LMFDB - Embedded newform 6786.2.a.br.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6786,2,Mod(1,6786)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6786, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6786.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6786 = 2 \cdot 3^{2} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6786.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:	$54.1864828116$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 27x^{5} - 12x^{4} + 119x^{3} - 77x - 20 $ x^7 - 27x^5 - 12x^4 + 119x^3 - 77x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2262)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.66249$ of defining polynomial
Character	$\chi$	$=$	6786.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.06890 q^{5} -2.55781 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.06890 q^{5} -2.55781 q^{7} -1.00000 q^{8} +2.06890 q^{10} -0.101709 q^{11} +1.00000 q^{13} +2.55781 q^{14} +1.00000 q^{16} -0.781872 q^{17} +3.60533 q^{19} -2.06890 q^{20} +0.101709 q^{22} -6.27747 q^{23} -0.719663 q^{25} -1.00000 q^{26} -2.55781 q^{28} +1.00000 q^{29} +4.08173 q^{31} -1.00000 q^{32} +0.781872 q^{34} +5.29184 q^{35} -5.20482 q^{37} -3.60533 q^{38} +2.06890 q^{40} -9.27153 q^{41} +1.21813 q^{43} -0.101709 q^{44} +6.27747 q^{46} -4.25234 q^{47} -0.457632 q^{49} +0.719663 q^{50} +1.00000 q^{52} -0.896414 q^{53} +0.210426 q^{55} +2.55781 q^{56} -1.00000 q^{58} +13.8548 q^{59} +13.5821 q^{61} -4.08173 q^{62} +1.00000 q^{64} -2.06890 q^{65} -7.74793 q^{67} -0.781872 q^{68} -5.29184 q^{70} -15.8160 q^{71} -3.68891 q^{73} +5.20482 q^{74} +3.60533 q^{76} +0.260153 q^{77} -1.57217 q^{79} -2.06890 q^{80} +9.27153 q^{82} -10.3009 q^{83} +1.61761 q^{85} -1.21813 q^{86} +0.101709 q^{88} +15.3478 q^{89} -2.55781 q^{91} -6.27747 q^{92} +4.25234 q^{94} -7.45905 q^{95} -0.742053 q^{97} +0.457632 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 4 q^{7} - 7 q^{8}+O(q^{10}) $ 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 4 * q^7 - 7 * q^8	$ 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 4 q^{7} - 7 q^{8} + 2 q^{10} - 3 q^{11} + 7 q^{13} - 4 q^{14} + 7 q^{16} - 5 q^{17} + 14 q^{19} - 2 q^{20} + 3 q^{22} + 4 q^{23} + 21 q^{25} - 7 q^{26} + 4 q^{28} + 7 q^{29} + 14 q^{31} - 7 q^{32} + 5 q^{34} - 11 q^{35} - q^{37} - 14 q^{38} + 2 q^{40} + 4 q^{41} + 9 q^{43} - 3 q^{44} - 4 q^{46} - 5 q^{47} + 19 q^{49} - 21 q^{50} + 7 q^{52} - 6 q^{53} - 17 q^{55} - 4 q^{56} - 7 q^{58} - 13 q^{59} - 3 q^{61} - 14 q^{62} + 7 q^{64} - 2 q^{65} + 18 q^{67} - 5 q^{68} + 11 q^{70} + 16 q^{71} - 17 q^{73} + q^{74} + 14 q^{76} - 4 q^{77} + 11 q^{79} - 2 q^{80} - 4 q^{82} - 14 q^{83} - 13 q^{85} - 9 q^{86} + 3 q^{88} + 7 q^{89} + 4 q^{91} + 4 q^{92} + 5 q^{94} + 29 q^{95} - 13 q^{97} - 19 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 4 * q^7 - 7 * q^8 + 2 * q^10 - 3 * q^11 + 7 * q^13 - 4 * q^14 + 7 * q^16 - 5 * q^17 + 14 * q^19 - 2 * q^20 + 3 * q^22 + 4 * q^23 + 21 * q^25 - 7 * q^26 + 4 * q^28 + 7 * q^29 + 14 * q^31 - 7 * q^32 + 5 * q^34 - 11 * q^35 - q^37 - 14 * q^38 + 2 * q^40 + 4 * q^41 + 9 * q^43 - 3 * q^44 - 4 * q^46 - 5 * q^47 + 19 * q^49 - 21 * q^50 + 7 * q^52 - 6 * q^53 - 17 * q^55 - 4 * q^56 - 7 * q^58 - 13 * q^59 - 3 * q^61 - 14 * q^62 + 7 * q^64 - 2 * q^65 + 18 * q^67 - 5 * q^68 + 11 * q^70 + 16 * q^71 - 17 * q^73 + q^74 + 14 * q^76 - 4 * q^77 + 11 * q^79 - 2 * q^80 - 4 * q^82 - 14 * q^83 - 13 * q^85 - 9 * q^86 + 3 * q^88 + 7 * q^89 + 4 * q^91 + 4 * q^92 + 5 * q^94 + 29 * q^95 - 13 * q^97 - 19 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−2.06890	−0.925239	−0.462620	−	0.886557i	$-0.653090\pi$
$5$	−2.06890	−0.925239	−0.462620	+	0.886557i	$0.653090\pi$
$6$	0	0
$7$	−2.55781	−0.966760	−0.483380	−	0.875411i	$-0.660591\pi$
$7$	−2.55781	−0.966760	−0.483380	+	0.875411i	$0.660591\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.06890	0.654243
$11$	−0.101709	−0.0306665	−0.0153333	−	0.999882i	$-0.504881\pi$
$11$	−0.101709	−0.0306665	−0.0153333	+	0.999882i	$0.504881\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.55781	0.683602
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.781872	−0.189632	−0.0948159	−	0.995495i	$-0.530226\pi$
$17$	−0.781872	−0.189632	−0.0948159	+	0.995495i	$0.530226\pi$
$18$	0	0
$19$	3.60533	0.827119	0.413559	−	0.910477i	$-0.364285\pi$
$19$	3.60533	0.827119	0.413559	+	0.910477i	$0.364285\pi$
$20$	−2.06890	−0.462620
$21$	0	0
$22$	0.101709	0.0216845
$23$	−6.27747	−1.30894	−0.654471	−	0.756087i	$-0.727109\pi$
$23$	−6.27747	−1.30894	−0.654471	+	0.756087i	$0.727109\pi$
$24$	0	0
$25$	−0.719663	−0.143933
$26$	−1.00000	−0.196116
$27$	0	0
$28$	−2.55781	−0.483380
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	4.08173	0.733100	0.366550	−	0.930398i	$-0.380539\pi$
$31$	4.08173	0.733100	0.366550	+	0.930398i	$0.380539\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.781872	0.134090
$35$	5.29184	0.894484
$36$	0	0
$37$	−5.20482	−0.855666	−0.427833	−	0.903858i	$-0.640723\pi$
$37$	−5.20482	−0.855666	−0.427833	+	0.903858i	$0.640723\pi$
$38$	−3.60533	−0.584861
$39$	0	0
$40$	2.06890	0.327121
$41$	−9.27153	−1.44797	−0.723985	−	0.689816i	$-0.757691\pi$
$41$	−9.27153	−1.44797	−0.723985	+	0.689816i	$0.757691\pi$
$42$	0	0
$43$	1.21813	0.185763	0.0928814	−	0.995677i	$-0.470392\pi$
$43$	1.21813	0.185763	0.0928814	+	0.995677i	$0.470392\pi$
$44$	−0.101709	−0.0153333
$45$	0	0
$46$	6.27747	0.925562
$47$	−4.25234	−0.620267	−0.310134	−	0.950693i	$-0.600374\pi$
$47$	−4.25234	−0.620267	−0.310134	+	0.950693i	$0.600374\pi$
$48$	0	0
$49$	−0.457632	−0.0653761
$50$	0.719663	0.101776
$51$	0	0
$52$	1.00000	0.138675
$53$	−0.896414	−0.123132	−0.0615660	−	0.998103i	$-0.519609\pi$
$53$	−0.896414	−0.123132	−0.0615660	+	0.998103i	$0.519609\pi$
$54$	0	0
$55$	0.210426	0.0283739
$56$	2.55781	0.341801
$57$	0	0
$58$	−1.00000	−0.131306
$59$	13.8548	1.80374	0.901870	−	0.432007i	$-0.142194\pi$
$59$	13.8548	1.80374	0.901870	+	0.432007i	$0.142194\pi$
$60$	0	0
$61$	13.5821	1.73901	0.869507	−	0.493920i	$-0.164436\pi$
$61$	13.5821	1.73901	0.869507	+	0.493920i	$0.164436\pi$
$62$	−4.08173	−0.518380
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.06890	−0.256615
$66$	0	0
$67$	−7.74793	−0.946561	−0.473280	−	0.880912i	$-0.656930\pi$
$67$	−7.74793	−0.946561	−0.473280	+	0.880912i	$0.656930\pi$
$68$	−0.781872	−0.0948159
$69$	0	0
$70$	−5.29184	−0.632496
$71$	−15.8160	−1.87702	−0.938508	−	0.345257i	$-0.887792\pi$
$71$	−15.8160	−1.87702	−0.938508	+	0.345257i	$0.887792\pi$
$72$	0	0
$73$	−3.68891	−0.431755	−0.215877	−	0.976421i	$-0.569261\pi$
$73$	−3.68891	−0.431755	−0.215877	+	0.976421i	$0.569261\pi$
$74$	5.20482	0.605048
$75$	0	0
$76$	3.60533	0.413559
$77$	0.260153	0.0296472
$78$	0	0
$79$	−1.57217	−0.176883	−0.0884417	−	0.996081i	$-0.528189\pi$
$79$	−1.57217	−0.176883	−0.0884417	+	0.996081i	$0.528189\pi$
$80$	−2.06890	−0.231310
$81$	0	0
$82$	9.27153	1.02387
$83$	−10.3009	−1.13067	−0.565337	−	0.824860i	$-0.691254\pi$
$83$	−10.3009	−1.13067	−0.565337	+	0.824860i	$0.691254\pi$
$84$	0	0
$85$	1.61761	0.175455
$86$	−1.21813	−0.131354
$87$	0	0
$88$	0.101709	0.0108423
$89$	15.3478	1.62686	0.813430	−	0.581663i	$-0.197598\pi$
$89$	15.3478	1.62686	0.813430	+	0.581663i	$0.197598\pi$
$90$	0	0
$91$	−2.55781	−0.268131
$92$	−6.27747	−0.654471
$93$	0	0
$94$	4.25234	0.438595
$95$	−7.45905	−0.765282
$96$	0	0
$97$	−0.742053	−0.0753440	−0.0376720	−	0.999290i	$-0.511994\pi$
$97$	−0.742053	−0.0753440	−0.0376720	+	0.999290i	$0.511994\pi$
$98$	0.457632	0.0462279
$99$	0	0
$100$	−0.719663	−0.0719663
$101$	18.1509	1.80608	0.903041	−	0.429555i	$-0.141329\pi$
$101$	18.1509	1.80608	0.903041	+	0.429555i	$0.141329\pi$
$102$	0	0
$103$	−17.1215	−1.68703	−0.843516	−	0.537104i	$-0.819518\pi$
$103$	−17.1215	−1.68703	−0.843516	+	0.537104i	$0.819518\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	0.896414	0.0870674
$107$	−3.79577	−0.366951	−0.183476	−	0.983024i	$-0.558735\pi$
$107$	−3.79577	−0.366951	−0.183476	+	0.983024i	$0.558735\pi$
$108$	0	0
$109$	−17.3724	−1.66398	−0.831989	−	0.554793i	$-0.812798\pi$
$109$	−17.3724	−1.66398	−0.831989	+	0.554793i	$0.812798\pi$
$110$	−0.210426	−0.0200634
$111$	0	0
$112$	−2.55781	−0.241690
$113$	−7.57137	−0.712254	−0.356127	−	0.934438i	$-0.615903\pi$
$113$	−7.57137	−0.712254	−0.356127	+	0.934438i	$0.615903\pi$
$114$	0	0
$115$	12.9874	1.21108
$116$	1.00000	0.0928477
$117$	0	0
$118$	−13.8548	−1.27544
$119$	1.99988	0.183328
$120$	0	0
$121$	−10.9897	−0.999060
$122$	−13.5821	−1.22967
$123$	0	0
$124$	4.08173	0.366550
$125$	11.8334	1.05841
$126$	0	0
$127$	20.5017	1.81923	0.909615	−	0.415452i	$-0.136377\pi$
$127$	20.5017	1.81923	0.909615	+	0.415452i	$0.136377\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.06890	0.181454
$131$	15.1897	1.32713	0.663563	−	0.748120i	$-0.269043\pi$
$131$	15.1897	1.32713	0.663563	+	0.748120i	$0.269043\pi$
$132$	0	0
$133$	−9.22172	−0.799625
$134$	7.74793	0.669319
$135$	0	0
$136$	0.781872	0.0670450
$137$	−20.7541	−1.77315	−0.886573	−	0.462589i	$-0.846921\pi$
$137$	−20.7541	−1.77315	−0.886573	+	0.462589i	$0.846921\pi$
$138$	0	0
$139$	−0.792827	−0.0672467	−0.0336234	−	0.999435i	$-0.510705\pi$
$139$	−0.792827	−0.0672467	−0.0336234	+	0.999435i	$0.510705\pi$
$140$	5.29184	0.447242
$141$	0	0
$142$	15.8160	1.32725
$143$	−0.101709	−0.00850536
$144$	0	0
$145$	−2.06890	−0.171813
$146$	3.68891	0.305297
$147$	0	0
$148$	−5.20482	−0.427833
$149$	6.42597	0.526436	0.263218	−	0.964736i	$-0.415216\pi$
$149$	6.42597	0.526436	0.263218	+	0.964736i	$0.415216\pi$
$150$	0	0
$151$	−5.82451	−0.473992	−0.236996	−	0.971511i	$-0.576163\pi$
$151$	−5.82451	−0.473992	−0.236996	+	0.971511i	$0.576163\pi$
$152$	−3.60533	−0.292431
$153$	0	0
$154$	−0.260153	−0.0209637
$155$	−8.44468	−0.678293
$156$	0	0
$157$	−4.38136	−0.349671	−0.174835	−	0.984598i	$-0.555939\pi$
$157$	−4.38136	−0.349671	−0.174835	+	0.984598i	$0.555939\pi$
$158$	1.57217	0.125075
$159$	0	0
$160$	2.06890	0.163561
$161$	16.0565	1.26543
$162$	0	0
$163$	3.02940	0.237281	0.118640	−	0.992937i	$-0.462146\pi$
$163$	3.02940	0.237281	0.118640	+	0.992937i	$0.462146\pi$
$164$	−9.27153	−0.723985
$165$	0	0
$166$	10.3009	0.799507
$167$	5.89233	0.455962	0.227981	−	0.973666i	$-0.426788\pi$
$167$	5.89233	0.455962	0.227981	+	0.973666i	$0.426788\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−1.61761	−0.124065
$171$	0	0
$172$	1.21813	0.0928814
$173$	23.6918	1.80125	0.900627	−	0.434592i	$-0.143107\pi$
$173$	23.6918	1.80125	0.900627	+	0.434592i	$0.143107\pi$
$174$	0	0
$175$	1.84076	0.139148
$176$	−0.101709	−0.00766663
$177$	0	0
$178$	−15.3478	−1.15036
$179$	−19.6543	−1.46903	−0.734516	−	0.678591i	$-0.762591\pi$
$179$	−19.6543	−1.46903	−0.734516	+	0.678591i	$0.762591\pi$
$180$	0	0
$181$	9.84121	0.731492	0.365746	−	0.930715i	$-0.380814\pi$
$181$	9.84121	0.731492	0.365746	+	0.930715i	$0.380814\pi$
$182$	2.55781	0.189597
$183$	0	0
$184$	6.27747	0.462781
$185$	10.7682	0.791696
$186$	0	0
$187$	0.0795237	0.00581535
$188$	−4.25234	−0.310134
$189$	0	0
$190$	7.45905	0.541136
$191$	1.18204	0.0855296	0.0427648	−	0.999085i	$-0.486383\pi$
$191$	1.18204	0.0855296	0.0427648	+	0.999085i	$0.486383\pi$
$192$	0	0
$193$	−21.3920	−1.53983	−0.769913	−	0.638149i	$-0.779700\pi$
$193$	−21.3920	−1.53983	−0.769913	+	0.638149i	$0.779700\pi$
$194$	0.742053	0.0532763
$195$	0	0
$196$	−0.457632	−0.0326880
$197$	24.4321	1.74071	0.870357	−	0.492420i	$-0.163888\pi$
$197$	24.4321	1.74071	0.870357	+	0.492420i	$0.163888\pi$
$198$	0	0
$199$	19.1314	1.35619	0.678094	−	0.734975i	$-0.262806\pi$
$199$	19.1314	1.35619	0.678094	+	0.734975i	$0.262806\pi$
$200$	0.719663	0.0508878
$201$	0	0
$202$	−18.1509	−1.27709
$203$	−2.55781	−0.179523
$204$	0	0
$205$	19.1818	1.33972
$206$	17.1215	1.19291
$207$	0	0
$208$	1.00000	0.0693375
$209$	−0.366695	−0.0253648
$210$	0	0
$211$	−0.942538	−0.0648870	−0.0324435	−	0.999474i	$-0.510329\pi$
$211$	−0.942538	−0.0648870	−0.0324435	+	0.999474i	$0.510329\pi$
$212$	−0.896414	−0.0615660
$213$	0	0
$214$	3.79577	0.259474
$215$	−2.52018	−0.171875
$216$	0	0
$217$	−10.4403	−0.708732
$218$	17.3724	1.17661
$219$	0	0
$220$	0.210426	0.0141869
$221$	−0.781872	−0.0525944
$222$	0	0
$223$	2.15810	0.144517	0.0722586	−	0.997386i	$-0.476979\pi$
$223$	2.15810	0.144517	0.0722586	+	0.997386i	$0.476979\pi$
$224$	2.55781	0.170901
$225$	0	0
$226$	7.57137	0.503640
$227$	9.96666	0.661511	0.330755	−	0.943717i	$-0.392696\pi$
$227$	9.96666	0.661511	0.330755	+	0.943717i	$0.392696\pi$
$228$	0	0
$229$	−12.4867	−0.825146	−0.412573	−	0.910925i	$-0.635370\pi$
$229$	−12.4867	−0.825146	−0.412573	+	0.910925i	$0.635370\pi$
$230$	−12.9874	−0.856366
$231$	0	0
$232$	−1.00000	−0.0656532
$233$	2.30359	0.150913	0.0754566	−	0.997149i	$-0.475959\pi$
$233$	2.30359	0.150913	0.0754566	+	0.997149i	$0.475959\pi$
$234$	0	0
$235$	8.79765	0.573895
$236$	13.8548	0.901870
$237$	0	0
$238$	−1.99988	−0.129633
$239$	12.7714	0.826113	0.413056	−	0.910705i	$-0.364461\pi$
$239$	12.7714	0.826113	0.413056	+	0.910705i	$0.364461\pi$
$240$	0	0
$241$	17.4820	1.12611	0.563056	−	0.826419i	$-0.309626\pi$
$241$	17.4820	1.12611	0.563056	+	0.826419i	$0.309626\pi$
$242$	10.9897	0.706442
$243$	0	0
$244$	13.5821	0.869507
$245$	0.946795	0.0604885
$246$	0	0
$247$	3.60533	0.229401
$248$	−4.08173	−0.259190
$249$	0	0
$250$	−11.8334	−0.748410
$251$	3.93438	0.248336	0.124168	−	0.992261i	$-0.460374\pi$
$251$	3.93438	0.248336	0.124168	+	0.992261i	$0.460374\pi$
$252$	0	0
$253$	0.638477	0.0401407
$254$	−20.5017	−1.28639
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.3153	−0.768206	−0.384103	−	0.923290i	$-0.625489\pi$
$257$	−12.3153	−0.768206	−0.384103	+	0.923290i	$0.625489\pi$
$258$	0	0
$259$	13.3129	0.827224
$260$	−2.06890	−0.128308
$261$	0	0
$262$	−15.1897	−0.938420
$263$	22.1750	1.36737	0.683683	−	0.729779i	$-0.260377\pi$
$263$	22.1750	1.36737	0.683683	+	0.729779i	$0.260377\pi$
$264$	0	0
$265$	1.85459	0.113926
$266$	9.22172	0.565420
$267$	0	0
$268$	−7.74793	−0.473280
$269$	−2.04811	−0.124875	−0.0624377	−	0.998049i	$-0.519887\pi$
$269$	−2.04811	−0.124875	−0.0624377	+	0.998049i	$0.519887\pi$
$270$	0	0
$271$	1.58720	0.0964157	0.0482079	−	0.998837i	$-0.484649\pi$
$271$	1.58720	0.0964157	0.0482079	+	0.998837i	$0.484649\pi$
$272$	−0.781872	−0.0474080
$273$	0	0
$274$	20.7541	1.25380
$275$	0.0731964	0.00441391
$276$	0	0
$277$	28.8443	1.73309	0.866543	−	0.499102i	$-0.166337\pi$
$277$	28.8443	1.73309	0.866543	+	0.499102i	$0.166337\pi$
$278$	0.792827	0.0475506
$279$	0	0
$280$	−5.29184	−0.316248
$281$	−15.0248	−0.896304	−0.448152	−	0.893957i	$-0.647918\pi$
$281$	−15.0248	−0.896304	−0.448152	+	0.893957i	$0.647918\pi$
$282$	0	0
$283$	−1.42549	−0.0847366	−0.0423683	−	0.999102i	$-0.513490\pi$
$283$	−1.42549	−0.0847366	−0.0423683	+	0.999102i	$0.513490\pi$
$284$	−15.8160	−0.938508
$285$	0	0
$286$	0.101709	0.00601420
$287$	23.7148	1.39984
$288$	0	0
$289$	−16.3887	−0.964040
$290$	2.06890	0.121490
$291$	0	0
$292$	−3.68891	−0.215877
$293$	15.2717	0.892183	0.446092	−	0.894987i	$-0.352816\pi$
$293$	15.2717	0.892183	0.446092	+	0.894987i	$0.352816\pi$
$294$	0	0
$295$	−28.6641	−1.66889
$296$	5.20482	0.302524
$297$	0	0
$298$	−6.42597	−0.372246
$299$	−6.27747	−0.363035
$300$	0	0
$301$	−3.11573	−0.179588
$302$	5.82451	0.335163
$303$	0	0
$304$	3.60533	0.206780
$305$	−28.1001	−1.60900
$306$	0	0
$307$	27.9743	1.59658	0.798288	−	0.602276i	$-0.205739\pi$
$307$	27.9743	1.59658	0.798288	+	0.602276i	$0.205739\pi$
$308$	0.260153	0.0148236
$309$	0	0
$310$	8.44468	0.479626
$311$	10.6280	0.602658	0.301329	−	0.953520i	$-0.402570\pi$
$311$	10.6280	0.602658	0.301329	+	0.953520i	$0.402570\pi$
$312$	0	0
$313$	22.4713	1.27015	0.635075	−	0.772450i	$-0.280969\pi$
$313$	22.4713	1.27015	0.635075	+	0.772450i	$0.280969\pi$
$314$	4.38136	0.247255
$315$	0	0
$316$	−1.57217	−0.0884417
$317$	14.7001	0.825639	0.412819	−	0.910813i	$-0.364544\pi$
$317$	14.7001	0.825639	0.412819	+	0.910813i	$0.364544\pi$
$318$	0	0
$319$	−0.101709	−0.00569463
$320$	−2.06890	−0.115655
$321$	0	0
$322$	−16.0565	−0.894796
$323$	−2.81890	−0.156848
$324$	0	0
$325$	−0.719663	−0.0399197
$326$	−3.02940	−0.167783
$327$	0	0
$328$	9.27153	0.511935
$329$	10.8766	0.599649
$330$	0	0
$331$	20.7754	1.14192	0.570959	−	0.820979i	$-0.306572\pi$
$331$	20.7754	1.14192	0.570959	+	0.820979i	$0.306572\pi$
$332$	−10.3009	−0.565337
$333$	0	0
$334$	−5.89233	−0.322414
$335$	16.0297	0.875795
$336$	0	0
$337$	7.87063	0.428741	0.214370	−	0.976752i	$-0.431230\pi$
$337$	7.87063	0.428741	0.214370	+	0.976752i	$0.431230\pi$
$338$	−1.00000	−0.0543928
$339$	0	0
$340$	1.61761	0.0877274
$341$	−0.415150	−0.0224816
$342$	0	0
$343$	19.0752	1.02996
$344$	−1.21813	−0.0656771
$345$	0	0
$346$	−23.6918	−1.27368
$347$	−22.0267	−1.18245	−0.591227	−	0.806505i	$-0.701356\pi$
$347$	−22.0267	−1.18245	−0.591227	+	0.806505i	$0.701356\pi$
$348$	0	0
$349$	−13.4586	−0.720422	−0.360211	−	0.932871i	$-0.617295\pi$
$349$	−13.4586	−0.720422	−0.360211	+	0.932871i	$0.617295\pi$
$350$	−1.84076	−0.0983926
$351$	0	0
$352$	0.101709	0.00542113
$353$	−9.60793	−0.511379	−0.255689	−	0.966759i	$-0.582302\pi$
$353$	−9.60793	−0.511379	−0.255689	+	0.966759i	$0.582302\pi$
$354$	0	0
$355$	32.7217	1.73669
$356$	15.3478	0.813430
$357$	0	0
$358$	19.6543	1.03876
$359$	27.5091	1.45188	0.725938	−	0.687760i	$-0.241406\pi$
$359$	27.5091	1.45188	0.725938	+	0.687760i	$0.241406\pi$
$360$	0	0
$361$	−6.00162	−0.315875
$362$	−9.84121	−0.517243
$363$	0	0
$364$	−2.55781	−0.134065
$365$	7.63198	0.399476
$366$	0	0
$367$	24.5874	1.28345	0.641726	−	0.766934i	$-0.278219\pi$
$367$	24.5874	1.28345	0.641726	+	0.766934i	$0.278219\pi$
$368$	−6.27747	−0.327236
$369$	0	0
$370$	−10.7682	−0.559814
$371$	2.29285	0.119039
$372$	0	0
$373$	−12.1712	−0.630201	−0.315101	−	0.949058i	$-0.602038\pi$
$373$	−12.1712	−0.630201	−0.315101	+	0.949058i	$0.602038\pi$
$374$	−0.0795237	−0.00411207
$375$	0	0
$376$	4.25234	0.219298
$377$	1.00000	0.0515026
$378$	0	0
$379$	5.59778	0.287539	0.143769	−	0.989611i	$-0.454078\pi$
$379$	5.59778	0.287539	0.143769	+	0.989611i	$0.454078\pi$
$380$	−7.45905	−0.382641
$381$	0	0
$382$	−1.18204	−0.0604785
$383$	32.6602	1.66886	0.834429	−	0.551116i	$-0.185798\pi$
$383$	32.6602	1.66886	0.834429	+	0.551116i	$0.185798\pi$
$384$	0	0
$385$	−0.538229	−0.0274307
$386$	21.3920	1.08882
$387$	0	0
$388$	−0.742053	−0.0376720
$389$	1.34497	0.0681926	0.0340963	−	0.999419i	$-0.489145\pi$
$389$	1.34497	0.0681926	0.0340963	+	0.999419i	$0.489145\pi$
$390$	0	0
$391$	4.90818	0.248217
$392$	0.457632	0.0231139
$393$	0	0
$394$	−24.4321	−1.23087
$395$	3.25267	0.163659
$396$	0	0
$397$	26.1137	1.31061	0.655304	−	0.755365i	$-0.272540\pi$
$397$	26.1137	1.31061	0.655304	+	0.755365i	$0.272540\pi$
$398$	−19.1314	−0.958970
$399$	0	0
$400$	−0.719663	−0.0359831
$401$	28.0758	1.40204	0.701020	−	0.713141i	$-0.252728\pi$
$401$	28.0758	1.40204	0.701020	+	0.713141i	$0.252728\pi$
$402$	0	0
$403$	4.08173	0.203325
$404$	18.1509	0.903041
$405$	0	0
$406$	2.55781	0.126942
$407$	0.529378	0.0262403
$408$	0	0
$409$	−33.9658	−1.67950	−0.839751	−	0.542971i	$-0.817299\pi$
$409$	−33.9658	−1.67950	−0.839751	+	0.542971i	$0.817299\pi$
$410$	−19.1818	−0.947324
$411$	0	0
$412$	−17.1215	−0.843516
$413$	−35.4379	−1.74378
$414$	0	0
$415$	21.3116	1.04614
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	0.366695	0.0179357
$419$	34.1568	1.66867	0.834333	−	0.551260i	$-0.185853\pi$
$419$	34.1568	1.66867	0.834333	+	0.551260i	$0.185853\pi$
$420$	0	0
$421$	−18.2931	−0.891549	−0.445774	−	0.895145i	$-0.647072\pi$
$421$	−18.2931	−0.891549	−0.445774	+	0.895145i	$0.647072\pi$
$422$	0.942538	0.0458820
$423$	0	0
$424$	0.896414	0.0435337
$425$	0.562684	0.0272942
$426$	0	0
$427$	−34.7405	−1.68121
$428$	−3.79577	−0.183476
$429$	0	0
$430$	2.52018	0.121534
$431$	−38.9306	−1.87522	−0.937612	−	0.347684i	$-0.886968\pi$
$431$	−38.9306	−1.87522	−0.937612	+	0.347684i	$0.886968\pi$
$432$	0	0
$433$	2.51376	0.120803	0.0604017	−	0.998174i	$-0.480762\pi$
$433$	2.51376	0.120803	0.0604017	+	0.998174i	$0.480762\pi$
$434$	10.4403	0.501149
$435$	0	0
$436$	−17.3724	−0.831989
$437$	−22.6323	−1.08265
$438$	0	0
$439$	−24.5708	−1.17270	−0.586351	−	0.810057i	$-0.699436\pi$
$439$	−24.5708	−1.17270	−0.586351	+	0.810057i	$0.699436\pi$
$440$	−0.210426	−0.0100317
$441$	0	0
$442$	0.781872	0.0371899
$443$	25.8803	1.22961	0.614804	−	0.788680i	$-0.289235\pi$
$443$	25.8803	1.22961	0.614804	+	0.788680i	$0.289235\pi$
$444$	0	0
$445$	−31.7529	−1.50523
$446$	−2.15810	−0.102189
$447$	0	0
$448$	−2.55781	−0.120845
$449$	7.71916	0.364290	0.182145	−	0.983272i	$-0.441696\pi$
$449$	7.71916	0.364290	0.182145	+	0.983272i	$0.441696\pi$
$450$	0	0
$451$	0.943001	0.0444042
$452$	−7.57137	−0.356127
$453$	0	0
$454$	−9.96666	−0.467759
$455$	5.29184	0.248085
$456$	0	0
$457$	−27.3070	−1.27737	−0.638684	−	0.769469i	$-0.720521\pi$
$457$	−27.3070	−1.27737	−0.638684	+	0.769469i	$0.720521\pi$
$458$	12.4867	0.583466
$459$	0	0
$460$	12.9874	0.605542
$461$	5.02954	0.234249	0.117124	−	0.993117i	$-0.462632\pi$
$461$	5.02954	0.234249	0.117124	+	0.993117i	$0.462632\pi$
$462$	0	0
$463$	−24.3838	−1.13321	−0.566605	−	0.823990i	$-0.691743\pi$
$463$	−24.3838	−1.13321	−0.566605	+	0.823990i	$0.691743\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−2.30359	−0.106712
$467$	−26.6527	−1.23334	−0.616671	−	0.787221i	$-0.711519\pi$
$467$	−26.6527	−1.23334	−0.616671	+	0.787221i	$0.711519\pi$
$468$	0	0
$469$	19.8177	0.915097
$470$	−8.79765	−0.405805
$471$	0	0
$472$	−13.8548	−0.637719
$473$	−0.123895	−0.00569670
$474$	0	0
$475$	−2.59462	−0.119049
$476$	1.99988	0.0916642
$477$	0	0
$478$	−12.7714	−0.584150
$479$	24.6977	1.12847	0.564233	−	0.825616i	$-0.309172\pi$
$479$	24.6977	1.12847	0.564233	+	0.825616i	$0.309172\pi$
$480$	0	0
$481$	−5.20482	−0.237319
$482$	−17.4820	−0.796281
$483$	0	0
$484$	−10.9897	−0.499530
$485$	1.53523	0.0697113
$486$	0	0
$487$	−21.8211	−0.988807	−0.494403	−	0.869233i	$-0.664613\pi$
$487$	−21.8211	−0.988807	−0.494403	+	0.869233i	$0.664613\pi$
$488$	−13.5821	−0.614834
$489$	0	0
$490$	−0.946795	−0.0427718
$491$	−2.27345	−0.102600	−0.0512998	−	0.998683i	$-0.516336\pi$
$491$	−2.27345	−0.102600	−0.0512998	+	0.998683i	$0.516336\pi$
$492$	0	0
$493$	−0.781872	−0.0352137
$494$	−3.60533	−0.162211
$495$	0	0
$496$	4.08173	0.183275
$497$	40.4543	1.81462
$498$	0	0
$499$	−3.35216	−0.150063	−0.0750316	−	0.997181i	$-0.523906\pi$
$499$	−3.35216	−0.150063	−0.0750316	+	0.997181i	$0.523906\pi$
$500$	11.8334	0.529206
$501$	0	0
$502$	−3.93438	−0.175600
$503$	−27.8521	−1.24187	−0.620933	−	0.783864i	$-0.713246\pi$
$503$	−27.8521	−1.24187	−0.620933	+	0.783864i	$0.713246\pi$
$504$	0	0
$505$	−37.5523	−1.67106
$506$	−0.638477	−0.0283838
$507$	0	0
$508$	20.5017	0.909615
$509$	−30.2212	−1.33953	−0.669765	−	0.742573i	$-0.733605\pi$
$509$	−30.2212	−1.33953	−0.669765	+	0.742573i	$0.733605\pi$
$510$	0	0
$511$	9.43552	0.417403
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	12.3153	0.543203
$515$	35.4227	1.56091
$516$	0	0
$517$	0.432502	0.0190214
$518$	−13.3129	−0.584935
$519$	0	0
$520$	2.06890	0.0907272
$521$	−0.642970	−0.0281690	−0.0140845	−	0.999901i	$-0.504483\pi$
$521$	−0.642970	−0.0281690	−0.0140845	+	0.999901i	$0.504483\pi$
$522$	0	0
$523$	−10.0670	−0.440201	−0.220101	−	0.975477i	$-0.570639\pi$
$523$	−10.0670	−0.440201	−0.220101	+	0.975477i	$0.570639\pi$
$524$	15.1897	0.663563
$525$	0	0
$526$	−22.1750	−0.966874
$527$	−3.19139	−0.139019
$528$	0	0
$529$	16.4066	0.713331
$530$	−1.85459	−0.0805582
$531$	0	0
$532$	−9.22172	−0.399812
$533$	−9.27153	−0.401595
$534$	0	0
$535$	7.85307	0.339518
$536$	7.74793	0.334660
$537$	0	0
$538$	2.04811	0.0883003
$539$	0.0465455	0.00200486
$540$	0	0
$541$	29.8203	1.28208	0.641038	−	0.767509i	$-0.278504\pi$
$541$	29.8203	1.28208	0.641038	+	0.767509i	$0.278504\pi$
$542$	−1.58720	−0.0681762
$543$	0	0
$544$	0.781872	0.0335225
$545$	35.9418	1.53958
$546$	0	0
$547$	31.3840	1.34188	0.670942	−	0.741509i	$-0.265890\pi$
$547$	31.3840	1.34188	0.670942	+	0.741509i	$0.265890\pi$
$548$	−20.7541	−0.886573
$549$	0	0
$550$	−0.0731964	−0.00312111
$551$	3.60533	0.153592
$552$	0	0
$553$	4.02132	0.171004
$554$	−28.8443	−1.22548
$555$	0	0
$556$	−0.792827	−0.0336234
$557$	14.6204	0.619484	0.309742	−	0.950821i	$-0.399757\pi$
$557$	14.6204	0.619484	0.309742	+	0.950821i	$0.399757\pi$
$558$	0	0
$559$	1.21813	0.0515213
$560$	5.29184	0.223621
$561$	0	0
$562$	15.0248	0.633783
$563$	0.554936	0.0233878	0.0116939	−	0.999932i	$-0.496278\pi$
$563$	0.554936	0.0233878	0.0116939	+	0.999932i	$0.496278\pi$
$564$	0	0
$565$	15.6644	0.659005
$566$	1.42549	0.0599178
$567$	0	0
$568$	15.8160	0.663626
$569$	36.2265	1.51870	0.759348	−	0.650685i	$-0.225518\pi$
$569$	36.2265	1.51870	0.759348	+	0.650685i	$0.225518\pi$
$570$	0	0
$571$	5.03812	0.210839	0.105419	−	0.994428i	$-0.466381\pi$
$571$	5.03812	0.210839	0.105419	+	0.994428i	$0.466381\pi$
$572$	−0.101709	−0.00425268
$573$	0	0
$574$	−23.7148	−0.989835
$575$	4.51766	0.188399
$576$	0	0
$577$	−25.0829	−1.04422	−0.522108	−	0.852879i	$-0.674854\pi$
$577$	−25.0829	−1.04422	−0.522108	+	0.852879i	$0.674854\pi$
$578$	16.3887	0.681679
$579$	0	0
$580$	−2.06890	−0.0859063
$581$	26.3478	1.09309
$582$	0	0
$583$	0.0911737	0.00377603
$584$	3.68891	0.152648
$585$	0	0
$586$	−15.2717	−0.630869
$587$	−17.5742	−0.725365	−0.362683	−	0.931913i	$-0.618139\pi$
$587$	−17.5742	−0.725365	−0.362683	+	0.931913i	$0.618139\pi$
$588$	0	0
$589$	14.7160	0.606361
$590$	28.6641	1.18008
$591$	0	0
$592$	−5.20482	−0.213917
$593$	−4.14375	−0.170164	−0.0850818	−	0.996374i	$-0.527115\pi$
$593$	−4.14375	−0.170164	−0.0850818	+	0.996374i	$0.527115\pi$
$594$	0	0
$595$	−4.13754	−0.169623
$596$	6.42597	0.263218
$597$	0	0
$598$	6.27747	0.256705
$599$	4.99034	0.203900	0.101950	−	0.994790i	$-0.467492\pi$
$599$	4.99034	0.203900	0.101950	+	0.994790i	$0.467492\pi$
$600$	0	0
$601$	31.4903	1.28452	0.642258	−	0.766489i	$-0.277998\pi$
$601$	31.4903	1.28452	0.642258	+	0.766489i	$0.277998\pi$
$602$	3.11573	0.126988
$603$	0	0
$604$	−5.82451	−0.236996
$605$	22.7365	0.924369
$606$	0	0
$607$	1.52544	0.0619157	0.0309579	−	0.999521i	$-0.490144\pi$
$607$	1.52544	0.0619157	0.0309579	+	0.999521i	$0.490144\pi$
$608$	−3.60533	−0.146215
$609$	0	0
$610$	28.1001	1.13774
$611$	−4.25234	−0.172031
$612$	0	0
$613$	32.6328	1.31803	0.659014	−	0.752131i	$-0.270974\pi$
$613$	32.6328	1.31803	0.659014	+	0.752131i	$0.270974\pi$
$614$	−27.9743	−1.12895
$615$	0	0
$616$	−0.260153	−0.0104819
$617$	12.9135	0.519880	0.259940	−	0.965625i	$-0.416297\pi$
$617$	12.9135	0.519880	0.259940	+	0.965625i	$0.416297\pi$
$618$	0	0
$619$	22.2938	0.896065	0.448033	−	0.894017i	$-0.352125\pi$
$619$	22.2938	0.896065	0.448033	+	0.894017i	$0.352125\pi$
$620$	−8.44468	−0.339147
$621$	0	0
$622$	−10.6280	−0.426143
$623$	−39.2566	−1.57278
$624$	0	0
$625$	−20.8838	−0.835351
$626$	−22.4713	−0.898132
$627$	0	0
$628$	−4.38136	−0.174835
$629$	4.06950	0.162262
$630$	0	0
$631$	24.0170	0.956101	0.478050	−	0.878332i	$-0.341344\pi$
$631$	24.0170	0.956101	0.478050	+	0.878332i	$0.341344\pi$
$632$	1.57217	0.0625377
$633$	0	0
$634$	−14.7001	−0.583815
$635$	−42.4159	−1.68322
$636$	0	0
$637$	−0.457632	−0.0181321
$638$	0.101709	0.00402671
$639$	0	0
$640$	2.06890	0.0817804
$641$	−16.3802	−0.646981	−0.323490	−	0.946231i	$-0.604856\pi$
$641$	−16.3802	−0.646981	−0.323490	+	0.946231i	$0.604856\pi$
$642$	0	0
$643$	6.57207	0.259177	0.129589	−	0.991568i	$-0.458634\pi$
$643$	6.57207	0.259177	0.129589	+	0.991568i	$0.458634\pi$
$644$	16.0565	0.632716
$645$	0	0
$646$	2.81890	0.110908
$647$	19.5565	0.768847	0.384423	−	0.923157i	$-0.374400\pi$
$647$	19.5565	0.768847	0.384423	+	0.923157i	$0.374400\pi$
$648$	0	0
$649$	−1.40916	−0.0553144
$650$	0.719663	0.0282275
$651$	0	0
$652$	3.02940	0.118640
$653$	6.50349	0.254501	0.127251	−	0.991871i	$-0.459385\pi$
$653$	6.50349	0.254501	0.127251	+	0.991871i	$0.459385\pi$
$654$	0	0
$655$	−31.4259	−1.22791
$656$	−9.27153	−0.361992
$657$	0	0
$658$	−10.8766	−0.424016
$659$	−3.74003	−0.145691	−0.0728455	−	0.997343i	$-0.523208\pi$
$659$	−3.74003	−0.145691	−0.0728455	+	0.997343i	$0.523208\pi$
$660$	0	0
$661$	17.9505	0.698192	0.349096	−	0.937087i	$-0.386489\pi$
$661$	17.9505	0.698192	0.349096	+	0.937087i	$0.386489\pi$
$662$	−20.7754	−0.807458
$663$	0	0
$664$	10.3009	0.399753
$665$	19.0788	0.739844
$666$	0	0
$667$	−6.27747	−0.243065
$668$	5.89233	0.227981
$669$	0	0
$670$	−16.0297	−0.619281
$671$	−1.38143	−0.0533295
$672$	0	0
$673$	45.3710	1.74892	0.874461	−	0.485095i	$-0.161215\pi$
$673$	45.3710	1.74892	0.874461	+	0.485095i	$0.161215\pi$
$674$	−7.87063	−0.303165
$675$	0	0
$676$	1.00000	0.0384615
$677$	35.1306	1.35018	0.675089	−	0.737736i	$-0.264105\pi$
$677$	35.1306	1.35018	0.675089	+	0.737736i	$0.264105\pi$
$678$	0	0
$679$	1.89803	0.0728396
$680$	−1.61761	−0.0620326
$681$	0	0
$682$	0.415150	0.0158969
$683$	−10.5951	−0.405410	−0.202705	−	0.979240i	$-0.564973\pi$
$683$	−10.5951	−0.405410	−0.202705	+	0.979240i	$0.564973\pi$
$684$	0	0
$685$	42.9382	1.64058
$686$	−19.0752	−0.728293
$687$	0	0
$688$	1.21813	0.0464407
$689$	−0.896414	−0.0341507
$690$	0	0
$691$	−38.6675	−1.47098	−0.735490	−	0.677535i	$-0.763048\pi$
$691$	−38.6675	−1.47098	−0.735490	+	0.677535i	$0.763048\pi$
$692$	23.6918	0.900627
$693$	0	0
$694$	22.0267	0.836122
$695$	1.64028	0.0622193
$696$	0	0
$697$	7.24915	0.274581
$698$	13.4586	0.509415
$699$	0	0
$700$	1.84076	0.0695741
$701$	39.8649	1.50568	0.752839	−	0.658205i	$-0.228684\pi$
$701$	39.8649	1.50568	0.752839	+	0.658205i	$0.228684\pi$
$702$	0	0
$703$	−18.7651	−0.707738
$704$	−0.101709	−0.00383332
$705$	0	0
$706$	9.60793	0.361599
$707$	−46.4265	−1.74605
$708$	0	0
$709$	−11.6437	−0.437290	−0.218645	−	0.975805i	$-0.570164\pi$
$709$	−11.6437	−0.437290	−0.218645	+	0.975805i	$0.570164\pi$
$710$	−32.7217	−1.22802
$711$	0	0
$712$	−15.3478	−0.575182
$713$	−25.6229	−0.959586
$714$	0	0
$715$	0.210426	0.00786949
$716$	−19.6543	−0.734516
$717$	0	0
$718$	−27.5091	−1.02663
$719$	12.4019	0.462512	0.231256	−	0.972893i	$-0.425717\pi$
$719$	12.4019	0.462512	0.231256	+	0.972893i	$0.425717\pi$
$720$	0	0
$721$	43.7935	1.63095
$722$	6.00162	0.223357
$723$	0	0
$724$	9.84121	0.365746
$725$	−0.719663	−0.0267276
$726$	0	0
$727$	12.3175	0.456832	0.228416	−	0.973564i	$-0.426645\pi$
$727$	12.3175	0.456832	0.228416	+	0.973564i	$0.426645\pi$
$728$	2.55781	0.0947986
$729$	0	0
$730$	−7.63198	−0.282472
$731$	−0.952420	−0.0352265
$732$	0	0
$733$	−32.9838	−1.21828	−0.609142	−	0.793061i	$-0.708486\pi$
$733$	−32.9838	−1.21828	−0.609142	+	0.793061i	$0.708486\pi$
$734$	−24.5874	−0.907538
$735$	0	0
$736$	6.27747	0.231391
$737$	0.788037	0.0290277
$738$	0	0
$739$	13.6260	0.501242	0.250621	−	0.968085i	$-0.419365\pi$
$739$	13.6260	0.501242	0.250621	+	0.968085i	$0.419365\pi$
$740$	10.7682	0.395848
$741$	0	0
$742$	−2.29285	−0.0841733
$743$	22.9226	0.840949	0.420474	−	0.907304i	$-0.361864\pi$
$743$	22.9226	0.840949	0.420474	+	0.907304i	$0.361864\pi$
$744$	0	0
$745$	−13.2947	−0.487079
$746$	12.1712	0.445620
$747$	0	0
$748$	0.0795237	0.00290767
$749$	9.70885	0.354754
$750$	0	0
$751$	41.0480	1.49786	0.748931	−	0.662648i	$-0.230567\pi$
$751$	41.0480	1.49786	0.748931	+	0.662648i	$0.230567\pi$
$752$	−4.25234	−0.155067
$753$	0	0
$754$	−1.00000	−0.0364179
$755$	12.0503	0.438556
$756$	0	0
$757$	16.0434	0.583109	0.291554	−	0.956554i	$-0.405828\pi$
$757$	16.0434	0.583109	0.291554	+	0.956554i	$0.405828\pi$
$758$	−5.59778	−0.203321
$759$	0	0
$760$	7.45905	0.270568
$761$	−48.9262	−1.77357	−0.886786	−	0.462180i	$-0.847067\pi$
$761$	−48.9262	−1.77357	−0.886786	+	0.462180i	$0.847067\pi$
$762$	0	0
$763$	44.4353	1.60867
$764$	1.18204	0.0427648
$765$	0	0
$766$	−32.6602	−1.18006
$767$	13.8548	0.500268
$768$	0	0
$769$	34.7245	1.25220	0.626099	−	0.779744i	$-0.284651\pi$
$769$	34.7245	1.25220	0.626099	+	0.779744i	$0.284651\pi$
$770$	0.538229	0.0193964
$771$	0	0
$772$	−21.3920	−0.769913
$773$	0.919502	0.0330722	0.0165361	−	0.999863i	$-0.494736\pi$
$773$	0.919502	0.0330722	0.0165361	+	0.999863i	$0.494736\pi$
$774$	0	0
$775$	−2.93747	−0.105517
$776$	0.742053	0.0266381
$777$	0	0
$778$	−1.34497	−0.0482194
$779$	−33.4269	−1.19764
$780$	0	0
$781$	1.60864	0.0575616
$782$	−4.90818	−0.175516
$783$	0	0
$784$	−0.457632	−0.0163440
$785$	9.06459	0.323529
$786$	0	0
$787$	−17.6308	−0.628468	−0.314234	−	0.949345i	$-0.601748\pi$
$787$	−17.6308	−0.628468	−0.314234	+	0.949345i	$0.601748\pi$
$788$	24.4321	0.870357
$789$	0	0
$790$	−3.25267	−0.115725
$791$	19.3661	0.688579
$792$	0	0
$793$	13.5821	0.482316
$794$	−26.1137	−0.926740
$795$	0	0
$796$	19.1314	0.678094
$797$	−17.8549	−0.632454	−0.316227	−	0.948684i	$-0.602416\pi$
$797$	−17.8549	−0.632454	−0.316227	+	0.948684i	$0.602416\pi$
$798$	0	0
$799$	3.32478	0.117622
$800$	0.719663	0.0254439
$801$	0	0
$802$	−28.0758	−0.991392
$803$	0.375197	0.0132404
$804$	0	0
$805$	−33.2193	−1.17083
$806$	−4.08173	−0.143773
$807$	0	0
$808$	−18.1509	−0.638546
$809$	47.8703	1.68303	0.841514	−	0.540235i	$-0.181665\pi$
$809$	47.8703	1.68303	0.841514	+	0.540235i	$0.181665\pi$
$810$	0	0
$811$	−50.3679	−1.76866	−0.884329	−	0.466865i	$-0.845384\pi$
$811$	−50.3679	−1.76866	−0.884329	+	0.466865i	$0.845384\pi$
$812$	−2.55781	−0.0897614
$813$	0	0
$814$	−0.529378	−0.0185547
$815$	−6.26751	−0.219541
$816$	0	0
$817$	4.39175	0.153648
$818$	33.9658	1.18759
$819$	0	0
$820$	19.1818	0.669859
$821$	−20.5700	−0.717898	−0.358949	−	0.933357i	$-0.616865\pi$
$821$	−20.5700	−0.717898	−0.358949	+	0.933357i	$0.616865\pi$
$822$	0	0
$823$	42.6012	1.48498	0.742492	−	0.669855i	$-0.233644\pi$
$823$	42.6012	1.48498	0.742492	+	0.669855i	$0.233644\pi$
$824$	17.1215	0.596456
$825$	0	0
$826$	35.4379	1.23304
$827$	4.48728	0.156038	0.0780190	−	0.996952i	$-0.475141\pi$
$827$	4.48728	0.156038	0.0780190	+	0.996952i	$0.475141\pi$
$828$	0	0
$829$	−25.4610	−0.884296	−0.442148	−	0.896942i	$-0.645783\pi$
$829$	−25.4610	−0.884296	−0.442148	+	0.896942i	$0.645783\pi$
$830$	−21.3116	−0.739735
$831$	0	0
$832$	1.00000	0.0346688
$833$	0.357810	0.0123974
$834$	0	0
$835$	−12.1906	−0.421874
$836$	−0.366695	−0.0126824
$837$	0	0
$838$	−34.1568	−1.17993
$839$	−38.6210	−1.33334	−0.666672	−	0.745351i	$-0.732282\pi$
$839$	−38.6210	−1.33334	−0.666672	+	0.745351i	$0.732282\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	18.2931	0.630420
$843$	0	0
$844$	−0.942538	−0.0324435
$845$	−2.06890	−0.0711722
$846$	0	0
$847$	28.1094	0.965850
$848$	−0.896414	−0.0307830
$849$	0	0
$850$	−0.562684	−0.0192999
$851$	32.6731	1.12002
$852$	0	0
$853$	−55.0128	−1.88360	−0.941801	−	0.336171i	$-0.890868\pi$
$853$	−55.0128	−1.88360	−0.941801	+	0.336171i	$0.890868\pi$
$854$	34.7405	1.18879
$855$	0	0
$856$	3.79577	0.129737
$857$	−21.3803	−0.730336	−0.365168	−	0.930942i	$-0.618988\pi$
$857$	−21.3803	−0.730336	−0.365168	+	0.930942i	$0.618988\pi$
$858$	0	0
$859$	13.4461	0.458776	0.229388	−	0.973335i	$-0.426328\pi$
$859$	13.4461	0.458776	0.229388	+	0.973335i	$0.426328\pi$
$860$	−2.52018	−0.0859375
$861$	0	0
$862$	38.9306	1.32598
$863$	14.4916	0.493300	0.246650	−	0.969105i	$-0.420670\pi$
$863$	14.4916	0.493300	0.246650	+	0.969105i	$0.420670\pi$
$864$	0	0
$865$	−49.0159	−1.66659
$866$	−2.51376	−0.0854209
$867$	0	0
$868$	−10.4403	−0.354366
$869$	0.159905	0.00542440
$870$	0	0
$871$	−7.74793	−0.262529
$872$	17.3724	0.588305
$873$	0	0
$874$	22.6323	0.765550
$875$	−30.2675	−1.02323
$876$	0	0
$877$	−34.8254	−1.17597	−0.587984	−	0.808872i	$-0.700078\pi$
$877$	−34.8254	−1.17597	−0.587984	+	0.808872i	$0.700078\pi$
$878$	24.5708	0.829225
$879$	0	0
$880$	0.210426	0.00709347
$881$	33.9159	1.14266	0.571329	−	0.820721i	$-0.306428\pi$
$881$	33.9159	1.14266	0.571329	+	0.820721i	$0.306428\pi$
$882$	0	0
$883$	31.1634	1.04873	0.524365	−	0.851494i	$-0.324303\pi$
$883$	31.1634	1.04873	0.524365	+	0.851494i	$0.324303\pi$
$884$	−0.781872	−0.0262972
$885$	0	0
$886$	−25.8803	−0.869464
$887$	30.1701	1.01301	0.506506	−	0.862236i	$-0.330937\pi$
$887$	30.1701	1.01301	0.506506	+	0.862236i	$0.330937\pi$
$888$	0	0
$889$	−52.4393	−1.75876
$890$	31.7529	1.06436
$891$	0	0
$892$	2.15810	0.0722586
$893$	−15.3311	−0.513034
$894$	0	0
$895$	40.6628	1.35921
$896$	2.55781	0.0854503
$897$	0	0
$898$	−7.71916	−0.257592
$899$	4.08173	0.136133
$900$	0	0
$901$	0.700881	0.0233497
$902$	−0.943001	−0.0313985
$903$	0	0
$904$	7.57137	0.251820
$905$	−20.3605	−0.676805
$906$	0	0
$907$	46.3265	1.53825	0.769124	−	0.639100i	$-0.220693\pi$
$907$	46.3265	1.53825	0.769124	+	0.639100i	$0.220693\pi$
$908$	9.96666	0.330755
$909$	0	0
$910$	−5.29184	−0.175423
$911$	−42.0322	−1.39259	−0.696295	−	0.717756i	$-0.745169\pi$
$911$	−42.0322	−1.39259	−0.696295	+	0.717756i	$0.745169\pi$
$912$	0	0
$913$	1.04770	0.0346738
$914$	27.3070	0.903235
$915$	0	0
$916$	−12.4867	−0.412573
$917$	−38.8522	−1.28301
$918$	0	0
$919$	−3.34093	−0.110207	−0.0551036	−	0.998481i	$-0.517549\pi$
$919$	−3.34093	−0.110207	−0.0551036	+	0.998481i	$0.517549\pi$
$920$	−12.9874	−0.428183
$921$	0	0
$922$	−5.02954	−0.165639
$923$	−15.8160	−0.520591
$924$	0	0
$925$	3.74571	0.123158
$926$	24.3838	0.801300
$927$	0	0
$928$	−1.00000	−0.0328266
$929$	10.3624	0.339978	0.169989	−	0.985446i	$-0.445627\pi$
$929$	10.3624	0.339978	0.169989	+	0.985446i	$0.445627\pi$
$930$	0	0
$931$	−1.64991	−0.0540738
$932$	2.30359	0.0754566
$933$	0	0
$934$	26.6527	0.872104
$935$	−0.164526	−0.00538059
$936$	0	0
$937$	51.1636	1.67144	0.835721	−	0.549155i	$-0.185050\pi$
$937$	51.1636	1.67144	0.835721	+	0.549155i	$0.185050\pi$
$938$	−19.8177	−0.647071
$939$	0	0
$940$	8.79765	0.286948
$941$	−25.3374	−0.825974	−0.412987	−	0.910737i	$-0.635515\pi$
$941$	−25.3374	−0.825974	−0.412987	+	0.910737i	$0.635515\pi$
$942$	0	0
$943$	58.2017	1.89531
$944$	13.8548	0.450935
$945$	0	0
$946$	0.123895	0.00402817
$947$	14.0947	0.458016	0.229008	−	0.973425i	$-0.426452\pi$
$947$	14.0947	0.458016	0.229008	+	0.973425i	$0.426452\pi$
$948$	0	0
$949$	−3.68891	−0.119747
$950$	2.59462	0.0841806
$951$	0	0
$952$	−1.99988	−0.0648164
$953$	−56.1370	−1.81846	−0.909228	−	0.416299i	$-0.863327\pi$
$953$	−56.1370	−1.81846	−0.909228	+	0.416299i	$0.863327\pi$
$954$	0	0
$955$	−2.44552	−0.0791353
$956$	12.7714	0.413056
$957$	0	0
$958$	−24.6977	−0.797946
$959$	53.0851	1.71421
$960$	0	0
$961$	−14.3395	−0.462564
$962$	5.20482	0.167810
$963$	0	0
$964$	17.4820	0.563056
$965$	44.2578	1.42471
$966$	0	0
$967$	9.91117	0.318722	0.159361	−	0.987220i	$-0.449057\pi$
$967$	9.91117	0.318722	0.159361	+	0.987220i	$0.449057\pi$
$968$	10.9897	0.353221
$969$	0	0
$970$	−1.53523	−0.0492933
$971$	41.0389	1.31700	0.658501	−	0.752580i	$-0.271191\pi$
$971$	41.0389	1.31700	0.658501	+	0.752580i	$0.271191\pi$
$972$	0	0
$973$	2.02790	0.0650114
$974$	21.8211	0.699192
$975$	0	0
$976$	13.5821	0.434754
$977$	−38.8849	−1.24404	−0.622019	−	0.783002i	$-0.713687\pi$
$977$	−38.8849	−1.24404	−0.622019	+	0.783002i	$0.713687\pi$
$978$	0	0
$979$	−1.56101	−0.0498901
$980$	0.946795	0.0302442
$981$	0	0
$982$	2.27345	0.0725488
$983$	−53.4483	−1.70474	−0.852368	−	0.522943i	$-0.824834\pi$
$983$	−53.4483	−1.70474	−0.852368	+	0.522943i	$0.824834\pi$
$984$	0	0
$985$	−50.5475	−1.61058
$986$	0.781872	0.0248999
$987$	0	0
$988$	3.60533	0.114701
$989$	−7.64676	−0.243153
$990$	0	0
$991$	−18.4908	−0.587380	−0.293690	−	0.955901i	$-0.594883\pi$
$991$	−18.4908	−0.587380	−0.293690	+	0.955901i	$0.594883\pi$
$992$	−4.08173	−0.129595
$993$	0	0
$994$	−40.4543	−1.28313
$995$	−39.5809	−1.25480
$996$	0	0
$997$	−9.63927	−0.305279	−0.152639	−	0.988282i	$-0.548777\pi$
$997$	−9.63927	−0.305279	−0.152639	+	0.988282i	$0.548777\pi$
$998$	3.35216	0.106111
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6786.2.a.br.1.3		7
3.2	odd	2		2262.2.a.z.1.5	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2262.2.a.z.1.5	✓	7	3.2	odd	2
6786.2.a.br.1.3		7	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:	\( N \)	\(=\)	\( 6786 = 2 \cdot 3^{2} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6786.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.06890 q^{5} -2.55781 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.06890 q^{5} -2.55781 q^{7} -1.00000 q^{8} +2.06890 q^{10} -0.101709 q^{11} +1.00000 q^{13} +2.55781 q^{14} +1.00000 q^{16} -0.781872 q^{17} +3.60533 q^{19} -2.06890 q^{20} +0.101709 q^{22} -6.27747 q^{23} -0.719663 q^{25} -1.00000 q^{26} -2.55781 q^{28} +1.00000 q^{29} +4.08173 q^{31} -1.00000 q^{32} +0.781872 q^{34} +5.29184 q^{35} -5.20482 q^{37} -3.60533 q^{38} +2.06890 q^{40} -9.27153 q^{41} +1.21813 q^{43} -0.101709 q^{44} +6.27747 q^{46} -4.25234 q^{47} -0.457632 q^{49} +0.719663 q^{50} +1.00000 q^{52} -0.896414 q^{53} +0.210426 q^{55} +2.55781 q^{56} -1.00000 q^{58} +13.8548 q^{59} +13.5821 q^{61} -4.08173 q^{62} +1.00000 q^{64} -2.06890 q^{65} -7.74793 q^{67} -0.781872 q^{68} -5.29184 q^{70} -15.8160 q^{71} -3.68891 q^{73} +5.20482 q^{74} +3.60533 q^{76} +0.260153 q^{77} -1.57217 q^{79} -2.06890 q^{80} +9.27153 q^{82} -10.3009 q^{83} +1.61761 q^{85} -1.21813 q^{86} +0.101709 q^{88} +15.3478 q^{89} -2.55781 q^{91} -6.27747 q^{92} +4.25234 q^{94} -7.45905 q^{95} -0.742053 q^{97} +0.457632 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 4 q^{7} - 7 q^{8}+O(q^{10}) \) 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 4 * q^7 - 7 * q^8	\( 7 q - 7 q^{2} + 7 q^{4} - 2 q^{5} + 4 q^{7} - 7 q^{8} + 2 q^{10} - 3 q^{11} + 7 q^{13} - 4 q^{14} + 7 q^{16} - 5 q^{17} + 14 q^{19} - 2 q^{20} + 3 q^{22} + 4 q^{23} + 21 q^{25} - 7 q^{26} + 4 q^{28} + 7 q^{29} + 14 q^{31} - 7 q^{32} + 5 q^{34} - 11 q^{35} - q^{37} - 14 q^{38} + 2 q^{40} + 4 q^{41} + 9 q^{43} - 3 q^{44} - 4 q^{46} - 5 q^{47} + 19 q^{49} - 21 q^{50} + 7 q^{52} - 6 q^{53} - 17 q^{55} - 4 q^{56} - 7 q^{58} - 13 q^{59} - 3 q^{61} - 14 q^{62} + 7 q^{64} - 2 q^{65} + 18 q^{67} - 5 q^{68} + 11 q^{70} + 16 q^{71} - 17 q^{73} + q^{74} + 14 q^{76} - 4 q^{77} + 11 q^{79} - 2 q^{80} - 4 q^{82} - 14 q^{83} - 13 q^{85} - 9 q^{86} + 3 q^{88} + 7 q^{89} + 4 q^{91} + 4 q^{92} + 5 q^{94} + 29 q^{95} - 13 q^{97} - 19 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 - 2 * q^5 + 4 * q^7 - 7 * q^8 + 2 * q^10 - 3 * q^11 + 7 * q^13 - 4 * q^14 + 7 * q^16 - 5 * q^17 + 14 * q^19 - 2 * q^20 + 3 * q^22 + 4 * q^23 + 21 * q^25 - 7 * q^26 + 4 * q^28 + 7 * q^29 + 14 * q^31 - 7 * q^32 + 5 * q^34 - 11 * q^35 - q^37 - 14 * q^38 + 2 * q^40 + 4 * q^41 + 9 * q^43 - 3 * q^44 - 4 * q^46 - 5 * q^47 + 19 * q^49 - 21 * q^50 + 7 * q^52 - 6 * q^53 - 17 * q^55 - 4 * q^56 - 7 * q^58 - 13 * q^59 - 3 * q^61 - 14 * q^62 + 7 * q^64 - 2 * q^65 + 18 * q^67 - 5 * q^68 + 11 * q^70 + 16 * q^71 - 17 * q^73 + q^74 + 14 * q^76 - 4 * q^77 + 11 * q^79 - 2 * q^80 - 4 * q^82 - 14 * q^83 - 13 * q^85 - 9 * q^86 + 3 * q^88 + 7 * q^89 + 4 * q^91 + 4 * q^92 + 5 * q^94 + 29 * q^95 - 13 * q^97 - 19 * q^98

Introduction

L-functions

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