LMFDB - Embedded newform 3696.2.a.bo.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3696,2,Mod(1,3696)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3696, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3696.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3696.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:	$29.5127085871$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	3696.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.58774 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.58774 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -5.87189 q^{13} +2.58774 q^{15} +7.51396 q^{17} +2.35793 q^{19} -1.00000 q^{21} -6.94567 q^{23} +1.69641 q^{25} -1.00000 q^{27} -5.87189 q^{29} +3.66152 q^{31} -1.00000 q^{33} -2.58774 q^{35} +3.30359 q^{37} +5.87189 q^{39} +5.28415 q^{41} -7.40530 q^{43} -2.58774 q^{45} -7.53341 q^{47} +1.00000 q^{49} -7.51396 q^{51} -4.22982 q^{53} -2.58774 q^{55} -2.35793 q^{57} +0.926221 q^{59} -2.00000 q^{61} +1.00000 q^{63} +15.1949 q^{65} +10.1017 q^{67} +6.94567 q^{69} -4.45963 q^{71} -2.12811 q^{73} -1.69641 q^{75} +1.00000 q^{77} +4.45963 q^{79} +1.00000 q^{81} +10.6894 q^{83} -19.4442 q^{85} +5.87189 q^{87} +15.8913 q^{89} -5.87189 q^{91} -3.66152 q^{93} -6.10170 q^{95} +10.1212 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} + 8 q^{17} + 8 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{35} + 4 q^{39} + 14 q^{41} + 14 q^{43} + 4 q^{45} + 3 q^{49} - 8 q^{51} + 4 q^{55} - 8 q^{57} - 6 q^{61} + 3 q^{63} + 14 q^{65} + 4 q^{67} + 10 q^{69} + 12 q^{71} - 20 q^{73} - 15 q^{75} + 3 q^{77} - 12 q^{79} + 3 q^{81} - 6 q^{83} - 6 q^{85} + 4 q^{87} + 26 q^{89} - 4 q^{91} - 2 q^{93} + 8 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 4 * q^15 + 8 * q^17 + 8 * q^19 - 3 * q^21 - 10 * q^23 + 15 * q^25 - 3 * q^27 - 4 * q^29 + 2 * q^31 - 3 * q^33 + 4 * q^35 + 4 * q^39 + 14 * q^41 + 14 * q^43 + 4 * q^45 + 3 * q^49 - 8 * q^51 + 4 * q^55 - 8 * q^57 - 6 * q^61 + 3 * q^63 + 14 * q^65 + 4 * q^67 + 10 * q^69 + 12 * q^71 - 20 * q^73 - 15 * q^75 + 3 * q^77 - 12 * q^79 + 3 * q^81 - 6 * q^83 - 6 * q^85 + 4 * q^87 + 26 * q^89 - 4 * q^91 - 2 * q^93 + 8 * q^95 - 4 * q^97 + 3 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.58774	−1.15727	−0.578637	−	0.815586i	$-0.696415\pi$
$5$	−2.58774	−1.15727	−0.578637	+	0.815586i	$0.696415\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.87189	−1.62857	−0.814284	−	0.580466i	$-0.802870\pi$
$13$	−5.87189	−1.62857	−0.814284	+	0.580466i	$0.802870\pi$
$14$	0	0
$15$	2.58774	0.668152
$16$	0	0
$17$	7.51396	1.82240	0.911202	−	0.411960i	$-0.135156\pi$
$17$	7.51396	1.82240	0.911202	+	0.411960i	$0.135156\pi$
$18$	0	0
$19$	2.35793	0.540945	0.270473	−	0.962728i	$-0.412820\pi$
$19$	2.35793	0.540945	0.270473	+	0.962728i	$0.412820\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−6.94567	−1.44827	−0.724136	−	0.689657i	$-0.757761\pi$
$23$	−6.94567	−1.44827	−0.724136	+	0.689657i	$0.757761\pi$
$24$	0	0
$25$	1.69641	0.339281
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−5.87189	−1.09038	−0.545191	−	0.838312i	$-0.683543\pi$
$29$	−5.87189	−1.09038	−0.545191	+	0.838312i	$0.683543\pi$
$30$	0	0
$31$	3.66152	0.657629	0.328814	−	0.944395i	$-0.393351\pi$
$31$	3.66152	0.657629	0.328814	+	0.944395i	$0.393351\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−2.58774	−0.437408
$36$	0	0
$37$	3.30359	0.543108	0.271554	−	0.962423i	$-0.412463\pi$
$37$	3.30359	0.543108	0.271554	+	0.962423i	$0.412463\pi$
$38$	0	0
$39$	5.87189	0.940255
$40$	0	0
$41$	5.28415	0.825245	0.412623	−	0.910902i	$-0.364613\pi$
$41$	5.28415	0.825245	0.412623	+	0.910902i	$0.364613\pi$
$42$	0	0
$43$	−7.40530	−1.12930	−0.564649	−	0.825331i	$-0.690988\pi$
$43$	−7.40530	−1.12930	−0.564649	+	0.825331i	$0.690988\pi$
$44$	0	0
$45$	−2.58774	−0.385758
$46$	0	0
$47$	−7.53341	−1.09886	−0.549430	−	0.835540i	$-0.685155\pi$
$47$	−7.53341	−1.09886	−0.549430	+	0.835540i	$0.685155\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.51396	−1.05217
$52$	0	0
$53$	−4.22982	−0.581010	−0.290505	−	0.956874i	$-0.593823\pi$
$53$	−4.22982	−0.581010	−0.290505	+	0.956874i	$0.593823\pi$
$54$	0	0
$55$	−2.58774	−0.348931
$56$	0	0
$57$	−2.35793	−0.312315
$58$	0	0
$59$	0.926221	0.120584	0.0602918	−	0.998181i	$-0.480797\pi$
$59$	0.926221	0.120584	0.0602918	+	0.998181i	$0.480797\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	15.1949	1.88470
$66$	0	0
$67$	10.1017	1.23412	0.617060	−	0.786916i	$-0.288324\pi$
$67$	10.1017	1.23412	0.617060	+	0.786916i	$0.288324\pi$
$68$	0	0
$69$	6.94567	0.836160
$70$	0	0
$71$	−4.45963	−0.529261	−0.264630	−	0.964350i	$-0.585250\pi$
$71$	−4.45963	−0.529261	−0.264630	+	0.964350i	$0.585250\pi$
$72$	0	0
$73$	−2.12811	−0.249077	−0.124538	−	0.992215i	$-0.539745\pi$
$73$	−2.12811	−0.249077	−0.124538	+	0.992215i	$0.539745\pi$
$74$	0	0
$75$	−1.69641	−0.195884
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	4.45963	0.501748	0.250874	−	0.968020i	$-0.419282\pi$
$79$	4.45963	0.501748	0.250874	+	0.968020i	$0.419282\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.6894	1.17332	0.586660	−	0.809834i	$-0.300443\pi$
$83$	10.6894	1.17332	0.586660	+	0.809834i	$0.300443\pi$
$84$	0	0
$85$	−19.4442	−2.10902
$86$	0	0
$87$	5.87189	0.629533
$88$	0	0
$89$	15.8913	1.68448	0.842239	−	0.539104i	$-0.181237\pi$
$89$	15.8913	1.68448	0.842239	+	0.539104i	$0.181237\pi$
$90$	0	0
$91$	−5.87189	−0.615541
$92$	0	0
$93$	−3.66152	−0.379682
$94$	0	0
$95$	−6.10170	−0.626022
$96$	0	0
$97$	10.1212	1.02765	0.513824	−	0.857896i	$-0.328229\pi$
$97$	10.1212	1.02765	0.513824	+	0.857896i	$0.328229\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−13.4053	−1.33388	−0.666939	−	0.745113i	$-0.732396\pi$
$101$	−13.4053	−1.33388	−0.666939	+	0.745113i	$0.732396\pi$
$102$	0	0
$103$	4.71585	0.464667	0.232333	−	0.972636i	$-0.425364\pi$
$103$	4.71585	0.464667	0.232333	+	0.972636i	$0.425364\pi$
$104$	0	0
$105$	2.58774	0.252538
$106$	0	0
$107$	−7.07378	−0.683848	−0.341924	−	0.939728i	$-0.611079\pi$
$107$	−7.07378	−0.683848	−0.341924	+	0.939728i	$0.611079\pi$
$108$	0	0
$109$	18.8370	1.80426	0.902129	−	0.431467i	$-0.142004\pi$
$109$	18.8370	1.80426	0.902129	+	0.431467i	$0.142004\pi$
$110$	0	0
$111$	−3.30359	−0.313563
$112$	0	0
$113$	−9.02792	−0.849276	−0.424638	−	0.905363i	$-0.639599\pi$
$113$	−9.02792	−0.849276	−0.424638	+	0.905363i	$0.639599\pi$
$114$	0	0
$115$	17.9736	1.67605
$116$	0	0
$117$	−5.87189	−0.542856
$118$	0	0
$119$	7.51396	0.688804
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−5.28415	−0.476456
$124$	0	0
$125$	8.54885	0.764632
$126$	0	0
$127$	10.2298	0.907749	0.453875	−	0.891066i	$-0.350041\pi$
$127$	10.2298	0.907749	0.453875	+	0.891066i	$0.350041\pi$
$128$	0	0
$129$	7.40530	0.652000
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	2.35793	0.204458
$134$	0	0
$135$	2.58774	0.222717
$136$	0	0
$137$	−10.8370	−0.925868	−0.462934	−	0.886393i	$-0.653203\pi$
$137$	−10.8370	−0.925868	−0.462934	+	0.886393i	$0.653203\pi$
$138$	0	0
$139$	−0.459630	−0.0389853	−0.0194927	−	0.999810i	$-0.506205\pi$
$139$	−0.459630	−0.0389853	−0.0194927	+	0.999810i	$0.506205\pi$
$140$	0	0
$141$	7.53341	0.634428
$142$	0	0
$143$	−5.87189	−0.491032
$144$	0	0
$145$	15.1949	1.26187
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	11.0474	0.905036	0.452518	−	0.891755i	$-0.350526\pi$
$149$	11.0474	0.905036	0.452518	+	0.891755i	$0.350526\pi$
$150$	0	0
$151$	16.1212	1.31192	0.655960	−	0.754795i	$-0.272264\pi$
$151$	16.1212	1.31192	0.655960	+	0.754795i	$0.272264\pi$
$152$	0	0
$153$	7.51396	0.607468
$154$	0	0
$155$	−9.47507	−0.761056
$156$	0	0
$157$	13.0668	1.04285	0.521423	−	0.853298i	$-0.325401\pi$
$157$	13.0668	1.04285	0.521423	+	0.853298i	$0.325401\pi$
$158$	0	0
$159$	4.22982	0.335446
$160$	0	0
$161$	−6.94567	−0.547395
$162$	0	0
$163$	10.1017	0.791227	0.395613	−	0.918417i	$-0.370532\pi$
$163$	10.1017	0.791227	0.395613	+	0.918417i	$0.370532\pi$
$164$	0	0
$165$	2.58774	0.201455
$166$	0	0
$167$	10.4860	0.811434	0.405717	−	0.913999i	$-0.367022\pi$
$167$	10.4860	0.811434	0.405717	+	0.913999i	$0.367022\pi$
$168$	0	0
$169$	21.4791	1.65224
$170$	0	0
$171$	2.35793	0.180315
$172$	0	0
$173$	14.4985	1.10230	0.551151	−	0.834405i	$-0.314189\pi$
$173$	14.4985	1.10230	0.551151	+	0.834405i	$0.314189\pi$
$174$	0	0
$175$	1.69641	0.128236
$176$	0	0
$177$	−0.926221	−0.0696190
$178$	0	0
$179$	−14.3510	−1.07264	−0.536321	−	0.844014i	$-0.680186\pi$
$179$	−14.3510	−1.07264	−0.536321	+	0.844014i	$0.680186\pi$
$180$	0	0
$181$	9.66152	0.718135	0.359068	−	0.933312i	$-0.383095\pi$
$181$	9.66152	0.718135	0.359068	+	0.933312i	$0.383095\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−8.54885	−0.628524
$186$	0	0
$187$	7.51396	0.549475
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	9.55286	0.691220	0.345610	−	0.938378i	$-0.387672\pi$
$191$	9.55286	0.691220	0.345610	+	0.938378i	$0.387672\pi$
$192$	0	0
$193$	−22.1212	−1.59232	−0.796158	−	0.605089i	$-0.793137\pi$
$193$	−22.1212	−1.59232	−0.796158	+	0.605089i	$0.793137\pi$
$194$	0	0
$195$	−15.1949	−1.08813
$196$	0	0
$197$	−26.2423	−1.86969	−0.934843	−	0.355061i	$-0.884460\pi$
$197$	−26.2423	−1.86969	−0.934843	+	0.355061i	$0.884460\pi$
$198$	0	0
$199$	−1.05433	−0.0747396	−0.0373698	−	0.999302i	$-0.511898\pi$
$199$	−1.05433	−0.0747396	−0.0373698	+	0.999302i	$0.511898\pi$
$200$	0	0
$201$	−10.1017	−0.712519
$202$	0	0
$203$	−5.87189	−0.412126
$204$	0	0
$205$	−13.6740	−0.955034
$206$	0	0
$207$	−6.94567	−0.482757
$208$	0	0
$209$	2.35793	0.163101
$210$	0	0
$211$	8.71585	0.600024	0.300012	−	0.953935i	$-0.403009\pi$
$211$	8.71585	0.600024	0.300012	+	0.953935i	$0.403009\pi$
$212$	0	0
$213$	4.45963	0.305569
$214$	0	0
$215$	19.1630	1.30691
$216$	0	0
$217$	3.66152	0.248560
$218$	0	0
$219$	2.12811	0.143804
$220$	0	0
$221$	−44.1212	−2.96791
$222$	0	0
$223$	23.7438	1.59000	0.795000	−	0.606609i	$-0.207471\pi$
$223$	23.7438	1.59000	0.795000	+	0.606609i	$0.207471\pi$
$224$	0	0
$225$	1.69641	0.113094
$226$	0	0
$227$	13.1755	0.874488	0.437244	−	0.899343i	$-0.355955\pi$
$227$	13.1755	0.874488	0.437244	+	0.899343i	$0.355955\pi$
$228$	0	0
$229$	13.3664	0.883277	0.441638	−	0.897193i	$-0.354397\pi$
$229$	13.3664	0.883277	0.441638	+	0.897193i	$0.354397\pi$
$230$	0	0
$231$	−1.00000	−0.0657952
$232$	0	0
$233$	7.13659	0.467533	0.233767	−	0.972293i	$-0.424895\pi$
$233$	7.13659	0.467533	0.233767	+	0.972293i	$0.424895\pi$
$234$	0	0
$235$	19.4945	1.27168
$236$	0	0
$237$	−4.45963	−0.289684
$238$	0	0
$239$	19.9930	1.29324	0.646621	−	0.762811i	$-0.276182\pi$
$239$	19.9930	1.29324	0.646621	+	0.762811i	$0.276182\pi$
$240$	0	0
$241$	−9.19493	−0.592297	−0.296149	−	0.955142i	$-0.595702\pi$
$241$	−9.19493	−0.592297	−0.296149	+	0.955142i	$0.595702\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.58774	−0.165325
$246$	0	0
$247$	−13.8455	−0.880967
$248$	0	0
$249$	−10.6894	−0.677416
$250$	0	0
$251$	9.42474	0.594885	0.297442	−	0.954740i	$-0.403866\pi$
$251$	9.42474	0.594885	0.297442	+	0.954740i	$0.403866\pi$
$252$	0	0
$253$	−6.94567	−0.436670
$254$	0	0
$255$	19.4442	1.21764
$256$	0	0
$257$	−0.0194469	−0.00121307	−0.000606533	−	1.00000i	$-0.500193\pi$
$257$	−0.0194469	−0.00121307	−0.000606533		1.00000i	$0.500193\pi$
$258$	0	0
$259$	3.30359	0.205275
$260$	0	0
$261$	−5.87189	−0.363461
$262$	0	0
$263$	−27.9930	−1.72612	−0.863062	−	0.505097i	$-0.831457\pi$
$263$	−27.9930	−1.72612	−0.863062	+	0.505097i	$0.831457\pi$
$264$	0	0
$265$	10.9457	0.672387
$266$	0	0
$267$	−15.8913	−0.972534
$268$	0	0
$269$	30.9193	1.88518	0.942590	−	0.333951i	$-0.108382\pi$
$269$	30.9193	1.88518	0.942590	+	0.333951i	$0.108382\pi$
$270$	0	0
$271$	22.3579	1.35815	0.679074	−	0.734070i	$-0.262382\pi$
$271$	22.3579	1.35815	0.679074	+	0.734070i	$0.262382\pi$
$272$	0	0
$273$	5.87189	0.355383
$274$	0	0
$275$	1.69641	0.102297
$276$	0	0
$277$	24.7717	1.48839	0.744194	−	0.667964i	$-0.232834\pi$
$277$	24.7717	1.48839	0.744194	+	0.667964i	$0.232834\pi$
$278$	0	0
$279$	3.66152	0.219210
$280$	0	0
$281$	−1.19493	−0.0712835	−0.0356418	−	0.999365i	$-0.511348\pi$
$281$	−1.19493	−0.0712835	−0.0356418	+	0.999365i	$0.511348\pi$
$282$	0	0
$283$	15.5723	0.925677	0.462839	−	0.886443i	$-0.346831\pi$
$283$	15.5723	0.925677	0.462839	+	0.886443i	$0.346831\pi$
$284$	0	0
$285$	6.10170	0.361434
$286$	0	0
$287$	5.28415	0.311913
$288$	0	0
$289$	39.4596	2.32115
$290$	0	0
$291$	−10.1212	−0.593312
$292$	0	0
$293$	−9.74378	−0.569238	−0.284619	−	0.958641i	$-0.591867\pi$
$293$	−9.74378	−0.569238	−0.284619	+	0.958641i	$0.591867\pi$
$294$	0	0
$295$	−2.39682	−0.139548
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	40.7842	2.35861
$300$	0	0
$301$	−7.40530	−0.426834
$302$	0	0
$303$	13.4053	0.770114
$304$	0	0
$305$	5.17548	0.296347
$306$	0	0
$307$	−6.35097	−0.362469	−0.181234	−	0.983440i	$-0.558009\pi$
$307$	−6.35097	−0.362469	−0.181234	+	0.983440i	$0.558009\pi$
$308$	0	0
$309$	−4.71585	−0.268275
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	16.9457	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$313$	16.9457	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$314$	0	0
$315$	−2.58774	−0.145803
$316$	0	0
$317$	20.0125	1.12401	0.562007	−	0.827133i	$-0.310030\pi$
$317$	20.0125	1.12401	0.562007	+	0.827133i	$0.310030\pi$
$318$	0	0
$319$	−5.87189	−0.328763
$320$	0	0
$321$	7.07378	0.394820
$322$	0	0
$323$	17.7174	0.985821
$324$	0	0
$325$	−9.96111	−0.552543
$326$	0	0
$327$	−18.8370	−1.04169
$328$	0	0
$329$	−7.53341	−0.415330
$330$	0	0
$331$	−7.54037	−0.414456	−0.207228	−	0.978293i	$-0.566444\pi$
$331$	−7.54037	−0.414456	−0.207228	+	0.978293i	$0.566444\pi$
$332$	0	0
$333$	3.30359	0.181036
$334$	0	0
$335$	−26.1406	−1.42821
$336$	0	0
$337$	−12.6894	−0.691238	−0.345619	−	0.938375i	$-0.612331\pi$
$337$	−12.6894	−0.691238	−0.345619	+	0.938375i	$0.612331\pi$
$338$	0	0
$339$	9.02792	0.490330
$340$	0	0
$341$	3.66152	0.198282
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−17.9736	−0.967666
$346$	0	0
$347$	−1.13659	−0.0610153	−0.0305076	−	0.999535i	$-0.509712\pi$
$347$	−1.13659	−0.0610153	−0.0305076	+	0.999535i	$0.509712\pi$
$348$	0	0
$349$	−14.7911	−0.791752	−0.395876	−	0.918304i	$-0.629559\pi$
$349$	−14.7911	−0.791752	−0.395876	+	0.918304i	$0.629559\pi$
$350$	0	0
$351$	5.87189	0.313418
$352$	0	0
$353$	−7.52092	−0.400298	−0.200149	−	0.979765i	$-0.564143\pi$
$353$	−7.52092	−0.400298	−0.200149	+	0.979765i	$0.564143\pi$
$354$	0	0
$355$	11.5404	0.612499
$356$	0	0
$357$	−7.51396	−0.397681
$358$	0	0
$359$	17.5962	0.928693	0.464346	−	0.885654i	$-0.346289\pi$
$359$	17.5962	0.928693	0.464346	+	0.885654i	$0.346289\pi$
$360$	0	0
$361$	−13.4402	−0.707378
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	5.50700	0.288250
$366$	0	0
$367$	−14.3121	−0.747084	−0.373542	−	0.927613i	$-0.621857\pi$
$367$	−14.3121	−0.747084	−0.373542	+	0.927613i	$0.621857\pi$
$368$	0	0
$369$	5.28415	0.275082
$370$	0	0
$371$	−4.22982	−0.219601
$372$	0	0
$373$	−16.4332	−0.850880	−0.425440	−	0.904987i	$-0.639881\pi$
$373$	−16.4332	−0.850880	−0.425440	+	0.904987i	$0.639881\pi$
$374$	0	0
$375$	−8.54885	−0.441461
$376$	0	0
$377$	34.4791	1.77576
$378$	0	0
$379$	−19.5334	−1.00336	−0.501682	−	0.865052i	$-0.667285\pi$
$379$	−19.5334	−1.00336	−0.501682	+	0.865052i	$0.667285\pi$
$380$	0	0
$381$	−10.2298	−0.524089
$382$	0	0
$383$	−2.35097	−0.120129	−0.0600644	−	0.998195i	$-0.519131\pi$
$383$	−2.35097	−0.120129	−0.0600644	+	0.998195i	$0.519131\pi$
$384$	0	0
$385$	−2.58774	−0.131884
$386$	0	0
$387$	−7.40530	−0.376432
$388$	0	0
$389$	−1.74378	−0.0884130	−0.0442065	−	0.999022i	$-0.514076\pi$
$389$	−1.74378	−0.0884130	−0.0442065	+	0.999022i	$0.514076\pi$
$390$	0	0
$391$	−52.1895	−2.63934
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	−11.5404	−0.580659
$396$	0	0
$397$	6.54189	0.328328	0.164164	−	0.986433i	$-0.447507\pi$
$397$	6.54189	0.328328	0.164164	+	0.986433i	$0.447507\pi$
$398$	0	0
$399$	−2.35793	−0.118044
$400$	0	0
$401$	5.66152	0.282723	0.141361	−	0.989958i	$-0.454852\pi$
$401$	5.66152	0.282723	0.141361	+	0.989958i	$0.454852\pi$
$402$	0	0
$403$	−21.5000	−1.07099
$404$	0	0
$405$	−2.58774	−0.128586
$406$	0	0
$407$	3.30359	0.163753
$408$	0	0
$409$	−5.48755	−0.271342	−0.135671	−	0.990754i	$-0.543319\pi$
$409$	−5.48755	−0.271342	−0.135671	+	0.990754i	$0.543319\pi$
$410$	0	0
$411$	10.8370	0.534550
$412$	0	0
$413$	0.926221	0.0455764
$414$	0	0
$415$	−27.6615	−1.35785
$416$	0	0
$417$	0.459630	0.0225082
$418$	0	0
$419$	−8.24926	−0.403003	−0.201501	−	0.979488i	$-0.564582\pi$
$419$	−8.24926	−0.403003	−0.201501	+	0.979488i	$0.564582\pi$
$420$	0	0
$421$	32.2617	1.57234	0.786171	−	0.618009i	$-0.212061\pi$
$421$	32.2617	1.57234	0.786171	+	0.618009i	$0.212061\pi$
$422$	0	0
$423$	−7.53341	−0.366287
$424$	0	0
$425$	12.7467	0.618307
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	5.87189	0.283497
$430$	0	0
$431$	23.2772	1.12122	0.560611	−	0.828079i	$-0.310566\pi$
$431$	23.2772	1.12122	0.560611	+	0.828079i	$0.310566\pi$
$432$	0	0
$433$	−15.9302	−0.765558	−0.382779	−	0.923840i	$-0.625033\pi$
$433$	−15.9302	−0.765558	−0.382779	+	0.923840i	$0.625033\pi$
$434$	0	0
$435$	−15.1949	−0.728541
$436$	0	0
$437$	−16.3774	−0.783436
$438$	0	0
$439$	−22.8565	−1.09088	−0.545439	−	0.838150i	$-0.683637\pi$
$439$	−22.8565	−1.09088	−0.545439	+	0.838150i	$0.683637\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−21.5962	−1.02607	−0.513034	−	0.858368i	$-0.671478\pi$
$443$	−21.5962	−1.02607	−0.513034	+	0.858368i	$0.671478\pi$
$444$	0	0
$445$	−41.1227	−1.94940
$446$	0	0
$447$	−11.0474	−0.522523
$448$	0	0
$449$	18.6630	0.880763	0.440382	−	0.897811i	$-0.354843\pi$
$449$	18.6630	0.880763	0.440382	+	0.897811i	$0.354843\pi$
$450$	0	0
$451$	5.28415	0.248821
$452$	0	0
$453$	−16.1212	−0.757438
$454$	0	0
$455$	15.1949	0.712349
$456$	0	0
$457$	36.4721	1.70609	0.853047	−	0.521834i	$-0.174752\pi$
$457$	36.4721	1.70609	0.853047	+	0.521834i	$0.174752\pi$
$458$	0	0
$459$	−7.51396	−0.350722
$460$	0	0
$461$	7.17548	0.334196	0.167098	−	0.985940i	$-0.446560\pi$
$461$	7.17548	0.334196	0.167098	+	0.985940i	$0.446560\pi$
$462$	0	0
$463$	0.452670	0.0210373	0.0105187	−	0.999945i	$-0.496652\pi$
$463$	0.452670	0.0210373	0.0105187	+	0.999945i	$0.496652\pi$
$464$	0	0
$465$	9.47507	0.439396
$466$	0	0
$467$	9.68097	0.447982	0.223991	−	0.974591i	$-0.428091\pi$
$467$	9.68097	0.447982	0.223991	+	0.974591i	$0.428091\pi$
$468$	0	0
$469$	10.1017	0.466453
$470$	0	0
$471$	−13.0668	−0.602087
$472$	0	0
$473$	−7.40530	−0.340496
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−4.22982	−0.193670
$478$	0	0
$479$	−24.3635	−1.11319	−0.556597	−	0.830782i	$-0.687893\pi$
$479$	−24.3635	−1.11319	−0.556597	+	0.830782i	$0.687893\pi$
$480$	0	0
$481$	−19.3983	−0.884488
$482$	0	0
$483$	6.94567	0.316039
$484$	0	0
$485$	−26.1909	−1.18927
$486$	0	0
$487$	19.2702	0.873217	0.436609	−	0.899652i	$-0.356179\pi$
$487$	19.2702	0.873217	0.436609	+	0.899652i	$0.356179\pi$
$488$	0	0
$489$	−10.1017	−0.456815
$490$	0	0
$491$	−29.6421	−1.33773	−0.668864	−	0.743385i	$-0.733219\pi$
$491$	−29.6421	−1.33773	−0.668864	+	0.743385i	$0.733219\pi$
$492$	0	0
$493$	−44.1212	−1.98712
$494$	0	0
$495$	−2.58774	−0.116310
$496$	0	0
$497$	−4.45963	−0.200042
$498$	0	0
$499$	−29.3719	−1.31487	−0.657434	−	0.753512i	$-0.728358\pi$
$499$	−29.3719	−1.31487	−0.657434	+	0.753512i	$0.728358\pi$
$500$	0	0
$501$	−10.4860	−0.468482
$502$	0	0
$503$	−12.5947	−0.561570	−0.280785	−	0.959771i	$-0.590595\pi$
$503$	−12.5947	−0.561570	−0.280785	+	0.959771i	$0.590595\pi$
$504$	0	0
$505$	34.6894	1.54366
$506$	0	0
$507$	−21.4791	−0.953919
$508$	0	0
$509$	−15.2144	−0.674365	−0.337183	−	0.941439i	$-0.609474\pi$
$509$	−15.2144	−0.674365	−0.337183	+	0.941439i	$0.609474\pi$
$510$	0	0
$511$	−2.12811	−0.0941421
$512$	0	0
$513$	−2.35793	−0.104105
$514$	0	0
$515$	−12.2034	−0.537746
$516$	0	0
$517$	−7.53341	−0.331319
$518$	0	0
$519$	−14.4985	−0.636415
$520$	0	0
$521$	20.5180	0.898909	0.449454	−	0.893303i	$-0.351618\pi$
$521$	20.5180	0.898909	0.449454	+	0.893303i	$0.351618\pi$
$522$	0	0
$523$	29.8844	1.30675	0.653376	−	0.757033i	$-0.273352\pi$
$523$	29.8844	1.30675	0.653376	+	0.757033i	$0.273352\pi$
$524$	0	0
$525$	−1.69641	−0.0740372
$526$	0	0
$527$	27.5125	1.19846
$528$	0	0
$529$	25.2423	1.09749
$530$	0	0
$531$	0.926221	0.0401946
$532$	0	0
$533$	−31.0279	−1.34397
$534$	0	0
$535$	18.3051	0.791399
$536$	0	0
$537$	14.3510	0.619290
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	26.9582	1.15902	0.579511	−	0.814965i	$-0.303244\pi$
$541$	26.9582	1.15902	0.579511	+	0.814965i	$0.303244\pi$
$542$	0	0
$543$	−9.66152	−0.414616
$544$	0	0
$545$	−48.7453	−2.08802
$546$	0	0
$547$	−26.0558	−1.11407	−0.557034	−	0.830490i	$-0.688061\pi$
$547$	−26.0558	−1.11407	−0.557034	+	0.830490i	$0.688061\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−13.8455	−0.589837
$552$	0	0
$553$	4.45963	0.189643
$554$	0	0
$555$	8.54885	0.362878
$556$	0	0
$557$	8.01945	0.339795	0.169897	−	0.985462i	$-0.445656\pi$
$557$	8.01945	0.339795	0.169897	+	0.985462i	$0.445656\pi$
$558$	0	0
$559$	43.4831	1.83914
$560$	0	0
$561$	−7.51396	−0.317240
$562$	0	0
$563$	−21.9736	−0.926077	−0.463038	−	0.886338i	$-0.653241\pi$
$563$	−21.9736	−0.926077	−0.463038	+	0.886338i	$0.653241\pi$
$564$	0	0
$565$	23.3619	0.982844
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−37.6740	−1.57661	−0.788304	−	0.615286i	$-0.789041\pi$
$571$	−37.6740	−1.57661	−0.788304	+	0.615286i	$0.789041\pi$
$572$	0	0
$573$	−9.55286	−0.399076
$574$	0	0
$575$	−11.7827	−0.491371
$576$	0	0
$577$	−7.80908	−0.325096	−0.162548	−	0.986701i	$-0.551971\pi$
$577$	−7.80908	−0.325096	−0.162548	+	0.986701i	$0.551971\pi$
$578$	0	0
$579$	22.1212	0.919324
$580$	0	0
$581$	10.6894	0.443473
$582$	0	0
$583$	−4.22982	−0.175181
$584$	0	0
$585$	15.1949	0.628233
$586$	0	0
$587$	43.2633	1.78567	0.892833	−	0.450388i	$-0.148714\pi$
$587$	43.2633	1.78567	0.892833	+	0.450388i	$0.148714\pi$
$588$	0	0
$589$	8.63360	0.355741
$590$	0	0
$591$	26.2423	1.07946
$592$	0	0
$593$	−29.7438	−1.22143	−0.610715	−	0.791850i	$-0.709118\pi$
$593$	−29.7438	−1.22143	−0.610715	+	0.791850i	$0.709118\pi$
$594$	0	0
$595$	−19.4442	−0.797134
$596$	0	0
$597$	1.05433	0.0431509
$598$	0	0
$599$	−11.5404	−0.471527	−0.235763	−	0.971810i	$-0.575759\pi$
$599$	−11.5404	−0.471527	−0.235763	+	0.971810i	$0.575759\pi$
$600$	0	0
$601$	7.06129	0.288036	0.144018	−	0.989575i	$-0.453998\pi$
$601$	7.06129	0.288036	0.144018	+	0.989575i	$0.453998\pi$
$602$	0	0
$603$	10.1017	0.411373
$604$	0	0
$605$	−2.58774	−0.105207
$606$	0	0
$607$	17.3859	0.705670	0.352835	−	0.935686i	$-0.385218\pi$
$607$	17.3859	0.705670	0.352835	+	0.935686i	$0.385218\pi$
$608$	0	0
$609$	5.87189	0.237941
$610$	0	0
$611$	44.2353	1.78957
$612$	0	0
$613$	39.4611	1.59382	0.796910	−	0.604098i	$-0.206466\pi$
$613$	39.4611	1.59382	0.796910	+	0.604098i	$0.206466\pi$
$614$	0	0
$615$	13.6740	0.551389
$616$	0	0
$617$	−4.40378	−0.177290	−0.0886448	−	0.996063i	$-0.528254\pi$
$617$	−4.40378	−0.177290	−0.0886448	+	0.996063i	$0.528254\pi$
$618$	0	0
$619$	−16.4163	−0.659825	−0.329913	−	0.944011i	$-0.607019\pi$
$619$	−16.4163	−0.659825	−0.329913	+	0.944011i	$0.607019\pi$
$620$	0	0
$621$	6.94567	0.278720
$622$	0	0
$623$	15.8913	0.636673
$624$	0	0
$625$	−30.6042	−1.22417
$626$	0	0
$627$	−2.35793	−0.0941665
$628$	0	0
$629$	24.8231	0.989761
$630$	0	0
$631$	−44.4596	−1.76991	−0.884955	−	0.465677i	$-0.845811\pi$
$631$	−44.4596	−1.76991	−0.884955	+	0.465677i	$0.845811\pi$
$632$	0	0
$633$	−8.71585	−0.346424
$634$	0	0
$635$	−26.4721	−1.05051
$636$	0	0
$637$	−5.87189	−0.232653
$638$	0	0
$639$	−4.45963	−0.176420
$640$	0	0
$641$	10.5419	0.416380	0.208190	−	0.978088i	$-0.433243\pi$
$641$	10.5419	0.416380	0.208190	+	0.978088i	$0.433243\pi$
$642$	0	0
$643$	23.4053	0.923015	0.461507	−	0.887136i	$-0.347309\pi$
$643$	23.4053	0.923015	0.461507	+	0.887136i	$0.347309\pi$
$644$	0	0
$645$	−19.1630	−0.754542
$646$	0	0
$647$	−11.4945	−0.451896	−0.225948	−	0.974139i	$-0.572548\pi$
$647$	−11.4945	−0.451896	−0.225948	+	0.974139i	$0.572548\pi$
$648$	0	0
$649$	0.926221	0.0363574
$650$	0	0
$651$	−3.66152	−0.143506
$652$	0	0
$653$	19.8774	0.777863	0.388932	−	0.921267i	$-0.372844\pi$
$653$	19.8774	0.777863	0.388932	+	0.921267i	$0.372844\pi$
$654$	0	0
$655$	10.3510	0.404446
$656$	0	0
$657$	−2.12811	−0.0830255
$658$	0	0
$659$	−15.0738	−0.587191	−0.293596	−	0.955930i	$-0.594852\pi$
$659$	−15.0738	−0.587191	−0.293596	+	0.955930i	$0.594852\pi$
$660$	0	0
$661$	−4.74226	−0.184453	−0.0922263	−	0.995738i	$-0.529398\pi$
$661$	−4.74226	−0.184453	−0.0922263	+	0.995738i	$0.529398\pi$
$662$	0	0
$663$	44.1212	1.71352
$664$	0	0
$665$	−6.10170	−0.236614
$666$	0	0
$667$	40.7842	1.57917
$668$	0	0
$669$	−23.7438	−0.917987
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	9.87885	0.380802	0.190401	−	0.981706i	$-0.439021\pi$
$673$	9.87885	0.380802	0.190401	+	0.981706i	$0.439021\pi$
$674$	0	0
$675$	−1.69641	−0.0652947
$676$	0	0
$677$	−29.0279	−1.11563	−0.557817	−	0.829964i	$-0.688361\pi$
$677$	−29.0279	−1.11563	−0.557817	+	0.829964i	$0.688361\pi$
$678$	0	0
$679$	10.1212	0.388414
$680$	0	0
$681$	−13.1755	−0.504886
$682$	0	0
$683$	37.9597	1.45249	0.726243	−	0.687438i	$-0.241265\pi$
$683$	37.9597	1.45249	0.726243	+	0.687438i	$0.241265\pi$
$684$	0	0
$685$	28.0434	1.07148
$686$	0	0
$687$	−13.3664	−0.509960
$688$	0	0
$689$	24.8370	0.946214
$690$	0	0
$691$	15.5264	0.590654	0.295327	−	0.955396i	$-0.404571\pi$
$691$	15.5264	0.590654	0.295327	+	0.955396i	$0.404571\pi$
$692$	0	0
$693$	1.00000	0.0379869
$694$	0	0
$695$	1.18940	0.0451167
$696$	0	0
$697$	39.7049	1.50393
$698$	0	0
$699$	−7.13659	−0.269931
$700$	0	0
$701$	−27.4317	−1.03608	−0.518041	−	0.855356i	$-0.673338\pi$
$701$	−27.4317	−1.03608	−0.518041	+	0.855356i	$0.673338\pi$
$702$	0	0
$703$	7.78963	0.293792
$704$	0	0
$705$	−19.4945	−0.734206
$706$	0	0
$707$	−13.4053	−0.504158
$708$	0	0
$709$	−44.5180	−1.67191	−0.835954	−	0.548800i	$-0.815085\pi$
$709$	−44.5180	−1.67191	−0.835954	+	0.548800i	$0.815085\pi$
$710$	0	0
$711$	4.45963	0.167249
$712$	0	0
$713$	−25.4317	−0.952425
$714$	0	0
$715$	15.1949	0.568258
$716$	0	0
$717$	−19.9930	−0.746654
$718$	0	0
$719$	−17.3859	−0.648383	−0.324191	−	0.945992i	$-0.605092\pi$
$719$	−17.3859	−0.648383	−0.324191	+	0.945992i	$0.605092\pi$
$720$	0	0
$721$	4.71585	0.175628
$722$	0	0
$723$	9.19493	0.341963
$724$	0	0
$725$	−9.96111	−0.369946
$726$	0	0
$727$	18.6506	0.691711	0.345855	−	0.938288i	$-0.387589\pi$
$727$	18.6506	0.691711	0.345855	+	0.938288i	$0.387589\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−55.6431	−2.05804
$732$	0	0
$733$	16.3510	0.603937	0.301968	−	0.953318i	$-0.402356\pi$
$733$	16.3510	0.603937	0.301968	+	0.953318i	$0.402356\pi$
$734$	0	0
$735$	2.58774	0.0954503
$736$	0	0
$737$	10.1017	0.372101
$738$	0	0
$739$	32.8370	1.20793	0.603964	−	0.797011i	$-0.293587\pi$
$739$	32.8370	1.20793	0.603964	+	0.797011i	$0.293587\pi$
$740$	0	0
$741$	13.8455	0.508626
$742$	0	0
$743$	−20.9123	−0.767198	−0.383599	−	0.923500i	$-0.625315\pi$
$743$	−20.9123	−0.767198	−0.383599	+	0.923500i	$0.625315\pi$
$744$	0	0
$745$	−28.5877	−1.04737
$746$	0	0
$747$	10.6894	0.391106
$748$	0	0
$749$	−7.07378	−0.258470
$750$	0	0
$751$	39.2633	1.43274	0.716368	−	0.697722i	$-0.245803\pi$
$751$	39.2633	1.43274	0.716368	+	0.697722i	$0.245803\pi$
$752$	0	0
$753$	−9.42474	−0.343457
$754$	0	0
$755$	−41.7174	−1.51825
$756$	0	0
$757$	−10.3454	−0.376011	−0.188006	−	0.982168i	$-0.560202\pi$
$757$	−10.3454	−0.376011	−0.188006	+	0.982168i	$0.560202\pi$
$758$	0	0
$759$	6.94567	0.252112
$760$	0	0
$761$	30.9582	1.12223	0.561116	−	0.827737i	$-0.310372\pi$
$761$	30.9582	1.12223	0.561116	+	0.827737i	$0.310372\pi$
$762$	0	0
$763$	18.8370	0.681945
$764$	0	0
$765$	−19.4442	−0.703006
$766$	0	0
$767$	−5.43867	−0.196379
$768$	0	0
$769$	−28.8998	−1.04215	−0.521077	−	0.853510i	$-0.674470\pi$
$769$	−28.8998	−1.04215	−0.521077	+	0.853510i	$0.674470\pi$
$770$	0	0
$771$	0.0194469	0.000700364 0
$772$	0	0
$773$	−26.7523	−0.962212	−0.481106	−	0.876663i	$-0.659765\pi$
$773$	−26.7523	−0.962212	−0.481106	+	0.876663i	$0.659765\pi$
$774$	0	0
$775$	6.21142	0.223121
$776$	0	0
$777$	−3.30359	−0.118516
$778$	0	0
$779$	12.4596	0.446413
$780$	0	0
$781$	−4.45963	−0.159578
$782$	0	0
$783$	5.87189	0.209844
$784$	0	0
$785$	−33.8135	−1.20686
$786$	0	0
$787$	−22.1406	−0.789227	−0.394614	−	0.918847i	$-0.629122\pi$
$787$	−22.1406	−0.789227	−0.394614	+	0.918847i	$0.629122\pi$
$788$	0	0
$789$	27.9930	0.996579
$790$	0	0
$791$	−9.02792	−0.320996
$792$	0	0
$793$	11.7438	0.417034
$794$	0	0
$795$	−10.9457	−0.388203
$796$	0	0
$797$	8.18396	0.289891	0.144945	−	0.989440i	$-0.453699\pi$
$797$	8.18396	0.289891	0.144945	+	0.989440i	$0.453699\pi$
$798$	0	0
$799$	−56.6058	−2.00257
$800$	0	0
$801$	15.8913	0.561493
$802$	0	0
$803$	−2.12811	−0.0750994
$804$	0	0
$805$	17.9736	0.633486
$806$	0	0
$807$	−30.9193	−1.08841
$808$	0	0
$809$	17.6685	0.621191	0.310595	−	0.950542i	$-0.399472\pi$
$809$	17.6685	0.621191	0.310595	+	0.950542i	$0.399472\pi$
$810$	0	0
$811$	43.0210	1.51067	0.755335	−	0.655339i	$-0.227474\pi$
$811$	43.0210	1.51067	0.755335	+	0.655339i	$0.227474\pi$
$812$	0	0
$813$	−22.3579	−0.784127
$814$	0	0
$815$	−26.1406	−0.915665
$816$	0	0
$817$	−17.4611	−0.610888
$818$	0	0
$819$	−5.87189	−0.205180
$820$	0	0
$821$	18.0364	0.629475	0.314737	−	0.949179i	$-0.398084\pi$
$821$	18.0364	0.629475	0.314737	+	0.949179i	$0.398084\pi$
$822$	0	0
$823$	8.21037	0.286195	0.143098	−	0.989709i	$-0.454294\pi$
$823$	8.21037	0.286195	0.143098	+	0.989709i	$0.454294\pi$
$824$	0	0
$825$	−1.69641	−0.0590613
$826$	0	0
$827$	−6.83148	−0.237554	−0.118777	−	0.992921i	$-0.537897\pi$
$827$	−6.83148	−0.237554	−0.118777	+	0.992921i	$0.537897\pi$
$828$	0	0
$829$	−25.7438	−0.894118	−0.447059	−	0.894504i	$-0.647529\pi$
$829$	−25.7438	−0.894118	−0.447059	+	0.894504i	$0.647529\pi$
$830$	0	0
$831$	−24.7717	−0.859321
$832$	0	0
$833$	7.51396	0.260343
$834$	0	0
$835$	−27.1352	−0.939051
$836$	0	0
$837$	−3.66152	−0.126561
$838$	0	0
$839$	30.6002	1.05644	0.528219	−	0.849108i	$-0.322860\pi$
$839$	30.6002	1.05644	0.528219	+	0.849108i	$0.322860\pi$
$840$	0	0
$841$	5.47908	0.188934
$842$	0	0
$843$	1.19493	0.0411556
$844$	0	0
$845$	−55.5823	−1.91209
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−15.5723	−0.534440
$850$	0	0
$851$	−22.9457	−0.786567
$852$	0	0
$853$	−30.4596	−1.04292	−0.521459	−	0.853276i	$-0.674612\pi$
$853$	−30.4596	−1.04292	−0.521459	+	0.853276i	$0.674612\pi$
$854$	0	0
$855$	−6.10170	−0.208674
$856$	0	0
$857$	−41.2702	−1.40976	−0.704882	−	0.709325i	$-0.749000\pi$
$857$	−41.2702	−1.40976	−0.704882	+	0.709325i	$0.749000\pi$
$858$	0	0
$859$	28.9193	0.986712	0.493356	−	0.869827i	$-0.335770\pi$
$859$	28.9193	0.986712	0.493356	+	0.869827i	$0.335770\pi$
$860$	0	0
$861$	−5.28415	−0.180083
$862$	0	0
$863$	−37.0015	−1.25955	−0.629773	−	0.776779i	$-0.716852\pi$
$863$	−37.0015	−1.25955	−0.629773	+	0.776779i	$0.716852\pi$
$864$	0	0
$865$	−37.5184	−1.27566
$866$	0	0
$867$	−39.4596	−1.34012
$868$	0	0
$869$	4.45963	0.151283
$870$	0	0
$871$	−59.3161	−2.00985
$872$	0	0
$873$	10.1212	0.342549
$874$	0	0
$875$	8.54885	0.289004
$876$	0	0
$877$	−6.12562	−0.206847	−0.103424	−	0.994637i	$-0.532980\pi$
$877$	−6.12562	−0.206847	−0.103424	+	0.994637i	$0.532980\pi$
$878$	0	0
$879$	9.74378	0.328649
$880$	0	0
$881$	25.4123	0.856161	0.428080	−	0.903741i	$-0.359190\pi$
$881$	25.4123	0.856161	0.428080	+	0.903741i	$0.359190\pi$
$882$	0	0
$883$	17.0040	0.572230	0.286115	−	0.958195i	$-0.407636\pi$
$883$	17.0040	0.572230	0.286115	+	0.958195i	$0.407636\pi$
$884$	0	0
$885$	2.39682	0.0805682
$886$	0	0
$887$	−35.2269	−1.18280	−0.591401	−	0.806378i	$-0.701425\pi$
$887$	−35.2269	−1.18280	−0.591401	+	0.806378i	$0.701425\pi$
$888$	0	0
$889$	10.2298	0.343097
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−17.7632	−0.594424
$894$	0	0
$895$	37.1366	1.24134
$896$	0	0
$897$	−40.7842	−1.36174
$898$	0	0
$899$	−21.5000	−0.717067
$900$	0	0
$901$	−31.7827	−1.05883
$902$	0	0
$903$	7.40530	0.246433
$904$	0	0
$905$	−25.0015	−0.831079
$906$	0	0
$907$	9.72682	0.322974	0.161487	−	0.986875i	$-0.448371\pi$
$907$	9.72682	0.322974	0.161487	+	0.986875i	$0.448371\pi$
$908$	0	0
$909$	−13.4053	−0.444626
$910$	0	0
$911$	42.7717	1.41709	0.708545	−	0.705666i	$-0.249352\pi$
$911$	42.7717	1.41709	0.708545	+	0.705666i	$0.249352\pi$
$912$	0	0
$913$	10.6894	0.353769
$914$	0	0
$915$	−5.17548	−0.171096
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−6.26871	−0.206786	−0.103393	−	0.994641i	$-0.532970\pi$
$919$	−6.26871	−0.206786	−0.103393	+	0.994641i	$0.532970\pi$
$920$	0	0
$921$	6.35097	0.209271
$922$	0	0
$923$	26.1865	0.861938
$924$	0	0
$925$	5.60424	0.184266
$926$	0	0
$927$	4.71585	0.154889
$928$	0	0
$929$	20.0194	0.656817	0.328408	−	0.944536i	$-0.393488\pi$
$929$	20.0194	0.656817	0.328408	+	0.944536i	$0.393488\pi$
$930$	0	0
$931$	2.35793	0.0772779
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−19.4442	−0.635893
$936$	0	0
$937$	−11.2144	−0.366358	−0.183179	−	0.983080i	$-0.558639\pi$
$937$	−11.2144	−0.366358	−0.183179	+	0.983080i	$0.558639\pi$
$938$	0	0
$939$	−16.9457	−0.553001
$940$	0	0
$941$	31.3400	1.02165	0.510827	−	0.859683i	$-0.329339\pi$
$941$	31.3400	1.02165	0.510827	+	0.859683i	$0.329339\pi$
$942$	0	0
$943$	−36.7019	−1.19518
$944$	0	0
$945$	2.58774	0.0841792
$946$	0	0
$947$	23.2841	0.756633	0.378317	−	0.925676i	$-0.376503\pi$
$947$	23.2841	0.756633	0.378317	+	0.925676i	$0.376503\pi$
$948$	0	0
$949$	12.4960	0.405638
$950$	0	0
$951$	−20.0125	−0.648949
$952$	0	0
$953$	26.8300	0.869110	0.434555	−	0.900645i	$-0.356906\pi$
$953$	26.8300	0.869110	0.434555	+	0.900645i	$0.356906\pi$
$954$	0	0
$955$	−24.7203	−0.799931
$956$	0	0
$957$	5.87189	0.189811
$958$	0	0
$959$	−10.8370	−0.349945
$960$	0	0
$961$	−17.5933	−0.567525
$962$	0	0
$963$	−7.07378	−0.227949
$964$	0	0
$965$	57.2438	1.84274
$966$	0	0
$967$	−36.2034	−1.16422	−0.582112	−	0.813109i	$-0.697773\pi$
$967$	−36.2034	−1.16422	−0.582112	+	0.813109i	$0.697773\pi$
$968$	0	0
$969$	−17.7174	−0.569164
$970$	0	0
$971$	−6.06281	−0.194565	−0.0972824	−	0.995257i	$-0.531015\pi$
$971$	−6.06281	−0.194565	−0.0972824	+	0.995257i	$0.531015\pi$
$972$	0	0
$973$	−0.459630	−0.0147351
$974$	0	0
$975$	9.96111	0.319011
$976$	0	0
$977$	−4.28263	−0.137013	−0.0685067	−	0.997651i	$-0.521823\pi$
$977$	−4.28263	−0.137013	−0.0685067	+	0.997651i	$0.521823\pi$
$978$	0	0
$979$	15.8913	0.507889
$980$	0	0
$981$	18.8370	0.601419
$982$	0	0
$983$	17.1894	0.548257	0.274128	−	0.961693i	$-0.411611\pi$
$983$	17.1894	0.548257	0.274128	+	0.961693i	$0.411611\pi$
$984$	0	0
$985$	67.9083	2.16374
$986$	0	0
$987$	7.53341	0.239791
$988$	0	0
$989$	51.4347	1.63553
$990$	0	0
$991$	−35.4945	−1.12752	−0.563760	−	0.825939i	$-0.690646\pi$
$991$	−35.4945	−1.12752	−0.563760	+	0.825939i	$0.690646\pi$
$992$	0	0
$993$	7.54037	0.239286
$994$	0	0
$995$	2.72834	0.0864942
$996$	0	0
$997$	30.4068	0.962993	0.481497	−	0.876448i	$-0.340093\pi$
$997$	30.4068	0.962993	0.481497	+	0.876448i	$0.340093\pi$
$998$	0	0
$999$	−3.30359	−0.104521

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3696.2.a.bo.1.1		3
4.3	odd	2		231.2.a.e.1.3	✓	3
12.11	even	2		693.2.a.l.1.1		3
20.19	odd	2		5775.2.a.bp.1.1		3
28.27	even	2		1617.2.a.t.1.3		3
44.43	even	2		2541.2.a.bg.1.1		3
84.83	odd	2		4851.2.a.bi.1.1		3
132.131	odd	2		7623.2.a.cd.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.e.1.3	✓	3	4.3	odd	2
693.2.a.l.1.1		3	12.11	even	2
1617.2.a.t.1.3		3	28.27	even	2
2541.2.a.bg.1.1		3	44.43	even	2
3696.2.a.bo.1.1		3	1.1	even	1	trivial
4851.2.a.bi.1.1		3	84.83	odd	2
5775.2.a.bp.1.1		3	20.19	odd	2
7623.2.a.cd.1.3		3	132.131	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3696.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.58774 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.58774 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} -5.87189 q^{13} +2.58774 q^{15} +7.51396 q^{17} +2.35793 q^{19} -1.00000 q^{21} -6.94567 q^{23} +1.69641 q^{25} -1.00000 q^{27} -5.87189 q^{29} +3.66152 q^{31} -1.00000 q^{33} -2.58774 q^{35} +3.30359 q^{37} +5.87189 q^{39} +5.28415 q^{41} -7.40530 q^{43} -2.58774 q^{45} -7.53341 q^{47} +1.00000 q^{49} -7.51396 q^{51} -4.22982 q^{53} -2.58774 q^{55} -2.35793 q^{57} +0.926221 q^{59} -2.00000 q^{61} +1.00000 q^{63} +15.1949 q^{65} +10.1017 q^{67} +6.94567 q^{69} -4.45963 q^{71} -2.12811 q^{73} -1.69641 q^{75} +1.00000 q^{77} +4.45963 q^{79} +1.00000 q^{81} +10.6894 q^{83} -19.4442 q^{85} +5.87189 q^{87} +15.8913 q^{89} -5.87189 q^{91} -3.66152 q^{93} -6.10170 q^{95} +10.1212 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} + 8 q^{17} + 8 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{35} + 4 q^{39} + 14 q^{41} + 14 q^{43} + 4 q^{45} + 3 q^{49} - 8 q^{51} + 4 q^{55} - 8 q^{57} - 6 q^{61} + 3 q^{63} + 14 q^{65} + 4 q^{67} + 10 q^{69} + 12 q^{71} - 20 q^{73} - 15 q^{75} + 3 q^{77} - 12 q^{79} + 3 q^{81} - 6 q^{83} - 6 q^{85} + 4 q^{87} + 26 q^{89} - 4 q^{91} - 2 q^{93} + 8 q^{95} - 4 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 4 * q^5 + 3 * q^7 + 3 * q^9 + 3 * q^11 - 4 * q^13 - 4 * q^15 + 8 * q^17 + 8 * q^19 - 3 * q^21 - 10 * q^23 + 15 * q^25 - 3 * q^27 - 4 * q^29 + 2 * q^31 - 3 * q^33 + 4 * q^35 + 4 * q^39 + 14 * q^41 + 14 * q^43 + 4 * q^45 + 3 * q^49 - 8 * q^51 + 4 * q^55 - 8 * q^57 - 6 * q^61 + 3 * q^63 + 14 * q^65 + 4 * q^67 + 10 * q^69 + 12 * q^71 - 20 * q^73 - 15 * q^75 + 3 * q^77 - 12 * q^79 + 3 * q^81 - 6 * q^83 - 6 * q^85 + 4 * q^87 + 26 * q^89 - 4 * q^91 - 2 * q^93 + 8 * q^95 - 4 * q^97 + 3 * q^99

Introduction

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