60.6530	−2.98816
$413$	−10.1055	−0.497261
$414$	0	0
$415$	11.3941	0.559317
$416$	−17.5917	−0.862504
$417$	0	0
$418$	24.9832	1.22197
$419$	18.3998	0.898889	0.449445	−	0.893308i	$-0.351622\pi$
$419$	18.3998	0.898889	0.449445	+	0.893308i	$0.351622\pi$
$420$	0	0
$421$	−35.4749	−1.72894	−0.864469	−	0.502685i	$-0.832345\pi$
$421$	−35.4749	−1.72894	−0.864469	+	0.502685i	$0.832345\pi$
$422$	6.63453	0.322964
$423$	0	0
$424$	−52.1076	−2.53057
$425$	−20.3013	−0.984758
$426$	0	0
$427$	−10.0000	−0.483934
$428$	27.3882	1.32386
$429$	0	0
$430$	5.80505	0.279945
$431$	−37.1684	−1.79034	−0.895170	−	0.445725i	$-0.852946\pi$
$431$	−37.1684	−1.79034	−0.895170	+	0.445725i	$0.852946\pi$
$432$	0	0
$433$	1.07057	0.0514482	0.0257241	−	0.999669i	$-0.491811\pi$
$433$	1.07057	0.0514482	0.0257241	+	0.999669i	$0.491811\pi$
$434$	26.9905	1.29558
$435$	0	0
$436$	52.4581	2.51229
$437$	−1.14071	−0.0545676
$438$	0	0
$439$	−13.4749	−0.643120	−0.321560	−	0.946889i	$-0.604207\pi$
$439$	−13.4749	−0.643120	−0.321560	+	0.946889i	$0.604207\pi$
$440$	74.1920	3.53697
$441$	0	0
$442$	−18.0538	−0.858732
$443$	−22.9680	−1.09124	−0.545621	−	0.838032i	$-0.683706\pi$
$443$	−22.9680	−1.09124	−0.545621	+	0.838032i	$0.683706\pi$
$444$	0	0
$445$	−1.74463	−0.0827035
$446$	−6.63453	−0.314154
$447$	0	0
$448$	−21.2996	−1.00631
$449$	−12.5896	−0.594139	−0.297070	−	0.954856i	$-0.596009\pi$
$449$	−12.5896	−0.594139	−0.297070	+	0.954856i	$0.596009\pi$
$450$	0	0
$451$	−45.5287	−2.14386
$452$	−32.1390	−1.51169
$453$	0	0
$454$	−42.3265	−1.98648
$455$	−1.40467	−0.0658519
$456$	0	0
$457$	−40.0203	−1.87207	−0.936035	−	0.351907i	$-0.885533\pi$
$457$	−40.0203	−1.87207	−0.936035	+	0.351907i	$0.885533\pi$
$458$	36.5068	1.70585
$459$	0	0
$460$	−5.47486	−0.255267
$461$	−7.20598	−0.335616	−0.167808	−	0.985820i	$-0.553669\pi$
$461$	−7.20598	−0.335616	−0.167808	+	0.985820i	$0.553669\pi$
$462$	0	0
$463$	0.491620	0.0228475	0.0114238	−	0.999935i	$-0.496364\pi$
$463$	0.491620	0.0228475	0.0114238	+	0.999935i	$0.496364\pi$
$464$	79.8107	3.70512
$465$	0	0
$466$	62.4581	2.89332
$467$	−21.3718	−0.988968	−0.494484	−	0.869187i	$-0.664643\pi$
$467$	−21.3718	−0.988968	−0.494484	+	0.869187i	$0.664643\pi$
$468$	0	0
$469$	4.49162	0.207404
$470$	20.0484	0.924766
$471$	0	0
$472$	−88.2930	−4.06402
$473$	9.28122	0.426751
$474$	0	0
$475$	−4.64716	−0.213226
$476$	−35.1834	−1.61263
$477$	0	0
$478$	−44.8181	−2.04993
$479$	−24.4975	−1.11932	−0.559660	−	0.828722i	$-0.689068\pi$
$479$	−24.4975	−1.11932	−0.559660	+	0.828722i	$0.689068\pi$
$480$	0	0
$481$	−6.49162	−0.295992
$482$	64.8206	2.95250
$483$	0	0
$484$	134.006	6.09120
$485$	−0.652772	−0.0296409
$486$	0	0
$487$	17.4749	0.791862	0.395931	−	0.918280i	$-0.370422\pi$
$487$	17.4749	0.791862	0.395931	+	0.918280i	$0.370422\pi$
$488$	−87.3709	−3.95509
$489$	0	0
$490$	−3.78109	−0.170812
$491$	−26.7554	−1.20745	−0.603727	−	0.797191i	$-0.706318\pi$
$491$	−26.7554	−1.20745	−0.603727	+	0.797191i	$0.706318\pi$
$492$	0	0
$493$	41.0909	1.85064
$494$	−4.13268	−0.185938
$495$	0	0
$496$	130.620	5.86499
$497$	−3.23594	−0.145152
$498$	0	0
$499$	2.14114	0.0958504	0.0479252	−	0.998851i	$-0.484739\pi$
$499$	2.14114	0.0958504	0.0479252	+	0.998851i	$0.484739\pi$
$500$	−59.1473	−2.64515
$501$	0	0
$502$	−14.6866	−0.655494
$503$	−37.7577	−1.68353	−0.841766	−	0.539843i	$-0.818484\pi$
$503$	−37.7577	−1.68353	−0.841766	+	0.539843i	$0.818484\pi$
$504$	0	0
$505$	20.5992	0.916654
$506$	−12.0906	−0.537491
$507$	0	0
$508$	20.9832	0.930981
$509$	11.0112	0.488062	0.244031	−	0.969767i	$-0.421530\pi$
$509$	11.0112	0.488062	0.244031	+	0.969767i	$0.421530\pi$
$510$	0	0
$511$	−8.02690	−0.355089
$512$	−1.53110	−0.0676659
$513$	0	0
$514$	−10.4916	−0.462766
$515$	−16.2410	−0.715666
$516$	0	0
$517$	32.0538	1.40972
$518$	−17.4742	−0.767771
$519$	0	0
$520$	−12.2727	−0.538194
$521$	−24.7415	−1.08394	−0.541972	−	0.840396i	$-0.682322\pi$
$521$	−24.7415	−1.08394	−0.541972	+	0.840396i	$0.682322\pi$
$522$	0	0
$523$	−11.5084	−0.503226	−0.251613	−	0.967828i	$-0.580961\pi$
$523$	−11.5084	−0.503226	−0.251613	+	0.967828i	$0.580961\pi$
$524$	−22.5325	−0.984336
$525$	0	0
$526$	−13.1244	−0.572250
$527$	67.2500	2.92946
$528$	0	0
$529$	−22.4480	−0.975998
$530$	−22.5503	−0.979522
$531$	0	0
$532$	−8.05381	−0.349177
$533$	7.53127	0.326216
$534$	0	0
$535$	7.33372	0.317065
$536$	39.2437	1.69507
$537$	0	0
$538$	21.6160	0.931932
$539$	−6.04528	−0.260389
$540$	0	0
$541$	38.4916	1.65488	0.827442	−	0.561551i	$-0.189795\pi$
$541$	38.4916	1.65488	0.827442	+	0.561551i	$0.189795\pi$
$542$	−87.6059	−3.76300
$543$	0	0
$544$	−117.987	−5.05864
$545$	14.0467	0.601694
$546$	0	0
$547$	40.4312	1.72871	0.864357	−	0.502879i	$-0.167726\pi$
$547$	40.4312	1.72871	0.864357	+	0.502879i	$0.167726\pi$
$548$	11.2662	0.481270
$549$	0	0
$550$	−49.2560	−2.10028
$551$	9.40608	0.400712
$552$	0	0
$553$	10.5185	0.447293
$554$	−31.0507	−1.31922
$555$	0	0
$556$	112.653	4.77755
$557$	26.0035	1.10180	0.550902	−	0.834570i	$-0.314284\pi$
$557$	26.0035	1.10180	0.550902	+	0.834570i	$0.314284\pi$
$558$	0	0
$559$	−1.53528	−0.0649356
$560$	−18.2985	−0.773252
$561$	0	0
$562$	12.0778	0.509470
$563$	37.9203	1.59815	0.799076	−	0.601231i	$-0.205323\pi$
$563$	37.9203	1.59815	0.799076	+	0.601231i	$0.205323\pi$
$564$	0	0
$565$	−8.60585	−0.362051
$566$	10.7672	0.452580
$567$	0	0
$568$	−28.2727	−1.18630
$569$	−12.9060	−0.541047	−0.270523	−	0.962713i	$-0.587197\pi$
$569$	−12.9060	−0.541047	−0.270523	+	0.962713i	$0.587197\pi$
$570$	0	0
$571$	−6.51853	−0.272792	−0.136396	−	0.990654i	$-0.543552\pi$
$571$	−6.51853	−0.272792	−0.136396	+	0.990654i	$0.543552\pi$
$572$	−31.7124	−1.32596
$573$	0	0
$574$	20.2727	0.846167
$575$	2.24898	0.0937889
$576$	0	0
$577$	−6.43781	−0.268010	−0.134005	−	0.990981i	$-0.542784\pi$
$577$	−6.43781	−0.268010	−0.134005	+	0.990981i	$0.542784\pi$
$578$	−75.3254	−3.13312
$579$	0	0
$580$	45.1447	1.87453
$581$	−8.11162	−0.336527
$582$	0	0
$583$	−36.0538	−1.49320
$584$	−70.1318	−2.90207
$585$	0	0
$586$	61.7476	2.55077
$587$	1.00696	0.0415615	0.0207807	−	0.999784i	$-0.493385\pi$
$587$	1.00696	0.0415615	0.0207807	+	0.999784i	$0.493385\pi$
$588$	0	0
$589$	15.3941	0.634305
$590$	−38.2100	−1.57308
$591$	0	0
$592$	−84.5657	−3.47563
$593$	21.1278	0.867616	0.433808	−	0.901005i	$-0.357170\pi$
$593$	21.1278	0.867616	0.433808	+	0.901005i	$0.357170\pi$
$594$	0	0
$595$	−9.42105	−0.386225
$596$	−82.4863	−3.37877
$597$	0	0
$598$	2.00000	0.0817861
$599$	37.8679	1.54724	0.773620	−	0.633650i	$-0.218444\pi$
$599$	37.8679	1.54724	0.773620	+	0.633650i	$0.218444\pi$
$600$	0	0
$601$	−44.9497	−1.83354	−0.916769	−	0.399418i	$-0.869212\pi$
$601$	−44.9497	−1.83354	−0.916769	+	0.399418i	$0.869212\pi$
$602$	−4.13268	−0.168436
$603$	0	0
$604$	60.6530	2.46794
$605$	35.8829	1.45885
$606$	0	0
$607$	−26.6328	−1.08099	−0.540495	−	0.841347i	$-0.681763\pi$
$607$	−26.6328	−1.08099	−0.540495	+	0.841347i	$0.681763\pi$
$608$	−27.0083	−1.09533
$609$	0	0
$610$	−37.8109	−1.53092
$611$	−5.30229	−0.214508
$612$	0	0
$613$	35.0371	1.41513	0.707567	−	0.706647i	$-0.249793\pi$
$613$	35.0371	1.41513	0.707567	+	0.706647i	$0.249793\pi$
$614$	33.9325	1.36940
$615$	0	0
$616$	−52.8181	−2.12810
$617$	41.5540	1.67290	0.836450	−	0.548043i	$-0.184627\pi$
$617$	41.5540	1.67290	0.836450	+	0.548043i	$0.184627\pi$
$618$	0	0
$619$	5.42105	0.217890	0.108945	−	0.994048i	$-0.465253\pi$
$619$	5.42105	0.217890	0.108945	+	0.994048i	$0.465253\pi$
$620$	73.8845	2.96727
$621$	0	0
$622$	50.4043	2.02103
$623$	1.24202	0.0497606
$624$	0	0
$625$	−0.703325	−0.0281330
$626$	−83.4008	−3.33337
$627$	0	0
$628$	5.61600	0.224103
$629$	−43.5390	−1.73601
$630$	0	0
$631$	22.8959	0.911472	0.455736	−	0.890115i	$-0.349376\pi$
$631$	22.8959	0.911472	0.455736	+	0.890115i	$0.349376\pi$
$632$	91.9013	3.65564
$633$	0	0
$634$	13.7811	0.547317
$635$	5.61868	0.222970
$636$	0	0
$637$	1.00000	0.0396214
$638$	99.6965	3.94702
$639$	0	0
$640$	−31.1148	−1.22992
$641$	12.2732	0.484762	0.242381	−	0.970181i	$-0.422072\pi$
$641$	12.2732	0.484762	0.242381	+	0.970181i	$0.422072\pi$
$642$	0	0
$643$	−19.0168	−0.749948	−0.374974	−	0.927035i	$-0.622348\pi$
$643$	−19.0168	−0.749948	−0.374974	+	0.927035i	$0.622348\pi$
$644$	3.89762	0.153588
$645$	0	0
$646$	−27.7177	−1.09054
$647$	1.48599	0.0584204	0.0292102	−	0.999573i	$-0.490701\pi$
$647$	1.48599	0.0584204	0.0292102	+	0.999573i	$0.490701\pi$
$648$	0	0
$649$	−61.0909	−2.39803
$650$	8.14783	0.319584
$651$	0	0
$652$	110.074	4.31083
$653$	−39.7138	−1.55412	−0.777061	−	0.629426i	$-0.783290\pi$
$653$	−39.7138	−1.55412	−0.777061	+	0.629426i	$0.783290\pi$
$654$	0	0
$655$	−6.03352	−0.235749
$656$	98.1092	3.83052
$657$	0	0
$658$	−14.2727	−0.556408
$659$	−0.580350	−0.0226072	−0.0113036	−	0.999936i	$-0.503598\pi$
$659$	−0.580350	−0.0226072	−0.0113036	+	0.999936i	$0.503598\pi$
$660$	0	0
$661$	−35.6429	−1.38635	−0.693174	−	0.720770i	$-0.743788\pi$
$661$	−35.6429	−1.38635	−0.693174	+	0.720770i	$0.743788\pi$
$662$	43.2137	1.67955
$663$	0	0
$664$	−70.8720	−2.75037
$665$	−2.15657	−0.0836281
$666$	0	0
$667$	−4.55204	−0.176256
$668$	−43.4052	−1.67940
$669$	0	0
$670$	16.9832	0.656120
$671$	−60.4528	−2.33376
$672$	0	0
$673$	−16.5185	−0.636742	−0.318371	−	0.947966i	$-0.603136\pi$
$673$	−16.5185	−0.636742	−0.318371	+	0.947966i	$0.603136\pi$
$674$	−5.69110	−0.219213
$675$	0	0
$676$	5.24581	0.201762
$677$	−16.4583	−0.632544	−0.316272	−	0.948668i	$-0.602431\pi$
$677$	−16.4583	−0.632544	−0.316272	+	0.948668i	$0.602431\pi$
$678$	0	0
$679$	0.464716	0.0178342
$680$	−82.3125	−3.15654
$681$	0	0
$682$	163.165	6.24791
$683$	35.5376	1.35981	0.679904	−	0.733301i	$-0.262021\pi$
$683$	35.5376	1.35981	0.679904	+	0.733301i	$0.262021\pi$
$684$	0	0
$685$	3.01676	0.115264
$686$	2.69180	0.102774
$687$	0	0
$688$	−20.0000	−0.762493
$689$	5.96396	0.227209
$690$	0	0
$691$	−18.9563	−0.721133	−0.360567	−	0.932734i	$-0.617417\pi$
$691$	−18.9563	−0.721133	−0.360567	+	0.932734i	$0.617417\pi$
$692$	−1.23312	−0.0468763
$693$	0	0
$694$	25.6160	0.972370
$695$	30.1651	1.14423
$696$	0	0
$697$	50.5119	1.91328
$698$	−81.9148	−3.10052
$699$	0	0
$700$	15.8786	0.600153
$701$	16.4059	0.619642	0.309821	−	0.950795i	$-0.399731\pi$
$701$	16.4059	0.619642	0.309821	+	0.950795i	$0.399731\pi$
$702$	0	0
$703$	−9.96648	−0.375893
$704$	−128.762	−4.85291
$705$	0	0
$706$	−91.2224	−3.43320
$707$	−14.6648	−0.551528
$708$	0	0
$709$	16.1411	0.606193	0.303097	−	0.952960i	$-0.401980\pi$
$709$	16.1411	0.606193	0.303097	+	0.952960i	$0.401980\pi$
$710$	−12.2354	−0.459187
$711$	0	0
$712$	10.8517	0.406683
$713$	−7.44995	−0.279003
$714$	0	0
$715$	−8.49162	−0.317569
$716$	3.04441	0.113775
$717$	0	0
$718$	26.1316	0.975222
$719$	−10.9299	−0.407615	−0.203808	−	0.979011i	$-0.565332\pi$
$719$	−10.9299	−0.407615	−0.203808	+	0.979011i	$0.565332\pi$
$720$	0	0
$721$	11.5622	0.430598
$722$	44.7994	1.66726
$723$	0	0
$724$	52.4581	1.94959
$725$	−18.5446	−0.688731
$726$	0	0
$727$	−8.92943	−0.331174	−0.165587	−	0.986195i	$-0.552952\pi$
$727$	−8.92943	−0.331174	−0.165587	+	0.986195i	$0.552952\pi$
$728$	8.73709	0.323818
$729$	0	0
$730$	−30.3505	−1.12332
$731$	−10.2971	−0.380851
$732$	0	0
$733$	21.0101	0.776027	0.388014	−	0.921654i	$-0.373161\pi$
$733$	21.0101	0.776027	0.388014	+	0.921654i	$0.373161\pi$
$734$	55.1594	2.03597
$735$	0	0
$736$	13.0706	0.481788
$737$	27.1531	1.00020
$738$	0	0
$739$	11.6160	0.427301	0.213651	−	0.976910i	$-0.431465\pi$
$739$	11.6160	0.427301	0.213651	+	0.976910i	$0.431465\pi$
$740$	−47.8343	−1.75843
$741$	0	0
$742$	16.0538	0.589354
$743$	11.8266	0.433876	0.216938	−	0.976185i	$-0.430393\pi$
$743$	11.8266	0.433876	0.216938	+	0.976185i	$0.430393\pi$
$744$	0	0
$745$	−22.0873	−0.809217
$746$	4.20511	0.153960
$747$	0	0
$748$	−212.694	−7.77685
$749$	−5.22096	−0.190770
$750$	0	0
$751$	−15.3941	−0.561740	−0.280870	−	0.959746i	$-0.590623\pi$
$751$	−15.3941	−0.561740	−0.280870	+	0.959746i	$0.590623\pi$
$752$	−69.0724	−2.51881
$753$	0	0
$754$	−16.4916	−0.600589
$755$	16.2410	0.591072
$756$	0	0
$757$	2.81520	0.102320	0.0511601	−	0.998690i	$-0.483708\pi$
$757$	2.81520	0.102320	0.0511601	+	0.998690i	$0.483708\pi$
$758$	76.9836	2.79617
$759$	0	0
$760$	−18.8421	−0.683475
$761$	35.4998	1.28687	0.643433	−	0.765502i	$-0.277509\pi$
$761$	35.4998	1.28687	0.643433	+	0.765502i	$0.277509\pi$
$762$	0	0
$763$	−10.0000	−0.362024
$764$	−106.403	−3.84954
$765$	0	0
$766$	−6.21891	−0.224698
$767$	10.1055	0.364890
$768$	0	0
$769$	4.41091	0.159061	0.0795307	−	0.996832i	$-0.474658\pi$
$769$	4.41091	0.159061	0.0795307	+	0.996832i	$0.474658\pi$
$770$	−22.8578	−0.823737
$771$	0	0
$772$	−113.111	−4.07096
$773$	2.38273	0.0857010	0.0428505	−	0.999081i	$-0.486356\pi$
$773$	2.38273	0.0857010	0.0428505	+	0.999081i	$0.486356\pi$
$774$	0	0
$775$	−30.3505	−1.09022
$776$	4.06026	0.145755
$777$	0	0
$778$	−57.9665	−2.07820
$779$	11.5626	0.414275
$780$	0	0
$781$	−19.5622	−0.699990
$782$	13.4139	0.479680
$783$	0	0
$784$	13.0269	0.465247
$785$	1.50379	0.0536727
$786$	0	0
$787$	8.11423	0.289241	0.144621	−	0.989487i	$-0.453804\pi$
$787$	8.11423	0.289241	0.144621	+	0.989487i	$0.453804\pi$
$788$	53.0118	1.88847
$789$	0	0
$790$	39.7715	1.41501
$791$	6.12660	0.217837
$792$	0	0
$793$	10.0000	0.355110
$794$	35.0208	1.24284
$795$	0	0
$796$	5.15790	0.182817
$797$	−13.4863	−0.477710	−0.238855	−	0.971055i	$-0.576772\pi$
$797$	−13.4863	−0.477710	−0.238855	+	0.971055i	$0.576772\pi$
$798$	0	0
$799$	−35.5622	−1.25810
$800$	53.2484	1.88262
$801$	0	0
$802$	13.7811	0.486627
$803$	−48.5249	−1.71241
$804$	0	0
$805$	1.04366	0.0367843
$806$	−26.9905	−0.950699
$807$	0	0
$808$	−128.128	−4.50752
$809$	−17.7292	−0.623327	−0.311663	−	0.950193i	$-0.600886\pi$
$809$	−17.7292	−0.623327	−0.311663	+	0.950193i	$0.600886\pi$
$810$	0	0
$811$	5.42105	0.190359	0.0951794	−	0.995460i	$-0.469658\pi$
$811$	5.42105	0.190359	0.0951794	+	0.995460i	$0.469658\pi$
$812$	−32.1390	−1.12786
$813$	0	0
$814$	−105.636	−3.70255
$815$	29.4745	1.03245
$816$	0	0
$817$	−2.35710	−0.0824644
$818$	−44.4646	−1.55467
$819$	0	0
$820$	55.4952	1.93797
$821$	−31.6400	−1.10424	−0.552121	−	0.833764i	$-0.686182\pi$
$821$	−31.6400	−1.10424	−0.552121	+	0.833764i	$0.686182\pi$
$822$	0	0
$823$	−39.1244	−1.36379	−0.681895	−	0.731450i	$-0.738844\pi$
$823$	−39.1244	−1.36379	−0.681895	+	0.731450i	$0.738844\pi$
$824$	101.020	3.51919
$825$	0	0
$826$	27.2021	0.946484
$827$	−23.2666	−0.809058	−0.404529	−	0.914525i	$-0.632565\pi$
$827$	−23.2666	−0.809058	−0.404529	+	0.914525i	$0.632565\pi$
$828$	0	0
$829$	−18.0000	−0.625166	−0.312583	−	0.949890i	$-0.601194\pi$
$829$	−18.0000	−0.625166	−0.312583	+	0.949890i	$0.601194\pi$
$830$	−30.6708	−1.06460
$831$	0	0
$832$	21.2996	0.738431
$833$	6.70695	0.232382
$834$	0	0
$835$	−11.6226	−0.402217
$836$	−48.6875	−1.68389
$837$	0	0
$838$	−49.5287	−1.71094
$839$	37.2587	1.28631	0.643156	−	0.765735i	$-0.277625\pi$
$839$	37.2587	1.28631	0.643156	+	0.765735i	$0.277625\pi$
$840$	0	0
$841$	8.53528	0.294320
$842$	95.4914	3.29085
$843$	0	0
$844$	−12.9294	−0.445049
$845$	1.40467	0.0483221
$846$	0	0
$847$	−25.5454	−0.877751
$848$	77.6919	2.66795
$849$	0	0
$850$	54.6472	1.87438
$851$	4.82325	0.165339
$852$	0	0
$853$	32.0269	1.09658	0.548290	−	0.836288i	$-0.315279\pi$
$853$	32.0269	1.09658	0.548290	+	0.836288i	$0.315279\pi$
$854$	26.9180	0.921116
$855$	0	0
$856$	−45.6160	−1.55912
$857$	−9.98643	−0.341130	−0.170565	−	0.985346i	$-0.554559\pi$
$857$	−9.98643	−0.341130	−0.170565	+	0.985346i	$0.554559\pi$
$858$	0	0
$859$	48.1614	1.64325	0.821623	−	0.570031i	$-0.193069\pi$
$859$	48.1614	1.64325	0.821623	+	0.570031i	$0.193069\pi$
$860$	−11.3129	−0.385768
$861$	0	0
$862$	100.050	3.40772
$863$	17.9732	0.611815	0.305907	−	0.952061i	$-0.401040\pi$
$863$	17.9732	0.611815	0.305907	+	0.952061i	$0.401040\pi$
$864$	0	0
$865$	−0.330193	−0.0112269
$866$	−2.88176	−0.0979262
$867$	0	0
$868$	−52.5992	−1.78533
$869$	63.5874	2.15706
$870$	0	0
$871$	−4.49162	−0.152193
$872$	−87.3709	−2.95875
$873$	0	0
$874$	3.07057	0.103864
$875$	11.2751	0.381169
$876$	0	0
$877$	18.9294	0.639201	0.319601	−	0.947552i	$-0.396451\pi$
$877$	18.9294	0.639201	0.319601	+	0.947552i	$0.396451\pi$
$878$	36.2717	1.22411
$879$	0	0
$880$	−110.620	−3.72898
$881$	38.5207	1.29779	0.648897	−	0.760876i	$-0.275231\pi$
$881$	38.5207	1.29779	0.648897	+	0.760876i	$0.275231\pi$
$882$	0	0
$883$	22.1411	0.745109	0.372554	−	0.928010i	$-0.378482\pi$
$883$	22.1411	0.745109	0.372554	+	0.928010i	$0.378482\pi$
$884$	35.1834	1.18335
$885$	0	0
$886$	61.8253	2.07706
$887$	27.6232	0.927498	0.463749	−	0.885967i	$-0.346504\pi$
$887$	27.6232	0.927498	0.463749	+	0.885967i	$0.346504\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	4.69621	0.157417
$891$	0	0
$892$	12.9294	0.432909
$893$	−8.14051	−0.272412
$894$	0	0
$895$	0.815200	0.0272491
$896$	22.1510	0.740013
$897$	0	0
$898$	33.8887	1.13088
$899$	61.4309	2.04883
$900$	0	0
$901$	40.0000	1.33259
$902$	122.554	4.08061
$903$	0	0
$904$	53.5287	1.78034
$905$	14.0467	0.466928
$906$	0	0
$907$	−39.7302	−1.31922	−0.659610	−	0.751608i	$-0.729279\pi$
$907$	−39.7302	−1.31922	−0.659610	+	0.751608i	$0.729279\pi$
$908$	82.4863	2.73740
$909$	0	0
$910$	3.78109	0.125342
$911$	17.2915	0.572894	0.286447	−	0.958096i	$-0.407526\pi$
$911$	17.2915	0.572894	0.286447	+	0.958096i	$0.407526\pi$
$912$	0	0
$913$	−49.0371	−1.62289
$914$	107.727	3.56329
$915$	0	0
$916$	−71.1447	−2.35069
$917$	4.29533	0.141844
$918$	0	0
$919$	34.1411	1.12621	0.563106	−	0.826385i	$-0.309606\pi$
$919$	34.1411	1.12621	0.563106	+	0.826385i	$0.309606\pi$
$920$	9.11858	0.300631
$921$	0	0
$922$	19.3971	0.638809
$923$	3.23594	0.106512
$924$	0	0
$925$	19.6495	0.646072
$926$	−1.32335	−0.0434879
$927$	0	0
$928$	−107.777	−3.53797
$929$	40.1782	1.31820	0.659102	−	0.752053i	$-0.270936\pi$
$929$	40.1782	1.31820	0.659102	+	0.752053i	$0.270936\pi$
$930$	0	0
$931$	1.53528	0.0503169
$932$	−121.719	−3.98703
$933$	0	0
$934$	57.5287	1.88240
$935$	−56.9529	−1.86256
$936$	0	0
$937$	−8.89591	−0.290617	−0.145308	−	0.989386i	$-0.546417\pi$
$937$	−8.89591	−0.290617	−0.145308	+	0.989386i	$0.546417\pi$
$938$	−12.0906	−0.394771
$939$	0	0
$940$	−39.0706	−1.27434
$941$	10.3784	0.338326	0.169163	−	0.985588i	$-0.445894\pi$
$941$	10.3784	0.338326	0.169163	+	0.985588i	$0.445894\pi$
$942$	0	0
$943$	−5.59571	−0.182221
$944$	131.644	4.28465
$945$	0	0
$946$	−24.9832	−0.812275
$947$	−7.36863	−0.239448	−0.119724	−	0.992807i	$-0.538201\pi$
$947$	−7.36863	−0.239448	−0.119724	+	0.992807i	$0.538201\pi$
$948$	0	0
$949$	8.02690	0.260564
$950$	12.5092	0.405853
$951$	0	0
$952$	58.5992	1.89921
$953$	11.7453	0.380467	0.190233	−	0.981739i	$-0.439075\pi$
$953$	11.7453	0.380467	0.190233	+	0.981739i	$0.439075\pi$
$954$	0	0
$955$	−28.4916	−0.921967
$956$	87.3420	2.82484
$957$	0	0
$958$	65.9425	2.13051
$959$	−2.14767	−0.0693517
$960$	0	0
$961$	69.5388	2.24319
$962$	17.4742	0.563390
$963$	0	0
$964$	−126.323	−4.06859
$965$	−30.2877	−0.974997
$966$	0	0
$967$	28.0538	0.902150	0.451075	−	0.892486i	$-0.351041\pi$
$967$	28.0538	0.902150	0.451075	+	0.892486i	$0.351041\pi$
$968$	−223.193	−7.17368
$969$	0	0
$970$	1.75713	0.0564182
$971$	−29.1092	−0.934160	−0.467080	−	0.884215i	$-0.654694\pi$
$971$	−29.1092	−0.934160	−0.467080	+	0.884215i	$0.654694\pi$
$972$	0	0
$973$	−21.4749	−0.688452
$974$	−47.0389	−1.50722
$975$	0	0
$976$	130.269	4.16981
$977$	−2.83823	−0.0908030	−0.0454015	−	0.998969i	$-0.514457\pi$
$977$	−2.83823	−0.0908030	−0.0454015	+	0.998969i	$0.514457\pi$
$978$	0	0
$979$	7.50838	0.239969
$980$	7.36863	0.235382
$981$	0	0
$982$	72.0203	2.29826
$983$	36.5881	1.16698	0.583489	−	0.812121i	$-0.301687\pi$
$983$	36.5881	1.16698	0.583489	+	0.812121i	$0.301687\pi$
$984$	0	0
$985$	14.1949	0.452289
$986$	−110.609	−3.52249
$987$	0	0
$988$	8.05381	0.256226
$989$	1.14071	0.0362725
$990$	0	0
$991$	−32.9832	−1.04775	−0.523874	−	0.851796i	$-0.675514\pi$
$991$	−32.9832	−1.04775	−0.523874	+	0.851796i	$0.675514\pi$
$992$	−176.390	−5.60040
$993$	0	0
$994$	8.71053	0.276281
$995$	1.38113	0.0437847
$996$	0	0
$997$	60.0203	1.90086	0.950431	−	0.310936i	$-0.100642\pi$
$997$	60.0203	1.90086	0.950431	+	0.310936i	$0.100642\pi$
$998$	−5.76352	−0.182441
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	819.2.a.m.1.1	✓	6
3.2	odd	2	inner	819.2.a.m.1.6	yes	6
7.6	odd	2		5733.2.a.bv.1.1		6
21.20	even	2		5733.2.a.bv.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
819.2.a.m.1.1	✓	6	1.1	even	1	trivial
819.2.a.m.1.6	yes	6	3.2	odd	2	inner
5733.2.a.bv.1.1		6	7.6	odd	2
5733.2.a.bv.1.6		6	21.20	even	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:	\( N \)	\(=\)	\( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	819.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69180 q^{2} +5.24581 q^{4} +1.40467 q^{5} -1.00000 q^{7} -8.73709 q^{8} +O(q^{10})\)	\(q-2.69180 q^{2} +5.24581 q^{4} +1.40467 q^{5} -1.00000 q^{7} -8.73709 q^{8} -3.78109 q^{10} -6.04528 q^{11} +1.00000 q^{13} +2.69180 q^{14} +13.0269 q^{16} +6.70695 q^{17} +1.53528 q^{19} +7.36863 q^{20} +16.2727 q^{22} -0.742996 q^{23} -3.02690 q^{25} -2.69180 q^{26} -5.24581 q^{28} +6.12660 q^{29} +10.0269 q^{31} -17.5917 q^{32} -18.0538 q^{34} -1.40467 q^{35} -6.49162 q^{37} -4.13268 q^{38} -12.2727 q^{40} +7.53127 q^{41} -1.53528 q^{43} -31.7124 q^{44} +2.00000 q^{46} -5.30229 q^{47} +1.00000 q^{49} +8.14783 q^{50} +5.24581 q^{52} +5.96396 q^{53} -8.49162 q^{55} +8.73709 q^{56} -16.4916 q^{58} +10.1055 q^{59} +10.0000 q^{61} -26.9905 q^{62} +21.2996 q^{64} +1.40467 q^{65} -4.49162 q^{67} +35.1834 q^{68} +3.78109 q^{70} +3.23594 q^{71} +8.02690 q^{73} +17.4742 q^{74} +8.05381 q^{76} +6.04528 q^{77} -10.5185 q^{79} +18.2985 q^{80} -20.2727 q^{82} +8.11162 q^{83} +9.42105 q^{85} +4.13268 q^{86} +52.8181 q^{88} -1.24202 q^{89} -1.00000 q^{91} -3.89762 q^{92} +14.2727 q^{94} +2.15657 q^{95} -0.464716 q^{97} -2.69180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4} - 6 q^{7}+O(q^{10}) \) 6 * q + 12 * q^4 - 6 * q^7	\( 6 q + 12 q^{4} - 6 q^{7} + 8 q^{10} + 6 q^{13} + 28 q^{16} - 2 q^{19} + 28 q^{22} + 32 q^{25} - 12 q^{28} + 10 q^{31} - 8 q^{34} - 4 q^{40} + 2 q^{43} + 12 q^{46} + 6 q^{49} + 12 q^{52} - 12 q^{55} - 60 q^{58} + 60 q^{61} + 8 q^{64} + 12 q^{67} - 8 q^{70} - 2 q^{73} - 52 q^{76} + 26 q^{79} - 52 q^{82} + 40 q^{85} + 108 q^{88} - 6 q^{91} + 16 q^{94} - 14 q^{97}+O(q^{100}) \) 6 * q + 12 * q^4 - 6 * q^7 + 8 * q^10 + 6 * q^13 + 28 * q^16 - 2 * q^19 + 28 * q^22 + 32 * q^25 - 12 * q^28 + 10 * q^31 - 8 * q^34 - 4 * q^40 + 2 * q^43 + 12 * q^46 + 6 * q^49 + 12 * q^52 - 12 * q^55 - 60 * q^58 + 60 * q^61 + 8 * q^64 + 12 * q^67 - 8 * q^70 - 2 * q^73 - 52 * q^76 + 26 * q^79 - 52 * q^82 + 40 * q^85 + 108 * q^88 - 6 * q^91 + 16 * q^94 - 14 * q^97

Introduction

L-functions

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