LMFDB - Embedded newform 1792.3.g.f.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(127,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1792.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$-1.92812i$ of defining polynomial
Character	$\chi$	$=$	1792.127
Dual form			1792.3.g.f.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.85623 q^{3} -0.490168i q^{5} -2.64575i q^{7} +5.87054 q^{9} +O(q^{10})$	$q-3.85623 q^{3} -0.490168i q^{5} -2.64575i q^{7} +5.87054 q^{9} +15.5633 q^{11} +3.50983i q^{13} +1.89020i q^{15} -24.1463 q^{17} -3.56870 q^{19} +10.2026i q^{21} +19.5741i q^{23} +24.7597 q^{25} +12.0679 q^{27} -10.9803i q^{29} +21.1767i q^{31} -60.0159 q^{33} -1.29686 q^{35} -58.4212i q^{37} -13.5347i q^{39} -54.1285 q^{41} -35.6420 q^{43} -2.87755i q^{45} +64.2248i q^{47} -7.00000 q^{49} +93.1140 q^{51} -87.4015i q^{53} -7.62865i q^{55} +13.7617 q^{57} +66.6954 q^{59} +16.8615i q^{61} -15.5320i q^{63} +1.72041 q^{65} -21.2420 q^{67} -75.4822i q^{69} +64.2140i q^{71} -99.4587 q^{73} -95.4793 q^{75} -41.1767i q^{77} -139.441i q^{79} -99.3716 q^{81} -6.03134 q^{83} +11.8358i q^{85} +42.3427i q^{87} +23.9821 q^{89} +9.28614 q^{91} -81.6624i q^{93} +1.74926i q^{95} +171.509 q^{97} +91.3652 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 40 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 40 * q^9	$ 8 q + 8 q^{3} + 40 q^{9} + 32 q^{11} - 16 q^{17} + 88 q^{19} - 104 q^{25} + 176 q^{27} + 56 q^{35} - 144 q^{41} - 224 q^{43} - 56 q^{49} + 16 q^{51} + 400 q^{57} + 232 q^{59} - 304 q^{65} - 368 q^{67} - 272 q^{73} - 664 q^{75} + 504 q^{81} - 424 q^{83} + 80 q^{89} + 56 q^{91} + 528 q^{97} + 544 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + 40 * q^9 + 32 * q^11 - 16 * q^17 + 88 * q^19 - 104 * q^25 + 176 * q^27 + 56 * q^35 - 144 * q^41 - 224 * q^43 - 56 * q^49 + 16 * q^51 + 400 * q^57 + 232 * q^59 - 304 * q^65 - 368 * q^67 - 272 * q^73 - 664 * q^75 + 504 * q^81 - 424 * q^83 + 80 * q^89 + 56 * q^91 + 528 * q^97 + 544 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−3.85623	−1.28541	−0.642706	−	0.766113i	$-0.722188\pi$
$3$	−3.85623	−1.28541	−0.642706	+	0.766113i	$0.722188\pi$
$4$	0	0
$5$	− 0.490168i	− 0.0980336i	−0.998798	−	0.0490168i	$-0.984391\pi$
$5$	− 0.490168i	− 0.0980336i	0.998798	−	0.0490168i	$-0.0156088\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	5.87054	0.652282
$10$	0	0
$11$	15.5633	1.41485	0.707425	−	0.706789i	$-0.249857\pi$
$11$	15.5633	1.41485	0.707425	+	0.706789i	$0.249857\pi$
$12$	0	0
$13$	3.50983i	0.269987i	0.990846	+	0.134994i	$0.0431014\pi$
$13$	3.50983i	0.269987i	−0.990846	+	0.134994i	$0.956899\pi$
$14$	0	0
$15$	1.89020i	0.126013i
$16$	0	0
$17$	−24.1463	−1.42037	−0.710187	−	0.704013i	$-0.751389\pi$
$17$	−24.1463	−1.42037	−0.710187	+	0.704013i	$0.751389\pi$
$18$	0	0
$19$	−3.56870	−0.187826	−0.0939132	−	0.995580i	$-0.529938\pi$
$19$	−3.56870	−0.187826	−0.0939132	+	0.995580i	$0.529938\pi$
$20$	0	0
$21$	10.2026i	0.485840i
$22$	0	0
$23$	19.5741i	0.851046i	0.904948	+	0.425523i	$0.139910\pi$
$23$	19.5741i	0.851046i	−0.904948	+	0.425523i	$0.860090\pi$
$24$	0	0
$25$	24.7597	0.990389
$26$	0	0
$27$	12.0679	0.446961
$28$	0	0
$29$	− 10.9803i	− 0.378632i	−0.981916	−	0.189316i	$-0.939373\pi$
$29$	− 10.9803i	− 0.378632i	0.981916	−	0.189316i	$-0.0606271\pi$
$30$	0	0
$31$	21.1767i	0.683120i	0.939860	+	0.341560i	$0.110955\pi$
$31$	21.1767i	0.683120i	−0.939860	+	0.341560i	$0.889045\pi$
$32$	0	0
$33$	−60.0159	−1.81866
$34$	0	0
$35$	−1.29686	−0.0370532
$36$	0	0
$37$	− 58.4212i	− 1.57895i	−0.613783	−	0.789475i	$-0.710353\pi$
$37$	− 58.4212i	− 1.57895i	0.613783	−	0.789475i	$-0.289647\pi$
$38$	0	0
$39$	− 13.5347i	− 0.347044i
$40$	0	0
$41$	−54.1285	−1.32021	−0.660103	−	0.751175i	$-0.729487\pi$
$41$	−54.1285	−1.32021	−0.660103	+	0.751175i	$0.729487\pi$
$42$	0	0
$43$	−35.6420	−0.828884	−0.414442	−	0.910076i	$-0.636023\pi$
$43$	−35.6420	−0.828884	−0.414442	+	0.910076i	$0.636023\pi$
$44$	0	0
$45$	− 2.87755i	− 0.0639455i
$46$	0	0
$47$	64.2248i	1.36648i	0.730192	+	0.683242i	$0.239431\pi$
$47$	64.2248i	1.36648i	−0.730192	+	0.683242i	$0.760569\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	93.1140	1.82576
$52$	0	0
$53$	− 87.4015i	− 1.64908i	−0.565800	−	0.824542i	$-0.691433\pi$
$53$	− 87.4015i	− 1.64908i	0.565800	−	0.824542i	$-0.308567\pi$
$54$	0	0
$55$	− 7.62865i	− 0.138703i
$56$	0	0
$57$	13.7617	0.241434
$58$	0	0
$59$	66.6954	1.13043	0.565215	−	0.824944i	$-0.308793\pi$
$59$	66.6954	1.13043	0.565215	+	0.824944i	$0.308793\pi$
$60$	0	0
$61$	16.8615i	0.276418i	0.990403	+	0.138209i	$0.0441346\pi$
$61$	16.8615i	0.276418i	−0.990403	+	0.138209i	$0.955865\pi$
$62$	0	0
$63$	− 15.5320i	− 0.246539i
$64$	0	0
$65$	1.72041	0.0264678
$66$	0	0
$67$	−21.2420	−0.317045	−0.158523	−	0.987355i	$-0.550673\pi$
$67$	−21.2420	−0.317045	−0.158523	+	0.987355i	$0.550673\pi$
$68$	0	0
$69$	− 75.4822i	− 1.09394i
$70$	0	0
$71$	64.2140i	0.904423i	0.891911	+	0.452212i	$0.149365\pi$
$71$	64.2140i	0.904423i	−0.891911	+	0.452212i	$0.850635\pi$
$72$	0	0
$73$	−99.4587	−1.36245	−0.681224	−	0.732075i	$-0.738552\pi$
$73$	−99.4587	−1.36245	−0.681224	+	0.732075i	$0.738552\pi$
$74$	0	0
$75$	−95.4793	−1.27306
$76$	0	0
$77$	− 41.1767i	− 0.534763i
$78$	0	0
$79$	− 139.441i	− 1.76507i	−0.470243	−	0.882537i	$-0.655834\pi$
$79$	− 139.441i	− 1.76507i	0.470243	−	0.882537i	$-0.344166\pi$
$80$	0	0
$81$	−99.3716	−1.22681
$82$	0	0
$83$	−6.03134	−0.0726668	−0.0363334	−	0.999340i	$-0.511568\pi$
$83$	−6.03134	−0.0726668	−0.0363334	+	0.999340i	$0.511568\pi$
$84$	0	0
$85$	11.8358i	0.139244i
$86$	0	0
$87$	42.3427i	0.486698i
$88$	0	0
$89$	23.9821	0.269462	0.134731	−	0.990882i	$-0.456983\pi$
$89$	23.9821	0.269462	0.134731	+	0.990882i	$0.456983\pi$
$90$	0	0
$91$	9.28614	0.102046
$92$	0	0
$93$	− 81.6624i	− 0.878091i
$94$	0	0
$95$	1.74926i	0.0184133i
$96$	0	0
$97$	171.509	1.76813	0.884064	−	0.467365i	$-0.154797\pi$
$97$	171.509	1.76813	0.884064	+	0.467365i	$0.154797\pi$
$98$	0	0
$99$	91.3652	0.922881
$100$	0	0
$101$	146.427i	1.44977i	0.688867	+	0.724887i	$0.258108\pi$
$101$	146.427i	1.44977i	−0.688867	+	0.724887i	$0.741892\pi$
$102$	0	0
$103$	118.849i	1.15388i	0.816787	+	0.576939i	$0.195753\pi$
$103$	118.849i	1.15388i	−0.816787	+	0.576939i	$0.804247\pi$
$104$	0	0
$105$	5.00100	0.0476286
$106$	0	0
$107$	142.434	1.33116	0.665581	−	0.746326i	$-0.268184\pi$
$107$	142.434	1.33116	0.665581	+	0.746326i	$0.268184\pi$
$108$	0	0
$109$	170.835i	1.56730i	0.621204	+	0.783649i	$0.286644\pi$
$109$	170.835i	1.56730i	−0.621204	+	0.783649i	$0.713356\pi$
$110$	0	0
$111$	225.286i	2.02960i
$112$	0	0
$113$	24.9436	0.220740	0.110370	−	0.993891i	$-0.464796\pi$
$113$	24.9436	0.220740	0.110370	+	0.993891i	$0.464796\pi$
$114$	0	0
$115$	9.59458	0.0834311
$116$	0	0
$117$	20.6046i	0.176108i
$118$	0	0
$119$	63.8852i	0.536851i
$120$	0	0
$121$	121.218	1.00180
$122$	0	0
$123$	208.732	1.69701
$124$	0	0
$125$	− 24.3906i	− 0.195125i
$126$	0	0
$127$	68.8755i	0.542326i	0.962533	+	0.271163i	$0.0874083\pi$
$127$	68.8755i	0.542326i	−0.962533	+	0.271163i	$0.912592\pi$
$128$	0	0
$129$	137.444	1.06546
$130$	0	0
$131$	101.326	0.773481	0.386741	−	0.922189i	$-0.373601\pi$
$131$	101.326	0.773481	0.386741	+	0.922189i	$0.373601\pi$
$132$	0	0
$133$	9.44190i	0.0709917i
$134$	0	0
$135$	− 5.91532i	− 0.0438172i
$136$	0	0
$137$	−63.8820	−0.466292	−0.233146	−	0.972442i	$-0.574902\pi$
$137$	−63.8820	−0.466292	−0.233146	+	0.972442i	$0.574902\pi$
$138$	0	0
$139$	199.256	1.43349	0.716747	−	0.697333i	$-0.245630\pi$
$139$	199.256	1.43349	0.716747	+	0.697333i	$0.245630\pi$
$140$	0	0
$141$	− 247.666i	− 1.75649i
$142$	0	0
$143$	54.6247i	0.381991i
$144$	0	0
$145$	−5.38221	−0.0371187
$146$	0	0
$147$	26.9936	0.183630
$148$	0	0
$149$	271.585i	1.82272i	0.411612	+	0.911359i	$0.364966\pi$
$149$	271.585i	1.82272i	−0.411612	+	0.911359i	$0.635034\pi$
$150$	0	0
$151$	131.329i	0.869729i	0.900496	+	0.434864i	$0.143204\pi$
$151$	131.329i	0.869729i	−0.900496	+	0.434864i	$0.856796\pi$
$152$	0	0
$153$	−141.752	−0.926484
$154$	0	0
$155$	10.3802	0.0669687
$156$	0	0
$157$	133.685i	0.851494i	0.904842	+	0.425747i	$0.139989\pi$
$157$	133.685i	0.851494i	−0.904842	+	0.425747i	$0.860011\pi$
$158$	0	0
$159$	337.041i	2.11975i
$160$	0	0
$161$	51.7881	0.321665
$162$	0	0
$163$	29.8167	0.182925	0.0914623	−	0.995809i	$-0.470846\pi$
$163$	29.8167	0.182925	0.0914623	+	0.995809i	$0.470846\pi$
$164$	0	0
$165$	29.4179i	0.178290i
$166$	0	0
$167$	− 259.708i	− 1.55514i	−0.628796	−	0.777570i	$-0.716452\pi$
$167$	− 259.708i	− 1.55514i	0.628796	−	0.777570i	$-0.283548\pi$
$168$	0	0
$169$	156.681	0.927107
$170$	0	0
$171$	−20.9502	−0.122516
$172$	0	0
$173$	− 84.9760i	− 0.491190i	−0.969372	−	0.245595i	$-0.921017\pi$
$173$	− 84.9760i	− 0.491190i	0.969372	−	0.245595i	$-0.0789834\pi$
$174$	0	0
$175$	− 65.5081i	− 0.374332i
$176$	0	0
$177$	−257.193	−1.45307
$178$	0	0
$179$	19.3648	0.108183	0.0540916	−	0.998536i	$-0.482774\pi$
$179$	19.3648	0.108183	0.0540916	+	0.998536i	$0.482774\pi$
$180$	0	0
$181$	183.350i	1.01298i	0.862245	+	0.506491i	$0.169058\pi$
$181$	183.350i	1.01298i	−0.862245	+	0.506491i	$0.830942\pi$
$182$	0	0
$183$	− 65.0219i	− 0.355311i
$184$	0	0
$185$	−28.6362	−0.154790
$186$	0	0
$187$	−375.798	−2.00961
$188$	0	0
$189$	− 31.9288i	− 0.168935i
$190$	0	0
$191$	− 93.1822i	− 0.487865i	−0.969792	−	0.243932i	$-0.921562\pi$
$191$	− 93.1822i	− 0.487865i	0.969792	−	0.243932i	$-0.0784375\pi$
$192$	0	0
$193$	276.855	1.43448	0.717242	−	0.696824i	$-0.245404\pi$
$193$	276.855	1.43448	0.717242	+	0.696824i	$0.245404\pi$
$194$	0	0
$195$	−6.63429	−0.0340220
$196$	0	0
$197$	177.712i	0.902089i	0.892501	+	0.451045i	$0.148948\pi$
$197$	177.712i	0.902089i	−0.892501	+	0.451045i	$0.851052\pi$
$198$	0	0
$199$	227.421i	1.14282i	0.820666	+	0.571408i	$0.193603\pi$
$199$	227.421i	1.14282i	−0.820666	+	0.571408i	$0.806397\pi$
$200$	0	0
$201$	81.9143	0.407534
$202$	0	0
$203$	−29.0512	−0.143110
$204$	0	0
$205$	26.5320i	0.129425i
$206$	0	0
$207$	114.910i	0.555122i
$208$	0	0
$209$	−55.5409	−0.265746
$210$	0	0
$211$	325.518	1.54274	0.771370	−	0.636387i	$-0.219572\pi$
$211$	325.518	1.54274	0.771370	+	0.636387i	$0.219572\pi$
$212$	0	0
$213$	− 247.624i	− 1.16256i
$214$	0	0
$215$	17.4706i	0.0812584i
$216$	0	0
$217$	56.0284	0.258195
$218$	0	0
$219$	383.536	1.75131
$220$	0	0
$221$	− 84.7496i	− 0.383482i
$222$	0	0
$223$	− 106.001i	− 0.475342i	−0.971346	−	0.237671i	$-0.923616\pi$
$223$	− 106.001i	− 0.475342i	0.971346	−	0.237671i	$-0.0763840\pi$
$224$	0	0
$225$	145.353	0.646013
$226$	0	0
$227$	−282.765	−1.24566	−0.622830	−	0.782357i	$-0.714017\pi$
$227$	−282.765	−1.24566	−0.622830	+	0.782357i	$0.714017\pi$
$228$	0	0
$229$	− 144.716i	− 0.631949i	−0.948768	−	0.315975i	$-0.897669\pi$
$229$	− 144.716i	− 0.631949i	0.948768	−	0.315975i	$-0.102331\pi$
$230$	0	0
$231$	158.787i	0.687390i
$232$	0	0
$233$	−36.9174	−0.158444	−0.0792219	−	0.996857i	$-0.525244\pi$
$233$	−36.9174	−0.158444	−0.0792219	+	0.996857i	$0.525244\pi$
$234$	0	0
$235$	31.4809	0.133961
$236$	0	0
$237$	537.716i	2.26885i
$238$	0	0
$239$	65.5138i	0.274116i	0.990563	+	0.137058i	$0.0437647\pi$
$239$	65.5138i	0.274116i	−0.990563	+	0.137058i	$0.956235\pi$
$240$	0	0
$241$	−155.844	−0.646656	−0.323328	−	0.946287i	$-0.604802\pi$
$241$	−155.844	−0.646656	−0.323328	+	0.946287i	$0.604802\pi$
$242$	0	0
$243$	274.589	1.12999
$244$	0	0
$245$	3.43118i	0.0140048i
$246$	0	0
$247$	− 12.5255i	− 0.0507107i
$248$	0	0
$249$	23.2583	0.0934067
$250$	0	0
$251$	230.946	0.920105	0.460052	−	0.887892i	$-0.347831\pi$
$251$	230.946	0.920105	0.460052	+	0.887892i	$0.347831\pi$
$252$	0	0
$253$	304.638i	1.20410i
$254$	0	0
$255$	− 45.6415i	− 0.178986i
$256$	0	0
$257$	337.195	1.31204	0.656022	−	0.754742i	$-0.272238\pi$
$257$	337.195	1.31204	0.656022	+	0.754742i	$0.272238\pi$
$258$	0	0
$259$	−154.568	−0.596787
$260$	0	0
$261$	− 64.4605i	− 0.246975i
$262$	0	0
$263$	395.803i	1.50495i	0.658619	+	0.752476i	$0.271141\pi$
$263$	395.803i	1.50495i	−0.658619	+	0.752476i	$0.728859\pi$
$264$	0	0
$265$	−42.8414	−0.161666
$266$	0	0
$267$	−92.4806	−0.346369
$268$	0	0
$269$	31.6881i	0.117800i	0.998264	+	0.0588998i	$0.0187592\pi$
$269$	31.6881i	0.117800i	−0.998264	+	0.0588998i	$0.981241\pi$
$270$	0	0
$271$	69.0163i	0.254673i	0.991860	+	0.127336i	$0.0406428\pi$
$271$	69.0163i	0.254673i	−0.991860	+	0.127336i	$0.959357\pi$
$272$	0	0
$273$	−35.8095	−0.131170
$274$	0	0
$275$	385.344	1.40125
$276$	0	0
$277$	− 54.5783i	− 0.197034i	−0.995135	−	0.0985168i	$-0.968590\pi$
$277$	− 54.5783i	− 0.197034i	0.995135	−	0.0985168i	$-0.0314098\pi$
$278$	0	0
$279$	124.319i	0.445587i
$280$	0	0
$281$	140.453	0.499831	0.249916	−	0.968268i	$-0.419597\pi$
$281$	140.453	0.499831	0.249916	+	0.968268i	$0.419597\pi$
$282$	0	0
$283$	−35.7205	−0.126221	−0.0631105	−	0.998007i	$-0.520102\pi$
$283$	−35.7205	−0.126221	−0.0631105	+	0.998007i	$0.520102\pi$
$284$	0	0
$285$	− 6.74557i	− 0.0236687i
$286$	0	0
$287$	143.210i	0.498991i
$288$	0	0
$289$	294.046	1.01746
$290$	0	0
$291$	−661.377	−2.27277
$292$	0	0
$293$	− 73.9179i	− 0.252279i	−0.992012	−	0.126140i	$-0.959741\pi$
$293$	− 73.9179i	− 0.252279i	0.992012	−	0.126140i	$-0.0402588\pi$
$294$	0	0
$295$	− 32.6919i	− 0.110820i
$296$	0	0
$297$	187.817	0.632382
$298$	0	0
$299$	−68.7017	−0.229771
$300$	0	0
$301$	94.2999i	0.313289i
$302$	0	0
$303$	− 564.658i	− 1.86356i
$304$	0	0
$305$	8.26498	0.0270983
$306$	0	0
$307$	507.360	1.65264	0.826319	−	0.563202i	$-0.190431\pi$
$307$	507.360	1.65264	0.826319	+	0.563202i	$0.190431\pi$
$308$	0	0
$309$	− 458.311i	− 1.48321i
$310$	0	0
$311$	242.948i	0.781185i	0.920564	+	0.390592i	$0.127730\pi$
$311$	242.948i	0.781185i	−0.920564	+	0.390592i	$0.872270\pi$
$312$	0	0
$313$	208.239	0.665301	0.332651	−	0.943050i	$-0.392057\pi$
$313$	208.239	0.665301	0.332651	+	0.943050i	$0.392057\pi$
$314$	0	0
$315$	−7.61328	−0.0241691
$316$	0	0
$317$	− 10.6307i	− 0.0335355i	−0.999859	−	0.0167677i	$-0.994662\pi$
$317$	− 10.6307i	− 0.0335355i	0.999859	−	0.0167677i	$-0.00533758\pi$
$318$	0	0
$319$	− 170.891i	− 0.535708i
$320$	0	0
$321$	−549.260	−1.71109
$322$	0	0
$323$	86.1711	0.266784
$324$	0	0
$325$	86.9025i	0.267392i
$326$	0	0
$327$	− 658.781i	− 2.01462i
$328$	0	0
$329$	169.923	0.516482
$330$	0	0
$331$	122.813	0.371035	0.185517	−	0.982641i	$-0.440604\pi$
$331$	122.813	0.371035	0.185517	+	0.982641i	$0.440604\pi$
$332$	0	0
$333$	− 342.964i	− 1.02992i
$334$	0	0
$335$	10.4122i	0.0310811i
$336$	0	0
$337$	−518.410	−1.53831	−0.769154	−	0.639063i	$-0.779322\pi$
$337$	−518.410	−1.53831	−0.769154	+	0.639063i	$0.779322\pi$
$338$	0	0
$339$	−96.1885	−0.283742
$340$	0	0
$341$	329.581i	0.966512i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	−36.9989	−0.107243
$346$	0	0
$347$	−231.720	−0.667780	−0.333890	−	0.942612i	$-0.608361\pi$
$347$	−231.720	−0.667780	−0.333890	+	0.942612i	$0.608361\pi$
$348$	0	0
$349$	170.720i	0.489169i	0.969628	+	0.244584i	$0.0786515\pi$
$349$	170.720i	0.489169i	−0.969628	+	0.244584i	$0.921348\pi$
$350$	0	0
$351$	42.3564i	0.120674i
$352$	0	0
$353$	−200.237	−0.567244	−0.283622	−	0.958936i	$-0.591536\pi$
$353$	−200.237	−0.567244	−0.283622	+	0.958936i	$0.591536\pi$
$354$	0	0
$355$	31.4757	0.0886638
$356$	0	0
$357$	− 246.356i	− 0.690074i
$358$	0	0
$359$	86.9601i	0.242229i	0.992639	+	0.121114i	$0.0386468\pi$
$359$	86.9601i	0.242229i	−0.992639	+	0.121114i	$0.961353\pi$
$360$	0	0
$361$	−348.264	−0.964721
$362$	0	0
$363$	−467.443	−1.28772
$364$	0	0
$365$	48.7515i	0.133566i
$366$	0	0
$367$	672.949i	1.83365i	0.399290	+	0.916825i	$0.369257\pi$
$367$	672.949i	1.83365i	−0.399290	+	0.916825i	$0.630743\pi$
$368$	0	0
$369$	−317.763	−0.861147
$370$	0	0
$371$	−231.243	−0.623295
$372$	0	0
$373$	− 692.328i	− 1.85611i	−0.372448	−	0.928053i	$-0.621481\pi$
$373$	− 692.328i	− 1.85611i	0.372448	−	0.928053i	$-0.378519\pi$
$374$	0	0
$375$	94.0560i	0.250816i
$376$	0	0
$377$	38.5391	0.102226
$378$	0	0
$379$	193.386	0.510253	0.255126	−	0.966908i	$-0.417883\pi$
$379$	193.386	0.510253	0.255126	+	0.966908i	$0.417883\pi$
$380$	0	0
$381$	− 265.600i	− 0.697112i
$382$	0	0
$383$	518.606i	1.35406i	0.735954	+	0.677031i	$0.236734\pi$
$383$	518.606i	1.35406i	−0.735954	+	0.677031i	$0.763266\pi$
$384$	0	0
$385$	−20.1835	−0.0524247
$386$	0	0
$387$	−209.238	−0.540666
$388$	0	0
$389$	531.999i	1.36761i	0.729666	+	0.683803i	$0.239675\pi$
$389$	531.999i	1.36761i	−0.729666	+	0.683803i	$0.760325\pi$
$390$	0	0
$391$	− 472.642i	− 1.20880i
$392$	0	0
$393$	−390.737	−0.994242
$394$	0	0
$395$	−68.3494	−0.173036
$396$	0	0
$397$	− 490.068i	− 1.23443i	−0.786795	−	0.617214i	$-0.788261\pi$
$397$	− 490.068i	− 1.23443i	0.786795	−	0.617214i	$-0.211739\pi$
$398$	0	0
$399$	− 36.4102i	− 0.0912535i
$400$	0	0
$401$	712.938	1.77790	0.888951	−	0.458003i	$-0.151435\pi$
$401$	712.938	1.77790	0.888951	+	0.458003i	$0.151435\pi$
$402$	0	0
$403$	−74.3268	−0.184434
$404$	0	0
$405$	48.7088i	0.120269i
$406$	0	0
$407$	− 909.228i	− 2.23398i
$408$	0	0
$409$	307.818	0.752612	0.376306	−	0.926495i	$-0.377194\pi$
$409$	307.818	0.752612	0.376306	+	0.926495i	$0.377194\pi$
$410$	0	0
$411$	246.344	0.599377
$412$	0	0
$413$	− 176.459i	− 0.427262i
$414$	0	0
$415$	2.95637i	0.00712378i
$416$	0	0
$417$	−768.377	−1.84263
$418$	0	0
$419$	176.023	0.420103	0.210051	−	0.977690i	$-0.432637\pi$
$419$	176.023	0.420103	0.210051	+	0.977690i	$0.432637\pi$
$420$	0	0
$421$	− 263.167i	− 0.625101i	−0.949901	−	0.312550i	$-0.898817\pi$
$421$	− 263.167i	− 0.625101i	0.949901	−	0.312550i	$-0.101183\pi$
$422$	0	0
$423$	377.034i	0.891333i
$424$	0	0
$425$	−597.857	−1.40672
$426$	0	0
$427$	44.6114	0.104476
$428$	0	0
$429$	− 210.646i	− 0.491016i
$430$	0	0
$431$	256.629i	0.595426i	0.954655	+	0.297713i	$0.0962238\pi$
$431$	256.629i	0.595426i	−0.954655	+	0.297713i	$0.903776\pi$
$432$	0	0
$433$	−636.797	−1.47066	−0.735331	−	0.677708i	$-0.762973\pi$
$433$	−636.797	−1.47066	−0.735331	+	0.677708i	$0.762973\pi$
$434$	0	0
$435$	20.7551	0.0477128
$436$	0	0
$437$	− 69.8540i	− 0.159849i
$438$	0	0
$439$	− 64.4215i	− 0.146746i	−0.997305	−	0.0733730i	$-0.976624\pi$
$439$	− 64.4215i	− 0.146746i	0.997305	−	0.0733730i	$-0.0233763\pi$
$440$	0	0
$441$	−41.0938	−0.0931831
$442$	0	0
$443$	449.411	1.01447	0.507236	−	0.861807i	$-0.330667\pi$
$443$	449.411	1.01447	0.507236	+	0.861807i	$0.330667\pi$
$444$	0	0
$445$	− 11.7553i	− 0.0264163i
$446$	0	0
$447$	− 1047.30i	− 2.34294i
$448$	0	0
$449$	362.900	0.808241	0.404120	−	0.914706i	$-0.367578\pi$
$449$	362.900	0.808241	0.404120	+	0.914706i	$0.367578\pi$
$450$	0	0
$451$	−842.420	−1.86789
$452$	0	0
$453$	− 506.435i	− 1.11796i
$454$	0	0
$455$	− 4.55177i	− 0.0100039i
$456$	0	0
$457$	61.1286	0.133761	0.0668803	−	0.997761i	$-0.478695\pi$
$457$	61.1286	0.133761	0.0668803	+	0.997761i	$0.478695\pi$
$458$	0	0
$459$	−291.397	−0.634851
$460$	0	0
$461$	− 564.752i	− 1.22506i	−0.790448	−	0.612530i	$-0.790152\pi$
$461$	− 564.752i	− 1.22506i	0.790448	−	0.612530i	$-0.209848\pi$
$462$	0	0
$463$	− 581.095i	− 1.25506i	−0.778590	−	0.627532i	$-0.784065\pi$
$463$	− 581.095i	− 1.25506i	0.778590	−	0.627532i	$-0.215935\pi$
$464$	0	0
$465$	−40.0283	−0.0860824
$466$	0	0
$467$	125.780	0.269336	0.134668	−	0.990891i	$-0.457003\pi$
$467$	125.780	0.269336	0.134668	+	0.990891i	$0.457003\pi$
$468$	0	0
$469$	56.2012i	0.119832i
$470$	0	0
$471$	− 515.519i	− 1.09452i
$472$	0	0
$473$	−554.709	−1.17275
$474$	0	0
$475$	−88.3601	−0.186021
$476$	0	0
$477$	− 513.094i	− 1.07567i
$478$	0	0
$479$	− 393.505i	− 0.821513i	−0.911745	−	0.410756i	$-0.865265\pi$
$479$	− 393.505i	− 0.821513i	0.911745	−	0.410756i	$-0.134735\pi$
$480$	0	0
$481$	205.048	0.426296
$482$	0	0
$483$	−199.707	−0.413472
$484$	0	0
$485$	− 84.0680i	− 0.173336i
$486$	0	0
$487$	149.606i	0.307198i	0.988133	+	0.153599i	$0.0490865\pi$
$487$	149.606i	0.307198i	−0.988133	+	0.153599i	$0.950914\pi$
$488$	0	0
$489$	−114.980	−0.235133
$490$	0	0
$491$	79.1826	0.161268	0.0806340	−	0.996744i	$-0.474305\pi$
$491$	79.1826	0.161268	0.0806340	+	0.996744i	$0.474305\pi$
$492$	0	0
$493$	265.135i	0.537799i
$494$	0	0
$495$	− 44.7843i	− 0.0904733i
$496$	0	0
$497$	169.894	0.341840
$498$	0	0
$499$	−244.391	−0.489762	−0.244881	−	0.969553i	$-0.578749\pi$
$499$	−244.391	−0.489762	−0.244881	+	0.969553i	$0.578749\pi$
$500$	0	0
$501$	1001.50i	1.99899i
$502$	0	0
$503$	278.539i	0.553755i	0.960905	+	0.276878i	$0.0892997\pi$
$503$	278.539i	0.553755i	−0.960905	+	0.276878i	$0.910700\pi$
$504$	0	0
$505$	71.7739	0.142127
$506$	0	0
$507$	−604.199	−1.19171
$508$	0	0
$509$	444.338i	0.872963i	0.899713	+	0.436481i	$0.143775\pi$
$509$	444.338i	0.872963i	−0.899713	+	0.436481i	$0.856225\pi$
$510$	0	0
$511$	263.143i	0.514957i
$512$	0	0
$513$	−43.0669	−0.0839510
$514$	0	0
$515$	58.2562	0.113119
$516$	0	0
$517$	999.552i	1.93337i
$518$	0	0
$519$	327.687i	0.631382i
$520$	0	0
$521$	−127.704	−0.245113	−0.122557	−	0.992462i	$-0.539109\pi$
$521$	−127.704	−0.245113	−0.122557	+	0.992462i	$0.539109\pi$
$522$	0	0
$523$	−937.135	−1.79184	−0.895922	−	0.444211i	$-0.853484\pi$
$523$	−937.135	−1.79184	−0.895922	+	0.444211i	$0.853484\pi$
$524$	0	0
$525$	252.615i	0.481171i
$526$	0	0
$527$	− 511.341i	− 0.970286i
$528$	0	0
$529$	145.856	0.275721
$530$	0	0
$531$	391.538	0.737359
$532$	0	0
$533$	− 189.982i	− 0.356439i
$534$	0	0
$535$	− 69.8167i	− 0.130499i
$536$	0	0
$537$	−74.6751	−0.139060
$538$	0	0
$539$	−108.943	−0.202121
$540$	0	0
$541$	− 540.845i	− 0.999713i	−0.866108	−	0.499857i	$-0.833386\pi$
$541$	− 540.845i	− 0.999713i	0.866108	−	0.499857i	$-0.166614\pi$
$542$	0	0
$543$	− 707.040i	− 1.30210i
$544$	0	0
$545$	83.7380	0.153648
$546$	0	0
$547$	−667.995	−1.22120	−0.610599	−	0.791940i	$-0.709071\pi$
$547$	−667.995	−1.22120	−0.610599	+	0.791940i	$0.709071\pi$
$548$	0	0
$549$	98.9862i	0.180303i
$550$	0	0
$551$	39.1855i	0.0711171i
$552$	0	0
$553$	−368.926	−0.667135
$554$	0	0
$555$	110.428	0.198969
$556$	0	0
$557$	746.470i	1.34016i	0.742288	+	0.670081i	$0.233741\pi$
$557$	746.470i	1.34016i	−0.742288	+	0.670081i	$0.766259\pi$
$558$	0	0
$559$	− 125.097i	− 0.223788i
$560$	0	0
$561$	1449.16	2.58318
$562$	0	0
$563$	518.437	0.920846	0.460423	−	0.887700i	$-0.347698\pi$
$563$	518.437	0.920846	0.460423	+	0.887700i	$0.347698\pi$
$564$	0	0
$565$	− 12.2266i	− 0.0216399i
$566$	0	0
$567$	262.913i	0.463691i
$568$	0	0
$569$	215.471	0.378683	0.189341	−	0.981911i	$-0.439365\pi$
$569$	215.471	0.378683	0.189341	+	0.981911i	$0.439365\pi$
$570$	0	0
$571$	−4.73397	−0.00829067	−0.00414534	−	0.999991i	$-0.501320\pi$
$571$	−4.73397	−0.00829067	−0.00414534	+	0.999991i	$0.501320\pi$
$572$	0	0
$573$	359.332i	0.627107i
$574$	0	0
$575$	484.649i	0.842867i
$576$	0	0
$577$	−72.0226	−0.124823	−0.0624113	−	0.998051i	$-0.519879\pi$
$577$	−72.0226	−0.124823	−0.0624113	+	0.998051i	$0.519879\pi$
$578$	0	0
$579$	−1067.62	−1.84390
$580$	0	0
$581$	15.9574i	0.0274655i
$582$	0	0
$583$	− 1360.26i	− 2.33321i
$584$	0	0
$585$	10.0997	0.0172645
$586$	0	0
$587$	−272.118	−0.463575	−0.231787	−	0.972766i	$-0.574457\pi$
$587$	−272.118	−0.463575	−0.231787	+	0.972766i	$0.574457\pi$
$588$	0	0
$589$	− 75.5734i	− 0.128308i
$590$	0	0
$591$	− 685.297i	− 1.15956i
$592$	0	0
$593$	842.019	1.41993	0.709966	−	0.704236i	$-0.248710\pi$
$593$	842.019	1.41993	0.709966	+	0.704236i	$0.248710\pi$
$594$	0	0
$595$	31.3145	0.0526294
$596$	0	0
$597$	− 876.987i	− 1.46899i
$598$	0	0
$599$	478.633i	0.799054i	0.916721	+	0.399527i	$0.130826\pi$
$599$	478.633i	0.799054i	−0.916721	+	0.399527i	$0.869174\pi$
$600$	0	0
$601$	−206.471	−0.343546	−0.171773	−	0.985137i	$-0.554950\pi$
$601$	−206.471	−0.343546	−0.171773	+	0.985137i	$0.554950\pi$
$602$	0	0
$603$	−124.702	−0.206803
$604$	0	0
$605$	− 59.4170i	− 0.0982099i
$606$	0	0
$607$	− 292.279i	− 0.481514i	−0.970585	−	0.240757i	$-0.922604\pi$
$607$	− 292.279i	− 0.481514i	0.970585	−	0.240757i	$-0.0773956\pi$
$608$	0	0
$609$	112.028	0.183955
$610$	0	0
$611$	−225.418	−0.368933
$612$	0	0
$613$	− 827.863i	− 1.35051i	−0.737584	−	0.675255i	$-0.764033\pi$
$613$	− 827.863i	− 1.35051i	0.737584	−	0.675255i	$-0.235967\pi$
$614$	0	0
$615$	− 102.314i	− 0.166364i
$616$	0	0
$617$	667.348	1.08160	0.540801	−	0.841151i	$-0.318121\pi$
$617$	667.348	1.08160	0.540801	+	0.841151i	$0.318121\pi$
$618$	0	0
$619$	−750.661	−1.21270	−0.606350	−	0.795198i	$-0.707367\pi$
$619$	−750.661	−1.21270	−0.606350	+	0.795198i	$0.707367\pi$
$620$	0	0
$621$	236.219i	0.380384i
$622$	0	0
$623$	− 63.4507i	− 0.101847i
$624$	0	0
$625$	607.038	0.971261
$626$	0	0
$627$	214.179	0.341593
$628$	0	0
$629$	1410.66i	2.24270i
$630$	0	0
$631$	− 1076.70i	− 1.70633i	−0.521637	−	0.853167i	$-0.674679\pi$
$631$	− 1076.70i	− 1.70633i	0.521637	−	0.853167i	$-0.325321\pi$
$632$	0	0
$633$	−1255.27	−1.98305
$634$	0	0
$635$	33.7605	0.0531662
$636$	0	0
$637$	− 24.5688i	− 0.0385696i
$638$	0	0
$639$	376.971i	0.589939i
$640$	0	0
$641$	−298.632	−0.465885	−0.232942	−	0.972491i	$-0.574835\pi$
$641$	−298.632	−0.465885	−0.232942	+	0.972491i	$0.574835\pi$
$642$	0	0
$643$	−504.242	−0.784202	−0.392101	−	0.919922i	$-0.628252\pi$
$643$	−504.242	−0.784202	−0.392101	+	0.919922i	$0.628252\pi$
$644$	0	0
$645$	− 67.3706i	− 0.104451i
$646$	0	0
$647$	76.2691i	0.117881i	0.998261	+	0.0589406i	$0.0187723\pi$
$647$	76.2691i	0.117881i	−0.998261	+	0.0589406i	$0.981228\pi$
$648$	0	0
$649$	1038.00	1.59939
$650$	0	0
$651$	−216.058	−0.331887
$652$	0	0
$653$	− 199.148i	− 0.304974i	−0.988305	−	0.152487i	$-0.951272\pi$
$653$	− 199.148i	− 0.304974i	0.988305	−	0.152487i	$-0.0487282\pi$
$654$	0	0
$655$	− 49.6668i	− 0.0758272i
$656$	0	0
$657$	−583.876	−0.888700
$658$	0	0
$659$	488.851	0.741807	0.370904	−	0.928671i	$-0.379048\pi$
$659$	488.851	0.741807	0.370904	+	0.928671i	$0.379048\pi$
$660$	0	0
$661$	477.111i	0.721802i	0.932604	+	0.360901i	$0.117531\pi$
$661$	477.111i	0.721802i	−0.932604	+	0.360901i	$0.882469\pi$
$662$	0	0
$663$	326.814i	0.492933i
$664$	0	0
$665$	4.62811	0.00695957
$666$	0	0
$667$	214.930	0.322234
$668$	0	0
$669$	408.765i	0.611010i
$670$	0	0
$671$	262.422i	0.391090i
$672$	0	0
$673$	−437.908	−0.650680	−0.325340	−	0.945597i	$-0.605479\pi$
$673$	−437.908	−0.650680	−0.325340	+	0.945597i	$0.605479\pi$
$674$	0	0
$675$	298.799	0.442665
$676$	0	0
$677$	− 509.653i	− 0.752810i	−0.926455	−	0.376405i	$-0.877160\pi$
$677$	− 509.653i	− 0.752810i	0.926455	−	0.376405i	$-0.122840\pi$
$678$	0	0
$679$	− 453.769i	− 0.668290i
$680$	0	0
$681$	1090.41	1.60119
$682$	0	0
$683$	−794.509	−1.16326	−0.581632	−	0.813452i	$-0.697586\pi$
$683$	−794.509	−1.16326	−0.581632	+	0.813452i	$0.697586\pi$
$684$	0	0
$685$	31.3129i	0.0457123i
$686$	0	0
$687$	558.060i	0.812315i
$688$	0	0
$689$	306.765	0.445232
$690$	0	0
$691$	19.0808	0.0276133	0.0138066	−	0.999905i	$-0.495605\pi$
$691$	19.0808	0.0276133	0.0138066	+	0.999905i	$0.495605\pi$
$692$	0	0
$693$	− 241.730i	− 0.348816i
$694$	0	0
$695$	− 97.6688i	− 0.140531i
$696$	0	0
$697$	1307.00	1.87519
$698$	0	0
$699$	142.362	0.203665
$700$	0	0
$701$	333.075i	0.475143i	0.971370	+	0.237572i	$0.0763514\pi$
$701$	333.075i	0.475143i	−0.971370	+	0.237572i	$0.923649\pi$
$702$	0	0
$703$	208.488i	0.296568i
$704$	0	0
$705$	−121.398	−0.172195
$706$	0	0
$707$	387.410	0.547963
$708$	0	0
$709$	692.721i	0.977039i	0.872553	+	0.488519i	$0.162463\pi$
$709$	692.721i	0.977039i	−0.872553	+	0.488519i	$0.837537\pi$
$710$	0	0
$711$	− 818.593i	− 1.15133i
$712$	0	0
$713$	−414.515	−0.581367
$714$	0	0
$715$	26.7753	0.0374479
$716$	0	0
$717$	− 252.636i	− 0.352352i
$718$	0	0
$719$	− 187.332i	− 0.260545i	−0.991478	−	0.130272i	$-0.958415\pi$
$719$	− 187.332i	− 0.260545i	0.991478	−	0.130272i	$-0.0415852\pi$
$720$	0	0
$721$	314.446	0.436125
$722$	0	0
$723$	600.971	0.831219
$724$	0	0
$725$	− 271.870i	− 0.374993i
$726$	0	0
$727$	669.583i	0.921022i	0.887654	+	0.460511i	$0.152334\pi$
$727$	669.583i	0.921022i	−0.887654	+	0.460511i	$0.847666\pi$
$728$	0	0
$729$	−164.534	−0.225698
$730$	0	0
$731$	860.624	1.17732
$732$	0	0
$733$	− 585.685i	− 0.799024i	−0.916728	−	0.399512i	$-0.869180\pi$
$733$	− 585.685i	− 0.799024i	0.916728	−	0.399512i	$-0.130820\pi$
$734$	0	0
$735$	− 13.2314i	− 0.0180019i
$736$	0	0
$737$	−330.597	−0.448571
$738$	0	0
$739$	−318.797	−0.431389	−0.215695	−	0.976461i	$-0.569202\pi$
$739$	−318.797	−0.431389	−0.215695	+	0.976461i	$0.569202\pi$
$740$	0	0
$741$	48.3014i	0.0651841i
$742$	0	0
$743$	1440.64i	1.93895i	0.245185	+	0.969476i	$0.421151\pi$
$743$	1440.64i	1.93895i	−0.245185	+	0.969476i	$0.578849\pi$
$744$	0	0
$745$	133.122	0.178688
$746$	0	0
$747$	−35.4072	−0.0473992
$748$	0	0
$749$	− 376.846i	− 0.503132i
$750$	0	0
$751$	238.758i	0.317920i	0.987285	+	0.158960i	$0.0508141\pi$
$751$	238.758i	0.317920i	−0.987285	+	0.158960i	$0.949186\pi$
$752$	0	0
$753$	−890.583	−1.18271
$754$	0	0
$755$	64.3733	0.0852626
$756$	0	0
$757$	975.208i	1.28825i	0.764919	+	0.644127i	$0.222779\pi$
$757$	975.208i	1.28825i	−0.764919	+	0.644127i	$0.777221\pi$
$758$	0	0
$759$	− 1174.75i	− 1.54777i
$760$	0	0
$761$	135.752	0.178386	0.0891929	−	0.996014i	$-0.471571\pi$
$761$	135.752	0.178386	0.0891929	+	0.996014i	$0.471571\pi$
$762$	0	0
$763$	451.988	0.592383
$764$	0	0
$765$	69.4823i	0.0908265i
$766$	0	0
$767$	234.090i	0.305202i
$768$	0	0
$769$	−1375.28	−1.78841	−0.894203	−	0.447661i	$-0.852257\pi$
$769$	−1375.28	−1.78841	−0.894203	+	0.447661i	$0.852257\pi$
$770$	0	0
$771$	−1300.30	−1.68652
$772$	0	0
$773$	844.987i	1.09313i	0.837418	+	0.546564i	$0.184064\pi$
$773$	844.987i	1.09313i	−0.837418	+	0.546564i	$0.815936\pi$
$774$	0	0
$775$	524.330i	0.676555i
$776$	0	0
$777$	596.050	0.767117
$778$	0	0
$779$	193.168	0.247970
$780$	0	0
$781$	999.385i	1.27962i
$782$	0	0
$783$	− 132.510i	− 0.169234i
$784$	0	0
$785$	65.5279	0.0834750
$786$	0	0
$787$	−706.124	−0.897235	−0.448617	−	0.893724i	$-0.648083\pi$
$787$	−706.124	−0.897235	−0.448617	+	0.893724i	$0.648083\pi$
$788$	0	0
$789$	− 1526.31i	− 1.93448i
$790$	0	0
$791$	− 65.9947i	− 0.0834319i
$792$	0	0
$793$	−59.1811	−0.0746294
$794$	0	0
$795$	165.206	0.207807
$796$	0	0
$797$	1103.57i	1.38465i	0.721586	+	0.692325i	$0.243414\pi$
$797$	1103.57i	1.38465i	−0.721586	+	0.692325i	$0.756586\pi$
$798$	0	0
$799$	− 1550.79i	− 1.94092i
$800$	0	0
$801$	140.788	0.175765
$802$	0	0
$803$	−1547.91	−1.92766
$804$	0	0
$805$	− 25.3849i	− 0.0315340i
$806$	0	0
$807$	− 122.197i	− 0.151421i
$808$	0	0
$809$	890.340	1.10054	0.550272	−	0.834986i	$-0.314524\pi$
$809$	890.340	1.10054	0.550272	+	0.834986i	$0.314524\pi$
$810$	0	0
$811$	326.603	0.402717	0.201358	−	0.979518i	$-0.435464\pi$
$811$	326.603	0.402717	0.201358	+	0.979518i	$0.435464\pi$
$812$	0	0
$813$	− 266.143i	− 0.327359i
$814$	0	0
$815$	− 14.6152i	− 0.0179328i
$816$	0	0
$817$	127.196	0.155686
$818$	0	0
$819$	54.5147	0.0665625
$820$	0	0
$821$	− 177.210i	− 0.215847i	−0.994159	−	0.107924i	$-0.965580\pi$
$821$	− 177.210i	− 0.215847i	0.994159	−	0.107924i	$-0.0344202\pi$
$822$	0	0
$823$	− 1474.36i	− 1.79144i	−0.444614	−	0.895722i	$-0.646659\pi$
$823$	− 1474.36i	− 1.79144i	0.444614	−	0.895722i	$-0.353341\pi$
$824$	0	0
$825$	−1485.98	−1.80118
$826$	0	0
$827$	339.749	0.410821	0.205411	−	0.978676i	$-0.434147\pi$
$827$	339.749	0.410821	0.205411	+	0.978676i	$0.434147\pi$
$828$	0	0
$829$	1063.05i	1.28233i	0.767402	+	0.641166i	$0.221549\pi$
$829$	1063.05i	1.28233i	−0.767402	+	0.641166i	$0.778451\pi$
$830$	0	0
$831$	210.467i	0.253269i
$832$	0	0
$833$	169.024	0.202910
$834$	0	0
$835$	−127.301	−0.152456
$836$	0	0
$837$	255.559i	0.305328i
$838$	0	0
$839$	− 670.497i	− 0.799162i	−0.916698	−	0.399581i	$-0.869156\pi$
$839$	− 670.497i	− 0.799162i	0.916698	−	0.399581i	$-0.130844\pi$
$840$	0	0
$841$	720.432	0.856638
$842$	0	0
$843$	−541.618	−0.642489
$844$	0	0
$845$	− 76.8000i	− 0.0908876i
$846$	0	0
$847$	− 320.712i	− 0.378644i
$848$	0	0
$849$	137.747	0.162246
$850$	0	0
$851$	1143.54	1.34376
$852$	0	0
$853$	165.395i	0.193898i	0.995289	+	0.0969489i	$0.0309083\pi$
$853$	165.395i	0.193898i	−0.995289	+	0.0969489i	$0.969092\pi$
$854$	0	0
$855$	10.2691i	0.0120107i
$856$	0	0
$857$	1088.22	1.26981	0.634904	−	0.772591i	$-0.281040\pi$
$857$	1088.22	1.26981	0.634904	+	0.772591i	$0.281040\pi$
$858$	0	0
$859$	335.213	0.390236	0.195118	−	0.980780i	$-0.437491\pi$
$859$	335.213	0.390236	0.195118	+	0.980780i	$0.437491\pi$
$860$	0	0
$861$	− 552.253i	− 0.641409i
$862$	0	0
$863$	− 527.699i	− 0.611471i	−0.952117	−	0.305735i	$-0.901098\pi$
$863$	− 527.699i	− 0.611471i	0.952117	−	0.305735i	$-0.0989023\pi$
$864$	0	0
$865$	−41.6525	−0.0481532
$866$	0	0
$867$	−1133.91	−1.30785
$868$	0	0
$869$	− 2170.16i	− 2.49731i
$870$	0	0
$871$	− 74.5560i	− 0.0855982i
$872$	0	0
$873$	1006.85	1.15332
$874$	0	0
$875$	−64.5315	−0.0737503
$876$	0	0
$877$	1257.89i	1.43431i	0.696915	+	0.717154i	$0.254556\pi$
$877$	1257.89i	1.43431i	−0.696915	+	0.717154i	$0.745444\pi$
$878$	0	0
$879$	285.045i	0.324283i
$880$	0	0
$881$	1111.12	1.26120	0.630599	−	0.776109i	$-0.282809\pi$
$881$	1111.12	1.26120	0.630599	+	0.776109i	$0.282809\pi$
$882$	0	0
$883$	358.772	0.406310	0.203155	−	0.979147i	$-0.434880\pi$
$883$	358.772	0.406310	0.203155	+	0.979147i	$0.434880\pi$
$884$	0	0
$885$	126.068i	0.142449i
$886$	0	0
$887$	1570.56i	1.77064i	0.464978	+	0.885322i	$0.346062\pi$
$887$	1570.56i	1.77064i	−0.464978	+	0.885322i	$0.653938\pi$
$888$	0	0
$889$	182.227	0.204980
$890$	0	0
$891$	−1546.55	−1.73575
$892$	0	0
$893$	− 229.199i	− 0.256662i
$894$	0	0
$895$	− 9.49200i	− 0.0106056i
$896$	0	0
$897$	264.930	0.295351
$898$	0	0
$899$	232.528	0.258651
$900$	0	0
$901$	2110.43i	2.34232i
$902$	0	0
$903$	− 363.642i	− 0.402705i
$904$	0	0
$905$	89.8722	0.0993063
$906$	0	0
$907$	901.249	0.993659	0.496829	−	0.867848i	$-0.334497\pi$
$907$	901.249	0.993659	0.496829	+	0.867848i	$0.334497\pi$
$908$	0	0
$909$	859.607i	0.945662i
$910$	0	0
$911$	901.747i	0.989843i	0.868938	+	0.494922i	$0.164803\pi$
$911$	901.747i	0.989843i	−0.868938	+	0.494922i	$0.835197\pi$
$912$	0	0
$913$	−93.8678	−0.102813
$914$	0	0
$915$	−31.8717	−0.0348324
$916$	0	0
$917$	− 268.084i	− 0.292348i
$918$	0	0
$919$	1165.98i	1.26875i	0.773025	+	0.634375i	$0.218743\pi$
$919$	1165.98i	1.26875i	−0.773025	+	0.634375i	$0.781257\pi$
$920$	0	0
$921$	−1956.50	−2.12432
$922$	0	0
$923$	−225.380	−0.244183
$924$	0	0
$925$	− 1446.49i	− 1.56378i
$926$	0	0
$927$	697.710i	0.752654i
$928$	0	0
$929$	−1400.58	−1.50762	−0.753811	−	0.657091i	$-0.771787\pi$
$929$	−1400.58	−1.50762	−0.753811	+	0.657091i	$0.771787\pi$
$930$	0	0
$931$	24.9809	0.0268323
$932$	0	0
$933$	− 936.866i	− 1.00414i
$934$	0	0
$935$	184.204i	0.197010i
$936$	0	0
$937$	−1252.37	−1.33657	−0.668285	−	0.743905i	$-0.732972\pi$
$937$	−1252.37	−1.33657	−0.668285	+	0.743905i	$0.732972\pi$
$938$	0	0
$939$	−803.020	−0.855186
$940$	0	0
$941$	− 335.582i	− 0.356622i	−0.983974	−	0.178311i	$-0.942937\pi$
$941$	− 335.582i	− 0.356622i	0.983974	−	0.178311i	$-0.0570634\pi$
$942$	0	0
$943$	− 1059.51i	− 1.12356i
$944$	0	0
$945$	−15.6505	−0.0165613
$946$	0	0
$947$	1588.48	1.67738	0.838689	−	0.544610i	$-0.183322\pi$
$947$	1588.48	1.67738	0.838689	+	0.544610i	$0.183322\pi$
$948$	0	0
$949$	− 349.083i	− 0.367843i
$950$	0	0
$951$	40.9946i	0.0431069i
$952$	0	0
$953$	99.1693	0.104060	0.0520301	−	0.998646i	$-0.483431\pi$
$953$	99.1693	0.104060	0.0520301	+	0.998646i	$0.483431\pi$
$954$	0	0
$955$	−45.6749	−0.0478271
$956$	0	0
$957$	658.995i	0.688605i
$958$	0	0
$959$	169.016i	0.176242i
$960$	0	0
$961$	512.546	0.533347
$962$	0	0
$963$	836.166	0.868292
$964$	0	0
$965$	− 135.706i	− 0.140628i
$966$	0	0
$967$	1322.78i	1.36792i	0.729518	+	0.683962i	$0.239744\pi$
$967$	1322.78i	1.36792i	−0.729518	+	0.683962i	$0.760256\pi$
$968$	0	0
$969$	−332.296	−0.342927
$970$	0	0
$971$	1571.76	1.61870	0.809350	−	0.587327i	$-0.199820\pi$
$971$	1571.76	1.61870	0.809350	+	0.587327i	$0.199820\pi$
$972$	0	0
$973$	− 527.181i	− 0.541810i
$974$	0	0
$975$	− 335.116i	− 0.343709i
$976$	0	0
$977$	−15.9043	−0.0162787	−0.00813936	−	0.999967i	$-0.502591\pi$
$977$	−15.9043	−0.0162787	−0.00813936	+	0.999967i	$0.502591\pi$
$978$	0	0
$979$	373.242	0.381248
$980$	0	0
$981$	1002.90i	1.02232i
$982$	0	0
$983$	− 202.919i	− 0.206428i	−0.994659	−	0.103214i	$-0.967087\pi$
$983$	− 202.919i	− 0.206428i	0.994659	−	0.103214i	$-0.0329127\pi$
$984$	0	0
$985$	87.1085	0.0884350
$986$	0	0
$987$	−655.262	−0.663892
$988$	0	0
$989$	− 697.659i	− 0.705418i
$990$	0	0
$991$	760.920i	0.767830i	0.923368	+	0.383915i	$0.125424\pi$
$991$	760.920i	0.767830i	−0.923368	+	0.383915i	$0.874576\pi$
$992$	0	0
$993$	−473.594	−0.476932
$994$	0	0
$995$	111.474	0.112034
$996$	0	0
$997$	− 860.114i	− 0.862702i	−0.902184	−	0.431351i	$-0.858037\pi$
$997$	− 860.114i	− 0.862702i	0.902184	−	0.431351i	$-0.141963\pi$
$998$	0	0
$999$	− 705.023i	− 0.705728i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.g.f.127.1		8
4.3	odd	2		1792.3.g.d.127.7		8
8.3	odd	2	inner	1792.3.g.f.127.2		8
8.5	even	2		1792.3.g.d.127.8		8
16.3	odd	4		448.3.d.e.127.7		8
16.5	even	4		224.3.d.b.127.7	yes	8
16.11	odd	4		224.3.d.b.127.2	✓	8
16.13	even	4		448.3.d.e.127.2		8
48.5	odd	4		2016.3.m.c.127.4		8
48.11	even	4		2016.3.m.c.127.3		8
112.27	even	4		1568.3.d.n.1471.7		8
112.69	odd	4		1568.3.d.n.1471.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.2	✓	8	16.11	odd	4
224.3.d.b.127.7	yes	8	16.5	even	4
448.3.d.e.127.2		8	16.13	even	4
448.3.d.e.127.7		8	16.3	odd	4
1568.3.d.n.1471.2		8	112.69	odd	4
1568.3.d.n.1471.7		8	112.27	even	4
1792.3.g.d.127.7		8	4.3	odd	2
1792.3.g.d.127.8		8	8.5	even	2
1792.3.g.f.127.1		8	1.1	even	1	trivial
1792.3.g.f.127.2		8	8.3	odd	2	inner
2016.3.m.c.127.3		8	48.11	even	4
2016.3.m.c.127.4		8	48.5	odd	4

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.85623 q^{3} -0.490168i q^{5} -2.64575i q^{7} +5.87054 q^{9} +O(q^{10})\)	\(q-3.85623 q^{3} -0.490168i q^{5} -2.64575i q^{7} +5.87054 q^{9} +15.5633 q^{11} +3.50983i q^{13} +1.89020i q^{15} -24.1463 q^{17} -3.56870 q^{19} +10.2026i q^{21} +19.5741i q^{23} +24.7597 q^{25} +12.0679 q^{27} -10.9803i q^{29} +21.1767i q^{31} -60.0159 q^{33} -1.29686 q^{35} -58.4212i q^{37} -13.5347i q^{39} -54.1285 q^{41} -35.6420 q^{43} -2.87755i q^{45} +64.2248i q^{47} -7.00000 q^{49} +93.1140 q^{51} -87.4015i q^{53} -7.62865i q^{55} +13.7617 q^{57} +66.6954 q^{59} +16.8615i q^{61} -15.5320i q^{63} +1.72041 q^{65} -21.2420 q^{67} -75.4822i q^{69} +64.2140i q^{71} -99.4587 q^{73} -95.4793 q^{75} -41.1767i q^{77} -139.441i q^{79} -99.3716 q^{81} -6.03134 q^{83} +11.8358i q^{85} +42.3427i q^{87} +23.9821 q^{89} +9.28614 q^{91} -81.6624i q^{93} +1.74926i q^{95} +171.509 q^{97} +91.3652 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 40 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 40 * q^9	\( 8 q + 8 q^{3} + 40 q^{9} + 32 q^{11} - 16 q^{17} + 88 q^{19} - 104 q^{25} + 176 q^{27} + 56 q^{35} - 144 q^{41} - 224 q^{43} - 56 q^{49} + 16 q^{51} + 400 q^{57} + 232 q^{59} - 304 q^{65} - 368 q^{67} - 272 q^{73} - 664 q^{75} + 504 q^{81} - 424 q^{83} + 80 q^{89} + 56 q^{91} + 528 q^{97} + 544 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + 40 * q^9 + 32 * q^11 - 16 * q^17 + 88 * q^19 - 104 * q^25 + 176 * q^27 + 56 * q^35 - 144 * q^41 - 224 * q^43 - 56 * q^49 + 16 * q^51 + 400 * q^57 + 232 * q^59 - 304 * q^65 - 368 * q^67 - 272 * q^73 - 664 * q^75 + 504 * q^81 - 424 * q^83 + 80 * q^89 + 56 * q^91 + 528 * q^97 + 544 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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