LMFDB - Embedded newform 7728.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1932)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.09017 q^{13} +3.61803 q^{15} -5.00000 q^{17} +3.47214 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.09017 q^{25} -1.00000 q^{27} -6.23607 q^{29} +8.70820 q^{31} -1.00000 q^{33} +3.61803 q^{35} +1.47214 q^{37} -5.09017 q^{39} -5.76393 q^{41} -6.32624 q^{43} -3.61803 q^{45} -3.70820 q^{47} +1.00000 q^{49} +5.00000 q^{51} +0.381966 q^{53} -3.61803 q^{55} -3.47214 q^{57} +3.61803 q^{59} -7.56231 q^{61} -1.00000 q^{63} -18.4164 q^{65} +7.32624 q^{67} +1.00000 q^{69} +4.32624 q^{71} -0.527864 q^{73} -8.09017 q^{75} -1.00000 q^{77} +10.7082 q^{79} +1.00000 q^{81} -9.18034 q^{83} +18.0902 q^{85} +6.23607 q^{87} -16.6180 q^{89} -5.09017 q^{91} -8.70820 q^{93} -12.5623 q^{95} -12.2361 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - q^{13} + 5 q^{15} - 10 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + q^{39} - 16 q^{41} + 3 q^{43} - 5 q^{45} + 6 q^{47} + 2 q^{49} + 10 q^{51} + 3 q^{53} - 5 q^{55} + 2 q^{57} + 5 q^{59} + 5 q^{61} - 2 q^{63} - 10 q^{65} - q^{67} + 2 q^{69} - 7 q^{71} - 10 q^{73} - 5 q^{75} - 2 q^{77} + 8 q^{79} + 2 q^{81} + 4 q^{83} + 25 q^{85} + 8 q^{87} - 31 q^{89} + q^{91} - 4 q^{93} - 5 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - q^13 + 5 * q^15 - 10 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 5 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 2 * q^33 + 5 * q^35 - 6 * q^37 + q^39 - 16 * q^41 + 3 * q^43 - 5 * q^45 + 6 * q^47 + 2 * q^49 + 10 * q^51 + 3 * q^53 - 5 * q^55 + 2 * q^57 + 5 * q^59 + 5 * q^61 - 2 * q^63 - 10 * q^65 - q^67 + 2 * q^69 - 7 * q^71 - 10 * q^73 - 5 * q^75 - 2 * q^77 + 8 * q^79 + 2 * q^81 + 4 * q^83 + 25 * q^85 + 8 * q^87 - 31 * q^89 + q^91 - 4 * q^93 - 5 * q^95 - 20 * q^97 + 2 * 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
$5$	−3.61803	−1.61803	−0.809017	−	0.587785i	$-0.800000\pi$
$5$	−3.61803	−1.61803	−0.809017	+	0.587785i	$0.800000\pi$
$6$	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	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	5.09017	1.41176	0.705880	−	0.708332i	$-0.250552\pi$
$13$	5.09017	1.41176	0.705880	+	0.708332i	$0.250552\pi$
$14$	0	0
$15$	3.61803	0.934172
$16$	0	0
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	0	0
$19$	3.47214	0.796563	0.398281	−	0.917263i	$-0.369607\pi$
$19$	3.47214	0.796563	0.398281	+	0.917263i	$0.369607\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	8.09017	1.61803
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.23607	−1.15801	−0.579004	−	0.815324i	$-0.696559\pi$
$29$	−6.23607	−1.15801	−0.579004	+	0.815324i	$0.696559\pi$
$30$	0	0
$31$	8.70820	1.56404	0.782020	−	0.623254i	$-0.214190\pi$
$31$	8.70820	1.56404	0.782020	+	0.623254i	$0.214190\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	3.61803	0.611559
$36$	0	0
$37$	1.47214	0.242018	0.121009	−	0.992651i	$-0.461387\pi$
$37$	1.47214	0.242018	0.121009	+	0.992651i	$0.461387\pi$
$38$	0	0
$39$	−5.09017	−0.815080
$40$	0	0
$41$	−5.76393	−0.900175	−0.450087	−	0.892984i	$-0.648607\pi$
$41$	−5.76393	−0.900175	−0.450087	+	0.892984i	$0.648607\pi$
$42$	0	0
$43$	−6.32624	−0.964742	−0.482371	−	0.875967i	$-0.660224\pi$
$43$	−6.32624	−0.964742	−0.482371	+	0.875967i	$0.660224\pi$
$44$	0	0
$45$	−3.61803	−0.539345
$46$	0	0
$47$	−3.70820	−0.540897	−0.270449	−	0.962734i	$-0.587172\pi$
$47$	−3.70820	−0.540897	−0.270449	+	0.962734i	$0.587172\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.00000	0.700140
$52$	0	0
$53$	0.381966	0.0524671	0.0262335	−	0.999656i	$-0.491649\pi$
$53$	0.381966	0.0524671	0.0262335	+	0.999656i	$0.491649\pi$
$54$	0	0
$55$	−3.61803	−0.487856
$56$	0	0
$57$	−3.47214	−0.459896
$58$	0	0
$59$	3.61803	0.471028	0.235514	−	0.971871i	$-0.424323\pi$
$59$	3.61803	0.471028	0.235514	+	0.971871i	$0.424323\pi$
$60$	0	0
$61$	−7.56231	−0.968254	−0.484127	−	0.874998i	$-0.660863\pi$
$61$	−7.56231	−0.968254	−0.484127	+	0.874998i	$0.660863\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−18.4164	−2.28427
$66$	0	0
$67$	7.32624	0.895042	0.447521	−	0.894273i	$-0.352307\pi$
$67$	7.32624	0.895042	0.447521	+	0.894273i	$0.352307\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	4.32624	0.513430	0.256715	−	0.966487i	$-0.417360\pi$
$71$	4.32624	0.513430	0.256715	+	0.966487i	$0.417360\pi$
$72$	0	0
$73$	−0.527864	−0.0617818	−0.0308909	−	0.999523i	$-0.509834\pi$
$73$	−0.527864	−0.0617818	−0.0308909	+	0.999523i	$0.509834\pi$
$74$	0	0
$75$	−8.09017	−0.934172
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	10.7082	1.20477	0.602384	−	0.798207i	$-0.294218\pi$
$79$	10.7082	1.20477	0.602384	+	0.798207i	$0.294218\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.18034	−1.00767	−0.503837	−	0.863799i	$-0.668079\pi$
$83$	−9.18034	−1.00767	−0.503837	+	0.863799i	$0.668079\pi$
$84$	0	0
$85$	18.0902	1.96215
$86$	0	0
$87$	6.23607	0.668577
$88$	0	0
$89$	−16.6180	−1.76151	−0.880754	−	0.473574i	$-0.842964\pi$
$89$	−16.6180	−1.76151	−0.880754	+	0.473574i	$0.842964\pi$
$90$	0	0
$91$	−5.09017	−0.533595
$92$	0	0
$93$	−8.70820	−0.902999
$94$	0	0
$95$	−12.5623	−1.28887
$96$	0	0
$97$	−12.2361	−1.24238	−0.621192	−	0.783658i	$-0.713351\pi$
$97$	−12.2361	−1.24238	−0.621192	+	0.783658i	$0.713351\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−5.61803	−0.559015	−0.279508	−	0.960143i	$-0.590171\pi$
$101$	−5.61803	−0.559015	−0.279508	+	0.960143i	$0.590171\pi$
$102$	0	0
$103$	−2.47214	−0.243587	−0.121793	−	0.992555i	$-0.538865\pi$
$103$	−2.47214	−0.243587	−0.121793	+	0.992555i	$0.538865\pi$
$104$	0	0
$105$	−3.61803	−0.353084
$106$	0	0
$107$	14.6180	1.41318	0.706589	−	0.707624i	$-0.250233\pi$
$107$	14.6180	1.41318	0.706589	+	0.707624i	$0.250233\pi$
$108$	0	0
$109$	−17.5066	−1.67683	−0.838413	−	0.545035i	$-0.816516\pi$
$109$	−17.5066	−1.67683	−0.838413	+	0.545035i	$0.816516\pi$
$110$	0	0
$111$	−1.47214	−0.139729
$112$	0	0
$113$	3.14590	0.295941	0.147971	−	0.988992i	$-0.452726\pi$
$113$	3.14590	0.295941	0.147971	+	0.988992i	$0.452726\pi$
$114$	0	0
$115$	3.61803	0.337383
$116$	0	0
$117$	5.09017	0.470586
$118$	0	0
$119$	5.00000	0.458349
$120$	0	0
$121$	−10.0000	−0.909091
$122$	0	0
$123$	5.76393	0.519716
$124$	0	0
$125$	−11.1803	−1.00000
$126$	0	0
$127$	−4.61803	−0.409784	−0.204892	−	0.978785i	$-0.565684\pi$
$127$	−4.61803	−0.409784	−0.204892	+	0.978785i	$0.565684\pi$
$128$	0	0
$129$	6.32624	0.556994
$130$	0	0
$131$	9.18034	0.802090	0.401045	−	0.916058i	$-0.368647\pi$
$131$	9.18034	0.802090	0.401045	+	0.916058i	$0.368647\pi$
$132$	0	0
$133$	−3.47214	−0.301072
$134$	0	0
$135$	3.61803	0.311391
$136$	0	0
$137$	−4.52786	−0.386842	−0.193421	−	0.981116i	$-0.561958\pi$
$137$	−4.52786	−0.386842	−0.193421	+	0.981116i	$0.561958\pi$
$138$	0	0
$139$	−0.673762	−0.0571478	−0.0285739	−	0.999592i	$-0.509097\pi$
$139$	−0.673762	−0.0571478	−0.0285739	+	0.999592i	$0.509097\pi$
$140$	0	0
$141$	3.70820	0.312287
$142$	0	0
$143$	5.09017	0.425661
$144$	0	0
$145$	22.5623	1.87370
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	19.1246	1.56675	0.783375	−	0.621550i	$-0.213497\pi$
$149$	19.1246	1.56675	0.783375	+	0.621550i	$0.213497\pi$
$150$	0	0
$151$	−5.23607	−0.426105	−0.213053	−	0.977041i	$-0.568341\pi$
$151$	−5.23607	−0.426105	−0.213053	+	0.977041i	$0.568341\pi$
$152$	0	0
$153$	−5.00000	−0.404226
$154$	0	0
$155$	−31.5066	−2.53067
$156$	0	0
$157$	14.6525	1.16939	0.584697	−	0.811251i	$-0.301213\pi$
$157$	14.6525	1.16939	0.584697	+	0.811251i	$0.301213\pi$
$158$	0	0
$159$	−0.381966	−0.0302919
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	14.0902	1.10363	0.551814	−	0.833967i	$-0.313936\pi$
$163$	14.0902	1.10363	0.551814	+	0.833967i	$0.313936\pi$
$164$	0	0
$165$	3.61803	0.281664
$166$	0	0
$167$	3.18034	0.246102	0.123051	−	0.992400i	$-0.460732\pi$
$167$	3.18034	0.246102	0.123051	+	0.992400i	$0.460732\pi$
$168$	0	0
$169$	12.9098	0.993064
$170$	0	0
$171$	3.47214	0.265521
$172$	0	0
$173$	9.18034	0.697968	0.348984	−	0.937129i	$-0.386527\pi$
$173$	9.18034	0.697968	0.348984	+	0.937129i	$0.386527\pi$
$174$	0	0
$175$	−8.09017	−0.611559
$176$	0	0
$177$	−3.61803	−0.271948
$178$	0	0
$179$	−10.2705	−0.767654	−0.383827	−	0.923405i	$-0.625394\pi$
$179$	−10.2705	−0.767654	−0.383827	+	0.923405i	$0.625394\pi$
$180$	0	0
$181$	−0.236068	−0.0175468	−0.00877340	−	0.999962i	$-0.502793\pi$
$181$	−0.236068	−0.0175468	−0.00877340	+	0.999962i	$0.502793\pi$
$182$	0	0
$183$	7.56231	0.559022
$184$	0	0
$185$	−5.32624	−0.391593
$186$	0	0
$187$	−5.00000	−0.365636
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−16.6525	−1.20493	−0.602465	−	0.798145i	$-0.705815\pi$
$191$	−16.6525	−1.20493	−0.602465	+	0.798145i	$0.705815\pi$
$192$	0	0
$193$	21.7082	1.56259	0.781295	−	0.624161i	$-0.214559\pi$
$193$	21.7082	1.56259	0.781295	+	0.624161i	$0.214559\pi$
$194$	0	0
$195$	18.4164	1.31883
$196$	0	0
$197$	22.6180	1.61147	0.805734	−	0.592277i	$-0.201771\pi$
$197$	22.6180	1.61147	0.805734	+	0.592277i	$0.201771\pi$
$198$	0	0
$199$	−3.79837	−0.269260	−0.134630	−	0.990896i	$-0.542985\pi$
$199$	−3.79837	−0.269260	−0.134630	+	0.990896i	$0.542985\pi$
$200$	0	0
$201$	−7.32624	−0.516753
$202$	0	0
$203$	6.23607	0.437686
$204$	0	0
$205$	20.8541	1.45651
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	3.47214	0.240173
$210$	0	0
$211$	−18.2361	−1.25542	−0.627711	−	0.778446i	$-0.716008\pi$
$211$	−18.2361	−1.25542	−0.627711	+	0.778446i	$0.716008\pi$
$212$	0	0
$213$	−4.32624	−0.296429
$214$	0	0
$215$	22.8885	1.56099
$216$	0	0
$217$	−8.70820	−0.591151
$218$	0	0
$219$	0.527864	0.0356697
$220$	0	0
$221$	−25.4508	−1.71201
$222$	0	0
$223$	−10.5623	−0.707304	−0.353652	−	0.935377i	$-0.615060\pi$
$223$	−10.5623	−0.707304	−0.353652	+	0.935377i	$0.615060\pi$
$224$	0	0
$225$	8.09017	0.539345
$226$	0	0
$227$	16.8541	1.11865	0.559323	−	0.828950i	$-0.311061\pi$
$227$	16.8541	1.11865	0.559323	+	0.828950i	$0.311061\pi$
$228$	0	0
$229$	−7.79837	−0.515331	−0.257666	−	0.966234i	$-0.582953\pi$
$229$	−7.79837	−0.515331	−0.257666	+	0.966234i	$0.582953\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	0	0
$233$	18.5623	1.21606	0.608029	−	0.793915i	$-0.291961\pi$
$233$	18.5623	1.21606	0.608029	+	0.793915i	$0.291961\pi$
$234$	0	0
$235$	13.4164	0.875190
$236$	0	0
$237$	−10.7082	−0.695573
$238$	0	0
$239$	−12.6738	−0.819798	−0.409899	−	0.912131i	$-0.634436\pi$
$239$	−12.6738	−0.819798	−0.409899	+	0.912131i	$0.634436\pi$
$240$	0	0
$241$	−25.6525	−1.65242	−0.826211	−	0.563361i	$-0.809508\pi$
$241$	−25.6525	−1.65242	−0.826211	+	0.563361i	$0.809508\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.61803	−0.231148
$246$	0	0
$247$	17.6738	1.12455
$248$	0	0
$249$	9.18034	0.581780
$250$	0	0
$251$	−3.81966	−0.241095	−0.120547	−	0.992708i	$-0.538465\pi$
$251$	−3.81966	−0.241095	−0.120547	+	0.992708i	$0.538465\pi$
$252$	0	0
$253$	−1.00000	−0.0628695
$254$	0	0
$255$	−18.0902	−1.13285
$256$	0	0
$257$	−0.291796	−0.0182017	−0.00910087	−	0.999959i	$-0.502897\pi$
$257$	−0.291796	−0.0182017	−0.00910087	+	0.999959i	$0.502897\pi$
$258$	0	0
$259$	−1.47214	−0.0914741
$260$	0	0
$261$	−6.23607	−0.386003
$262$	0	0
$263$	27.1803	1.67601	0.838006	−	0.545661i	$-0.183721\pi$
$263$	27.1803	1.67601	0.838006	+	0.545661i	$0.183721\pi$
$264$	0	0
$265$	−1.38197	−0.0848935
$266$	0	0
$267$	16.6180	1.01701
$268$	0	0
$269$	19.8541	1.21053	0.605263	−	0.796026i	$-0.293068\pi$
$269$	19.8541	1.21053	0.605263	+	0.796026i	$0.293068\pi$
$270$	0	0
$271$	6.52786	0.396540	0.198270	−	0.980147i	$-0.436468\pi$
$271$	6.52786	0.396540	0.198270	+	0.980147i	$0.436468\pi$
$272$	0	0
$273$	5.09017	0.308071
$274$	0	0
$275$	8.09017	0.487856
$276$	0	0
$277$	31.0902	1.86803	0.934014	−	0.357237i	$-0.116281\pi$
$277$	31.0902	1.86803	0.934014	+	0.357237i	$0.116281\pi$
$278$	0	0
$279$	8.70820	0.521347
$280$	0	0
$281$	8.18034	0.487998	0.243999	−	0.969775i	$-0.421541\pi$
$281$	8.18034	0.487998	0.243999	+	0.969775i	$0.421541\pi$
$282$	0	0
$283$	−2.96556	−0.176284	−0.0881421	−	0.996108i	$-0.528093\pi$
$283$	−2.96556	−0.176284	−0.0881421	+	0.996108i	$0.528093\pi$
$284$	0	0
$285$	12.5623	0.744127
$286$	0	0
$287$	5.76393	0.340234
$288$	0	0
$289$	8.00000	0.470588
$290$	0	0
$291$	12.2361	0.717291
$292$	0	0
$293$	2.47214	0.144424	0.0722119	−	0.997389i	$-0.476994\pi$
$293$	2.47214	0.144424	0.0722119	+	0.997389i	$0.476994\pi$
$294$	0	0
$295$	−13.0902	−0.762139
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−5.09017	−0.294372
$300$	0	0
$301$	6.32624	0.364638
$302$	0	0
$303$	5.61803	0.322748
$304$	0	0
$305$	27.3607	1.56667
$306$	0	0
$307$	−29.0689	−1.65905	−0.829524	−	0.558470i	$-0.811388\pi$
$307$	−29.0689	−1.65905	−0.829524	+	0.558470i	$0.811388\pi$
$308$	0	0
$309$	2.47214	0.140635
$310$	0	0
$311$	15.0902	0.855685	0.427843	−	0.903853i	$-0.359274\pi$
$311$	15.0902	0.855685	0.427843	+	0.903853i	$0.359274\pi$
$312$	0	0
$313$	−0.583592	−0.0329866	−0.0164933	−	0.999864i	$-0.505250\pi$
$313$	−0.583592	−0.0329866	−0.0164933	+	0.999864i	$0.505250\pi$
$314$	0	0
$315$	3.61803	0.203853
$316$	0	0
$317$	26.3820	1.48176	0.740879	−	0.671638i	$-0.234409\pi$
$317$	26.3820	1.48176	0.740879	+	0.671638i	$0.234409\pi$
$318$	0	0
$319$	−6.23607	−0.349153
$320$	0	0
$321$	−14.6180	−0.815899
$322$	0	0
$323$	−17.3607	−0.965974
$324$	0	0
$325$	41.1803	2.28427
$326$	0	0
$327$	17.5066	0.968116
$328$	0	0
$329$	3.70820	0.204440
$330$	0	0
$331$	33.4164	1.83673	0.918366	−	0.395732i	$-0.129509\pi$
$331$	33.4164	1.83673	0.918366	+	0.395732i	$0.129509\pi$
$332$	0	0
$333$	1.47214	0.0806726
$334$	0	0
$335$	−26.5066	−1.44821
$336$	0	0
$337$	−12.9787	−0.706996	−0.353498	−	0.935435i	$-0.615008\pi$
$337$	−12.9787	−0.706996	−0.353498	+	0.935435i	$0.615008\pi$
$338$	0	0
$339$	−3.14590	−0.170862
$340$	0	0
$341$	8.70820	0.471576
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.61803	−0.194788
$346$	0	0
$347$	34.3050	1.84159	0.920793	−	0.390051i	$-0.127543\pi$
$347$	34.3050	1.84159	0.920793	+	0.390051i	$0.127543\pi$
$348$	0	0
$349$	2.96556	0.158743	0.0793713	−	0.996845i	$-0.474709\pi$
$349$	2.96556	0.158743	0.0793713	+	0.996845i	$0.474709\pi$
$350$	0	0
$351$	−5.09017	−0.271693
$352$	0	0
$353$	−15.6525	−0.833097	−0.416549	−	0.909113i	$-0.636760\pi$
$353$	−15.6525	−0.833097	−0.416549	+	0.909113i	$0.636760\pi$
$354$	0	0
$355$	−15.6525	−0.830747
$356$	0	0
$357$	−5.00000	−0.264628
$358$	0	0
$359$	16.5623	0.874125	0.437063	−	0.899431i	$-0.356019\pi$
$359$	16.5623	0.874125	0.437063	+	0.899431i	$0.356019\pi$
$360$	0	0
$361$	−6.94427	−0.365488
$362$	0	0
$363$	10.0000	0.524864
$364$	0	0
$365$	1.90983	0.0999651
$366$	0	0
$367$	27.2705	1.42351	0.711755	−	0.702428i	$-0.247901\pi$
$367$	27.2705	1.42351	0.711755	+	0.702428i	$0.247901\pi$
$368$	0	0
$369$	−5.76393	−0.300058
$370$	0	0
$371$	−0.381966	−0.0198307
$372$	0	0
$373$	32.8885	1.70290	0.851452	−	0.524432i	$-0.175722\pi$
$373$	32.8885	1.70290	0.851452	+	0.524432i	$0.175722\pi$
$374$	0	0
$375$	11.1803	0.577350
$376$	0	0
$377$	−31.7426	−1.63483
$378$	0	0
$379$	17.8885	0.918873	0.459436	−	0.888211i	$-0.348051\pi$
$379$	17.8885	0.918873	0.459436	+	0.888211i	$0.348051\pi$
$380$	0	0
$381$	4.61803	0.236589
$382$	0	0
$383$	16.2361	0.829624	0.414812	−	0.909907i	$-0.363847\pi$
$383$	16.2361	0.829624	0.414812	+	0.909907i	$0.363847\pi$
$384$	0	0
$385$	3.61803	0.184392
$386$	0	0
$387$	−6.32624	−0.321581
$388$	0	0
$389$	20.1246	1.02036	0.510179	−	0.860068i	$-0.329579\pi$
$389$	20.1246	1.02036	0.510179	+	0.860068i	$0.329579\pi$
$390$	0	0
$391$	5.00000	0.252861
$392$	0	0
$393$	−9.18034	−0.463087
$394$	0	0
$395$	−38.7426	−1.94935
$396$	0	0
$397$	−2.18034	−0.109428	−0.0547141	−	0.998502i	$-0.517425\pi$
$397$	−2.18034	−0.109428	−0.0547141	+	0.998502i	$0.517425\pi$
$398$	0	0
$399$	3.47214	0.173824
$400$	0	0
$401$	−0.708204	−0.0353660	−0.0176830	−	0.999844i	$-0.505629\pi$
$401$	−0.708204	−0.0353660	−0.0176830	+	0.999844i	$0.505629\pi$
$402$	0	0
$403$	44.3262	2.20805
$404$	0	0
$405$	−3.61803	−0.179782
$406$	0	0
$407$	1.47214	0.0729711
$408$	0	0
$409$	25.1803	1.24509	0.622544	−	0.782585i	$-0.286099\pi$
$409$	25.1803	1.24509	0.622544	+	0.782585i	$0.286099\pi$
$410$	0	0
$411$	4.52786	0.223343
$412$	0	0
$413$	−3.61803	−0.178032
$414$	0	0
$415$	33.2148	1.63045
$416$	0	0
$417$	0.673762	0.0329943
$418$	0	0
$419$	3.14590	0.153687	0.0768436	−	0.997043i	$-0.475516\pi$
$419$	3.14590	0.153687	0.0768436	+	0.997043i	$0.475516\pi$
$420$	0	0
$421$	38.0344	1.85369	0.926843	−	0.375450i	$-0.122512\pi$
$421$	38.0344	1.85369	0.926843	+	0.375450i	$0.122512\pi$
$422$	0	0
$423$	−3.70820	−0.180299
$424$	0	0
$425$	−40.4508	−1.96215
$426$	0	0
$427$	7.56231	0.365966
$428$	0	0
$429$	−5.09017	−0.245756
$430$	0	0
$431$	−18.0344	−0.868688	−0.434344	−	0.900747i	$-0.643020\pi$
$431$	−18.0344	−0.868688	−0.434344	+	0.900747i	$0.643020\pi$
$432$	0	0
$433$	−5.70820	−0.274319	−0.137159	−	0.990549i	$-0.543797\pi$
$433$	−5.70820	−0.274319	−0.137159	+	0.990549i	$0.543797\pi$
$434$	0	0
$435$	−22.5623	−1.08178
$436$	0	0
$437$	−3.47214	−0.166095
$438$	0	0
$439$	28.7082	1.37017	0.685084	−	0.728464i	$-0.259766\pi$
$439$	28.7082	1.37017	0.685084	+	0.728464i	$0.259766\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	8.94427	0.424955	0.212478	−	0.977166i	$-0.431847\pi$
$443$	8.94427	0.424955	0.212478	+	0.977166i	$0.431847\pi$
$444$	0	0
$445$	60.1246	2.85018
$446$	0	0
$447$	−19.1246	−0.904563
$448$	0	0
$449$	−19.1459	−0.903551	−0.451775	−	0.892132i	$-0.649209\pi$
$449$	−19.1459	−0.903551	−0.451775	+	0.892132i	$0.649209\pi$
$450$	0	0
$451$	−5.76393	−0.271413
$452$	0	0
$453$	5.23607	0.246012
$454$	0	0
$455$	18.4164	0.863375
$456$	0	0
$457$	−6.27051	−0.293322	−0.146661	−	0.989187i	$-0.546853\pi$
$457$	−6.27051	−0.293322	−0.146661	+	0.989187i	$0.546853\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	−3.72949	−0.173700	−0.0868498	−	0.996221i	$-0.527680\pi$
$461$	−3.72949	−0.173700	−0.0868498	+	0.996221i	$0.527680\pi$
$462$	0	0
$463$	−22.5279	−1.04696	−0.523479	−	0.852038i	$-0.675366\pi$
$463$	−22.5279	−1.04696	−0.523479	+	0.852038i	$0.675366\pi$
$464$	0	0
$465$	31.5066	1.46108
$466$	0	0
$467$	−34.5967	−1.60095	−0.800473	−	0.599368i	$-0.795418\pi$
$467$	−34.5967	−1.60095	−0.800473	+	0.599368i	$0.795418\pi$
$468$	0	0
$469$	−7.32624	−0.338294
$470$	0	0
$471$	−14.6525	−0.675150
$472$	0	0
$473$	−6.32624	−0.290881
$474$	0	0
$475$	28.0902	1.28887
$476$	0	0
$477$	0.381966	0.0174890
$478$	0	0
$479$	24.8885	1.13719	0.568593	−	0.822619i	$-0.307488\pi$
$479$	24.8885	1.13719	0.568593	+	0.822619i	$0.307488\pi$
$480$	0	0
$481$	7.49342	0.341671
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	44.2705	2.01022
$486$	0	0
$487$	−42.7771	−1.93841	−0.969207	−	0.246246i	$-0.920803\pi$
$487$	−42.7771	−1.93841	−0.969207	+	0.246246i	$0.920803\pi$
$488$	0	0
$489$	−14.0902	−0.637180
$490$	0	0
$491$	−5.09017	−0.229716	−0.114858	−	0.993382i	$-0.536641\pi$
$491$	−5.09017	−0.229716	−0.114858	+	0.993382i	$0.536641\pi$
$492$	0	0
$493$	31.1803	1.40429
$494$	0	0
$495$	−3.61803	−0.162619
$496$	0	0
$497$	−4.32624	−0.194058
$498$	0	0
$499$	21.0344	0.941631	0.470815	−	0.882232i	$-0.343960\pi$
$499$	21.0344	0.941631	0.470815	+	0.882232i	$0.343960\pi$
$500$	0	0
$501$	−3.18034	−0.142087
$502$	0	0
$503$	19.3262	0.861714	0.430857	−	0.902420i	$-0.358211\pi$
$503$	19.3262	0.861714	0.430857	+	0.902420i	$0.358211\pi$
$504$	0	0
$505$	20.3262	0.904506
$506$	0	0
$507$	−12.9098	−0.573346
$508$	0	0
$509$	6.76393	0.299806	0.149903	−	0.988701i	$-0.452104\pi$
$509$	6.76393	0.299806	0.149903	+	0.988701i	$0.452104\pi$
$510$	0	0
$511$	0.527864	0.0233513
$512$	0	0
$513$	−3.47214	−0.153299
$514$	0	0
$515$	8.94427	0.394132
$516$	0	0
$517$	−3.70820	−0.163087
$518$	0	0
$519$	−9.18034	−0.402972
$520$	0	0
$521$	31.3050	1.37149	0.685747	−	0.727840i	$-0.259475\pi$
$521$	31.3050	1.37149	0.685747	+	0.727840i	$0.259475\pi$
$522$	0	0
$523$	8.47214	0.370461	0.185230	−	0.982695i	$-0.440697\pi$
$523$	8.47214	0.370461	0.185230	+	0.982695i	$0.440697\pi$
$524$	0	0
$525$	8.09017	0.353084
$526$	0	0
$527$	−43.5410	−1.89668
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.61803	0.157009
$532$	0	0
$533$	−29.3394	−1.27083
$534$	0	0
$535$	−52.8885	−2.28657
$536$	0	0
$537$	10.2705	0.443205
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−5.23607	−0.225116	−0.112558	−	0.993645i	$-0.535904\pi$
$541$	−5.23607	−0.225116	−0.112558	+	0.993645i	$0.535904\pi$
$542$	0	0
$543$	0.236068	0.0101306
$544$	0	0
$545$	63.3394	2.71316
$546$	0	0
$547$	32.4508	1.38750	0.693749	−	0.720217i	$-0.255958\pi$
$547$	32.4508	1.38750	0.693749	+	0.720217i	$0.255958\pi$
$548$	0	0
$549$	−7.56231	−0.322751
$550$	0	0
$551$	−21.6525	−0.922426
$552$	0	0
$553$	−10.7082	−0.455359
$554$	0	0
$555$	5.32624	0.226086
$556$	0	0
$557$	−24.1803	−1.02455	−0.512277	−	0.858820i	$-0.671198\pi$
$557$	−24.1803	−1.02455	−0.512277	+	0.858820i	$0.671198\pi$
$558$	0	0
$559$	−32.2016	−1.36198
$560$	0	0
$561$	5.00000	0.211100
$562$	0	0
$563$	−37.5623	−1.58306	−0.791531	−	0.611129i	$-0.790716\pi$
$563$	−37.5623	−1.58306	−0.791531	+	0.611129i	$0.790716\pi$
$564$	0	0
$565$	−11.3820	−0.478843
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−41.8885	−1.75606	−0.878030	−	0.478606i	$-0.841142\pi$
$569$	−41.8885	−1.75606	−0.878030	+	0.478606i	$0.841142\pi$
$570$	0	0
$571$	−21.1803	−0.886370	−0.443185	−	0.896430i	$-0.646151\pi$
$571$	−21.1803	−0.886370	−0.443185	+	0.896430i	$0.646151\pi$
$572$	0	0
$573$	16.6525	0.695667
$574$	0	0
$575$	−8.09017	−0.337383
$576$	0	0
$577$	−5.41641	−0.225488	−0.112744	−	0.993624i	$-0.535964\pi$
$577$	−5.41641	−0.225488	−0.112744	+	0.993624i	$0.535964\pi$
$578$	0	0
$579$	−21.7082	−0.902162
$580$	0	0
$581$	9.18034	0.380865
$582$	0	0
$583$	0.381966	0.0158194
$584$	0	0
$585$	−18.4164	−0.761425
$586$	0	0
$587$	42.6180	1.75903	0.879517	−	0.475867i	$-0.157866\pi$
$587$	42.6180	1.75903	0.879517	+	0.475867i	$0.157866\pi$
$588$	0	0
$589$	30.2361	1.24586
$590$	0	0
$591$	−22.6180	−0.930382
$592$	0	0
$593$	−35.0132	−1.43782	−0.718909	−	0.695104i	$-0.755358\pi$
$593$	−35.0132	−1.43782	−0.718909	+	0.695104i	$0.755358\pi$
$594$	0	0
$595$	−18.0902	−0.741625
$596$	0	0
$597$	3.79837	0.155457
$598$	0	0
$599$	4.79837	0.196056	0.0980281	−	0.995184i	$-0.468746\pi$
$599$	4.79837	0.196056	0.0980281	+	0.995184i	$0.468746\pi$
$600$	0	0
$601$	−29.9098	−1.22005	−0.610024	−	0.792383i	$-0.708840\pi$
$601$	−29.9098	−1.22005	−0.610024	+	0.792383i	$0.708840\pi$
$602$	0	0
$603$	7.32624	0.298347
$604$	0	0
$605$	36.1803	1.47094
$606$	0	0
$607$	−45.3394	−1.84027	−0.920135	−	0.391602i	$-0.871921\pi$
$607$	−45.3394	−1.84027	−0.920135	+	0.391602i	$0.871921\pi$
$608$	0	0
$609$	−6.23607	−0.252698
$610$	0	0
$611$	−18.8754	−0.763616
$612$	0	0
$613$	37.0689	1.49720	0.748599	−	0.663023i	$-0.230727\pi$
$613$	37.0689	1.49720	0.748599	+	0.663023i	$0.230727\pi$
$614$	0	0
$615$	−20.8541	−0.840919
$616$	0	0
$617$	−30.0344	−1.20914	−0.604571	−	0.796552i	$-0.706655\pi$
$617$	−30.0344	−1.20914	−0.604571	+	0.796552i	$0.706655\pi$
$618$	0	0
$619$	−19.9230	−0.800772	−0.400386	−	0.916346i	$-0.631124\pi$
$619$	−19.9230	−0.800772	−0.400386	+	0.916346i	$0.631124\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	16.6180	0.665787
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.47214	−0.138664
$628$	0	0
$629$	−7.36068	−0.293490
$630$	0	0
$631$	−0.527864	−0.0210139	−0.0105070	−	0.999945i	$-0.503345\pi$
$631$	−0.527864	−0.0210139	−0.0105070	+	0.999945i	$0.503345\pi$
$632$	0	0
$633$	18.2361	0.724819
$634$	0	0
$635$	16.7082	0.663045
$636$	0	0
$637$	5.09017	0.201680
$638$	0	0
$639$	4.32624	0.171143
$640$	0	0
$641$	−16.6180	−0.656373	−0.328186	−	0.944613i	$-0.606437\pi$
$641$	−16.6180	−0.656373	−0.328186	+	0.944613i	$0.606437\pi$
$642$	0	0
$643$	10.9098	0.430242	0.215121	−	0.976587i	$-0.430985\pi$
$643$	10.9098	0.430242	0.215121	+	0.976587i	$0.430985\pi$
$644$	0	0
$645$	−22.8885	−0.901236
$646$	0	0
$647$	−15.9098	−0.625480	−0.312740	−	0.949839i	$-0.601247\pi$
$647$	−15.9098	−0.625480	−0.312740	+	0.949839i	$0.601247\pi$
$648$	0	0
$649$	3.61803	0.142020
$650$	0	0
$651$	8.70820	0.341301
$652$	0	0
$653$	34.5623	1.35253	0.676264	−	0.736660i	$-0.263598\pi$
$653$	34.5623	1.35253	0.676264	+	0.736660i	$0.263598\pi$
$654$	0	0
$655$	−33.2148	−1.29781
$656$	0	0
$657$	−0.527864	−0.0205939
$658$	0	0
$659$	1.00000	0.0389545	0.0194772	−	0.999810i	$-0.493800\pi$
$659$	1.00000	0.0389545	0.0194772	+	0.999810i	$0.493800\pi$
$660$	0	0
$661$	−38.7082	−1.50557	−0.752787	−	0.658264i	$-0.771291\pi$
$661$	−38.7082	−1.50557	−0.752787	+	0.658264i	$0.771291\pi$
$662$	0	0
$663$	25.4508	0.988429
$664$	0	0
$665$	12.5623	0.487145
$666$	0	0
$667$	6.23607	0.241462
$668$	0	0
$669$	10.5623	0.408362
$670$	0	0
$671$	−7.56231	−0.291940
$672$	0	0
$673$	29.9443	1.15427	0.577133	−	0.816650i	$-0.304171\pi$
$673$	29.9443	1.15427	0.577133	+	0.816650i	$0.304171\pi$
$674$	0	0
$675$	−8.09017	−0.311391
$676$	0	0
$677$	−0.437694	−0.0168220	−0.00841098	−	0.999965i	$-0.502677\pi$
$677$	−0.437694	−0.0168220	−0.00841098	+	0.999965i	$0.502677\pi$
$678$	0	0
$679$	12.2361	0.469577
$680$	0	0
$681$	−16.8541	−0.645851
$682$	0	0
$683$	15.9443	0.610091	0.305045	−	0.952338i	$-0.401328\pi$
$683$	15.9443	0.610091	0.305045	+	0.952338i	$0.401328\pi$
$684$	0	0
$685$	16.3820	0.625923
$686$	0	0
$687$	7.79837	0.297527
$688$	0	0
$689$	1.94427	0.0740709
$690$	0	0
$691$	−14.2148	−0.540756	−0.270378	−	0.962754i	$-0.587149\pi$
$691$	−14.2148	−0.540756	−0.270378	+	0.962754i	$0.587149\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	2.43769	0.0924670
$696$	0	0
$697$	28.8197	1.09162
$698$	0	0
$699$	−18.5623	−0.702091
$700$	0	0
$701$	−25.9098	−0.978601	−0.489300	−	0.872115i	$-0.662748\pi$
$701$	−25.9098	−0.978601	−0.489300	+	0.872115i	$0.662748\pi$
$702$	0	0
$703$	5.11146	0.192782
$704$	0	0
$705$	−13.4164	−0.505291
$706$	0	0
$707$	5.61803	0.211288
$708$	0	0
$709$	27.1591	1.01998	0.509990	−	0.860180i	$-0.329649\pi$
$709$	27.1591	1.01998	0.509990	+	0.860180i	$0.329649\pi$
$710$	0	0
$711$	10.7082	0.401589
$712$	0	0
$713$	−8.70820	−0.326125
$714$	0	0
$715$	−18.4164	−0.688735
$716$	0	0
$717$	12.6738	0.473310
$718$	0	0
$719$	20.7771	0.774855	0.387427	−	0.921900i	$-0.373364\pi$
$719$	20.7771	0.774855	0.387427	+	0.921900i	$0.373364\pi$
$720$	0	0
$721$	2.47214	0.0920672
$722$	0	0
$723$	25.6525	0.954026
$724$	0	0
$725$	−50.4508	−1.87370
$726$	0	0
$727$	37.2492	1.38150	0.690749	−	0.723095i	$-0.257281\pi$
$727$	37.2492	1.38150	0.690749	+	0.723095i	$0.257281\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.6312	1.16992
$732$	0	0
$733$	2.29180	0.0846494	0.0423247	−	0.999104i	$-0.486524\pi$
$733$	2.29180	0.0846494	0.0423247	+	0.999104i	$0.486524\pi$
$734$	0	0
$735$	3.61803	0.133453
$736$	0	0
$737$	7.32624	0.269865
$738$	0	0
$739$	20.2361	0.744396	0.372198	−	0.928153i	$-0.378604\pi$
$739$	20.2361	0.744396	0.372198	+	0.928153i	$0.378604\pi$
$740$	0	0
$741$	−17.6738	−0.649262
$742$	0	0
$743$	−6.85410	−0.251453	−0.125726	−	0.992065i	$-0.540126\pi$
$743$	−6.85410	−0.251453	−0.125726	+	0.992065i	$0.540126\pi$
$744$	0	0
$745$	−69.1935	−2.53505
$746$	0	0
$747$	−9.18034	−0.335891
$748$	0	0
$749$	−14.6180	−0.534131
$750$	0	0
$751$	22.6738	0.827377	0.413689	−	0.910418i	$-0.364240\pi$
$751$	22.6738	0.827377	0.413689	+	0.910418i	$0.364240\pi$
$752$	0	0
$753$	3.81966	0.139196
$754$	0	0
$755$	18.9443	0.689453
$756$	0	0
$757$	16.4164	0.596664	0.298332	−	0.954462i	$-0.403570\pi$
$757$	16.4164	0.596664	0.298332	+	0.954462i	$0.403570\pi$
$758$	0	0
$759$	1.00000	0.0362977
$760$	0	0
$761$	−7.05573	−0.255770	−0.127885	−	0.991789i	$-0.540819\pi$
$761$	−7.05573	−0.255770	−0.127885	+	0.991789i	$0.540819\pi$
$762$	0	0
$763$	17.5066	0.633781
$764$	0	0
$765$	18.0902	0.654051
$766$	0	0
$767$	18.4164	0.664978
$768$	0	0
$769$	−13.3475	−0.481324	−0.240662	−	0.970609i	$-0.577365\pi$
$769$	−13.3475	−0.481324	−0.240662	+	0.970609i	$0.577365\pi$
$770$	0	0
$771$	0.291796	0.0105088
$772$	0	0
$773$	−1.29180	−0.0464627	−0.0232313	−	0.999730i	$-0.507395\pi$
$773$	−1.29180	−0.0464627	−0.0232313	+	0.999730i	$0.507395\pi$
$774$	0	0
$775$	70.4508	2.53067
$776$	0	0
$777$	1.47214	0.0528126
$778$	0	0
$779$	−20.0132	−0.717046
$780$	0	0
$781$	4.32624	0.154805
$782$	0	0
$783$	6.23607	0.222859
$784$	0	0
$785$	−53.0132	−1.89212
$786$	0	0
$787$	−24.7426	−0.881980	−0.440990	−	0.897512i	$-0.645373\pi$
$787$	−24.7426	−0.881980	−0.440990	+	0.897512i	$0.645373\pi$
$788$	0	0
$789$	−27.1803	−0.967646
$790$	0	0
$791$	−3.14590	−0.111855
$792$	0	0
$793$	−38.4934	−1.36694
$794$	0	0
$795$	1.38197	0.0490133
$796$	0	0
$797$	40.0689	1.41931	0.709656	−	0.704548i	$-0.248850\pi$
$797$	40.0689	1.41931	0.709656	+	0.704548i	$0.248850\pi$
$798$	0	0
$799$	18.5410	0.655934
$800$	0	0
$801$	−16.6180	−0.587169
$802$	0	0
$803$	−0.527864	−0.0186279
$804$	0	0
$805$	−3.61803	−0.127519
$806$	0	0
$807$	−19.8541	−0.698897
$808$	0	0
$809$	47.3820	1.66586	0.832931	−	0.553377i	$-0.186661\pi$
$809$	47.3820	1.66586	0.832931	+	0.553377i	$0.186661\pi$
$810$	0	0
$811$	47.3607	1.66306	0.831529	−	0.555481i	$-0.187466\pi$
$811$	47.3607	1.66306	0.831529	+	0.555481i	$0.187466\pi$
$812$	0	0
$813$	−6.52786	−0.228942
$814$	0	0
$815$	−50.9787	−1.78571
$816$	0	0
$817$	−21.9656	−0.768478
$818$	0	0
$819$	−5.09017	−0.177865
$820$	0	0
$821$	7.34752	0.256430	0.128215	−	0.991746i	$-0.459075\pi$
$821$	7.34752	0.256430	0.128215	+	0.991746i	$0.459075\pi$
$822$	0	0
$823$	13.7984	0.480981	0.240491	−	0.970651i	$-0.422692\pi$
$823$	13.7984	0.480981	0.240491	+	0.970651i	$0.422692\pi$
$824$	0	0
$825$	−8.09017	−0.281664
$826$	0	0
$827$	−53.7984	−1.87075	−0.935376	−	0.353654i	$-0.884939\pi$
$827$	−53.7984	−1.87075	−0.935376	+	0.353654i	$0.884939\pi$
$828$	0	0
$829$	−4.23607	−0.147125	−0.0735624	−	0.997291i	$-0.523437\pi$
$829$	−4.23607	−0.147125	−0.0735624	+	0.997291i	$0.523437\pi$
$830$	0	0
$831$	−31.0902	−1.07851
$832$	0	0
$833$	−5.00000	−0.173240
$834$	0	0
$835$	−11.5066	−0.398202
$836$	0	0
$837$	−8.70820	−0.301000
$838$	0	0
$839$	7.27051	0.251006	0.125503	−	0.992093i	$-0.459946\pi$
$839$	7.27051	0.251006	0.125503	+	0.992093i	$0.459946\pi$
$840$	0	0
$841$	9.88854	0.340984
$842$	0	0
$843$	−8.18034	−0.281746
$844$	0	0
$845$	−46.7082	−1.60681
$846$	0	0
$847$	10.0000	0.343604
$848$	0	0
$849$	2.96556	0.101778
$850$	0	0
$851$	−1.47214	−0.0504642
$852$	0	0
$853$	−44.5410	−1.52506	−0.762528	−	0.646956i	$-0.776042\pi$
$853$	−44.5410	−1.52506	−0.762528	+	0.646956i	$0.776042\pi$
$854$	0	0
$855$	−12.5623	−0.429622
$856$	0	0
$857$	22.6525	0.773794	0.386897	−	0.922123i	$-0.373547\pi$
$857$	22.6525	0.773794	0.386897	+	0.922123i	$0.373547\pi$
$858$	0	0
$859$	−23.8885	−0.815067	−0.407533	−	0.913190i	$-0.633611\pi$
$859$	−23.8885	−0.815067	−0.407533	+	0.913190i	$0.633611\pi$
$860$	0	0
$861$	−5.76393	−0.196434
$862$	0	0
$863$	53.4853	1.82066	0.910330	−	0.413883i	$-0.135828\pi$
$863$	53.4853	1.82066	0.910330	+	0.413883i	$0.135828\pi$
$864$	0	0
$865$	−33.2148	−1.12934
$866$	0	0
$867$	−8.00000	−0.271694
$868$	0	0
$869$	10.7082	0.363251
$870$	0	0
$871$	37.2918	1.26358
$872$	0	0
$873$	−12.2361	−0.414128
$874$	0	0
$875$	11.1803	0.377964
$876$	0	0
$877$	−57.3050	−1.93505	−0.967525	−	0.252774i	$-0.918657\pi$
$877$	−57.3050	−1.93505	−0.967525	+	0.252774i	$0.918657\pi$
$878$	0	0
$879$	−2.47214	−0.0833831
$880$	0	0
$881$	−10.8754	−0.366401	−0.183201	−	0.983076i	$-0.558646\pi$
$881$	−10.8754	−0.366401	−0.183201	+	0.983076i	$0.558646\pi$
$882$	0	0
$883$	−26.3262	−0.885948	−0.442974	−	0.896534i	$-0.646077\pi$
$883$	−26.3262	−0.885948	−0.442974	+	0.896534i	$0.646077\pi$
$884$	0	0
$885$	13.0902	0.440021
$886$	0	0
$887$	14.7295	0.494568	0.247284	−	0.968943i	$-0.420462\pi$
$887$	14.7295	0.494568	0.247284	+	0.968943i	$0.420462\pi$
$888$	0	0
$889$	4.61803	0.154884
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−12.8754	−0.430858
$894$	0	0
$895$	37.1591	1.24209
$896$	0	0
$897$	5.09017	0.169956
$898$	0	0
$899$	−54.3050	−1.81117
$900$	0	0
$901$	−1.90983	−0.0636257
$902$	0	0
$903$	−6.32624	−0.210524
$904$	0	0
$905$	0.854102	0.0283913
$906$	0	0
$907$	48.9230	1.62446	0.812231	−	0.583337i	$-0.198253\pi$
$907$	48.9230	1.62446	0.812231	+	0.583337i	$0.198253\pi$
$908$	0	0
$909$	−5.61803	−0.186338
$910$	0	0
$911$	−42.5410	−1.40945	−0.704723	−	0.709482i	$-0.748929\pi$
$911$	−42.5410	−1.40945	−0.704723	+	0.709482i	$0.748929\pi$
$912$	0	0
$913$	−9.18034	−0.303825
$914$	0	0
$915$	−27.3607	−0.904516
$916$	0	0
$917$	−9.18034	−0.303162
$918$	0	0
$919$	−19.0000	−0.626752	−0.313376	−	0.949629i	$-0.601460\pi$
$919$	−19.0000	−0.626752	−0.313376	+	0.949629i	$0.601460\pi$
$920$	0	0
$921$	29.0689	0.957852
$922$	0	0
$923$	22.0213	0.724839
$924$	0	0
$925$	11.9098	0.391593
$926$	0	0
$927$	−2.47214	−0.0811956
$928$	0	0
$929$	39.0344	1.28068	0.640339	−	0.768092i	$-0.278794\pi$
$929$	39.0344	1.28068	0.640339	+	0.768092i	$0.278794\pi$
$930$	0	0
$931$	3.47214	0.113795
$932$	0	0
$933$	−15.0902	−0.494030
$934$	0	0
$935$	18.0902	0.591612
$936$	0	0
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	0	0
$939$	0.583592	0.0190448
$940$	0	0
$941$	−10.8328	−0.353140	−0.176570	−	0.984288i	$-0.556500\pi$
$941$	−10.8328	−0.353140	−0.176570	+	0.984288i	$0.556500\pi$
$942$	0	0
$943$	5.76393	0.187699
$944$	0	0
$945$	−3.61803	−0.117695
$946$	0	0
$947$	−2.36068	−0.0767118	−0.0383559	−	0.999264i	$-0.512212\pi$
$947$	−2.36068	−0.0767118	−0.0383559	+	0.999264i	$0.512212\pi$
$948$	0	0
$949$	−2.68692	−0.0872210
$950$	0	0
$951$	−26.3820	−0.855494
$952$	0	0
$953$	23.6312	0.765489	0.382745	−	0.923854i	$-0.374979\pi$
$953$	23.6312	0.765489	0.382745	+	0.923854i	$0.374979\pi$
$954$	0	0
$955$	60.2492	1.94962
$956$	0	0
$957$	6.23607	0.201583
$958$	0	0
$959$	4.52786	0.146212
$960$	0	0
$961$	44.8328	1.44622
$962$	0	0
$963$	14.6180	0.471060
$964$	0	0
$965$	−78.5410	−2.52832
$966$	0	0
$967$	46.9443	1.50963	0.754813	−	0.655940i	$-0.227728\pi$
$967$	46.9443	1.50963	0.754813	+	0.655940i	$0.227728\pi$
$968$	0	0
$969$	17.3607	0.557705
$970$	0	0
$971$	−15.7426	−0.505206	−0.252603	−	0.967570i	$-0.581287\pi$
$971$	−15.7426	−0.505206	−0.252603	+	0.967570i	$0.581287\pi$
$972$	0	0
$973$	0.673762	0.0215998
$974$	0	0
$975$	−41.1803	−1.31883
$976$	0	0
$977$	59.7426	1.91134	0.955668	−	0.294445i	$-0.0951349\pi$
$977$	59.7426	1.91134	0.955668	+	0.294445i	$0.0951349\pi$
$978$	0	0
$979$	−16.6180	−0.531115
$980$	0	0
$981$	−17.5066	−0.558942
$982$	0	0
$983$	−8.70820	−0.277749	−0.138874	−	0.990310i	$-0.544348\pi$
$983$	−8.70820	−0.277749	−0.138874	+	0.990310i	$0.544348\pi$
$984$	0	0
$985$	−81.8328	−2.60741
$986$	0	0
$987$	−3.70820	−0.118033
$988$	0	0
$989$	6.32624	0.201163
$990$	0	0
$991$	10.7295	0.340833	0.170417	−	0.985372i	$-0.445489\pi$
$991$	10.7295	0.340833	0.170417	+	0.985372i	$0.445489\pi$
$992$	0	0
$993$	−33.4164	−1.06044
$994$	0	0
$995$	13.7426	0.435671
$996$	0	0
$997$	−17.7639	−0.562589	−0.281295	−	0.959621i	$-0.590764\pi$
$997$	−17.7639	−0.562589	−0.281295	+	0.959621i	$0.590764\pi$
$998$	0	0
$999$	−1.47214	−0.0465763

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.w.1.1		2
4.3	odd	2		1932.2.a.f.1.1	✓	2
12.11	even	2		5796.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.2.a.f.1.1	✓	2	4.3	odd	2
5796.2.a.n.1.2		2	12.11	even	2
7728.2.a.w.1.1		2	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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.61803 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.00000 q^{11} +5.09017 q^{13} +3.61803 q^{15} -5.00000 q^{17} +3.47214 q^{19} +1.00000 q^{21} -1.00000 q^{23} +8.09017 q^{25} -1.00000 q^{27} -6.23607 q^{29} +8.70820 q^{31} -1.00000 q^{33} +3.61803 q^{35} +1.47214 q^{37} -5.09017 q^{39} -5.76393 q^{41} -6.32624 q^{43} -3.61803 q^{45} -3.70820 q^{47} +1.00000 q^{49} +5.00000 q^{51} +0.381966 q^{53} -3.61803 q^{55} -3.47214 q^{57} +3.61803 q^{59} -7.56231 q^{61} -1.00000 q^{63} -18.4164 q^{65} +7.32624 q^{67} +1.00000 q^{69} +4.32624 q^{71} -0.527864 q^{73} -8.09017 q^{75} -1.00000 q^{77} +10.7082 q^{79} +1.00000 q^{81} -9.18034 q^{83} +18.0902 q^{85} +6.23607 q^{87} -16.6180 q^{89} -5.09017 q^{91} -8.70820 q^{93} -12.5623 q^{95} -12.2361 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 5 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - q^{13} + 5 q^{15} - 10 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 5 q^{25} - 2 q^{27} - 8 q^{29} + 4 q^{31} - 2 q^{33} + 5 q^{35} - 6 q^{37} + q^{39} - 16 q^{41} + 3 q^{43} - 5 q^{45} + 6 q^{47} + 2 q^{49} + 10 q^{51} + 3 q^{53} - 5 q^{55} + 2 q^{57} + 5 q^{59} + 5 q^{61} - 2 q^{63} - 10 q^{65} - q^{67} + 2 q^{69} - 7 q^{71} - 10 q^{73} - 5 q^{75} - 2 q^{77} + 8 q^{79} + 2 q^{81} + 4 q^{83} + 25 q^{85} + 8 q^{87} - 31 q^{89} + q^{91} - 4 q^{93} - 5 q^{95} - 20 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 5 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - q^13 + 5 * q^15 - 10 * q^17 - 2 * q^19 + 2 * q^21 - 2 * q^23 + 5 * q^25 - 2 * q^27 - 8 * q^29 + 4 * q^31 - 2 * q^33 + 5 * q^35 - 6 * q^37 + q^39 - 16 * q^41 + 3 * q^43 - 5 * q^45 + 6 * q^47 + 2 * q^49 + 10 * q^51 + 3 * q^53 - 5 * q^55 + 2 * q^57 + 5 * q^59 + 5 * q^61 - 2 * q^63 - 10 * q^65 - q^67 + 2 * q^69 - 7 * q^71 - 10 * q^73 - 5 * q^75 - 2 * q^77 + 8 * q^79 + 2 * q^81 + 4 * q^83 + 25 * q^85 + 8 * q^87 - 31 * q^89 + q^91 - 4 * q^93 - 5 * q^95 - 20 * q^97 + 2 * q^99

Introduction

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