LMFDB - Embedded newform 6032.2.a.be.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 159 x^{9} - 827 x^{8} - 362 x^{7} + 3029 x^{6} - 142 x^{5} + \cdots + 16 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 159x^9 - 827x^8 - 362x^7 + 3029x^6 - 142x^5 - 4316x^4 + 208x^3 + 2228x^2 + 376x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$1.82663$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82663 q^{3} +2.84424 q^{5} +3.25048 q^{7} +0.336589 q^{9} +O(q^{10})$	$q+1.82663 q^{3} +2.84424 q^{5} +3.25048 q^{7} +0.336589 q^{9} -0.241451 q^{11} -1.00000 q^{13} +5.19538 q^{15} +5.04383 q^{17} -1.20799 q^{19} +5.93743 q^{21} +7.89305 q^{23} +3.08968 q^{25} -4.86507 q^{27} +1.00000 q^{29} +1.85543 q^{31} -0.441042 q^{33} +9.24513 q^{35} +0.584712 q^{37} -1.82663 q^{39} -4.04962 q^{41} +1.75999 q^{43} +0.957339 q^{45} -7.07705 q^{47} +3.56561 q^{49} +9.21322 q^{51} +7.29349 q^{53} -0.686743 q^{55} -2.20655 q^{57} -7.06212 q^{59} +11.9489 q^{61} +1.09408 q^{63} -2.84424 q^{65} -4.31366 q^{67} +14.4177 q^{69} -3.00413 q^{71} -14.0367 q^{73} +5.64371 q^{75} -0.784831 q^{77} -8.80130 q^{79} -9.89648 q^{81} +4.97663 q^{83} +14.3458 q^{85} +1.82663 q^{87} +10.6314 q^{89} -3.25048 q^{91} +3.38918 q^{93} -3.43581 q^{95} +18.3661 q^{97} -0.0812698 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.82663	1.05461	0.527304	−	0.849677i	$-0.323203\pi$
$3$	1.82663	1.05461	0.527304	+	0.849677i	$0.323203\pi$
$4$	0	0
$5$	2.84424	1.27198	0.635990	−	0.771697i	$-0.280592\pi$
$5$	2.84424	1.27198	0.635990	+	0.771697i	$0.280592\pi$
$6$	0	0
$7$	3.25048	1.22857	0.614283	−	0.789086i	$-0.289445\pi$
$7$	3.25048	1.22857	0.614283	+	0.789086i	$0.289445\pi$
$8$	0	0
$9$	0.336589	0.112196
$10$	0	0
$11$	−0.241451	−0.0728002	−0.0364001	−	0.999337i	$-0.511589\pi$
$11$	−0.241451	−0.0728002	−0.0364001	+	0.999337i	$0.511589\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	5.19538	1.34144
$16$	0	0
$17$	5.04383	1.22331	0.611654	−	0.791126i	$-0.290505\pi$
$17$	5.04383	1.22331	0.611654	+	0.791126i	$0.290505\pi$
$18$	0	0
$19$	−1.20799	−0.277132	−0.138566	−	0.990353i	$-0.544249\pi$
$19$	−1.20799	−0.277132	−0.138566	+	0.990353i	$0.544249\pi$
$20$	0	0
$21$	5.93743	1.29565
$22$	0	0
$23$	7.89305	1.64581	0.822907	−	0.568176i	$-0.192351\pi$
$23$	7.89305	1.64581	0.822907	+	0.568176i	$0.192351\pi$
$24$	0	0
$25$	3.08968	0.617935
$26$	0	0
$27$	−4.86507	−0.936284
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	1.85543	0.333244	0.166622	−	0.986021i	$-0.446714\pi$
$31$	1.85543	0.333244	0.166622	+	0.986021i	$0.446714\pi$
$32$	0	0
$33$	−0.441042	−0.0767756
$34$	0	0
$35$	9.24513	1.56271
$36$	0	0
$37$	0.584712	0.0961260	0.0480630	−	0.998844i	$-0.484695\pi$
$37$	0.584712	0.0961260	0.0480630	+	0.998844i	$0.484695\pi$
$38$	0	0
$39$	−1.82663	−0.292495
$40$	0	0
$41$	−4.04962	−0.632444	−0.316222	−	0.948685i	$-0.602414\pi$
$41$	−4.04962	−0.632444	−0.316222	+	0.948685i	$0.602414\pi$
$42$	0	0
$43$	1.75999	0.268396	0.134198	−	0.990955i	$-0.457154\pi$
$43$	1.75999	0.268396	0.134198	+	0.990955i	$0.457154\pi$
$44$	0	0
$45$	0.957339	0.142712
$46$	0	0
$47$	−7.07705	−1.03229	−0.516147	−	0.856500i	$-0.672634\pi$
$47$	−7.07705	−1.03229	−0.516147	+	0.856500i	$0.672634\pi$
$48$	0	0
$49$	3.56561	0.509373
$50$	0	0
$51$	9.21322	1.29011
$52$	0	0
$53$	7.29349	1.00184	0.500919	−	0.865494i	$-0.332996\pi$
$53$	7.29349	1.00184	0.500919	+	0.865494i	$0.332996\pi$
$54$	0	0
$55$	−0.686743	−0.0926004
$56$	0	0
$57$	−2.20655	−0.292265
$58$	0	0
$59$	−7.06212	−0.919410	−0.459705	−	0.888072i	$-0.652045\pi$
$59$	−7.06212	−0.919410	−0.459705	+	0.888072i	$0.652045\pi$
$60$	0	0
$61$	11.9489	1.52990	0.764952	−	0.644088i	$-0.222763\pi$
$61$	11.9489	1.52990	0.764952	+	0.644088i	$0.222763\pi$
$62$	0	0
$63$	1.09408	0.137841
$64$	0	0
$65$	−2.84424	−0.352784
$66$	0	0
$67$	−4.31366	−0.526997	−0.263499	−	0.964660i	$-0.584876\pi$
$67$	−4.31366	−0.526997	−0.263499	+	0.964660i	$0.584876\pi$
$68$	0	0
$69$	14.4177	1.73569
$70$	0	0
$71$	−3.00413	−0.356524	−0.178262	−	0.983983i	$-0.557048\pi$
$71$	−3.00413	−0.356524	−0.178262	+	0.983983i	$0.557048\pi$
$72$	0	0
$73$	−14.0367	−1.64288	−0.821438	−	0.570297i	$-0.806828\pi$
$73$	−14.0367	−1.64288	−0.821438	+	0.570297i	$0.806828\pi$
$74$	0	0
$75$	5.64371	0.651679
$76$	0	0
$77$	−0.784831	−0.0894398
$78$	0	0
$79$	−8.80130	−0.990224	−0.495112	−	0.868829i	$-0.664873\pi$
$79$	−8.80130	−0.990224	−0.495112	+	0.868829i	$0.664873\pi$
$80$	0	0
$81$	−9.89648	−1.09961
$82$	0	0
$83$	4.97663	0.546256	0.273128	−	0.961978i	$-0.411942\pi$
$83$	4.97663	0.546256	0.273128	+	0.961978i	$0.411942\pi$
$84$	0	0
$85$	14.3458	1.55602
$86$	0	0
$87$	1.82663	0.195836
$88$	0	0
$89$	10.6314	1.12693	0.563465	−	0.826140i	$-0.309468\pi$
$89$	10.6314	1.12693	0.563465	+	0.826140i	$0.309468\pi$
$90$	0	0
$91$	−3.25048	−0.340743
$92$	0	0
$93$	3.38918	0.351442
$94$	0	0
$95$	−3.43581	−0.352506
$96$	0	0
$97$	18.3661	1.86480	0.932399	−	0.361432i	$-0.117712\pi$
$97$	18.3661	1.86480	0.932399	+	0.361432i	$0.117712\pi$
$98$	0	0
$99$	−0.0812698	−0.00816792
$100$	0	0
$101$	0.0331318	0.00329673	0.00164837	−	0.999999i	$-0.499475\pi$
$101$	0.0331318	0.00329673	0.00164837	+	0.999999i	$0.499475\pi$
$102$	0	0
$103$	−2.07655	−0.204608	−0.102304	−	0.994753i	$-0.532621\pi$
$103$	−2.07655	−0.204608	−0.102304	+	0.994753i	$0.532621\pi$
$104$	0	0
$105$	16.8875	1.64805
$106$	0	0
$107$	10.8554	1.04943	0.524716	−	0.851277i	$-0.324171\pi$
$107$	10.8554	1.04943	0.524716	+	0.851277i	$0.324171\pi$
$108$	0	0
$109$	−4.01330	−0.384405	−0.192202	−	0.981355i	$-0.561563\pi$
$109$	−4.01330	−0.384405	−0.192202	+	0.981355i	$0.561563\pi$
$110$	0	0
$111$	1.06805	0.101375
$112$	0	0
$113$	−14.5318	−1.36704	−0.683518	−	0.729934i	$-0.739551\pi$
$113$	−14.5318	−1.36704	−0.683518	+	0.729934i	$0.739551\pi$
$114$	0	0
$115$	22.4497	2.09344
$116$	0	0
$117$	−0.336589	−0.0311177
$118$	0	0
$119$	16.3949	1.50291
$120$	0	0
$121$	−10.9417	−0.994700
$122$	0	0
$123$	−7.39717	−0.666980
$124$	0	0
$125$	−5.43341	−0.485979
$126$	0	0
$127$	−0.806518	−0.0715669	−0.0357835	−	0.999360i	$-0.511393\pi$
$127$	−0.806518	−0.0715669	−0.0357835	+	0.999360i	$0.511393\pi$
$128$	0	0
$129$	3.21486	0.283052
$130$	0	0
$131$	1.91953	0.167710	0.0838550	−	0.996478i	$-0.473277\pi$
$131$	1.91953	0.167710	0.0838550	+	0.996478i	$0.473277\pi$
$132$	0	0
$133$	−3.92654	−0.340475
$134$	0	0
$135$	−13.8374	−1.19094
$136$	0	0
$137$	6.47850	0.553496	0.276748	−	0.960943i	$-0.410743\pi$
$137$	6.47850	0.553496	0.276748	+	0.960943i	$0.410743\pi$
$138$	0	0
$139$	−18.7056	−1.58659	−0.793296	−	0.608836i	$-0.791637\pi$
$139$	−18.7056	−1.58659	−0.793296	+	0.608836i	$0.791637\pi$
$140$	0	0
$141$	−12.9272	−1.08867
$142$	0	0
$143$	0.241451	0.0201911
$144$	0	0
$145$	2.84424	0.236201
$146$	0	0
$147$	6.51307	0.537189
$148$	0	0
$149$	15.3158	1.25472	0.627358	−	0.778731i	$-0.284136\pi$
$149$	15.3158	1.25472	0.627358	+	0.778731i	$0.284136\pi$
$150$	0	0
$151$	−5.90564	−0.480594	−0.240297	−	0.970699i	$-0.577245\pi$
$151$	−5.90564	−0.480594	−0.240297	+	0.970699i	$0.577245\pi$
$152$	0	0
$153$	1.69770	0.137251
$154$	0	0
$155$	5.27727	0.423880
$156$	0	0
$157$	12.8942	1.02907	0.514535	−	0.857469i	$-0.327965\pi$
$157$	12.8942	1.02907	0.514535	+	0.857469i	$0.327965\pi$
$158$	0	0
$159$	13.3225	1.05655
$160$	0	0
$161$	25.6562	2.02199
$162$	0	0
$163$	−13.4366	−1.05244	−0.526219	−	0.850349i	$-0.676391\pi$
$163$	−13.4366	−1.05244	−0.526219	+	0.850349i	$0.676391\pi$
$164$	0	0
$165$	−1.25443	−0.0976571
$166$	0	0
$167$	8.99850	0.696325	0.348162	−	0.937434i	$-0.386806\pi$
$167$	8.99850	0.696325	0.348162	+	0.937434i	$0.386806\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.406596	−0.0310932
$172$	0	0
$173$	25.6345	1.94896	0.974479	−	0.224480i	$-0.0720683\pi$
$173$	25.6345	1.94896	0.974479	+	0.224480i	$0.0720683\pi$
$174$	0	0
$175$	10.0429	0.759174
$176$	0	0
$177$	−12.8999	−0.969616
$178$	0	0
$179$	−4.97204	−0.371627	−0.185814	−	0.982585i	$-0.559492\pi$
$179$	−4.97204	−0.371627	−0.185814	+	0.982585i	$0.559492\pi$
$180$	0	0
$181$	26.5328	1.97217	0.986083	−	0.166256i	$-0.0531677\pi$
$181$	26.5328	1.97217	0.986083	+	0.166256i	$0.0531677\pi$
$182$	0	0
$183$	21.8263	1.61345
$184$	0	0
$185$	1.66306	0.122270
$186$	0	0
$187$	−1.21784	−0.0890570
$188$	0	0
$189$	−15.8138	−1.15029
$190$	0	0
$191$	6.17709	0.446959	0.223479	−	0.974709i	$-0.428258\pi$
$191$	6.17709	0.446959	0.223479	+	0.974709i	$0.428258\pi$
$192$	0	0
$193$	19.9703	1.43750	0.718749	−	0.695270i	$-0.244715\pi$
$193$	19.9703	1.43750	0.718749	+	0.695270i	$0.244715\pi$
$194$	0	0
$195$	−5.19538	−0.372049
$196$	0	0
$197$	−22.3761	−1.59423	−0.797114	−	0.603828i	$-0.793641\pi$
$197$	−22.3761	−1.59423	−0.797114	+	0.603828i	$0.793641\pi$
$198$	0	0
$199$	−20.7179	−1.46865	−0.734325	−	0.678798i	$-0.762501\pi$
$199$	−20.7179	−1.46865	−0.734325	+	0.678798i	$0.762501\pi$
$200$	0	0
$201$	−7.87947	−0.555775
$202$	0	0
$203$	3.25048	0.228139
$204$	0	0
$205$	−11.5181	−0.804457
$206$	0	0
$207$	2.65672	0.184654
$208$	0	0
$209$	0.291670	0.0201752
$210$	0	0
$211$	−2.25412	−0.155180	−0.0775900	−	0.996985i	$-0.524723\pi$
$211$	−2.25412	−0.155180	−0.0775900	+	0.996985i	$0.524723\pi$
$212$	0	0
$213$	−5.48744	−0.375993
$214$	0	0
$215$	5.00583	0.341395
$216$	0	0
$217$	6.03102	0.409413
$218$	0	0
$219$	−25.6400	−1.73259
$220$	0	0
$221$	−5.04383	−0.339285
$222$	0	0
$223$	−15.2412	−1.02063	−0.510313	−	0.859988i	$-0.670471\pi$
$223$	−15.2412	−1.02063	−0.510313	+	0.859988i	$0.670471\pi$
$224$	0	0
$225$	1.03995	0.0693301
$226$	0	0
$227$	−26.0829	−1.73118	−0.865591	−	0.500752i	$-0.833057\pi$
$227$	−26.0829	−1.73118	−0.865591	+	0.500752i	$0.833057\pi$
$228$	0	0
$229$	2.33901	0.154566	0.0772831	−	0.997009i	$-0.475375\pi$
$229$	2.33901	0.154566	0.0772831	+	0.997009i	$0.475375\pi$
$230$	0	0
$231$	−1.43360	−0.0943238
$232$	0	0
$233$	18.9396	1.24078	0.620388	−	0.784295i	$-0.286975\pi$
$233$	18.9396	1.24078	0.620388	+	0.784295i	$0.286975\pi$
$234$	0	0
$235$	−20.1288	−1.31306
$236$	0	0
$237$	−16.0768	−1.04430
$238$	0	0
$239$	−24.3911	−1.57773	−0.788866	−	0.614565i	$-0.789331\pi$
$239$	−24.3911	−1.57773	−0.788866	+	0.614565i	$0.789331\pi$
$240$	0	0
$241$	1.22075	0.0786356	0.0393178	−	0.999227i	$-0.487482\pi$
$241$	1.22075	0.0786356	0.0393178	+	0.999227i	$0.487482\pi$
$242$	0	0
$243$	−3.48201	−0.223371
$244$	0	0
$245$	10.1414	0.647913
$246$	0	0
$247$	1.20799	0.0768625
$248$	0	0
$249$	9.09047	0.576085
$250$	0	0
$251$	6.51532	0.411244	0.205622	−	0.978632i	$-0.434078\pi$
$251$	6.51532	0.411244	0.205622	+	0.978632i	$0.434078\pi$
$252$	0	0
$253$	−1.90578	−0.119816
$254$	0	0
$255$	26.2046	1.64099
$256$	0	0
$257$	−10.8054	−0.674025	−0.337013	−	0.941500i	$-0.609416\pi$
$257$	−10.8054	−0.674025	−0.337013	+	0.941500i	$0.609416\pi$
$258$	0	0
$259$	1.90059	0.118097
$260$	0	0
$261$	0.336589	0.0208344
$262$	0	0
$263$	18.8198	1.16048	0.580240	−	0.814445i	$-0.302959\pi$
$263$	18.8198	1.16048	0.580240	+	0.814445i	$0.302959\pi$
$264$	0	0
$265$	20.7444	1.27432
$266$	0	0
$267$	19.4197	1.18847
$268$	0	0
$269$	−21.4728	−1.30922	−0.654610	−	0.755967i	$-0.727167\pi$
$269$	−21.4728	−1.30922	−0.654610	+	0.755967i	$0.727167\pi$
$270$	0	0
$271$	−18.4789	−1.12251	−0.561257	−	0.827642i	$-0.689682\pi$
$271$	−18.4789	−1.12251	−0.561257	+	0.827642i	$0.689682\pi$
$272$	0	0
$273$	−5.93743	−0.359350
$274$	0	0
$275$	−0.746005	−0.0449858
$276$	0	0
$277$	−11.8023	−0.709133	−0.354567	−	0.935031i	$-0.615372\pi$
$277$	−11.8023	−0.709133	−0.354567	+	0.935031i	$0.615372\pi$
$278$	0	0
$279$	0.624517	0.0373888
$280$	0	0
$281$	−1.86238	−0.111100	−0.0555502	−	0.998456i	$-0.517691\pi$
$281$	−1.86238	−0.111100	−0.0555502	+	0.998456i	$0.517691\pi$
$282$	0	0
$283$	−23.8966	−1.42051	−0.710253	−	0.703947i	$-0.751419\pi$
$283$	−23.8966	−1.42051	−0.710253	+	0.703947i	$0.751419\pi$
$284$	0	0
$285$	−6.27596	−0.371756
$286$	0	0
$287$	−13.1632	−0.776999
$288$	0	0
$289$	8.44019	0.496482
$290$	0	0
$291$	33.5482	1.96663
$292$	0	0
$293$	−11.2701	−0.658404	−0.329202	−	0.944260i	$-0.606780\pi$
$293$	−11.2701	−0.658404	−0.329202	+	0.944260i	$0.606780\pi$
$294$	0	0
$295$	−20.0863	−1.16947
$296$	0	0
$297$	1.17468	0.0681616
$298$	0	0
$299$	−7.89305	−0.456467
$300$	0	0
$301$	5.72081	0.329742
$302$	0	0
$303$	0.0605196	0.00347676
$304$	0	0
$305$	33.9856	1.94601
$306$	0	0
$307$	18.3044	1.04469	0.522345	−	0.852735i	$-0.325057\pi$
$307$	18.3044	1.04469	0.522345	+	0.852735i	$0.325057\pi$
$308$	0	0
$309$	−3.79309	−0.215781
$310$	0	0
$311$	−26.9659	−1.52909	−0.764547	−	0.644568i	$-0.777037\pi$
$311$	−26.9659	−1.52909	−0.764547	+	0.644568i	$0.777037\pi$
$312$	0	0
$313$	−7.71273	−0.435950	−0.217975	−	0.975954i	$-0.569945\pi$
$313$	−7.71273	−0.435950	−0.217975	+	0.975954i	$0.569945\pi$
$314$	0	0
$315$	3.11181	0.175331
$316$	0	0
$317$	−4.66655	−0.262099	−0.131050	−	0.991376i	$-0.541835\pi$
$317$	−4.66655	−0.262099	−0.131050	+	0.991376i	$0.541835\pi$
$318$	0	0
$319$	−0.241451	−0.0135186
$320$	0	0
$321$	19.8289	1.10674
$322$	0	0
$323$	−6.09289	−0.339017
$324$	0	0
$325$	−3.08968	−0.171384
$326$	0	0
$327$	−7.33083	−0.405396
$328$	0	0
$329$	−23.0038	−1.26824
$330$	0	0
$331$	28.7108	1.57809	0.789045	−	0.614335i	$-0.210576\pi$
$331$	28.7108	1.57809	0.789045	+	0.614335i	$0.210576\pi$
$332$	0	0
$333$	0.196808	0.0107850
$334$	0	0
$335$	−12.2691	−0.670330
$336$	0	0
$337$	9.76070	0.531699	0.265850	−	0.964015i	$-0.414348\pi$
$337$	9.76070	0.531699	0.265850	+	0.964015i	$0.414348\pi$
$338$	0	0
$339$	−26.5442	−1.44169
$340$	0	0
$341$	−0.447994	−0.0242602
$342$	0	0
$343$	−11.1634	−0.602767
$344$	0	0
$345$	41.0073	2.20776
$346$	0	0
$347$	11.1557	0.598871	0.299435	−	0.954117i	$-0.403202\pi$
$347$	11.1557	0.598871	0.299435	+	0.954117i	$0.403202\pi$
$348$	0	0
$349$	2.98831	0.159961	0.0799803	−	0.996796i	$-0.474514\pi$
$349$	2.98831	0.159961	0.0799803	+	0.996796i	$0.474514\pi$
$350$	0	0
$351$	4.86507	0.259678
$352$	0	0
$353$	−15.3152	−0.815144	−0.407572	−	0.913173i	$-0.633625\pi$
$353$	−15.3152	−0.815144	−0.407572	+	0.913173i	$0.633625\pi$
$354$	0	0
$355$	−8.54445	−0.453492
$356$	0	0
$357$	29.9474	1.58498
$358$	0	0
$359$	16.5557	0.873774	0.436887	−	0.899516i	$-0.356081\pi$
$359$	16.5557	0.873774	0.436887	+	0.899516i	$0.356081\pi$
$360$	0	0
$361$	−17.5408	−0.923198
$362$	0	0
$363$	−19.9865	−1.04902
$364$	0	0
$365$	−39.9238	−2.08971
$366$	0	0
$367$	6.74472	0.352072	0.176036	−	0.984384i	$-0.443673\pi$
$367$	6.74472	0.352072	0.176036	+	0.984384i	$0.443673\pi$
$368$	0	0
$369$	−1.36306	−0.0709580
$370$	0	0
$371$	23.7073	1.23082
$372$	0	0
$373$	−37.2939	−1.93101	−0.965503	−	0.260392i	$-0.916148\pi$
$373$	−37.2939	−1.93101	−0.965503	+	0.260392i	$0.916148\pi$
$374$	0	0
$375$	−9.92485	−0.512517
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	22.9762	1.18021	0.590103	−	0.807328i	$-0.299087\pi$
$379$	22.9762	1.18021	0.590103	+	0.807328i	$0.299087\pi$
$380$	0	0
$381$	−1.47321	−0.0754750
$382$	0	0
$383$	−15.2284	−0.778133	−0.389066	−	0.921210i	$-0.627202\pi$
$383$	−15.2284	−0.778133	−0.389066	+	0.921210i	$0.627202\pi$
$384$	0	0
$385$	−2.23224	−0.113766
$386$	0	0
$387$	0.592394	0.0301131
$388$	0	0
$389$	28.5322	1.44664	0.723320	−	0.690513i	$-0.242615\pi$
$389$	28.5322	1.44664	0.723320	+	0.690513i	$0.242615\pi$
$390$	0	0
$391$	39.8112	2.01334
$392$	0	0
$393$	3.50627	0.176868
$394$	0	0
$395$	−25.0330	−1.25955
$396$	0	0
$397$	−0.418469	−0.0210024	−0.0105012	−	0.999945i	$-0.503343\pi$
$397$	−0.418469	−0.0210024	−0.0105012	+	0.999945i	$0.503343\pi$
$398$	0	0
$399$	−7.17236	−0.359067
$400$	0	0
$401$	38.0856	1.90190	0.950952	−	0.309337i	$-0.100107\pi$
$401$	38.0856	1.90190	0.950952	+	0.309337i	$0.100107\pi$
$402$	0	0
$403$	−1.85543	−0.0924254
$404$	0	0
$405$	−28.1479	−1.39868
$406$	0	0
$407$	−0.141179	−0.00699799
$408$	0	0
$409$	10.3328	0.510925	0.255463	−	0.966819i	$-0.417772\pi$
$409$	10.3328	0.510925	0.255463	+	0.966819i	$0.417772\pi$
$410$	0	0
$411$	11.8339	0.583721
$412$	0	0
$413$	−22.9553	−1.12955
$414$	0	0
$415$	14.1547	0.694827
$416$	0	0
$417$	−34.1684	−1.67323
$418$	0	0
$419$	−5.41047	−0.264319	−0.132159	−	0.991228i	$-0.542191\pi$
$419$	−5.41047	−0.264319	−0.132159	+	0.991228i	$0.542191\pi$
$420$	0	0
$421$	28.6607	1.39684	0.698419	−	0.715689i	$-0.253887\pi$
$421$	28.6607	1.39684	0.698419	+	0.715689i	$0.253887\pi$
$422$	0	0
$423$	−2.38206	−0.115820
$424$	0	0
$425$	15.5838	0.755925
$426$	0	0
$427$	38.8397	1.87959
$428$	0	0
$429$	0.441042	0.0212937
$430$	0	0
$431$	−12.1264	−0.584106	−0.292053	−	0.956402i	$-0.594338\pi$
$431$	−12.1264	−0.584106	−0.292053	+	0.956402i	$0.594338\pi$
$432$	0	0
$433$	−6.82265	−0.327876	−0.163938	−	0.986471i	$-0.552420\pi$
$433$	−6.82265	−0.327876	−0.163938	+	0.986471i	$0.552420\pi$
$434$	0	0
$435$	5.19538	0.249099
$436$	0	0
$437$	−9.53472	−0.456107
$438$	0	0
$439$	−16.1345	−0.770056	−0.385028	−	0.922905i	$-0.625808\pi$
$439$	−16.1345	−0.770056	−0.385028	+	0.922905i	$0.625808\pi$
$440$	0	0
$441$	1.20015	0.0571499
$442$	0	0
$443$	6.87865	0.326815	0.163407	−	0.986559i	$-0.447752\pi$
$443$	6.87865	0.326815	0.163407	+	0.986559i	$0.447752\pi$
$444$	0	0
$445$	30.2383	1.43343
$446$	0	0
$447$	27.9763	1.32323
$448$	0	0
$449$	8.04263	0.379555	0.189778	−	0.981827i	$-0.439223\pi$
$449$	8.04263	0.379555	0.189778	+	0.981827i	$0.439223\pi$
$450$	0	0
$451$	0.977784	0.0460420
$452$	0	0
$453$	−10.7874	−0.506838
$454$	0	0
$455$	−9.24513	−0.433418
$456$	0	0
$457$	−30.8529	−1.44324	−0.721618	−	0.692291i	$-0.756601\pi$
$457$	−30.8529	−1.44324	−0.721618	+	0.692291i	$0.756601\pi$
$458$	0	0
$459$	−24.5386	−1.14536
$460$	0	0
$461$	−5.36616	−0.249927	−0.124964	−	0.992161i	$-0.539881\pi$
$461$	−5.36616	−0.249927	−0.124964	+	0.992161i	$0.539881\pi$
$462$	0	0
$463$	31.4057	1.45954	0.729772	−	0.683690i	$-0.239626\pi$
$463$	31.4057	1.45954	0.729772	+	0.683690i	$0.239626\pi$
$464$	0	0
$465$	9.63964	0.447027
$466$	0	0
$467$	−23.1937	−1.07327	−0.536637	−	0.843813i	$-0.680306\pi$
$467$	−23.1937	−1.07327	−0.536637	+	0.843813i	$0.680306\pi$
$468$	0	0
$469$	−14.0215	−0.647451
$470$	0	0
$471$	23.5530	1.08526
$472$	0	0
$473$	−0.424951	−0.0195393
$474$	0	0
$475$	−3.73230	−0.171250
$476$	0	0
$477$	2.45491	0.112403
$478$	0	0
$479$	12.8745	0.588249	0.294124	−	0.955767i	$-0.404972\pi$
$479$	12.8745	0.588249	0.294124	+	0.955767i	$0.404972\pi$
$480$	0	0
$481$	−0.584712	−0.0266606
$482$	0	0
$483$	46.8644	2.13241
$484$	0	0
$485$	52.2376	2.37199
$486$	0	0
$487$	−9.89269	−0.448281	−0.224140	−	0.974557i	$-0.571957\pi$
$487$	−9.89269	−0.448281	−0.224140	+	0.974557i	$0.571957\pi$
$488$	0	0
$489$	−24.5438	−1.10991
$490$	0	0
$491$	−21.7166	−0.980057	−0.490029	−	0.871706i	$-0.663014\pi$
$491$	−21.7166	−0.980057	−0.490029	+	0.871706i	$0.663014\pi$
$492$	0	0
$493$	5.04383	0.227163
$494$	0	0
$495$	−0.231150	−0.0103894
$496$	0	0
$497$	−9.76485	−0.438013
$498$	0	0
$499$	22.3652	1.00120	0.500602	−	0.865677i	$-0.333112\pi$
$499$	22.3652	1.00120	0.500602	+	0.865677i	$0.333112\pi$
$500$	0	0
$501$	16.4370	0.734349
$502$	0	0
$503$	4.28605	0.191106	0.0955528	−	0.995424i	$-0.469538\pi$
$503$	4.28605	0.191106	0.0955528	+	0.995424i	$0.469538\pi$
$504$	0	0
$505$	0.0942346	0.00419338
$506$	0	0
$507$	1.82663	0.0811236
$508$	0	0
$509$	6.18182	0.274004	0.137002	−	0.990571i	$-0.456253\pi$
$509$	6.18182	0.274004	0.137002	+	0.990571i	$0.456253\pi$
$510$	0	0
$511$	−45.6261	−2.01838
$512$	0	0
$513$	5.87696	0.259474
$514$	0	0
$515$	−5.90618	−0.260258
$516$	0	0
$517$	1.70876	0.0751512
$518$	0	0
$519$	46.8249	2.05538
$520$	0	0
$521$	7.54079	0.330368	0.165184	−	0.986263i	$-0.447178\pi$
$521$	7.54079	0.330368	0.165184	+	0.986263i	$0.447178\pi$
$522$	0	0
$523$	8.46770	0.370267	0.185133	−	0.982713i	$-0.440728\pi$
$523$	8.46770	0.370267	0.185133	+	0.982713i	$0.440728\pi$
$524$	0	0
$525$	18.3447	0.800630
$526$	0	0
$527$	9.35845	0.407660
$528$	0	0
$529$	39.3002	1.70870
$530$	0	0
$531$	−2.37703	−0.103154
$532$	0	0
$533$	4.04962	0.175408
$534$	0	0
$535$	30.8754	1.33486
$536$	0	0
$537$	−9.08209	−0.391921
$538$	0	0
$539$	−0.860920	−0.0370825
$540$	0	0
$541$	−28.2726	−1.21553	−0.607767	−	0.794115i	$-0.707935\pi$
$541$	−28.2726	−1.21553	−0.607767	+	0.794115i	$0.707935\pi$
$542$	0	0
$543$	48.4656	2.07986
$544$	0	0
$545$	−11.4148	−0.488956
$546$	0	0
$547$	24.1808	1.03390	0.516948	−	0.856017i	$-0.327068\pi$
$547$	24.1808	1.03390	0.516948	+	0.856017i	$0.327068\pi$
$548$	0	0
$549$	4.02188	0.171650
$550$	0	0
$551$	−1.20799	−0.0514621
$552$	0	0
$553$	−28.6084	−1.21655
$554$	0	0
$555$	3.03780	0.128947
$556$	0	0
$557$	−27.5093	−1.16561	−0.582803	−	0.812614i	$-0.698044\pi$
$557$	−27.5093	−1.16561	−0.582803	+	0.812614i	$0.698044\pi$
$558$	0	0
$559$	−1.75999	−0.0744397
$560$	0	0
$561$	−2.22454	−0.0939201
$562$	0	0
$563$	6.74122	0.284108	0.142054	−	0.989859i	$-0.454629\pi$
$563$	6.74122	0.284108	0.142054	+	0.989859i	$0.454629\pi$
$564$	0	0
$565$	−41.3318	−1.73884
$566$	0	0
$567$	−32.1683	−1.35094
$568$	0	0
$569$	−18.2095	−0.763383	−0.381692	−	0.924290i	$-0.624658\pi$
$569$	−18.2095	−0.763383	−0.381692	+	0.924290i	$0.624658\pi$
$570$	0	0
$571$	18.2065	0.761917	0.380958	−	0.924592i	$-0.375594\pi$
$571$	18.2065	0.761917	0.380958	+	0.924592i	$0.375594\pi$
$572$	0	0
$573$	11.2833	0.471366
$574$	0	0
$575$	24.3870	1.01701
$576$	0	0
$577$	−16.5465	−0.688838	−0.344419	−	0.938816i	$-0.611924\pi$
$577$	−16.5465	−0.688838	−0.344419	+	0.938816i	$0.611924\pi$
$578$	0	0
$579$	36.4785	1.51599
$580$	0	0
$581$	16.1764	0.671111
$582$	0	0
$583$	−1.76102	−0.0729339
$584$	0	0
$585$	−0.957339	−0.0395811
$586$	0	0
$587$	−7.99915	−0.330160	−0.165080	−	0.986280i	$-0.552788\pi$
$587$	−7.99915	−0.330160	−0.165080	+	0.986280i	$0.552788\pi$
$588$	0	0
$589$	−2.24134	−0.0923526
$590$	0	0
$591$	−40.8729	−1.68129
$592$	0	0
$593$	−0.816893	−0.0335458	−0.0167729	−	0.999859i	$-0.505339\pi$
$593$	−0.816893	−0.0335458	−0.0167729	+	0.999859i	$0.505339\pi$
$594$	0	0
$595$	46.6308	1.91168
$596$	0	0
$597$	−37.8439	−1.54885
$598$	0	0
$599$	−0.961504	−0.0392860	−0.0196430	−	0.999807i	$-0.506253\pi$
$599$	−0.961504	−0.0392860	−0.0196430	+	0.999807i	$0.506253\pi$
$600$	0	0
$601$	7.10733	0.289914	0.144957	−	0.989438i	$-0.453696\pi$
$601$	7.10733	0.289914	0.144957	+	0.989438i	$0.453696\pi$
$602$	0	0
$603$	−1.45193	−0.0591272
$604$	0	0
$605$	−31.1208	−1.26524
$606$	0	0
$607$	17.2118	0.698605	0.349302	−	0.937010i	$-0.386419\pi$
$607$	17.2118	0.698605	0.349302	+	0.937010i	$0.386419\pi$
$608$	0	0
$609$	5.93743	0.240597
$610$	0	0
$611$	7.07705	0.286307
$612$	0	0
$613$	10.2267	0.413051	0.206526	−	0.978441i	$-0.433784\pi$
$613$	10.2267	0.413051	0.206526	+	0.978441i	$0.433784\pi$
$614$	0	0
$615$	−21.0393	−0.848386
$616$	0	0
$617$	−29.7847	−1.19909	−0.599543	−	0.800343i	$-0.704651\pi$
$617$	−29.7847	−1.19909	−0.599543	+	0.800343i	$0.704651\pi$
$618$	0	0
$619$	30.3343	1.21924	0.609619	−	0.792695i	$-0.291323\pi$
$619$	30.3343	1.21924	0.609619	+	0.792695i	$0.291323\pi$
$620$	0	0
$621$	−38.4003	−1.54095
$622$	0	0
$623$	34.5573	1.38451
$624$	0	0
$625$	−30.9023	−1.23609
$626$	0	0
$627$	0.532774	0.0212770
$628$	0	0
$629$	2.94919	0.117592
$630$	0	0
$631$	−4.21278	−0.167708	−0.0838541	−	0.996478i	$-0.526723\pi$
$631$	−4.21278	−0.167708	−0.0838541	+	0.996478i	$0.526723\pi$
$632$	0	0
$633$	−4.11745	−0.163654
$634$	0	0
$635$	−2.29393	−0.0910317
$636$	0	0
$637$	−3.56561	−0.141275
$638$	0	0
$639$	−1.01116	−0.0400008
$640$	0	0
$641$	−8.77408	−0.346555	−0.173278	−	0.984873i	$-0.555436\pi$
$641$	−8.77408	−0.346555	−0.173278	+	0.984873i	$0.555436\pi$
$642$	0	0
$643$	11.0741	0.436721	0.218360	−	0.975868i	$-0.429929\pi$
$643$	11.0741	0.436721	0.218360	+	0.975868i	$0.429929\pi$
$644$	0	0
$645$	9.14381	0.360037
$646$	0	0
$647$	−1.79590	−0.0706041	−0.0353021	−	0.999377i	$-0.511239\pi$
$647$	−1.79590	−0.0706041	−0.0353021	+	0.999377i	$0.511239\pi$
$648$	0	0
$649$	1.70515	0.0669332
$650$	0	0
$651$	11.0165	0.431769
$652$	0	0
$653$	23.2678	0.910539	0.455269	−	0.890354i	$-0.349543\pi$
$653$	23.2678	0.910539	0.455269	+	0.890354i	$0.349543\pi$
$654$	0	0
$655$	5.45959	0.213324
$656$	0	0
$657$	−4.72462	−0.184325
$658$	0	0
$659$	−15.2257	−0.593109	−0.296555	−	0.955016i	$-0.595838\pi$
$659$	−15.2257	−0.593109	−0.296555	+	0.955016i	$0.595838\pi$
$660$	0	0
$661$	−6.14485	−0.239007	−0.119503	−	0.992834i	$-0.538130\pi$
$661$	−6.14485	−0.239007	−0.119503	+	0.992834i	$0.538130\pi$
$662$	0	0
$663$	−9.21322	−0.357812
$664$	0	0
$665$	−11.1680	−0.433077
$666$	0	0
$667$	7.89305	0.305620
$668$	0	0
$669$	−27.8401	−1.07636
$670$	0	0
$671$	−2.88508	−0.111377
$672$	0	0
$673$	23.8730	0.920237	0.460119	−	0.887857i	$-0.347807\pi$
$673$	23.8730	0.920237	0.460119	+	0.887857i	$0.347807\pi$
$674$	0	0
$675$	−15.0315	−0.578563
$676$	0	0
$677$	−19.2536	−0.739975	−0.369987	−	0.929037i	$-0.620638\pi$
$677$	−19.2536	−0.739975	−0.369987	+	0.929037i	$0.620638\pi$
$678$	0	0
$679$	59.6987	2.29103
$680$	0	0
$681$	−47.6439	−1.82572
$682$	0	0
$683$	49.9296	1.91050	0.955252	−	0.295795i	$-0.0955844\pi$
$683$	49.9296	1.91050	0.955252	+	0.295795i	$0.0955844\pi$
$684$	0	0
$685$	18.4264	0.704036
$686$	0	0
$687$	4.27251	0.163007
$688$	0	0
$689$	−7.29349	−0.277860
$690$	0	0
$691$	−9.22703	−0.351013	−0.175506	−	0.984478i	$-0.556156\pi$
$691$	−9.22703	−0.351013	−0.175506	+	0.984478i	$0.556156\pi$
$692$	0	0
$693$	−0.264166	−0.0100348
$694$	0	0
$695$	−53.2033	−2.01811
$696$	0	0
$697$	−20.4256	−0.773674
$698$	0	0
$699$	34.5957	1.30853
$700$	0	0
$701$	24.4618	0.923910	0.461955	−	0.886903i	$-0.347148\pi$
$701$	24.4618	0.923910	0.461955	+	0.886903i	$0.347148\pi$
$702$	0	0
$703$	−0.706326	−0.0266396
$704$	0	0
$705$	−36.7680	−1.38476
$706$	0	0
$707$	0.107694	0.00405025
$708$	0	0
$709$	0.152400	0.00572351	0.00286176	−	0.999996i	$-0.499089\pi$
$709$	0.152400	0.00572351	0.00286176	+	0.999996i	$0.499089\pi$
$710$	0	0
$711$	−2.96242	−0.111100
$712$	0	0
$713$	14.6450	0.548458
$714$	0	0
$715$	0.686743	0.0256827
$716$	0	0
$717$	−44.5537	−1.66389
$718$	0	0
$719$	−35.4138	−1.32071	−0.660356	−	0.750952i	$-0.729595\pi$
$719$	−35.4138	−1.32071	−0.660356	+	0.750952i	$0.729595\pi$
$720$	0	0
$721$	−6.74977	−0.251374
$722$	0	0
$723$	2.22987	0.0829296
$724$	0	0
$725$	3.08968	0.114748
$726$	0	0
$727$	−31.5101	−1.16864	−0.584322	−	0.811522i	$-0.698640\pi$
$727$	−31.5101	−1.16864	−0.584322	+	0.811522i	$0.698640\pi$
$728$	0	0
$729$	23.3291	0.864040
$730$	0	0
$731$	8.87709	0.328331
$732$	0	0
$733$	−21.5025	−0.794212	−0.397106	−	0.917773i	$-0.629985\pi$
$733$	−21.5025	−0.794212	−0.397106	+	0.917773i	$0.629985\pi$
$734$	0	0
$735$	18.5247	0.683294
$736$	0	0
$737$	1.04154	0.0383655
$738$	0	0
$739$	26.0327	0.957629	0.478814	−	0.877916i	$-0.341067\pi$
$739$	26.0327	0.957629	0.478814	+	0.877916i	$0.341067\pi$
$740$	0	0
$741$	2.20655	0.0810598
$742$	0	0
$743$	−40.9133	−1.50096	−0.750482	−	0.660891i	$-0.770178\pi$
$743$	−40.9133	−1.50096	−0.750482	+	0.660891i	$0.770178\pi$
$744$	0	0
$745$	43.5616	1.59598
$746$	0	0
$747$	1.67508	0.0612880
$748$	0	0
$749$	35.2853	1.28930
$750$	0	0
$751$	3.81700	0.139284	0.0696422	−	0.997572i	$-0.477814\pi$
$751$	3.81700	0.139284	0.0696422	+	0.997572i	$0.477814\pi$
$752$	0	0
$753$	11.9011	0.433700
$754$	0	0
$755$	−16.7970	−0.611307
$756$	0	0
$757$	−12.0690	−0.438657	−0.219328	−	0.975651i	$-0.570387\pi$
$757$	−12.0690	−0.438657	−0.219328	+	0.975651i	$0.570387\pi$
$758$	0	0
$759$	−3.48117	−0.126358
$760$	0	0
$761$	5.71300	0.207096	0.103548	−	0.994624i	$-0.466980\pi$
$761$	5.71300	0.207096	0.103548	+	0.994624i	$0.466980\pi$
$762$	0	0
$763$	−13.0452	−0.472267
$764$	0	0
$765$	4.82865	0.174580
$766$	0	0
$767$	7.06212	0.254998
$768$	0	0
$769$	4.76288	0.171754	0.0858768	−	0.996306i	$-0.472631\pi$
$769$	4.76288	0.171754	0.0858768	+	0.996306i	$0.472631\pi$
$770$	0	0
$771$	−19.7376	−0.710832
$772$	0	0
$773$	−42.9739	−1.54566	−0.772831	−	0.634612i	$-0.781160\pi$
$773$	−42.9739	−1.54566	−0.772831	+	0.634612i	$0.781160\pi$
$774$	0	0
$775$	5.73267	0.205923
$776$	0	0
$777$	3.47169	0.124546
$778$	0	0
$779$	4.89190	0.175270
$780$	0	0
$781$	0.725349	0.0259550
$782$	0	0
$783$	−4.86507	−0.173864
$784$	0	0
$785$	36.6742	1.30896
$786$	0	0
$787$	28.7164	1.02363	0.511814	−	0.859096i	$-0.328974\pi$
$787$	28.7164	1.02363	0.511814	+	0.859096i	$0.328974\pi$
$788$	0	0
$789$	34.3769	1.22385
$790$	0	0
$791$	−47.2353	−1.67949
$792$	0	0
$793$	−11.9489	−0.424319
$794$	0	0
$795$	37.8924	1.34391
$796$	0	0
$797$	−20.9853	−0.743339	−0.371670	−	0.928365i	$-0.621215\pi$
$797$	−20.9853	−0.743339	−0.371670	+	0.928365i	$0.621215\pi$
$798$	0	0
$799$	−35.6954	−1.26281
$800$	0	0
$801$	3.57843	0.126438
$802$	0	0
$803$	3.38918	0.119602
$804$	0	0
$805$	72.9722	2.57193
$806$	0	0
$807$	−39.2230	−1.38071
$808$	0	0
$809$	48.0239	1.68843	0.844215	−	0.536005i	$-0.180067\pi$
$809$	48.0239	1.68843	0.844215	+	0.536005i	$0.180067\pi$
$810$	0	0
$811$	−32.8713	−1.15427	−0.577133	−	0.816650i	$-0.695829\pi$
$811$	−32.8713	−1.15427	−0.577133	+	0.816650i	$0.695829\pi$
$812$	0	0
$813$	−33.7542	−1.18381
$814$	0	0
$815$	−38.2169	−1.33868
$816$	0	0
$817$	−2.12605	−0.0743811
$818$	0	0
$819$	−1.09408	−0.0382301
$820$	0	0
$821$	−18.6296	−0.650176	−0.325088	−	0.945684i	$-0.605394\pi$
$821$	−18.6296	−0.650176	−0.325088	+	0.945684i	$0.605394\pi$
$822$	0	0
$823$	−24.3560	−0.848998	−0.424499	−	0.905429i	$-0.639550\pi$
$823$	−24.3560	−0.848998	−0.424499	+	0.905429i	$0.639550\pi$
$824$	0	0
$825$	−1.36268	−0.0474423
$826$	0	0
$827$	−26.2927	−0.914286	−0.457143	−	0.889393i	$-0.651127\pi$
$827$	−26.2927	−0.914286	−0.457143	+	0.889393i	$0.651127\pi$
$828$	0	0
$829$	−45.0054	−1.56310	−0.781551	−	0.623841i	$-0.785571\pi$
$829$	−45.0054	−1.56310	−0.781551	+	0.623841i	$0.785571\pi$
$830$	0	0
$831$	−21.5585	−0.747857
$832$	0	0
$833$	17.9843	0.623120
$834$	0	0
$835$	25.5939	0.885712
$836$	0	0
$837$	−9.02679	−0.312011
$838$	0	0
$839$	12.0776	0.416966	0.208483	−	0.978026i	$-0.433147\pi$
$839$	12.0776	0.416966	0.208483	+	0.978026i	$0.433147\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−3.40189	−0.117167
$844$	0	0
$845$	2.84424	0.0978447
$846$	0	0
$847$	−35.5658	−1.22205
$848$	0	0
$849$	−43.6503	−1.49808
$850$	0	0
$851$	4.61516	0.158206
$852$	0	0
$853$	53.6252	1.83609	0.918047	−	0.396472i	$-0.129766\pi$
$853$	53.6252	1.83609	0.918047	+	0.396472i	$0.129766\pi$
$854$	0	0
$855$	−1.15646	−0.0395500
$856$	0	0
$857$	32.6025	1.11368	0.556841	−	0.830619i	$-0.312013\pi$
$857$	32.6025	1.11368	0.556841	+	0.830619i	$0.312013\pi$
$858$	0	0
$859$	45.6878	1.55885	0.779423	−	0.626498i	$-0.215512\pi$
$859$	45.6878	1.55885	0.779423	+	0.626498i	$0.215512\pi$
$860$	0	0
$861$	−24.0443	−0.819429
$862$	0	0
$863$	−36.0842	−1.22832	−0.614160	−	0.789182i	$-0.710505\pi$
$863$	−36.0842	−1.22832	−0.614160	+	0.789182i	$0.710505\pi$
$864$	0	0
$865$	72.9107	2.47904
$866$	0	0
$867$	15.4171	0.523593
$868$	0	0
$869$	2.12508	0.0720884
$870$	0	0
$871$	4.31366	0.146163
$872$	0	0
$873$	6.18184	0.209224
$874$	0	0
$875$	−17.6612	−0.597057
$876$	0	0
$877$	−3.47241	−0.117255	−0.0586275	−	0.998280i	$-0.518672\pi$
$877$	−3.47241	−0.117255	−0.0586275	+	0.998280i	$0.518672\pi$
$878$	0	0
$879$	−20.5863	−0.694358
$880$	0	0
$881$	−4.68261	−0.157761	−0.0788806	−	0.996884i	$-0.525135\pi$
$881$	−4.68261	−0.157761	−0.0788806	+	0.996884i	$0.525135\pi$
$882$	0	0
$883$	−5.69355	−0.191603	−0.0958016	−	0.995400i	$-0.530541\pi$
$883$	−5.69355	−0.191603	−0.0958016	+	0.995400i	$0.530541\pi$
$884$	0	0
$885$	−36.6904	−1.23333
$886$	0	0
$887$	41.8486	1.40514	0.702569	−	0.711616i	$-0.252036\pi$
$887$	41.8486	1.40514	0.702569	+	0.711616i	$0.252036\pi$
$888$	0	0
$889$	−2.62157	−0.0879246
$890$	0	0
$891$	2.38951	0.0800517
$892$	0	0
$893$	8.54901	0.286082
$894$	0	0
$895$	−14.1416	−0.472703
$896$	0	0
$897$	−14.4177	−0.481393
$898$	0	0
$899$	1.85543	0.0618819
$900$	0	0
$901$	36.7871	1.22556
$902$	0	0
$903$	10.4498	0.347749
$904$	0	0
$905$	75.4655	2.50856
$906$	0	0
$907$	−33.4859	−1.11188	−0.555941	−	0.831222i	$-0.687642\pi$
$907$	−33.4859	−1.11188	−0.555941	+	0.831222i	$0.687642\pi$
$908$	0	0
$909$	0.0111518	0.000369882 0
$910$	0	0
$911$	−2.35687	−0.0780866	−0.0390433	−	0.999238i	$-0.512431\pi$
$911$	−2.35687	−0.0780866	−0.0390433	+	0.999238i	$0.512431\pi$
$912$	0	0
$913$	−1.20161	−0.0397675
$914$	0	0
$915$	62.0792	2.05227
$916$	0	0
$917$	6.23939	0.206043
$918$	0	0
$919$	2.73048	0.0900701	0.0450351	−	0.998985i	$-0.485660\pi$
$919$	2.73048	0.0900701	0.0450351	+	0.998985i	$0.485660\pi$
$920$	0	0
$921$	33.4355	1.10174
$922$	0	0
$923$	3.00413	0.0988821
$924$	0	0
$925$	1.80657	0.0593997
$926$	0	0
$927$	−0.698943	−0.0229563
$928$	0	0
$929$	−16.6144	−0.545101	−0.272551	−	0.962141i	$-0.587867\pi$
$929$	−16.6144	−0.545101	−0.272551	+	0.962141i	$0.587867\pi$
$930$	0	0
$931$	−4.30722	−0.141164
$932$	0	0
$933$	−49.2568	−1.61259
$934$	0	0
$935$	−3.46381	−0.113279
$936$	0	0
$937$	59.1816	1.93338	0.966690	−	0.255950i	$-0.0823883\pi$
$937$	59.1816	1.93338	0.966690	+	0.255950i	$0.0823883\pi$
$938$	0	0
$939$	−14.0883	−0.459756
$940$	0	0
$941$	18.0514	0.588458	0.294229	−	0.955735i	$-0.404937\pi$
$941$	18.0514	0.588458	0.294229	+	0.955735i	$0.404937\pi$
$942$	0	0
$943$	−31.9638	−1.04089
$944$	0	0
$945$	−44.9782	−1.46314
$946$	0	0
$947$	−55.2965	−1.79689	−0.898447	−	0.439082i	$-0.855304\pi$
$947$	−55.2965	−1.79689	−0.898447	+	0.439082i	$0.855304\pi$
$948$	0	0
$949$	14.0367	0.455652
$950$	0	0
$951$	−8.52407	−0.276412
$952$	0	0
$953$	−15.6365	−0.506515	−0.253258	−	0.967399i	$-0.581502\pi$
$953$	−15.6365	−0.506515	−0.253258	+	0.967399i	$0.581502\pi$
$954$	0	0
$955$	17.5691	0.568523
$956$	0	0
$957$	−0.441042	−0.0142569
$958$	0	0
$959$	21.0582	0.680006
$960$	0	0
$961$	−27.5574	−0.888948
$962$	0	0
$963$	3.65382	0.117743
$964$	0	0
$965$	56.8004	1.82847
$966$	0	0
$967$	24.4701	0.786907	0.393453	−	0.919345i	$-0.371280\pi$
$967$	24.4701	0.786907	0.393453	+	0.919345i	$0.371280\pi$
$968$	0	0
$969$	−11.1295	−0.357530
$970$	0	0
$971$	32.7225	1.05012	0.525058	−	0.851066i	$-0.324044\pi$
$971$	32.7225	1.05012	0.525058	+	0.851066i	$0.324044\pi$
$972$	0	0
$973$	−60.8023	−1.94923
$974$	0	0
$975$	−5.64371	−0.180743
$976$	0	0
$977$	−28.4852	−0.911322	−0.455661	−	0.890154i	$-0.650597\pi$
$977$	−28.4852	−0.911322	−0.455661	+	0.890154i	$0.650597\pi$
$978$	0	0
$979$	−2.56697	−0.0820407
$980$	0	0
$981$	−1.35084	−0.0431288
$982$	0	0
$983$	−37.6413	−1.20057	−0.600285	−	0.799786i	$-0.704946\pi$
$983$	−37.6413	−1.20057	−0.600285	+	0.799786i	$0.704946\pi$
$984$	0	0
$985$	−63.6428	−2.02783
$986$	0	0
$987$	−42.0195	−1.33750
$988$	0	0
$989$	13.8917	0.441730
$990$	0	0
$991$	−3.08976	−0.0981495	−0.0490748	−	0.998795i	$-0.515627\pi$
$991$	−3.08976	−0.0981495	−0.0490748	+	0.998795i	$0.515627\pi$
$992$	0	0
$993$	52.4442	1.66427
$994$	0	0
$995$	−58.9265	−1.86810
$996$	0	0
$997$	−27.7525	−0.878930	−0.439465	−	0.898260i	$-0.644832\pi$
$997$	−27.7525	−0.878930	−0.439465	+	0.898260i	$0.644832\pi$
$998$	0	0
$999$	−2.84467	−0.0900013

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.be.1.9		13
4.3	odd	2		3016.2.a.k.1.5	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.k.1.5	✓	13	4.3	odd	2
6032.2.a.be.1.9		13	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82663 q^{3} +2.84424 q^{5} +3.25048 q^{7} +0.336589 q^{9} +O(q^{10})\)	\(q+1.82663 q^{3} +2.84424 q^{5} +3.25048 q^{7} +0.336589 q^{9} -0.241451 q^{11} -1.00000 q^{13} +5.19538 q^{15} +5.04383 q^{17} -1.20799 q^{19} +5.93743 q^{21} +7.89305 q^{23} +3.08968 q^{25} -4.86507 q^{27} +1.00000 q^{29} +1.85543 q^{31} -0.441042 q^{33} +9.24513 q^{35} +0.584712 q^{37} -1.82663 q^{39} -4.04962 q^{41} +1.75999 q^{43} +0.957339 q^{45} -7.07705 q^{47} +3.56561 q^{49} +9.21322 q^{51} +7.29349 q^{53} -0.686743 q^{55} -2.20655 q^{57} -7.06212 q^{59} +11.9489 q^{61} +1.09408 q^{63} -2.84424 q^{65} -4.31366 q^{67} +14.4177 q^{69} -3.00413 q^{71} -14.0367 q^{73} +5.64371 q^{75} -0.784831 q^{77} -8.80130 q^{79} -9.89648 q^{81} +4.97663 q^{83} +14.3458 q^{85} +1.82663 q^{87} +10.6314 q^{89} -3.25048 q^{91} +3.38918 q^{93} -3.43581 q^{95} +18.3661 q^{97} -0.0812698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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