LMFDB - Embedded newform 6032.2.a.z.1.10

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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3x^{9} - 17x^{8} + 47x^{7} + 104x^{6} - 235x^{5} - 283x^{4} + 364x^{3} + 330x^{2} + 12x - 8 $ x^10 - 3x^9 - 17x^8 + 47x^7 + 104x^6 - 235x^5 - 283x^4 + 364x^3 + 330x^2 + 12*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-2.55589$ 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)

$f(q)$	$=$	$q+2.55589 q^{3} -2.69451 q^{5} +1.01002 q^{7} +3.53259 q^{9} +O(q^{10})$	$q+2.55589 q^{3} -2.69451 q^{5} +1.01002 q^{7} +3.53259 q^{9} -2.93594 q^{11} -1.00000 q^{13} -6.88688 q^{15} -4.17439 q^{17} +1.18543 q^{19} +2.58149 q^{21} +6.85409 q^{23} +2.26039 q^{25} +1.36123 q^{27} -1.00000 q^{29} +4.62351 q^{31} -7.50394 q^{33} -2.72150 q^{35} -4.42704 q^{37} -2.55589 q^{39} +5.14943 q^{41} -3.08591 q^{43} -9.51859 q^{45} +9.02225 q^{47} -5.97987 q^{49} -10.6693 q^{51} -6.85687 q^{53} +7.91091 q^{55} +3.02983 q^{57} -4.90603 q^{59} -7.80816 q^{61} +3.56797 q^{63} +2.69451 q^{65} +1.60652 q^{67} +17.5183 q^{69} -1.51530 q^{71} -14.7228 q^{73} +5.77732 q^{75} -2.96534 q^{77} -7.59173 q^{79} -7.11859 q^{81} -1.23411 q^{83} +11.2479 q^{85} -2.55589 q^{87} +8.99598 q^{89} -1.01002 q^{91} +11.8172 q^{93} -3.19416 q^{95} -14.5581 q^{97} -10.3714 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) $ 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9	$ 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) $ 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9 - 14 * q^11 - 10 * q^13 - 7 * q^15 + 5 * q^17 - 11 * q^19 - 7 * q^23 + 10 * q^25 - 21 * q^27 - 10 * q^29 - 5 * q^31 + 5 * q^33 - 11 * q^35 + 8 * q^37 + 3 * q^39 + 14 * q^41 - 35 * q^43 + 7 * q^45 - 7 * q^49 - 20 * q^51 - 11 * q^53 - 8 * q^55 + 4 * q^57 - 23 * q^59 - 8 * q^61 - 43 * q^63 - 4 * q^65 - 27 * q^67 + 10 * q^69 - 3 * q^71 + 7 * q^73 - 23 * q^75 + 2 * q^77 - 9 * q^79 - 6 * q^81 - 48 * q^83 - 6 * q^85 + 3 * q^87 + 20 * q^89 - 3 * q^91 - 11 * q^93 - 11 * q^95 + q^97 - 54 * 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
$3$	2.55589	1.47565	0.737823	−	0.674995i	$-0.235854\pi$
$3$	2.55589	1.47565	0.737823	+	0.674995i	$0.235854\pi$
$4$	0	0
$5$	−2.69451	−1.20502	−0.602511	−	0.798111i	$-0.705833\pi$
$5$	−2.69451	−1.20502	−0.602511	+	0.798111i	$0.705833\pi$
$6$	0	0
$7$	1.01002	0.381750	0.190875	−	0.981614i	$-0.438867\pi$
$7$	1.01002	0.381750	0.190875	+	0.981614i	$0.438867\pi$
$8$	0	0
$9$	3.53259	1.17753
$10$	0	0
$11$	−2.93594	−0.885218	−0.442609	−	0.896715i	$-0.645947\pi$
$11$	−2.93594	−0.885218	−0.442609	+	0.896715i	$0.645947\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−6.88688	−1.77819
$16$	0	0
$17$	−4.17439	−1.01244	−0.506219	−	0.862405i	$-0.668957\pi$
$17$	−4.17439	−1.01244	−0.506219	+	0.862405i	$0.668957\pi$
$18$	0	0
$19$	1.18543	0.271957	0.135978	−	0.990712i	$-0.456582\pi$
$19$	1.18543	0.271957	0.135978	+	0.990712i	$0.456582\pi$
$20$	0	0
$21$	2.58149	0.563328
$22$	0	0
$23$	6.85409	1.42918	0.714588	−	0.699545i	$-0.246614\pi$
$23$	6.85409	1.42918	0.714588	+	0.699545i	$0.246614\pi$
$24$	0	0
$25$	2.26039	0.452078
$26$	0	0
$27$	1.36123	0.261969
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.62351	0.830407	0.415203	−	0.909729i	$-0.363710\pi$
$31$	4.62351	0.830407	0.415203	+	0.909729i	$0.363710\pi$
$32$	0	0
$33$	−7.50394	−1.30627
$34$	0	0
$35$	−2.72150	−0.460017
$36$	0	0
$37$	−4.42704	−0.727800	−0.363900	−	0.931438i	$-0.618555\pi$
$37$	−4.42704	−0.727800	−0.363900	+	0.931438i	$0.618555\pi$
$38$	0	0
$39$	−2.55589	−0.409270
$40$	0	0
$41$	5.14943	0.804205	0.402103	−	0.915595i	$-0.368280\pi$
$41$	5.14943	0.804205	0.402103	+	0.915595i	$0.368280\pi$
$42$	0	0
$43$	−3.08591	−0.470597	−0.235299	−	0.971923i	$-0.575607\pi$
$43$	−3.08591	−0.470597	−0.235299	+	0.971923i	$0.575607\pi$
$44$	0	0
$45$	−9.51859	−1.41895
$46$	0	0
$47$	9.02225	1.31603	0.658015	−	0.753004i	$-0.271396\pi$
$47$	9.02225	1.31603	0.658015	+	0.753004i	$0.271396\pi$
$48$	0	0
$49$	−5.97987	−0.854267
$50$	0	0
$51$	−10.6693	−1.49400
$52$	0	0
$53$	−6.85687	−0.941864	−0.470932	−	0.882170i	$-0.656082\pi$
$53$	−6.85687	−0.941864	−0.470932	+	0.882170i	$0.656082\pi$
$54$	0	0
$55$	7.91091	1.06671
$56$	0	0
$57$	3.02983	0.401311
$58$	0	0
$59$	−4.90603	−0.638710	−0.319355	−	0.947635i	$-0.603466\pi$
$59$	−4.90603	−0.638710	−0.319355	+	0.947635i	$0.603466\pi$
$60$	0	0
$61$	−7.80816	−0.999733	−0.499866	−	0.866103i	$-0.666618\pi$
$61$	−7.80816	−0.999733	−0.499866	+	0.866103i	$0.666618\pi$
$62$	0	0
$63$	3.56797	0.449522
$64$	0	0
$65$	2.69451	0.334213
$66$	0	0
$67$	1.60652	0.196268	0.0981338	−	0.995173i	$-0.468713\pi$
$67$	1.60652	0.196268	0.0981338	+	0.995173i	$0.468713\pi$
$68$	0	0
$69$	17.5183	2.10896
$70$	0	0
$71$	−1.51530	−0.179832	−0.0899162	−	0.995949i	$-0.528660\pi$
$71$	−1.51530	−0.179832	−0.0899162	+	0.995949i	$0.528660\pi$
$72$	0	0
$73$	−14.7228	−1.72317	−0.861587	−	0.507610i	$-0.830529\pi$
$73$	−14.7228	−1.72317	−0.861587	+	0.507610i	$0.830529\pi$
$74$	0	0
$75$	5.77732	0.667107
$76$	0	0
$77$	−2.96534	−0.337932
$78$	0	0
$79$	−7.59173	−0.854137	−0.427068	−	0.904219i	$-0.640454\pi$
$79$	−7.59173	−0.854137	−0.427068	+	0.904219i	$0.640454\pi$
$80$	0	0
$81$	−7.11859	−0.790955
$82$	0	0
$83$	−1.23411	−0.135461	−0.0677306	−	0.997704i	$-0.521576\pi$
$83$	−1.23411	−0.135461	−0.0677306	+	0.997704i	$0.521576\pi$
$84$	0	0
$85$	11.2479	1.22001
$86$	0	0
$87$	−2.55589	−0.274020
$88$	0	0
$89$	8.99598	0.953572	0.476786	−	0.879019i	$-0.341802\pi$
$89$	8.99598	0.953572	0.476786	+	0.879019i	$0.341802\pi$
$90$	0	0
$91$	−1.01002	−0.105878
$92$	0	0
$93$	11.8172	1.22539
$94$	0	0
$95$	−3.19416	−0.327714
$96$	0	0
$97$	−14.5581	−1.47815	−0.739075	−	0.673623i	$-0.764737\pi$
$97$	−14.5581	−1.47815	−0.739075	+	0.673623i	$0.764737\pi$
$98$	0	0
$99$	−10.3714	−1.04237
$100$	0	0
$101$	−3.90294	−0.388358	−0.194179	−	0.980966i	$-0.562204\pi$
$101$	−3.90294	−0.388358	−0.194179	+	0.980966i	$0.562204\pi$
$102$	0	0
$103$	−14.2890	−1.40794	−0.703968	−	0.710231i	$-0.748590\pi$
$103$	−14.2890	−1.40794	−0.703968	+	0.710231i	$0.748590\pi$
$104$	0	0
$105$	−6.95586	−0.678823
$106$	0	0
$107$	−19.1315	−1.84951	−0.924754	−	0.380566i	$-0.875729\pi$
$107$	−19.1315	−1.84951	−0.924754	+	0.380566i	$0.875729\pi$
$108$	0	0
$109$	0.177883	0.0170381	0.00851904	−	0.999964i	$-0.497288\pi$
$109$	0.177883	0.0170381	0.00851904	+	0.999964i	$0.497288\pi$
$110$	0	0
$111$	−11.3150	−1.07398
$112$	0	0
$113$	−0.838401	−0.0788701	−0.0394351	−	0.999222i	$-0.512556\pi$
$113$	−0.838401	−0.0788701	−0.0394351	+	0.999222i	$0.512556\pi$
$114$	0	0
$115$	−18.4684	−1.72219
$116$	0	0
$117$	−3.53259	−0.326588
$118$	0	0
$119$	−4.21620	−0.386499
$120$	0	0
$121$	−2.38028	−0.216389
$122$	0	0
$123$	13.1614	1.18672
$124$	0	0
$125$	7.38191	0.660258
$126$	0	0
$127$	−14.5882	−1.29449	−0.647245	−	0.762282i	$-0.724079\pi$
$127$	−14.5882	−1.29449	−0.647245	+	0.762282i	$0.724079\pi$
$128$	0	0
$129$	−7.88726	−0.694434
$130$	0	0
$131$	−16.7848	−1.46649	−0.733245	−	0.679964i	$-0.761995\pi$
$131$	−16.7848	−1.46649	−0.733245	+	0.679964i	$0.761995\pi$
$132$	0	0
$133$	1.19730	0.103819
$134$	0	0
$135$	−3.66786	−0.315679
$136$	0	0
$137$	13.8591	1.18406	0.592030	−	0.805916i	$-0.298327\pi$
$137$	13.8591	1.18406	0.592030	+	0.805916i	$0.298327\pi$
$138$	0	0
$139$	−7.92266	−0.671991	−0.335996	−	0.941864i	$-0.609073\pi$
$139$	−7.92266	−0.671991	−0.335996	+	0.941864i	$0.609073\pi$
$140$	0	0
$141$	23.0599	1.94199
$142$	0	0
$143$	2.93594	0.245515
$144$	0	0
$145$	2.69451	0.223767
$146$	0	0
$147$	−15.2839	−1.26059
$148$	0	0
$149$	−3.23005	−0.264616	−0.132308	−	0.991209i	$-0.542239\pi$
$149$	−3.23005	−0.264616	−0.132308	+	0.991209i	$0.542239\pi$
$150$	0	0
$151$	6.63746	0.540149	0.270075	−	0.962839i	$-0.412952\pi$
$151$	6.63746	0.540149	0.270075	+	0.962839i	$0.412952\pi$
$152$	0	0
$153$	−14.7464	−1.19218
$154$	0	0
$155$	−12.4581	−1.00066
$156$	0	0
$157$	−21.5298	−1.71827	−0.859134	−	0.511750i	$-0.828997\pi$
$157$	−21.5298	−1.71827	−0.859134	+	0.511750i	$0.828997\pi$
$158$	0	0
$159$	−17.5254	−1.38986
$160$	0	0
$161$	6.92274	0.545588
$162$	0	0
$163$	−7.90878	−0.619463	−0.309732	−	0.950824i	$-0.600239\pi$
$163$	−7.90878	−0.619463	−0.309732	+	0.950824i	$0.600239\pi$
$164$	0	0
$165$	20.2194	1.57408
$166$	0	0
$167$	14.1043	1.09142	0.545712	−	0.837973i	$-0.316259\pi$
$167$	14.1043	1.09142	0.545712	+	0.837973i	$0.316259\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.18764	0.320237
$172$	0	0
$173$	13.6873	1.04062	0.520311	−	0.853977i	$-0.325816\pi$
$173$	13.6873	1.04062	0.520311	+	0.853977i	$0.325816\pi$
$174$	0	0
$175$	2.28303	0.172581
$176$	0	0
$177$	−12.5393	−0.942510
$178$	0	0
$179$	−1.52838	−0.114237	−0.0571183	−	0.998367i	$-0.518191\pi$
$179$	−1.52838	−0.114237	−0.0571183	+	0.998367i	$0.518191\pi$
$180$	0	0
$181$	23.5935	1.75369	0.876844	−	0.480775i	$-0.159645\pi$
$181$	23.5935	1.75369	0.876844	+	0.480775i	$0.159645\pi$
$182$	0	0
$183$	−19.9568	−1.47525
$184$	0	0
$185$	11.9287	0.877016
$186$	0	0
$187$	12.2557	0.896229
$188$	0	0
$189$	1.37487	0.100007
$190$	0	0
$191$	12.9763	0.938935	0.469468	−	0.882950i	$-0.344446\pi$
$191$	12.9763	0.938935	0.469468	+	0.882950i	$0.344446\pi$
$192$	0	0
$193$	7.63479	0.549564	0.274782	−	0.961507i	$-0.411394\pi$
$193$	7.63479	0.549564	0.274782	+	0.961507i	$0.411394\pi$
$194$	0	0
$195$	6.88688	0.493180
$196$	0	0
$197$	−0.481698	−0.0343195	−0.0171598	−	0.999853i	$-0.505462\pi$
$197$	−0.481698	−0.0343195	−0.0171598	+	0.999853i	$0.505462\pi$
$198$	0	0
$199$	−3.56641	−0.252816	−0.126408	−	0.991978i	$-0.540345\pi$
$199$	−3.56641	−0.252816	−0.126408	+	0.991978i	$0.540345\pi$
$200$	0	0
$201$	4.10609	0.289621
$202$	0	0
$203$	−1.01002	−0.0708892
$204$	0	0
$205$	−13.8752	−0.969085
$206$	0	0
$207$	24.2127	1.68290
$208$	0	0
$209$	−3.48035	−0.240741
$210$	0	0
$211$	−16.0822	−1.10714	−0.553572	−	0.832801i	$-0.686736\pi$
$211$	−16.0822	−1.10714	−0.553572	+	0.832801i	$0.686736\pi$
$212$	0	0
$213$	−3.87293	−0.265369
$214$	0	0
$215$	8.31502	0.567080
$216$	0	0
$217$	4.66982	0.317008
$218$	0	0
$219$	−37.6299	−2.54279
$220$	0	0
$221$	4.17439	0.280800
$222$	0	0
$223$	17.5830	1.17744	0.588721	−	0.808336i	$-0.299632\pi$
$223$	17.5830	1.17744	0.588721	+	0.808336i	$0.299632\pi$
$224$	0	0
$225$	7.98503	0.532335
$226$	0	0
$227$	−1.61299	−0.107058	−0.0535289	−	0.998566i	$-0.517047\pi$
$227$	−1.61299	−0.107058	−0.0535289	+	0.998566i	$0.517047\pi$
$228$	0	0
$229$	12.7885	0.845086	0.422543	−	0.906343i	$-0.361138\pi$
$229$	12.7885	0.845086	0.422543	+	0.906343i	$0.361138\pi$
$230$	0	0
$231$	−7.57910	−0.498668
$232$	0	0
$233$	19.0699	1.24931	0.624655	−	0.780901i	$-0.285240\pi$
$233$	19.0699	1.24931	0.624655	+	0.780901i	$0.285240\pi$
$234$	0	0
$235$	−24.3106	−1.58585
$236$	0	0
$237$	−19.4037	−1.26040
$238$	0	0
$239$	−5.43278	−0.351418	−0.175709	−	0.984442i	$-0.556222\pi$
$239$	−5.43278	−0.351418	−0.175709	+	0.984442i	$0.556222\pi$
$240$	0	0
$241$	−10.2922	−0.662982	−0.331491	−	0.943458i	$-0.607552\pi$
$241$	−10.2922	−0.662982	−0.331491	+	0.943458i	$0.607552\pi$
$242$	0	0
$243$	−22.2781	−1.42914
$244$	0	0
$245$	16.1128	1.02941
$246$	0	0
$247$	−1.18543	−0.0754272
$248$	0	0
$249$	−3.15425	−0.199893
$250$	0	0
$251$	25.8290	1.63032	0.815158	−	0.579239i	$-0.196650\pi$
$251$	25.8290	1.63032	0.815158	+	0.579239i	$0.196650\pi$
$252$	0	0
$253$	−20.1232	−1.26513
$254$	0	0
$255$	28.7485	1.80030
$256$	0	0
$257$	−13.3686	−0.833908	−0.416954	−	0.908928i	$-0.636902\pi$
$257$	−13.3686	−0.833908	−0.416954	+	0.908928i	$0.636902\pi$
$258$	0	0
$259$	−4.47138	−0.277838
$260$	0	0
$261$	−3.53259	−0.218662
$262$	0	0
$263$	7.41273	0.457088	0.228544	−	0.973534i	$-0.426603\pi$
$263$	7.41273	0.457088	0.228544	+	0.973534i	$0.426603\pi$
$264$	0	0
$265$	18.4759	1.13497
$266$	0	0
$267$	22.9928	1.40713
$268$	0	0
$269$	2.30010	0.140239	0.0701197	−	0.997539i	$-0.477662\pi$
$269$	2.30010	0.140239	0.0701197	+	0.997539i	$0.477662\pi$
$270$	0	0
$271$	0.322992	0.0196204	0.00981019	−	0.999952i	$-0.496877\pi$
$271$	0.322992	0.0196204	0.00981019	+	0.999952i	$0.496877\pi$
$272$	0	0
$273$	−2.58149	−0.156239
$274$	0	0
$275$	−6.63636	−0.400188
$276$	0	0
$277$	9.55984	0.574395	0.287198	−	0.957871i	$-0.407276\pi$
$277$	9.55984	0.574395	0.287198	+	0.957871i	$0.407276\pi$
$278$	0	0
$279$	16.3329	0.977828
$280$	0	0
$281$	3.15752	0.188362	0.0941809	−	0.995555i	$-0.469977\pi$
$281$	3.15752	0.188362	0.0941809	+	0.995555i	$0.469977\pi$
$282$	0	0
$283$	19.4124	1.15395	0.576975	−	0.816762i	$-0.304233\pi$
$283$	19.4124	1.15395	0.576975	+	0.816762i	$0.304233\pi$
$284$	0	0
$285$	−8.16392	−0.483589
$286$	0	0
$287$	5.20100	0.307005
$288$	0	0
$289$	0.425532	0.0250313
$290$	0	0
$291$	−37.2089	−2.18122
$292$	0	0
$293$	−8.25226	−0.482102	−0.241051	−	0.970512i	$-0.577492\pi$
$293$	−8.25226	−0.482102	−0.241051	+	0.970512i	$0.577492\pi$
$294$	0	0
$295$	13.2193	0.769660
$296$	0	0
$297$	−3.99649	−0.231900
$298$	0	0
$299$	−6.85409	−0.396382
$300$	0	0
$301$	−3.11682	−0.179651
$302$	0	0
$303$	−9.97551	−0.573078
$304$	0	0
$305$	21.0392	1.20470
$306$	0	0
$307$	−25.4901	−1.45479	−0.727397	−	0.686217i	$-0.759270\pi$
$307$	−25.4901	−1.45479	−0.727397	+	0.686217i	$0.759270\pi$
$308$	0	0
$309$	−36.5211	−2.07762
$310$	0	0
$311$	30.6232	1.73648	0.868240	−	0.496144i	$-0.165251\pi$
$311$	30.6232	1.73648	0.868240	+	0.496144i	$0.165251\pi$
$312$	0	0
$313$	−25.2463	−1.42700	−0.713502	−	0.700654i	$-0.752892\pi$
$313$	−25.2463	−1.42700	−0.713502	+	0.700654i	$0.752892\pi$
$314$	0	0
$315$	−9.61393	−0.541684
$316$	0	0
$317$	13.9075	0.781124	0.390562	−	0.920577i	$-0.372281\pi$
$317$	13.9075	0.781124	0.390562	+	0.920577i	$0.372281\pi$
$318$	0	0
$319$	2.93594	0.164381
$320$	0	0
$321$	−48.8979	−2.72922
$322$	0	0
$323$	−4.94845	−0.275339
$324$	0	0
$325$	−2.26039	−0.125384
$326$	0	0
$327$	0.454649	0.0251422
$328$	0	0
$329$	9.11262	0.502395
$330$	0	0
$331$	−8.88832	−0.488546	−0.244273	−	0.969706i	$-0.578549\pi$
$331$	−8.88832	−0.488546	−0.244273	+	0.969706i	$0.578549\pi$
$332$	0	0
$333$	−15.6389	−0.857006
$334$	0	0
$335$	−4.32878	−0.236507
$336$	0	0
$337$	24.8423	1.35325	0.676624	−	0.736329i	$-0.263442\pi$
$337$	24.8423	1.35325	0.676624	+	0.736329i	$0.263442\pi$
$338$	0	0
$339$	−2.14286	−0.116384
$340$	0	0
$341$	−13.5743	−0.735091
$342$	0	0
$343$	−13.1099	−0.707867
$344$	0	0
$345$	−47.2033	−2.54134
$346$	0	0
$347$	−19.6883	−1.05692	−0.528462	−	0.848957i	$-0.677231\pi$
$347$	−19.6883	−1.05692	−0.528462	+	0.848957i	$0.677231\pi$
$348$	0	0
$349$	3.39063	0.181496	0.0907481	−	0.995874i	$-0.471074\pi$
$349$	3.39063	0.181496	0.0907481	+	0.995874i	$0.471074\pi$
$350$	0	0
$351$	−1.36123	−0.0726572
$352$	0	0
$353$	6.13333	0.326444	0.163222	−	0.986589i	$-0.447811\pi$
$353$	6.13333	0.326444	0.163222	+	0.986589i	$0.447811\pi$
$354$	0	0
$355$	4.08298	0.216702
$356$	0	0
$357$	−10.7762	−0.570335
$358$	0	0
$359$	−22.4517	−1.18496	−0.592478	−	0.805586i	$-0.701850\pi$
$359$	−22.4517	−1.18496	−0.592478	+	0.805586i	$0.701850\pi$
$360$	0	0
$361$	−17.5948	−0.926040
$362$	0	0
$363$	−6.08374	−0.319313
$364$	0	0
$365$	39.6708	2.07646
$366$	0	0
$367$	−3.06414	−0.159947	−0.0799735	−	0.996797i	$-0.525484\pi$
$367$	−3.06414	−0.159947	−0.0799735	+	0.996797i	$0.525484\pi$
$368$	0	0
$369$	18.1908	0.946975
$370$	0	0
$371$	−6.92555	−0.359557
$372$	0	0
$373$	10.4430	0.540718	0.270359	−	0.962760i	$-0.412858\pi$
$373$	10.4430	0.540718	0.270359	+	0.962760i	$0.412858\pi$
$374$	0	0
$375$	18.8674	0.974306
$376$	0	0
$377$	1.00000	0.0515026
$378$	0	0
$379$	23.8191	1.22350	0.611752	−	0.791050i	$-0.290465\pi$
$379$	23.8191	1.22350	0.611752	+	0.791050i	$0.290465\pi$
$380$	0	0
$381$	−37.2857	−1.91021
$382$	0	0
$383$	−32.1364	−1.64209	−0.821046	−	0.570862i	$-0.806609\pi$
$383$	−32.1364	−1.64209	−0.821046	+	0.570862i	$0.806609\pi$
$384$	0	0
$385$	7.99015	0.407216
$386$	0	0
$387$	−10.9012	−0.554142
$388$	0	0
$389$	10.0919	0.511677	0.255839	−	0.966720i	$-0.417648\pi$
$389$	10.0919	0.511677	0.255839	+	0.966720i	$0.417648\pi$
$390$	0	0
$391$	−28.6116	−1.44695
$392$	0	0
$393$	−42.9000	−2.16402
$394$	0	0
$395$	20.4560	1.02925
$396$	0	0
$397$	−12.1266	−0.608616	−0.304308	−	0.952574i	$-0.598425\pi$
$397$	−12.1266	−0.608616	−0.304308	+	0.952574i	$0.598425\pi$
$398$	0	0
$399$	3.06018	0.153201
$400$	0	0
$401$	4.96698	0.248039	0.124020	−	0.992280i	$-0.460421\pi$
$401$	4.96698	0.248039	0.124020	+	0.992280i	$0.460421\pi$
$402$	0	0
$403$	−4.62351	−0.230313
$404$	0	0
$405$	19.1811	0.953118
$406$	0	0
$407$	12.9975	0.644262
$408$	0	0
$409$	1.31709	0.0651259	0.0325629	−	0.999470i	$-0.489633\pi$
$409$	1.31709	0.0651259	0.0325629	+	0.999470i	$0.489633\pi$
$410$	0	0
$411$	35.4223	1.74725
$412$	0	0
$413$	−4.95517	−0.243828
$414$	0	0
$415$	3.32532	0.163234
$416$	0	0
$417$	−20.2495	−0.991620
$418$	0	0
$419$	−9.34401	−0.456485	−0.228242	−	0.973604i	$-0.573298\pi$
$419$	−9.34401	−0.456485	−0.228242	+	0.973604i	$0.573298\pi$
$420$	0	0
$421$	19.0853	0.930161	0.465080	−	0.885268i	$-0.346025\pi$
$421$	19.0853	0.930161	0.465080	+	0.885268i	$0.346025\pi$
$422$	0	0
$423$	31.8719	1.54966
$424$	0	0
$425$	−9.43575	−0.457701
$426$	0	0
$427$	−7.88637	−0.381648
$428$	0	0
$429$	7.50394	0.362293
$430$	0	0
$431$	−23.1453	−1.11487	−0.557434	−	0.830221i	$-0.688214\pi$
$431$	−23.1453	−1.11487	−0.557434	+	0.830221i	$0.688214\pi$
$432$	0	0
$433$	−27.4401	−1.31869	−0.659343	−	0.751842i	$-0.729166\pi$
$433$	−27.4401	−1.31869	−0.659343	+	0.751842i	$0.729166\pi$
$434$	0	0
$435$	6.88688	0.330201
$436$	0	0
$437$	8.12505	0.388674
$438$	0	0
$439$	24.9817	1.19231	0.596157	−	0.802868i	$-0.296694\pi$
$439$	24.9817	1.19231	0.596157	+	0.802868i	$0.296694\pi$
$440$	0	0
$441$	−21.1244	−1.00592
$442$	0	0
$443$	41.5544	1.97431	0.987154	−	0.159774i	$-0.0510767\pi$
$443$	41.5544	1.97431	0.987154	+	0.159774i	$0.0510767\pi$
$444$	0	0
$445$	−24.2398	−1.14908
$446$	0	0
$447$	−8.25567	−0.390480
$448$	0	0
$449$	34.3081	1.61910	0.809551	−	0.587050i	$-0.199711\pi$
$449$	34.3081	1.61910	0.809551	+	0.587050i	$0.199711\pi$
$450$	0	0
$451$	−15.1184	−0.711897
$452$	0	0
$453$	16.9646	0.797069
$454$	0	0
$455$	2.72150	0.127586
$456$	0	0
$457$	−32.6139	−1.52561	−0.762807	−	0.646627i	$-0.776179\pi$
$457$	−32.6139	−1.52561	−0.762807	+	0.646627i	$0.776179\pi$
$458$	0	0
$459$	−5.68232	−0.265228
$460$	0	0
$461$	−15.6421	−0.728527	−0.364264	−	0.931296i	$-0.618679\pi$
$461$	−15.6421	−0.728527	−0.364264	+	0.931296i	$0.618679\pi$
$462$	0	0
$463$	18.9745	0.881821	0.440910	−	0.897551i	$-0.354656\pi$
$463$	18.9745	0.881821	0.440910	+	0.897551i	$0.354656\pi$
$464$	0	0
$465$	−31.8416	−1.47662
$466$	0	0
$467$	−5.19785	−0.240528	−0.120264	−	0.992742i	$-0.538374\pi$
$467$	−5.19785	−0.240528	−0.120264	+	0.992742i	$0.538374\pi$
$468$	0	0
$469$	1.62261	0.0749252
$470$	0	0
$471$	−55.0280	−2.53555
$472$	0	0
$473$	9.06004	0.416581
$474$	0	0
$475$	2.67954	0.122946
$476$	0	0
$477$	−24.2225	−1.10907
$478$	0	0
$479$	−9.50865	−0.434461	−0.217231	−	0.976120i	$-0.569702\pi$
$479$	−9.50865	−0.434461	−0.217231	+	0.976120i	$0.569702\pi$
$480$	0	0
$481$	4.42704	0.201856
$482$	0	0
$483$	17.6938	0.805095
$484$	0	0
$485$	39.2269	1.78120
$486$	0	0
$487$	−3.08270	−0.139691	−0.0698453	−	0.997558i	$-0.522251\pi$
$487$	−3.08270	−0.139691	−0.0698453	+	0.997558i	$0.522251\pi$
$488$	0	0
$489$	−20.2140	−0.914108
$490$	0	0
$491$	22.1802	1.00098	0.500489	−	0.865743i	$-0.333154\pi$
$491$	22.1802	1.00098	0.500489	+	0.865743i	$0.333154\pi$
$492$	0	0
$493$	4.17439	0.188005
$494$	0	0
$495$	27.9460	1.25608
$496$	0	0
$497$	−1.53047	−0.0686511
$498$	0	0
$499$	−1.91259	−0.0856193	−0.0428096	−	0.999083i	$-0.513631\pi$
$499$	−1.91259	−0.0856193	−0.0428096	+	0.999083i	$0.513631\pi$
$500$	0	0
$501$	36.0491	1.61055
$502$	0	0
$503$	−24.7511	−1.10359	−0.551797	−	0.833978i	$-0.686058\pi$
$503$	−24.7511	−1.10359	−0.551797	+	0.833978i	$0.686058\pi$
$504$	0	0
$505$	10.5165	0.467979
$506$	0	0
$507$	2.55589	0.113511
$508$	0	0
$509$	−4.15216	−0.184041	−0.0920207	−	0.995757i	$-0.529333\pi$
$509$	−4.15216	−0.184041	−0.0920207	+	0.995757i	$0.529333\pi$
$510$	0	0
$511$	−14.8703	−0.657822
$512$	0	0
$513$	1.61365	0.0712443
$514$	0	0
$515$	38.5019	1.69659
$516$	0	0
$517$	−26.4888	−1.16497
$518$	0	0
$519$	34.9832	1.53559
$520$	0	0
$521$	9.43920	0.413539	0.206770	−	0.978390i	$-0.433705\pi$
$521$	9.43920	0.413539	0.206770	+	0.978390i	$0.433705\pi$
$522$	0	0
$523$	5.74077	0.251027	0.125513	−	0.992092i	$-0.459942\pi$
$523$	5.74077	0.251027	0.125513	+	0.992092i	$0.459942\pi$
$524$	0	0
$525$	5.83518	0.254668
$526$	0	0
$527$	−19.3003	−0.840736
$528$	0	0
$529$	23.9785	1.04255
$530$	0	0
$531$	−17.3310	−0.752100
$532$	0	0
$533$	−5.14943	−0.223046
$534$	0	0
$535$	51.5499	2.22870
$536$	0	0
$537$	−3.90638	−0.168573
$538$	0	0
$539$	17.5565	0.756212
$540$	0	0
$541$	31.6163	1.35929	0.679646	−	0.733540i	$-0.262133\pi$
$541$	31.6163	1.35929	0.679646	+	0.733540i	$0.262133\pi$
$542$	0	0
$543$	60.3023	2.58782
$544$	0	0
$545$	−0.479307	−0.0205313
$546$	0	0
$547$	−8.31551	−0.355546	−0.177773	−	0.984072i	$-0.556889\pi$
$547$	−8.31551	−0.355546	−0.177773	+	0.984072i	$0.556889\pi$
$548$	0	0
$549$	−27.5830	−1.17721
$550$	0	0
$551$	−1.18543	−0.0505011
$552$	0	0
$553$	−7.66777	−0.326067
$554$	0	0
$555$	30.4885	1.29416
$556$	0	0
$557$	25.5381	1.08209	0.541043	−	0.840995i	$-0.318030\pi$
$557$	25.5381	1.08209	0.541043	+	0.840995i	$0.318030\pi$
$558$	0	0
$559$	3.08591	0.130520
$560$	0	0
$561$	31.3244	1.32252
$562$	0	0
$563$	1.28078	0.0539785	0.0269892	−	0.999636i	$-0.491408\pi$
$563$	1.28078	0.0539785	0.0269892	+	0.999636i	$0.491408\pi$
$564$	0	0
$565$	2.25908	0.0950402
$566$	0	0
$567$	−7.18989	−0.301947
$568$	0	0
$569$	13.1726	0.552223	0.276111	−	0.961126i	$-0.410954\pi$
$569$	13.1726	0.552223	0.276111	+	0.961126i	$0.410954\pi$
$570$	0	0
$571$	18.4477	0.772011	0.386006	−	0.922496i	$-0.373854\pi$
$571$	18.4477	0.772011	0.386006	+	0.922496i	$0.373854\pi$
$572$	0	0
$573$	33.1661	1.38553
$574$	0	0
$575$	15.4929	0.646100
$576$	0	0
$577$	6.92568	0.288320	0.144160	−	0.989554i	$-0.453952\pi$
$577$	6.92568	0.288320	0.144160	+	0.989554i	$0.453952\pi$
$578$	0	0
$579$	19.5137	0.810962
$580$	0	0
$581$	−1.24647	−0.0517123
$582$	0	0
$583$	20.1313	0.833755
$584$	0	0
$585$	9.51859	0.393545
$586$	0	0
$587$	−19.1456	−0.790223	−0.395111	−	0.918633i	$-0.629294\pi$
$587$	−19.1456	−0.790223	−0.395111	+	0.918633i	$0.629294\pi$
$588$	0	0
$589$	5.48085	0.225835
$590$	0	0
$591$	−1.23117	−0.0506435
$592$	0	0
$593$	26.0348	1.06912	0.534562	−	0.845130i	$-0.320477\pi$
$593$	26.0348	1.06912	0.534562	+	0.845130i	$0.320477\pi$
$594$	0	0
$595$	11.3606	0.465739
$596$	0	0
$597$	−9.11535	−0.373067
$598$	0	0
$599$	−6.16441	−0.251871	−0.125936	−	0.992038i	$-0.540193\pi$
$599$	−6.16441	−0.251871	−0.125936	+	0.992038i	$0.540193\pi$
$600$	0	0
$601$	43.0266	1.75509	0.877545	−	0.479494i	$-0.159180\pi$
$601$	43.0266	1.75509	0.877545	+	0.479494i	$0.159180\pi$
$602$	0	0
$603$	5.67517	0.231111
$604$	0	0
$605$	6.41369	0.260754
$606$	0	0
$607$	22.7193	0.922146	0.461073	−	0.887362i	$-0.347465\pi$
$607$	22.7193	0.922146	0.461073	+	0.887362i	$0.347465\pi$
$608$	0	0
$609$	−2.58149	−0.104607
$610$	0	0
$611$	−9.02225	−0.365001
$612$	0	0
$613$	−4.15871	−0.167969	−0.0839844	−	0.996467i	$-0.526765\pi$
$613$	−4.15871	−0.167969	−0.0839844	+	0.996467i	$0.526765\pi$
$614$	0	0
$615$	−35.4635	−1.43003
$616$	0	0
$617$	21.2139	0.854039	0.427020	−	0.904242i	$-0.359564\pi$
$617$	21.2139	0.854039	0.427020	+	0.904242i	$0.359564\pi$
$618$	0	0
$619$	−25.9915	−1.04468	−0.522342	−	0.852736i	$-0.674942\pi$
$619$	−25.9915	−1.04468	−0.522342	+	0.852736i	$0.674942\pi$
$620$	0	0
$621$	9.33001	0.374400
$622$	0	0
$623$	9.08609	0.364026
$624$	0	0
$625$	−31.1926	−1.24770
$626$	0	0
$627$	−8.89540	−0.355248
$628$	0	0
$629$	18.4802	0.736853
$630$	0	0
$631$	−2.54329	−0.101247	−0.0506234	−	0.998718i	$-0.516121\pi$
$631$	−2.54329	−0.101247	−0.0506234	+	0.998718i	$0.516121\pi$
$632$	0	0
$633$	−41.1044	−1.63375
$634$	0	0
$635$	39.3079	1.55989
$636$	0	0
$637$	5.97987	0.236931
$638$	0	0
$639$	−5.35291	−0.211758
$640$	0	0
$641$	−30.1041	−1.18904	−0.594519	−	0.804081i	$-0.702658\pi$
$641$	−30.1041	−1.18904	−0.594519	+	0.804081i	$0.702658\pi$
$642$	0	0
$643$	31.0277	1.22361	0.611806	−	0.791008i	$-0.290443\pi$
$643$	31.0277	1.22361	0.611806	+	0.791008i	$0.290443\pi$
$644$	0	0
$645$	21.2523	0.836809
$646$	0	0
$647$	15.6470	0.615148	0.307574	−	0.951524i	$-0.400483\pi$
$647$	15.6470	0.615148	0.307574	+	0.951524i	$0.400483\pi$
$648$	0	0
$649$	14.4038	0.565398
$650$	0	0
$651$	11.9356	0.467791
$652$	0	0
$653$	−14.8117	−0.579627	−0.289813	−	0.957083i	$-0.593593\pi$
$653$	−14.8117	−0.579627	−0.289813	+	0.957083i	$0.593593\pi$
$654$	0	0
$655$	45.2267	1.76715
$656$	0	0
$657$	−52.0096	−2.02909
$658$	0	0
$659$	−10.7770	−0.419813	−0.209907	−	0.977721i	$-0.567316\pi$
$659$	−10.7770	−0.419813	−0.209907	+	0.977721i	$0.567316\pi$
$660$	0	0
$661$	−39.8984	−1.55187	−0.775935	−	0.630813i	$-0.782721\pi$
$661$	−39.8984	−1.55187	−0.775935	+	0.630813i	$0.782721\pi$
$662$	0	0
$663$	10.6693	0.414361
$664$	0	0
$665$	−3.22615	−0.125105
$666$	0	0
$667$	−6.85409	−0.265391
$668$	0	0
$669$	44.9402	1.73749
$670$	0	0
$671$	22.9243	0.884982
$672$	0	0
$673$	1.96425	0.0757164	0.0378582	−	0.999283i	$-0.487946\pi$
$673$	1.96425	0.0757164	0.0378582	+	0.999283i	$0.487946\pi$
$674$	0	0
$675$	3.07692	0.118431
$676$	0	0
$677$	16.6271	0.639030	0.319515	−	0.947581i	$-0.396480\pi$
$677$	16.6271	0.639030	0.319515	+	0.947581i	$0.396480\pi$
$678$	0	0
$679$	−14.7039	−0.564284
$680$	0	0
$681$	−4.12263	−0.157979
$682$	0	0
$683$	−30.2080	−1.15588	−0.577939	−	0.816080i	$-0.696143\pi$
$683$	−30.2080	−1.15588	−0.577939	+	0.816080i	$0.696143\pi$
$684$	0	0
$685$	−37.3434	−1.42682
$686$	0	0
$687$	32.6859	1.24705
$688$	0	0
$689$	6.85687	0.261226
$690$	0	0
$691$	46.5558	1.77107	0.885534	−	0.464575i	$-0.153793\pi$
$691$	46.5558	1.77107	0.885534	+	0.464575i	$0.153793\pi$
$692$	0	0
$693$	−10.4753	−0.397925
$694$	0	0
$695$	21.3477	0.809764
$696$	0	0
$697$	−21.4957	−0.814208
$698$	0	0
$699$	48.7406	1.84354
$700$	0	0
$701$	1.28618	0.0485784	0.0242892	−	0.999705i	$-0.492268\pi$
$701$	1.28618	0.0485784	0.0242892	+	0.999705i	$0.492268\pi$
$702$	0	0
$703$	−5.24795	−0.197930
$704$	0	0
$705$	−62.1352	−2.34015
$706$	0	0
$707$	−3.94204	−0.148256
$708$	0	0
$709$	−15.6551	−0.587939	−0.293969	−	0.955815i	$-0.594976\pi$
$709$	−15.6551	−0.587939	−0.293969	+	0.955815i	$0.594976\pi$
$710$	0	0
$711$	−26.8185	−1.00577
$712$	0	0
$713$	31.6899	1.18680
$714$	0	0
$715$	−7.91091	−0.295851
$716$	0	0
$717$	−13.8856	−0.518568
$718$	0	0
$719$	−37.9724	−1.41613	−0.708065	−	0.706147i	$-0.750432\pi$
$719$	−37.9724	−1.41613	−0.708065	+	0.706147i	$0.750432\pi$
$720$	0	0
$721$	−14.4321	−0.537480
$722$	0	0
$723$	−26.3059	−0.978326
$724$	0	0
$725$	−2.26039	−0.0839488
$726$	0	0
$727$	−19.0330	−0.705896	−0.352948	−	0.935643i	$-0.614821\pi$
$727$	−19.0330	−0.705896	−0.352948	+	0.935643i	$0.614821\pi$
$728$	0	0
$729$	−35.5845	−1.31795
$730$	0	0
$731$	12.8818	0.476451
$732$	0	0
$733$	28.3270	1.04628	0.523140	−	0.852247i	$-0.324760\pi$
$733$	28.3270	1.04628	0.523140	+	0.852247i	$0.324760\pi$
$734$	0	0
$735$	41.1826	1.51904
$736$	0	0
$737$	−4.71664	−0.173740
$738$	0	0
$739$	−29.2890	−1.07741	−0.538706	−	0.842494i	$-0.681087\pi$
$739$	−29.2890	−1.07741	−0.538706	+	0.842494i	$0.681087\pi$
$740$	0	0
$741$	−3.02983	−0.111304
$742$	0	0
$743$	−18.9690	−0.695906	−0.347953	−	0.937512i	$-0.613123\pi$
$743$	−18.9690	−0.695906	−0.347953	+	0.937512i	$0.613123\pi$
$744$	0	0
$745$	8.70342	0.318869
$746$	0	0
$747$	−4.35960	−0.159509
$748$	0	0
$749$	−19.3231	−0.706050
$750$	0	0
$751$	54.1556	1.97617	0.988084	−	0.153917i	$-0.0491890\pi$
$751$	54.1556	1.97617	0.988084	+	0.153917i	$0.0491890\pi$
$752$	0	0
$753$	66.0163	2.40577
$754$	0	0
$755$	−17.8847	−0.650892
$756$	0	0
$757$	−48.2570	−1.75393	−0.876965	−	0.480555i	$-0.840435\pi$
$757$	−48.2570	−1.75393	−0.876965	+	0.480555i	$0.840435\pi$
$758$	0	0
$759$	−51.4327	−1.86689
$760$	0	0
$761$	−8.03503	−0.291270	−0.145635	−	0.989338i	$-0.546522\pi$
$761$	−8.03503	−0.291270	−0.145635	+	0.989338i	$0.546522\pi$
$762$	0	0
$763$	0.179664	0.00650429
$764$	0	0
$765$	39.7343	1.43660
$766$	0	0
$767$	4.90603	0.177146
$768$	0	0
$769$	−13.7927	−0.497377	−0.248688	−	0.968584i	$-0.579999\pi$
$769$	−13.7927	−0.497377	−0.248688	+	0.968584i	$0.579999\pi$
$770$	0	0
$771$	−34.1686	−1.23055
$772$	0	0
$773$	−27.4204	−0.986242	−0.493121	−	0.869961i	$-0.664144\pi$
$773$	−27.4204	−0.986242	−0.493121	+	0.869961i	$0.664144\pi$
$774$	0	0
$775$	10.4509	0.375409
$776$	0	0
$777$	−11.4284	−0.409990
$778$	0	0
$779$	6.10429	0.218709
$780$	0	0
$781$	4.44881	0.159191
$782$	0	0
$783$	−1.36123	−0.0486465
$784$	0	0
$785$	58.0124	2.07055
$786$	0	0
$787$	4.51939	0.161099	0.0805495	−	0.996751i	$-0.474332\pi$
$787$	4.51939	0.161099	0.0805495	+	0.996751i	$0.474332\pi$
$788$	0	0
$789$	18.9461	0.674500
$790$	0	0
$791$	−0.846799	−0.0301087
$792$	0	0
$793$	7.80816	0.277276
$794$	0	0
$795$	47.2225	1.67481
$796$	0	0
$797$	−1.66259	−0.0588919	−0.0294460	−	0.999566i	$-0.509374\pi$
$797$	−1.66259	−0.0588919	−0.0294460	+	0.999566i	$0.509374\pi$
$798$	0	0
$799$	−37.6624	−1.33240
$800$	0	0
$801$	31.7791	1.12286
$802$	0	0
$803$	43.2252	1.52538
$804$	0	0
$805$	−18.6534	−0.657446
$806$	0	0
$807$	5.87880	0.206944
$808$	0	0
$809$	−0.0953465	−0.00335220	−0.00167610	−	0.999999i	$-0.500534\pi$
$809$	−0.0953465	−0.00335220	−0.00167610	+	0.999999i	$0.500534\pi$
$810$	0	0
$811$	4.44527	0.156094	0.0780472	−	0.996950i	$-0.475132\pi$
$811$	4.44527	0.156094	0.0780472	+	0.996950i	$0.475132\pi$
$812$	0	0
$813$	0.825533	0.0289527
$814$	0	0
$815$	21.3103	0.746467
$816$	0	0
$817$	−3.65814	−0.127982
$818$	0	0
$819$	−3.56797	−0.124675
$820$	0	0
$821$	−43.2761	−1.51035	−0.755173	−	0.655526i	$-0.772447\pi$
$821$	−43.2761	−1.51035	−0.755173	+	0.655526i	$0.772447\pi$
$822$	0	0
$823$	37.5274	1.30812	0.654062	−	0.756441i	$-0.273064\pi$
$823$	37.5274	1.30812	0.654062	+	0.756441i	$0.273064\pi$
$824$	0	0
$825$	−16.9618	−0.590535
$826$	0	0
$827$	−46.5089	−1.61727	−0.808637	−	0.588309i	$-0.799794\pi$
$827$	−46.5089	−1.61727	−0.808637	+	0.588309i	$0.799794\pi$
$828$	0	0
$829$	28.8891	1.00336	0.501680	−	0.865053i	$-0.332715\pi$
$829$	28.8891	1.00336	0.501680	+	0.865053i	$0.332715\pi$
$830$	0	0
$831$	24.4339	0.847603
$832$	0	0
$833$	24.9623	0.864892
$834$	0	0
$835$	−38.0042	−1.31519
$836$	0	0
$837$	6.29367	0.217541
$838$	0	0
$839$	−13.3590	−0.461203	−0.230602	−	0.973048i	$-0.574069\pi$
$839$	−13.3590	−0.461203	−0.230602	+	0.973048i	$0.574069\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	8.07028	0.277955
$844$	0	0
$845$	−2.69451	−0.0926940
$846$	0	0
$847$	−2.40412	−0.0826066
$848$	0	0
$849$	49.6161	1.70282
$850$	0	0
$851$	−30.3433	−1.04016
$852$	0	0
$853$	−8.54676	−0.292635	−0.146318	−	0.989238i	$-0.546742\pi$
$853$	−8.54676	−0.292635	−0.146318	+	0.989238i	$0.546742\pi$
$854$	0	0
$855$	−11.2836	−0.385892
$856$	0	0
$857$	−20.4484	−0.698503	−0.349251	−	0.937029i	$-0.613564\pi$
$857$	−20.4484	−0.698503	−0.349251	+	0.937029i	$0.613564\pi$
$858$	0	0
$859$	−15.9895	−0.545556	−0.272778	−	0.962077i	$-0.587942\pi$
$859$	−15.9895	−0.545556	−0.272778	+	0.962077i	$0.587942\pi$
$860$	0	0
$861$	13.2932	0.453031
$862$	0	0
$863$	21.5850	0.734762	0.367381	−	0.930070i	$-0.380254\pi$
$863$	21.5850	0.734762	0.367381	+	0.930070i	$0.380254\pi$
$864$	0	0
$865$	−36.8805	−1.25397
$866$	0	0
$867$	1.08761	0.0369373
$868$	0	0
$869$	22.2888	0.756097
$870$	0	0
$871$	−1.60652	−0.0544348
$872$	0	0
$873$	−51.4277	−1.74056
$874$	0	0
$875$	7.45585	0.252054
$876$	0	0
$877$	−17.2156	−0.581330	−0.290665	−	0.956825i	$-0.593876\pi$
$877$	−17.2156	−0.581330	−0.290665	+	0.956825i	$0.593876\pi$
$878$	0	0
$879$	−21.0919	−0.711412
$880$	0	0
$881$	−14.1481	−0.476662	−0.238331	−	0.971184i	$-0.576600\pi$
$881$	−14.1481	−0.476662	−0.238331	+	0.971184i	$0.576600\pi$
$882$	0	0
$883$	12.8569	0.432669	0.216334	−	0.976319i	$-0.430590\pi$
$883$	12.8569	0.432669	0.216334	+	0.976319i	$0.430590\pi$
$884$	0	0
$885$	33.7872	1.13575
$886$	0	0
$887$	13.9845	0.469554	0.234777	−	0.972049i	$-0.424564\pi$
$887$	13.9845	0.469554	0.234777	+	0.972049i	$0.424564\pi$
$888$	0	0
$889$	−14.7343	−0.494172
$890$	0	0
$891$	20.8997	0.700168
$892$	0	0
$893$	10.6953	0.357903
$894$	0	0
$895$	4.11824	0.137658
$896$	0	0
$897$	−17.5183	−0.584920
$898$	0	0
$899$	−4.62351	−0.154203
$900$	0	0
$901$	28.6233	0.953579
$902$	0	0
$903$	−7.96626	−0.265100
$904$	0	0
$905$	−63.5728	−2.11323
$906$	0	0
$907$	20.3838	0.676833	0.338416	−	0.940997i	$-0.390109\pi$
$907$	20.3838	0.676833	0.338416	+	0.940997i	$0.390109\pi$
$908$	0	0
$909$	−13.7875	−0.457302
$910$	0	0
$911$	−26.2823	−0.870770	−0.435385	−	0.900244i	$-0.643388\pi$
$911$	−26.2823	−0.870770	−0.435385	+	0.900244i	$0.643388\pi$
$912$	0	0
$913$	3.62327	0.119913
$914$	0	0
$915$	53.7739	1.77771
$916$	0	0
$917$	−16.9529	−0.559833
$918$	0	0
$919$	−16.6324	−0.548654	−0.274327	−	0.961637i	$-0.588455\pi$
$919$	−16.6324	−0.548654	−0.274327	+	0.961637i	$0.588455\pi$
$920$	0	0
$921$	−65.1498	−2.14676
$922$	0	0
$923$	1.51530	0.0498766
$924$	0	0
$925$	−10.0068	−0.329023
$926$	0	0
$927$	−50.4771	−1.65789
$928$	0	0
$929$	20.7912	0.682136	0.341068	−	0.940039i	$-0.389211\pi$
$929$	20.7912	0.682136	0.341068	+	0.940039i	$0.389211\pi$
$930$	0	0
$931$	−7.08872	−0.232323
$932$	0	0
$933$	78.2695	2.56243
$934$	0	0
$935$	−33.0232	−1.07998
$936$	0	0
$937$	−49.6745	−1.62280	−0.811398	−	0.584494i	$-0.801293\pi$
$937$	−49.6745	−1.62280	−0.811398	+	0.584494i	$0.801293\pi$
$938$	0	0
$939$	−64.5267	−2.10575
$940$	0	0
$941$	4.23406	0.138026	0.0690132	−	0.997616i	$-0.478015\pi$
$941$	4.23406	0.138026	0.0690132	+	0.997616i	$0.478015\pi$
$942$	0	0
$943$	35.2946	1.14935
$944$	0	0
$945$	−3.70460	−0.120510
$946$	0	0
$947$	−14.8760	−0.483405	−0.241703	−	0.970350i	$-0.577706\pi$
$947$	−14.8760	−0.483405	−0.241703	+	0.970350i	$0.577706\pi$
$948$	0	0
$949$	14.7228	0.477922
$950$	0	0
$951$	35.5461	1.15266
$952$	0	0
$953$	37.4722	1.21384	0.606922	−	0.794761i	$-0.292404\pi$
$953$	37.4722	1.21384	0.606922	+	0.794761i	$0.292404\pi$
$954$	0	0
$955$	−34.9649	−1.13144
$956$	0	0
$957$	7.50394	0.242568
$958$	0	0
$959$	13.9979	0.452015
$960$	0	0
$961$	−9.62316	−0.310424
$962$	0	0
$963$	−67.5835	−2.17785
$964$	0	0
$965$	−20.5720	−0.662237
$966$	0	0
$967$	−42.7873	−1.37595	−0.687973	−	0.725736i	$-0.741499\pi$
$967$	−42.7873	−1.37595	−0.687973	+	0.725736i	$0.741499\pi$
$968$	0	0
$969$	−12.6477	−0.406303
$970$	0	0
$971$	−26.2803	−0.843376	−0.421688	−	0.906741i	$-0.638562\pi$
$971$	−26.2803	−0.843376	−0.421688	+	0.906741i	$0.638562\pi$
$972$	0	0
$973$	−8.00201	−0.256533
$974$	0	0
$975$	−5.77732	−0.185022
$976$	0	0
$977$	43.1943	1.38191	0.690954	−	0.722899i	$-0.257191\pi$
$977$	43.1943	1.38191	0.690954	+	0.722899i	$0.257191\pi$
$978$	0	0
$979$	−26.4116	−0.844119
$980$	0	0
$981$	0.628386	0.0200628
$982$	0	0
$983$	−9.43785	−0.301021	−0.150510	−	0.988608i	$-0.548092\pi$
$983$	−9.43785	−0.301021	−0.150510	+	0.988608i	$0.548092\pi$
$984$	0	0
$985$	1.29794	0.0413558
$986$	0	0
$987$	23.2909	0.741357
$988$	0	0
$989$	−21.1511	−0.672566
$990$	0	0
$991$	−42.0236	−1.33492	−0.667462	−	0.744644i	$-0.732619\pi$
$991$	−42.0236	−1.33492	−0.667462	+	0.744644i	$0.732619\pi$
$992$	0	0
$993$	−22.7176	−0.720921
$994$	0	0
$995$	9.60973	0.304649
$996$	0	0
$997$	40.3710	1.27856	0.639281	−	0.768973i	$-0.279232\pi$
$997$	40.3710	1.27856	0.639281	+	0.768973i	$0.279232\pi$
$998$	0	0
$999$	−6.02623	−0.190661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.z.1.10		10
4.3	odd	2		3016.2.a.h.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.h.1.1	✓	10	4.3	odd	2
6032.2.a.z.1.10		10	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)

\(f(q)\)	\(=\)	\(q+2.55589 q^{3} -2.69451 q^{5} +1.01002 q^{7} +3.53259 q^{9} +O(q^{10})\)	\(q+2.55589 q^{3} -2.69451 q^{5} +1.01002 q^{7} +3.53259 q^{9} -2.93594 q^{11} -1.00000 q^{13} -6.88688 q^{15} -4.17439 q^{17} +1.18543 q^{19} +2.58149 q^{21} +6.85409 q^{23} +2.26039 q^{25} +1.36123 q^{27} -1.00000 q^{29} +4.62351 q^{31} -7.50394 q^{33} -2.72150 q^{35} -4.42704 q^{37} -2.55589 q^{39} +5.14943 q^{41} -3.08591 q^{43} -9.51859 q^{45} +9.02225 q^{47} -5.97987 q^{49} -10.6693 q^{51} -6.85687 q^{53} +7.91091 q^{55} +3.02983 q^{57} -4.90603 q^{59} -7.80816 q^{61} +3.56797 q^{63} +2.69451 q^{65} +1.60652 q^{67} +17.5183 q^{69} -1.51530 q^{71} -14.7228 q^{73} +5.77732 q^{75} -2.96534 q^{77} -7.59173 q^{79} -7.11859 q^{81} -1.23411 q^{83} +11.2479 q^{85} -2.55589 q^{87} +8.99598 q^{89} -1.01002 q^{91} +11.8172 q^{93} -3.19416 q^{95} -14.5581 q^{97} -10.3714 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9}+O(q^{10}) \) 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9	\( 10 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 13 q^{9} - 14 q^{11} - 10 q^{13} - 7 q^{15} + 5 q^{17} - 11 q^{19} - 7 q^{23} + 10 q^{25} - 21 q^{27} - 10 q^{29} - 5 q^{31} + 5 q^{33} - 11 q^{35} + 8 q^{37} + 3 q^{39} + 14 q^{41} - 35 q^{43} + 7 q^{45} - 7 q^{49} - 20 q^{51} - 11 q^{53} - 8 q^{55} + 4 q^{57} - 23 q^{59} - 8 q^{61} - 43 q^{63} - 4 q^{65} - 27 q^{67} + 10 q^{69} - 3 q^{71} + 7 q^{73} - 23 q^{75} + 2 q^{77} - 9 q^{79} - 6 q^{81} - 48 q^{83} - 6 q^{85} + 3 q^{87} + 20 q^{89} - 3 q^{91} - 11 q^{93} - 11 q^{95} + q^{97} - 54 q^{99}+O(q^{100}) \) 10 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 13 * q^9 - 14 * q^11 - 10 * q^13 - 7 * q^15 + 5 * q^17 - 11 * q^19 - 7 * q^23 + 10 * q^25 - 21 * q^27 - 10 * q^29 - 5 * q^31 + 5 * q^33 - 11 * q^35 + 8 * q^37 + 3 * q^39 + 14 * q^41 - 35 * q^43 + 7 * q^45 - 7 * q^49 - 20 * q^51 - 11 * q^53 - 8 * q^55 + 4 * q^57 - 23 * q^59 - 8 * q^61 - 43 * q^63 - 4 * q^65 - 27 * q^67 + 10 * q^69 - 3 * q^71 + 7 * q^73 - 23 * q^75 + 2 * q^77 - 9 * q^79 - 6 * q^81 - 48 * q^83 - 6 * q^85 + 3 * q^87 + 20 * q^89 - 3 * q^91 - 11 * q^93 - 11 * q^95 + q^97 - 54 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more