LMFDB - Embedded newform 4784.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4784,2,Mod(1,4784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4784, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4784.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4784 = 2^{4} \cdot 13 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4784.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:	$38.2004323270$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 19x^{8} + 18x^{7} + 127x^{6} - 109x^{5} - 357x^{4} + 252x^{3} + 400x^{2} - 192x - 128 $ x^10 - x^9 - 19x^8 + 18x^7 + 127x^6 - 109x^5 - 357x^4 + 252x^3 + 400x^2 - 192x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 299)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.26436$ of defining polynomial
Character	$\chi$	$=$	4784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37123 q^{3} -3.16568 q^{5} -1.09348 q^{7} +2.62273 q^{9} +O(q^{10})$	$q-2.37123 q^{3} -3.16568 q^{5} -1.09348 q^{7} +2.62273 q^{9} +5.61162 q^{11} -1.00000 q^{13} +7.50657 q^{15} -0.216286 q^{17} -6.03266 q^{19} +2.59290 q^{21} -1.00000 q^{23} +5.02155 q^{25} +0.894583 q^{27} +9.11937 q^{29} -8.04245 q^{31} -13.3064 q^{33} +3.46162 q^{35} -2.55717 q^{37} +2.37123 q^{39} -7.13855 q^{41} -4.22532 q^{43} -8.30275 q^{45} +1.07565 q^{47} -5.80429 q^{49} +0.512864 q^{51} +0.577213 q^{53} -17.7646 q^{55} +14.3048 q^{57} +4.96068 q^{59} +8.94721 q^{61} -2.86792 q^{63} +3.16568 q^{65} -7.26955 q^{67} +2.37123 q^{69} -8.41151 q^{71} -9.27194 q^{73} -11.9073 q^{75} -6.13621 q^{77} -4.24311 q^{79} -9.98947 q^{81} +6.74657 q^{83} +0.684694 q^{85} -21.6241 q^{87} -15.0002 q^{89} +1.09348 q^{91} +19.0705 q^{93} +19.0975 q^{95} -15.7654 q^{97} +14.7178 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 3 q^{3} + 3 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) $ 10 * q - 3 * q^3 + 3 * q^5 + 2 * q^7 + 17 * q^9	$ 10 q - 3 q^{3} + 3 q^{5} + 2 q^{7} + 17 q^{9} - 3 q^{11} - 10 q^{13} - 2 q^{15} - 3 q^{17} - 2 q^{19} + 21 q^{21} - 10 q^{23} + 33 q^{25} - 6 q^{27} + 17 q^{29} - 5 q^{31} - 23 q^{33} - 3 q^{35} + 16 q^{37} + 3 q^{39} - 16 q^{41} + 9 q^{43} + 32 q^{45} + 11 q^{47} + 40 q^{49} + 31 q^{51} + 8 q^{53} + 14 q^{55} - 35 q^{57} - 2 q^{59} + 48 q^{61} + 15 q^{63} - 3 q^{65} + 6 q^{67} + 3 q^{69} - 24 q^{71} - 33 q^{73} + 22 q^{75} + 15 q^{77} - 17 q^{79} + 30 q^{81} + 21 q^{83} + 58 q^{85} - 23 q^{87} - 16 q^{89} - 2 q^{91} + 15 q^{93} + 27 q^{95} - 40 q^{97} + 17 q^{99}+O(q^{100}) $ 10 * q - 3 * q^3 + 3 * q^5 + 2 * q^7 + 17 * q^9 - 3 * q^11 - 10 * q^13 - 2 * q^15 - 3 * q^17 - 2 * q^19 + 21 * q^21 - 10 * q^23 + 33 * q^25 - 6 * q^27 + 17 * q^29 - 5 * q^31 - 23 * q^33 - 3 * q^35 + 16 * q^37 + 3 * q^39 - 16 * q^41 + 9 * q^43 + 32 * q^45 + 11 * q^47 + 40 * q^49 + 31 * q^51 + 8 * q^53 + 14 * q^55 - 35 * q^57 - 2 * q^59 + 48 * q^61 + 15 * q^63 - 3 * q^65 + 6 * q^67 + 3 * q^69 - 24 * q^71 - 33 * q^73 + 22 * q^75 + 15 * q^77 - 17 * q^79 + 30 * q^81 + 21 * q^83 + 58 * q^85 - 23 * q^87 - 16 * q^89 - 2 * q^91 + 15 * q^93 + 27 * q^95 - 40 * q^97 + 17 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.37123	−1.36903	−0.684515	−	0.728998i	$-0.739986\pi$
$3$	−2.37123	−1.36903	−0.684515	+	0.728998i	$0.739986\pi$
$4$	0	0
$5$	−3.16568	−1.41574	−0.707868	−	0.706344i	$-0.750343\pi$
$5$	−3.16568	−1.41574	−0.707868	+	0.706344i	$0.750343\pi$
$6$	0	0
$7$	−1.09348	−0.413298	−0.206649	−	0.978415i	$-0.566256\pi$
$7$	−1.09348	−0.413298	−0.206649	+	0.978415i	$0.566256\pi$
$8$	0	0
$9$	2.62273	0.874245
$10$	0	0
$11$	5.61162	1.69197	0.845984	−	0.533209i	$-0.179014\pi$
$11$	5.61162	1.69197	0.845984	+	0.533209i	$0.179014\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	7.50657	1.93819
$16$	0	0
$17$	−0.216286	−0.0524571	−0.0262286	−	0.999656i	$-0.508350\pi$
$17$	−0.216286	−0.0524571	−0.0262286	+	0.999656i	$0.508350\pi$
$18$	0	0
$19$	−6.03266	−1.38399	−0.691994	−	0.721904i	$-0.743267\pi$
$19$	−6.03266	−1.38399	−0.691994	+	0.721904i	$0.743267\pi$
$20$	0	0
$21$	2.59290	0.565817
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	5.02155	1.00431
$26$	0	0
$27$	0.894583	0.172163
$28$	0	0
$29$	9.11937	1.69342	0.846712	−	0.532051i	$-0.178578\pi$
$29$	9.11937	1.69342	0.846712	+	0.532051i	$0.178578\pi$
$30$	0	0
$31$	−8.04245	−1.44447	−0.722234	−	0.691649i	$-0.756885\pi$
$31$	−8.04245	−1.44447	−0.722234	+	0.691649i	$0.756885\pi$
$32$	0	0
$33$	−13.3064	−2.31636
$34$	0	0
$35$	3.46162	0.585121
$36$	0	0
$37$	−2.55717	−0.420396	−0.210198	−	0.977659i	$-0.567411\pi$
$37$	−2.55717	−0.420396	−0.210198	+	0.977659i	$0.567411\pi$
$38$	0	0
$39$	2.37123	0.379701
$40$	0	0
$41$	−7.13855	−1.11485	−0.557427	−	0.830226i	$-0.688211\pi$
$41$	−7.13855	−1.11485	−0.557427	+	0.830226i	$0.688211\pi$
$42$	0	0
$43$	−4.22532	−0.644355	−0.322178	−	0.946679i	$-0.604415\pi$
$43$	−4.22532	−0.644355	−0.322178	+	0.946679i	$0.604415\pi$
$44$	0	0
$45$	−8.30275	−1.23770
$46$	0	0
$47$	1.07565	0.156900	0.0784500	−	0.996918i	$-0.475003\pi$
$47$	1.07565	0.156900	0.0784500	+	0.996918i	$0.475003\pi$
$48$	0	0
$49$	−5.80429	−0.829185
$50$	0	0
$51$	0.512864	0.0718154
$52$	0	0
$53$	0.577213	0.0792863	0.0396431	−	0.999214i	$-0.487378\pi$
$53$	0.577213	0.0792863	0.0396431	+	0.999214i	$0.487378\pi$
$54$	0	0
$55$	−17.7646	−2.39538
$56$	0	0
$57$	14.3048	1.89472
$58$	0	0
$59$	4.96068	0.645825	0.322913	−	0.946429i	$-0.395338\pi$
$59$	4.96068	0.645825	0.322913	+	0.946429i	$0.395338\pi$
$60$	0	0
$61$	8.94721	1.14557	0.572787	−	0.819704i	$-0.305862\pi$
$61$	8.94721	1.14557	0.572787	+	0.819704i	$0.305862\pi$
$62$	0	0
$63$	−2.86792	−0.361323
$64$	0	0
$65$	3.16568	0.392655
$66$	0	0
$67$	−7.26955	−0.888117	−0.444059	−	0.895998i	$-0.646462\pi$
$67$	−7.26955	−0.888117	−0.444059	+	0.895998i	$0.646462\pi$
$68$	0	0
$69$	2.37123	0.285463
$70$	0	0
$71$	−8.41151	−0.998263	−0.499131	−	0.866526i	$-0.666347\pi$
$71$	−8.41151	−0.998263	−0.499131	+	0.866526i	$0.666347\pi$
$72$	0	0
$73$	−9.27194	−1.08520	−0.542599	−	0.839992i	$-0.682560\pi$
$73$	−9.27194	−1.08520	−0.542599	+	0.839992i	$0.682560\pi$
$74$	0	0
$75$	−11.9073	−1.37493
$76$	0	0
$77$	−6.13621	−0.699286
$78$	0	0
$79$	−4.24311	−0.477387	−0.238693	−	0.971095i	$-0.576719\pi$
$79$	−4.24311	−0.477387	−0.238693	+	0.971095i	$0.576719\pi$
$80$	0	0
$81$	−9.98947	−1.10994
$82$	0	0
$83$	6.74657	0.740533	0.370266	−	0.928926i	$-0.379266\pi$
$83$	6.74657	0.740533	0.370266	+	0.928926i	$0.379266\pi$
$84$	0	0
$85$	0.684694	0.0742655
$86$	0	0
$87$	−21.6241	−2.31835
$88$	0	0
$89$	−15.0002	−1.59002	−0.795008	−	0.606599i	$-0.792534\pi$
$89$	−15.0002	−1.59002	−0.795008	+	0.606599i	$0.792534\pi$
$90$	0	0
$91$	1.09348	0.114628
$92$	0	0
$93$	19.0705	1.97752
$94$	0	0
$95$	19.0975	1.95936
$96$	0	0
$97$	−15.7654	−1.60073	−0.800366	−	0.599512i	$-0.795361\pi$
$97$	−15.7654	−1.60073	−0.800366	+	0.599512i	$0.795361\pi$
$98$	0	0
$99$	14.7178	1.47919
$100$	0	0
$101$	4.71172	0.468833	0.234417	−	0.972136i	$-0.424682\pi$
$101$	4.71172	0.468833	0.234417	+	0.972136i	$0.424682\pi$
$102$	0	0
$103$	10.9592	1.07984	0.539921	−	0.841716i	$-0.318454\pi$
$103$	10.9592	1.07984	0.539921	+	0.841716i	$0.318454\pi$
$104$	0	0
$105$	−8.20830	−0.801048
$106$	0	0
$107$	−15.2331	−1.47264	−0.736319	−	0.676635i	$-0.763438\pi$
$107$	−15.2331	−1.47264	−0.736319	+	0.676635i	$0.763438\pi$
$108$	0	0
$109$	0.662195	0.0634268	0.0317134	−	0.999497i	$-0.489904\pi$
$109$	0.662195	0.0634268	0.0317134	+	0.999497i	$0.489904\pi$
$110$	0	0
$111$	6.06364	0.575535
$112$	0	0
$113$	4.97000	0.467538	0.233769	−	0.972292i	$-0.424894\pi$
$113$	4.97000	0.467538	0.233769	+	0.972292i	$0.424894\pi$
$114$	0	0
$115$	3.16568	0.295202
$116$	0	0
$117$	−2.62273	−0.242472
$118$	0	0
$119$	0.236505	0.0216804
$120$	0	0
$121$	20.4903	1.86275
$122$	0	0
$123$	16.9271	1.52627
$124$	0	0
$125$	−0.0682287	−0.00610256
$126$	0	0
$127$	3.66248	0.324992	0.162496	−	0.986709i	$-0.448045\pi$
$127$	3.66248	0.324992	0.162496	+	0.986709i	$0.448045\pi$
$128$	0	0
$129$	10.0192	0.882142
$130$	0	0
$131$	−9.96833	−0.870937	−0.435468	−	0.900204i	$-0.643417\pi$
$131$	−9.96833	−0.870937	−0.435468	+	0.900204i	$0.643417\pi$
$132$	0	0
$133$	6.59661	0.571999
$134$	0	0
$135$	−2.83197	−0.243737
$136$	0	0
$137$	−1.03707	−0.0886028	−0.0443014	−	0.999018i	$-0.514106\pi$
$137$	−1.03707	−0.0886028	−0.0443014	+	0.999018i	$0.514106\pi$
$138$	0	0
$139$	14.0681	1.19324	0.596620	−	0.802524i	$-0.296510\pi$
$139$	14.0681	1.19324	0.596620	+	0.802524i	$0.296510\pi$
$140$	0	0
$141$	−2.55062	−0.214801
$142$	0	0
$143$	−5.61162	−0.469267
$144$	0	0
$145$	−28.8690	−2.39744
$146$	0	0
$147$	13.7633	1.13518
$148$	0	0
$149$	4.42571	0.362568	0.181284	−	0.983431i	$-0.441975\pi$
$149$	4.42571	0.362568	0.181284	+	0.983431i	$0.441975\pi$
$150$	0	0
$151$	11.0342	0.897950	0.448975	−	0.893544i	$-0.351789\pi$
$151$	11.0342	0.897950	0.448975	+	0.893544i	$0.351789\pi$
$152$	0	0
$153$	−0.567261	−0.0458604
$154$	0	0
$155$	25.4599	2.04499
$156$	0	0
$157$	6.92247	0.552473	0.276237	−	0.961090i	$-0.410913\pi$
$157$	6.92247	0.552473	0.276237	+	0.961090i	$0.410913\pi$
$158$	0	0
$159$	−1.36870	−0.108545
$160$	0	0
$161$	1.09348	0.0861785
$162$	0	0
$163$	−14.4924	−1.13513	−0.567566	−	0.823328i	$-0.692115\pi$
$163$	−14.4924	−1.13513	−0.567566	+	0.823328i	$0.692115\pi$
$164$	0	0
$165$	42.1240	3.27935
$166$	0	0
$167$	−0.492391	−0.0381024	−0.0190512	−	0.999819i	$-0.506065\pi$
$167$	−0.492391	−0.0381024	−0.0190512	+	0.999819i	$0.506065\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−15.8221	−1.20994
$172$	0	0
$173$	−6.92186	−0.526259	−0.263129	−	0.964761i	$-0.584755\pi$
$173$	−6.92186	−0.526259	−0.263129	+	0.964761i	$0.584755\pi$
$174$	0	0
$175$	−5.49098	−0.415079
$176$	0	0
$177$	−11.7629	−0.884154
$178$	0	0
$179$	−24.1437	−1.80458	−0.902292	−	0.431126i	$-0.858116\pi$
$179$	−24.1437	−1.80458	−0.902292	+	0.431126i	$0.858116\pi$
$180$	0	0
$181$	14.6257	1.08712	0.543560	−	0.839370i	$-0.317076\pi$
$181$	14.6257	1.08712	0.543560	+	0.839370i	$0.317076\pi$
$182$	0	0
$183$	−21.2159	−1.56832
$184$	0	0
$185$	8.09519	0.595170
$186$	0	0
$187$	−1.21372	−0.0887557
$188$	0	0
$189$	−0.978212	−0.0711544
$190$	0	0
$191$	21.7537	1.57404	0.787021	−	0.616927i	$-0.211623\pi$
$191$	21.7537	1.57404	0.787021	+	0.616927i	$0.211623\pi$
$192$	0	0
$193$	5.42681	0.390630	0.195315	−	0.980741i	$-0.437427\pi$
$193$	5.42681	0.390630	0.195315	+	0.980741i	$0.437427\pi$
$194$	0	0
$195$	−7.50657	−0.537556
$196$	0	0
$197$	19.1789	1.36644	0.683221	−	0.730211i	$-0.260578\pi$
$197$	19.1789	1.36644	0.683221	+	0.730211i	$0.260578\pi$
$198$	0	0
$199$	15.0120	1.06417	0.532086	−	0.846690i	$-0.321408\pi$
$199$	15.0120	1.06417	0.532086	+	0.846690i	$0.321408\pi$
$200$	0	0
$201$	17.2378	1.21586
$202$	0	0
$203$	−9.97188	−0.699889
$204$	0	0
$205$	22.5984	1.57834
$206$	0	0
$207$	−2.62273	−0.182293
$208$	0	0
$209$	−33.8530	−2.34166
$210$	0	0
$211$	25.3580	1.74571	0.872857	−	0.487976i	$-0.162265\pi$
$211$	25.3580	1.74571	0.872857	+	0.487976i	$0.162265\pi$
$212$	0	0
$213$	19.9456	1.36665
$214$	0	0
$215$	13.3760	0.912238
$216$	0	0
$217$	8.79429	0.596995
$218$	0	0
$219$	21.9859	1.48567
$220$	0	0
$221$	0.216286	0.0145490
$222$	0	0
$223$	8.19092	0.548505	0.274252	−	0.961658i	$-0.411570\pi$
$223$	8.19092	0.548505	0.274252	+	0.961658i	$0.411570\pi$
$224$	0	0
$225$	13.1702	0.878013
$226$	0	0
$227$	−18.0533	−1.19824	−0.599120	−	0.800659i	$-0.704483\pi$
$227$	−18.0533	−1.19824	−0.599120	+	0.800659i	$0.704483\pi$
$228$	0	0
$229$	3.12577	0.206557	0.103278	−	0.994652i	$-0.467067\pi$
$229$	3.12577	0.206557	0.103278	+	0.994652i	$0.467067\pi$
$230$	0	0
$231$	14.5504	0.957344
$232$	0	0
$233$	16.0182	1.04938	0.524692	−	0.851292i	$-0.324180\pi$
$233$	16.0182	1.04938	0.524692	+	0.851292i	$0.324180\pi$
$234$	0	0
$235$	−3.40517	−0.222129
$236$	0	0
$237$	10.0614	0.653557
$238$	0	0
$239$	22.1469	1.43256	0.716282	−	0.697811i	$-0.245843\pi$
$239$	22.1469	1.43256	0.716282	+	0.697811i	$0.245843\pi$
$240$	0	0
$241$	29.1271	1.87624	0.938120	−	0.346309i	$-0.112565\pi$
$241$	29.1271	1.87624	0.938120	+	0.346309i	$0.112565\pi$
$242$	0	0
$243$	21.0036	1.34738
$244$	0	0
$245$	18.3746	1.17391
$246$	0	0
$247$	6.03266	0.383849
$248$	0	0
$249$	−15.9977	−1.01381
$250$	0	0
$251$	16.6635	1.05179	0.525895	−	0.850549i	$-0.323730\pi$
$251$	16.6635	1.05179	0.525895	+	0.850549i	$0.323730\pi$
$252$	0	0
$253$	−5.61162	−0.352800
$254$	0	0
$255$	−1.62357	−0.101672
$256$	0	0
$257$	−20.0539	−1.25093	−0.625464	−	0.780253i	$-0.715090\pi$
$257$	−20.0539	−1.25093	−0.625464	+	0.780253i	$0.715090\pi$
$258$	0	0
$259$	2.79622	0.173749
$260$	0	0
$261$	23.9177	1.48047
$262$	0	0
$263$	9.00900	0.555519	0.277759	−	0.960651i	$-0.410408\pi$
$263$	9.00900	0.555519	0.277759	+	0.960651i	$0.410408\pi$
$264$	0	0
$265$	−1.82727	−0.112248
$266$	0	0
$267$	35.5689	2.17678
$268$	0	0
$269$	−19.1642	−1.16846	−0.584230	−	0.811588i	$-0.698603\pi$
$269$	−19.1642	−1.16846	−0.584230	+	0.811588i	$0.698603\pi$
$270$	0	0
$271$	−15.6869	−0.952911	−0.476456	−	0.879199i	$-0.658079\pi$
$271$	−15.6869	−0.952911	−0.476456	+	0.879199i	$0.658079\pi$
$272$	0	0
$273$	−2.59290	−0.156929
$274$	0	0
$275$	28.1791	1.69926
$276$	0	0
$277$	22.6425	1.36046	0.680229	−	0.733000i	$-0.261881\pi$
$277$	22.6425	1.36046	0.680229	+	0.733000i	$0.261881\pi$
$278$	0	0
$279$	−21.0932	−1.26282
$280$	0	0
$281$	30.3083	1.80804	0.904021	−	0.427488i	$-0.140601\pi$
$281$	30.3083	1.80804	0.904021	+	0.427488i	$0.140601\pi$
$282$	0	0
$283$	−4.31430	−0.256459	−0.128229	−	0.991745i	$-0.540929\pi$
$283$	−4.31430	−0.256459	−0.128229	+	0.991745i	$0.540929\pi$
$284$	0	0
$285$	−45.2846	−2.68243
$286$	0	0
$287$	7.80588	0.460767
$288$	0	0
$289$	−16.9532	−0.997248
$290$	0	0
$291$	37.3833	2.19145
$292$	0	0
$293$	−26.1259	−1.52629	−0.763145	−	0.646228i	$-0.776346\pi$
$293$	−26.1259	−1.52629	−0.763145	+	0.646228i	$0.776346\pi$
$294$	0	0
$295$	−15.7039	−0.914318
$296$	0	0
$297$	5.02006	0.291294
$298$	0	0
$299$	1.00000	0.0578315
$300$	0	0
$301$	4.62032	0.266311
$302$	0	0
$303$	−11.1726	−0.641847
$304$	0	0
$305$	−28.3240	−1.62183
$306$	0	0
$307$	15.1631	0.865404	0.432702	−	0.901537i	$-0.357560\pi$
$307$	15.1631	0.865404	0.432702	+	0.901537i	$0.357560\pi$
$308$	0	0
$309$	−25.9868	−1.47834
$310$	0	0
$311$	7.65322	0.433975	0.216987	−	0.976174i	$-0.430377\pi$
$311$	7.65322	0.433975	0.216987	+	0.976174i	$0.430377\pi$
$312$	0	0
$313$	16.6191	0.939365	0.469683	−	0.882835i	$-0.344368\pi$
$313$	16.6191	0.939365	0.469683	+	0.882835i	$0.344368\pi$
$314$	0	0
$315$	9.07891	0.511539
$316$	0	0
$317$	−21.0679	−1.18329	−0.591646	−	0.806198i	$-0.701522\pi$
$317$	−21.0679	−1.18329	−0.591646	+	0.806198i	$0.701522\pi$
$318$	0	0
$319$	51.1745	2.86522
$320$	0	0
$321$	36.1211	2.01609
$322$	0	0
$323$	1.30478	0.0726000
$324$	0	0
$325$	−5.02155	−0.278546
$326$	0	0
$327$	−1.57022	−0.0868332
$328$	0	0
$329$	−1.17621	−0.0648464
$330$	0	0
$331$	−22.6982	−1.24761	−0.623803	−	0.781581i	$-0.714413\pi$
$331$	−22.6982	−1.24761	−0.623803	+	0.781581i	$0.714413\pi$
$332$	0	0
$333$	−6.70677	−0.367529
$334$	0	0
$335$	23.0131	1.25734
$336$	0	0
$337$	−36.5071	−1.98867	−0.994334	−	0.106302i	$-0.966099\pi$
$337$	−36.5071	−1.98867	−0.994334	+	0.106302i	$0.966099\pi$
$338$	0	0
$339$	−11.7850	−0.640074
$340$	0	0
$341$	−45.1312	−2.44399
$342$	0	0
$343$	14.0013	0.755998
$344$	0	0
$345$	−7.50657	−0.404140
$346$	0	0
$347$	−0.190187	−0.0102098	−0.00510488	−	0.999987i	$-0.501625\pi$
$347$	−0.190187	−0.0102098	−0.00510488	+	0.999987i	$0.501625\pi$
$348$	0	0
$349$	11.6025	0.621067	0.310533	−	0.950563i	$-0.399492\pi$
$349$	11.6025	0.621067	0.310533	+	0.950563i	$0.399492\pi$
$350$	0	0
$351$	−0.894583	−0.0477493
$352$	0	0
$353$	10.1717	0.541388	0.270694	−	0.962666i	$-0.412747\pi$
$353$	10.1717	0.541388	0.270694	+	0.962666i	$0.412747\pi$
$354$	0	0
$355$	26.6282	1.41328
$356$	0	0
$357$	−0.560809	−0.0296811
$358$	0	0
$359$	9.74914	0.514540	0.257270	−	0.966340i	$-0.417177\pi$
$359$	9.74914	0.514540	0.257270	+	0.966340i	$0.417177\pi$
$360$	0	0
$361$	17.3930	0.915420
$362$	0	0
$363$	−48.5872	−2.55017
$364$	0	0
$365$	29.3520	1.53636
$366$	0	0
$367$	−21.6515	−1.13020	−0.565101	−	0.825022i	$-0.691163\pi$
$367$	−21.6515	−1.13020	−0.565101	+	0.825022i	$0.691163\pi$
$368$	0	0
$369$	−18.7225	−0.974655
$370$	0	0
$371$	−0.631172	−0.0327688
$372$	0	0
$373$	−4.61039	−0.238717	−0.119359	−	0.992851i	$-0.538084\pi$
$373$	−4.61039	−0.238717	−0.119359	+	0.992851i	$0.538084\pi$
$374$	0	0
$375$	0.161786	0.00835459
$376$	0	0
$377$	−9.11937	−0.469672
$378$	0	0
$379$	16.8903	0.867594	0.433797	−	0.901011i	$-0.357173\pi$
$379$	16.8903	0.867594	0.433797	+	0.901011i	$0.357173\pi$
$380$	0	0
$381$	−8.68458	−0.444925
$382$	0	0
$383$	−21.1619	−1.08132	−0.540661	−	0.841240i	$-0.681826\pi$
$383$	−21.1619	−1.08132	−0.540661	+	0.841240i	$0.681826\pi$
$384$	0	0
$385$	19.4253	0.990005
$386$	0	0
$387$	−11.0819	−0.563324
$388$	0	0
$389$	−16.3200	−0.827455	−0.413727	−	0.910401i	$-0.635773\pi$
$389$	−16.3200	−0.827455	−0.413727	+	0.910401i	$0.635773\pi$
$390$	0	0
$391$	0.216286	0.0109381
$392$	0	0
$393$	23.6372	1.19234
$394$	0	0
$395$	13.4323	0.675854
$396$	0	0
$397$	8.97905	0.450646	0.225323	−	0.974284i	$-0.427656\pi$
$397$	8.97905	0.450646	0.225323	+	0.974284i	$0.427656\pi$
$398$	0	0
$399$	−15.6421	−0.783084
$400$	0	0
$401$	23.0991	1.15351	0.576757	−	0.816916i	$-0.304318\pi$
$401$	23.0991	1.15351	0.576757	+	0.816916i	$0.304318\pi$
$402$	0	0
$403$	8.04245	0.400623
$404$	0	0
$405$	31.6235	1.57138
$406$	0	0
$407$	−14.3499	−0.711296
$408$	0	0
$409$	5.51078	0.272491	0.136245	−	0.990675i	$-0.456496\pi$
$409$	5.51078	0.272491	0.136245	+	0.990675i	$0.456496\pi$
$410$	0	0
$411$	2.45913	0.121300
$412$	0	0
$413$	−5.42442	−0.266918
$414$	0	0
$415$	−21.3575	−1.04840
$416$	0	0
$417$	−33.3587	−1.63358
$418$	0	0
$419$	−7.11440	−0.347561	−0.173781	−	0.984784i	$-0.555598\pi$
$419$	−7.11440	−0.347561	−0.173781	+	0.984784i	$0.555598\pi$
$420$	0	0
$421$	−1.96792	−0.0959107	−0.0479554	−	0.998849i	$-0.515271\pi$
$421$	−1.96792	−0.0959107	−0.0479554	+	0.998849i	$0.515271\pi$
$422$	0	0
$423$	2.82115	0.137169
$424$	0	0
$425$	−1.08609	−0.0526832
$426$	0	0
$427$	−9.78363	−0.473463
$428$	0	0
$429$	13.3064	0.642441
$430$	0	0
$431$	1.92926	0.0929293	0.0464647	−	0.998920i	$-0.485205\pi$
$431$	1.92926	0.0929293	0.0464647	+	0.998920i	$0.485205\pi$
$432$	0	0
$433$	−16.8576	−0.810124	−0.405062	−	0.914289i	$-0.632750\pi$
$433$	−16.8576	−0.810124	−0.405062	+	0.914289i	$0.632750\pi$
$434$	0	0
$435$	68.4552	3.28217
$436$	0	0
$437$	6.03266	0.288581
$438$	0	0
$439$	−17.6365	−0.841746	−0.420873	−	0.907120i	$-0.638276\pi$
$439$	−17.6365	−0.841746	−0.420873	+	0.907120i	$0.638276\pi$
$440$	0	0
$441$	−15.2231	−0.724911
$442$	0	0
$443$	−21.6353	−1.02792	−0.513961	−	0.857813i	$-0.671822\pi$
$443$	−21.6353	−1.02792	−0.513961	+	0.857813i	$0.671822\pi$
$444$	0	0
$445$	47.4858	2.25104
$446$	0	0
$447$	−10.4944	−0.496367
$448$	0	0
$449$	−14.4722	−0.682984	−0.341492	−	0.939885i	$-0.610932\pi$
$449$	−14.4722	−0.682984	−0.341492	+	0.939885i	$0.610932\pi$
$450$	0	0
$451$	−40.0588	−1.88630
$452$	0	0
$453$	−26.1646	−1.22932
$454$	0	0
$455$	−3.46162	−0.162283
$456$	0	0
$457$	−15.1384	−0.708143	−0.354071	−	0.935218i	$-0.615203\pi$
$457$	−15.1384	−0.708143	−0.354071	+	0.935218i	$0.615203\pi$
$458$	0	0
$459$	−0.193486	−0.00903115
$460$	0	0
$461$	−11.9706	−0.557524	−0.278762	−	0.960360i	$-0.589924\pi$
$461$	−11.9706	−0.557524	−0.278762	+	0.960360i	$0.589924\pi$
$462$	0	0
$463$	−17.4422	−0.810608	−0.405304	−	0.914182i	$-0.632834\pi$
$463$	−17.4422	−0.810608	−0.405304	+	0.914182i	$0.632834\pi$
$464$	0	0
$465$	−60.3712	−2.79965
$466$	0	0
$467$	−1.56771	−0.0725450	−0.0362725	−	0.999342i	$-0.511548\pi$
$467$	−1.56771	−0.0725450	−0.0362725	+	0.999342i	$0.511548\pi$
$468$	0	0
$469$	7.94913	0.367057
$470$	0	0
$471$	−16.4148	−0.756353
$472$	0	0
$473$	−23.7109	−1.09023
$474$	0	0
$475$	−30.2933	−1.38995
$476$	0	0
$477$	1.51388	0.0693156
$478$	0	0
$479$	12.2507	0.559749	0.279875	−	0.960037i	$-0.409707\pi$
$479$	12.2507	0.559749	0.279875	+	0.960037i	$0.409707\pi$
$480$	0	0
$481$	2.55717	0.116597
$482$	0	0
$483$	−2.59290	−0.117981
$484$	0	0
$485$	49.9082	2.26621
$486$	0	0
$487$	−27.1802	−1.23165	−0.615827	−	0.787881i	$-0.711178\pi$
$487$	−27.1802	−1.23165	−0.615827	+	0.787881i	$0.711178\pi$
$488$	0	0
$489$	34.3648	1.55403
$490$	0	0
$491$	18.1400	0.818648	0.409324	−	0.912389i	$-0.365765\pi$
$491$	18.1400	0.818648	0.409324	+	0.912389i	$0.365765\pi$
$492$	0	0
$493$	−1.97239	−0.0888322
$494$	0	0
$495$	−46.5919	−2.09415
$496$	0	0
$497$	9.19785	0.412580
$498$	0	0
$499$	41.6502	1.86452	0.932260	−	0.361789i	$-0.117834\pi$
$499$	41.6502	1.86452	0.932260	+	0.361789i	$0.117834\pi$
$500$	0	0
$501$	1.16757	0.0521633
$502$	0	0
$503$	−21.7307	−0.968925	−0.484463	−	0.874812i	$-0.660985\pi$
$503$	−21.7307	−0.968925	−0.484463	+	0.874812i	$0.660985\pi$
$504$	0	0
$505$	−14.9158	−0.663745
$506$	0	0
$507$	−2.37123	−0.105310
$508$	0	0
$509$	10.7617	0.477006	0.238503	−	0.971142i	$-0.423343\pi$
$509$	10.7617	0.477006	0.238503	+	0.971142i	$0.423343\pi$
$510$	0	0
$511$	10.1387	0.448510
$512$	0	0
$513$	−5.39672	−0.238271
$514$	0	0
$515$	−34.6934	−1.52877
$516$	0	0
$517$	6.03615	0.265470
$518$	0	0
$519$	16.4133	0.720465
$520$	0	0
$521$	−3.82763	−0.167691	−0.0838457	−	0.996479i	$-0.526720\pi$
$521$	−3.82763	−0.167691	−0.0838457	+	0.996479i	$0.526720\pi$
$522$	0	0
$523$	41.9169	1.83290	0.916450	−	0.400150i	$-0.131042\pi$
$523$	41.9169	1.83290	0.916450	+	0.400150i	$0.131042\pi$
$524$	0	0
$525$	13.0204	0.568256
$526$	0	0
$527$	1.73947	0.0757726
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	13.0105	0.564609
$532$	0	0
$533$	7.13855	0.309205
$534$	0	0
$535$	48.2231	2.08487
$536$	0	0
$537$	57.2502	2.47053
$538$	0	0
$539$	−32.5715	−1.40295
$540$	0	0
$541$	21.6940	0.932699	0.466349	−	0.884601i	$-0.345569\pi$
$541$	21.6940	0.932699	0.466349	+	0.884601i	$0.345569\pi$
$542$	0	0
$543$	−34.6809	−1.48830
$544$	0	0
$545$	−2.09630	−0.0897956
$546$	0	0
$547$	39.3753	1.68356	0.841782	−	0.539817i	$-0.181507\pi$
$547$	39.3753	1.68356	0.841782	+	0.539817i	$0.181507\pi$
$548$	0	0
$549$	23.4662	1.00151
$550$	0	0
$551$	−55.0141	−2.34368
$552$	0	0
$553$	4.63977	0.197303
$554$	0	0
$555$	−19.1956	−0.814806
$556$	0	0
$557$	34.6203	1.46691	0.733455	−	0.679738i	$-0.237907\pi$
$557$	34.6203	1.46691	0.733455	+	0.679738i	$0.237907\pi$
$558$	0	0
$559$	4.22532	0.178712
$560$	0	0
$561$	2.87800	0.121509
$562$	0	0
$563$	4.66039	0.196412	0.0982059	−	0.995166i	$-0.468690\pi$
$563$	4.66039	0.196412	0.0982059	+	0.995166i	$0.468690\pi$
$564$	0	0
$565$	−15.7334	−0.661911
$566$	0	0
$567$	10.9233	0.458736
$568$	0	0
$569$	41.2171	1.72791	0.863956	−	0.503567i	$-0.167979\pi$
$569$	41.2171	1.72791	0.863956	+	0.503567i	$0.167979\pi$
$570$	0	0
$571$	32.8618	1.37522	0.687612	−	0.726078i	$-0.258659\pi$
$571$	32.8618	1.37522	0.687612	+	0.726078i	$0.258659\pi$
$572$	0	0
$573$	−51.5830	−2.15491
$574$	0	0
$575$	−5.02155	−0.209413
$576$	0	0
$577$	12.9817	0.540434	0.270217	−	0.962800i	$-0.412905\pi$
$577$	12.9817	0.540434	0.270217	+	0.962800i	$0.412905\pi$
$578$	0	0
$579$	−12.8682	−0.534785
$580$	0	0
$581$	−7.37727	−0.306061
$582$	0	0
$583$	3.23910	0.134150
$584$	0	0
$585$	8.30275	0.343276
$586$	0	0
$587$	36.3803	1.50158	0.750788	−	0.660543i	$-0.229674\pi$
$587$	36.3803	1.50158	0.750788	+	0.660543i	$0.229674\pi$
$588$	0	0
$589$	48.5174	1.99912
$590$	0	0
$591$	−45.4777	−1.87070
$592$	0	0
$593$	−17.3748	−0.713499	−0.356750	−	0.934200i	$-0.616115\pi$
$593$	−17.3748	−0.713499	−0.356750	+	0.934200i	$0.616115\pi$
$594$	0	0
$595$	−0.748701	−0.0306937
$596$	0	0
$597$	−35.5969	−1.45688
$598$	0	0
$599$	−1.05120	−0.0429507	−0.0214754	−	0.999769i	$-0.506836\pi$
$599$	−1.05120	−0.0429507	−0.0214754	+	0.999769i	$0.506836\pi$
$600$	0	0
$601$	11.2683	0.459643	0.229821	−	0.973233i	$-0.426186\pi$
$601$	11.2683	0.459643	0.229821	+	0.973233i	$0.426186\pi$
$602$	0	0
$603$	−19.0661	−0.776432
$604$	0	0
$605$	−64.8658	−2.63717
$606$	0	0
$607$	−25.1906	−1.02245	−0.511227	−	0.859445i	$-0.670809\pi$
$607$	−25.1906	−1.02245	−0.511227	+	0.859445i	$0.670809\pi$
$608$	0	0
$609$	23.6456	0.958169
$610$	0	0
$611$	−1.07565	−0.0435162
$612$	0	0
$613$	9.21738	0.372287	0.186143	−	0.982523i	$-0.440401\pi$
$613$	9.21738	0.372287	0.186143	+	0.982523i	$0.440401\pi$
$614$	0	0
$615$	−53.5860	−2.16080
$616$	0	0
$617$	−21.4518	−0.863618	−0.431809	−	0.901965i	$-0.642124\pi$
$617$	−21.4518	−0.863618	−0.431809	+	0.901965i	$0.642124\pi$
$618$	0	0
$619$	−19.8781	−0.798969	−0.399485	−	0.916740i	$-0.630811\pi$
$619$	−19.8781	−0.798969	−0.399485	+	0.916740i	$0.630811\pi$
$620$	0	0
$621$	−0.894583	−0.0358984
$622$	0	0
$623$	16.4024	0.657150
$624$	0	0
$625$	−24.8918	−0.995671
$626$	0	0
$627$	80.2733	3.20581
$628$	0	0
$629$	0.553080	0.0220528
$630$	0	0
$631$	2.22972	0.0887638	0.0443819	−	0.999015i	$-0.485868\pi$
$631$	2.22972	0.0887638	0.0443819	+	0.999015i	$0.485868\pi$
$632$	0	0
$633$	−60.1296	−2.38994
$634$	0	0
$635$	−11.5942	−0.460104
$636$	0	0
$637$	5.80429	0.229975
$638$	0	0
$639$	−22.0612	−0.872726
$640$	0	0
$641$	1.43174	0.0565503	0.0282751	−	0.999600i	$-0.490999\pi$
$641$	1.43174	0.0565503	0.0282751	+	0.999600i	$0.490999\pi$
$642$	0	0
$643$	26.2134	1.03375	0.516877	−	0.856060i	$-0.327094\pi$
$643$	26.2134	1.03375	0.516877	+	0.856060i	$0.327094\pi$
$644$	0	0
$645$	−31.7177	−1.24888
$646$	0	0
$647$	20.6640	0.812386	0.406193	−	0.913787i	$-0.366856\pi$
$647$	20.6640	0.812386	0.406193	+	0.913787i	$0.366856\pi$
$648$	0	0
$649$	27.8374	1.09272
$650$	0	0
$651$	−20.8533	−0.817305
$652$	0	0
$653$	10.8658	0.425212	0.212606	−	0.977138i	$-0.431805\pi$
$653$	10.8658	0.425212	0.212606	+	0.977138i	$0.431805\pi$
$654$	0	0
$655$	31.5566	1.23302
$656$	0	0
$657$	−24.3178	−0.948729
$658$	0	0
$659$	−28.6913	−1.11765	−0.558827	−	0.829284i	$-0.688749\pi$
$659$	−28.6913	−1.11765	−0.558827	+	0.829284i	$0.688749\pi$
$660$	0	0
$661$	22.5301	0.876318	0.438159	−	0.898898i	$-0.355631\pi$
$661$	22.5301	0.876318	0.438159	+	0.898898i	$0.355631\pi$
$662$	0	0
$663$	−0.512864	−0.0199180
$664$	0	0
$665$	−20.8828	−0.809800
$666$	0	0
$667$	−9.11937	−0.353104
$668$	0	0
$669$	−19.4226	−0.750920
$670$	0	0
$671$	50.2084	1.93827
$672$	0	0
$673$	31.5184	1.21495	0.607473	−	0.794341i	$-0.292183\pi$
$673$	31.5184	1.21495	0.607473	+	0.794341i	$0.292183\pi$
$674$	0	0
$675$	4.49220	0.172905
$676$	0	0
$677$	26.4770	1.01759	0.508797	−	0.860887i	$-0.330090\pi$
$677$	26.4770	1.01759	0.508797	+	0.860887i	$0.330090\pi$
$678$	0	0
$679$	17.2392	0.661579
$680$	0	0
$681$	42.8085	1.64043
$682$	0	0
$683$	−17.3961	−0.665643	−0.332821	−	0.942990i	$-0.608001\pi$
$683$	−17.3961	−0.665643	−0.332821	+	0.942990i	$0.608001\pi$
$684$	0	0
$685$	3.28303	0.125438
$686$	0	0
$687$	−7.41193	−0.282783
$688$	0	0
$689$	−0.577213	−0.0219901
$690$	0	0
$691$	−51.1248	−1.94488	−0.972440	−	0.233153i	$-0.925096\pi$
$691$	−51.1248	−1.94488	−0.972440	+	0.233153i	$0.925096\pi$
$692$	0	0
$693$	−16.0937	−0.611348
$694$	0	0
$695$	−44.5351	−1.68931
$696$	0	0
$697$	1.54397	0.0584820
$698$	0	0
$699$	−37.9828	−1.43664
$700$	0	0
$701$	2.98710	0.112821	0.0564105	−	0.998408i	$-0.482034\pi$
$701$	2.98710	0.112821	0.0564105	+	0.998408i	$0.482034\pi$
$702$	0	0
$703$	15.4265	0.581823
$704$	0	0
$705$	8.07445	0.304101
$706$	0	0
$707$	−5.15218	−0.193768
$708$	0	0
$709$	22.8486	0.858096	0.429048	−	0.903282i	$-0.358849\pi$
$709$	22.8486	0.858096	0.429048	+	0.903282i	$0.358849\pi$
$710$	0	0
$711$	−11.1285	−0.417353
$712$	0	0
$713$	8.04245	0.301192
$714$	0	0
$715$	17.7646	0.664359
$716$	0	0
$717$	−52.5154	−1.96122
$718$	0	0
$719$	−13.0998	−0.488540	−0.244270	−	0.969707i	$-0.578548\pi$
$719$	−13.0998	−0.488540	−0.244270	+	0.969707i	$0.578548\pi$
$720$	0	0
$721$	−11.9837	−0.446296
$722$	0	0
$723$	−69.0670	−2.56863
$724$	0	0
$725$	45.7934	1.70072
$726$	0	0
$727$	−6.86149	−0.254479	−0.127239	−	0.991872i	$-0.540612\pi$
$727$	−6.86149	−0.254479	−0.127239	+	0.991872i	$0.540612\pi$
$728$	0	0
$729$	−19.8359	−0.734664
$730$	0	0
$731$	0.913879	0.0338010
$732$	0	0
$733$	−25.5999	−0.945553	−0.472776	−	0.881182i	$-0.656748\pi$
$733$	−25.5999	−0.945553	−0.472776	+	0.881182i	$0.656748\pi$
$734$	0	0
$735$	−43.5703	−1.60712
$736$	0	0
$737$	−40.7940	−1.50267
$738$	0	0
$739$	−10.4709	−0.385177	−0.192589	−	0.981280i	$-0.561688\pi$
$739$	−10.4709	−0.385177	−0.192589	+	0.981280i	$0.561688\pi$
$740$	0	0
$741$	−14.3048	−0.525501
$742$	0	0
$743$	−3.87862	−0.142293	−0.0711463	−	0.997466i	$-0.522666\pi$
$743$	−3.87862	−0.142293	−0.0711463	+	0.997466i	$0.522666\pi$
$744$	0	0
$745$	−14.0104	−0.513301
$746$	0	0
$747$	17.6945	0.647407
$748$	0	0
$749$	16.6571	0.608638
$750$	0	0
$751$	−27.3306	−0.997309	−0.498654	−	0.866801i	$-0.666172\pi$
$751$	−27.3306	−0.997309	−0.498654	+	0.866801i	$0.666172\pi$
$752$	0	0
$753$	−39.5130	−1.43993
$754$	0	0
$755$	−34.9308	−1.27126
$756$	0	0
$757$	−1.67111	−0.0607375	−0.0303687	−	0.999539i	$-0.509668\pi$
$757$	−1.67111	−0.0607375	−0.0303687	+	0.999539i	$0.509668\pi$
$758$	0	0
$759$	13.3064	0.482993
$760$	0	0
$761$	31.3220	1.13542	0.567711	−	0.823228i	$-0.307829\pi$
$761$	31.3220	1.13542	0.567711	+	0.823228i	$0.307829\pi$
$762$	0	0
$763$	−0.724099	−0.0262141
$764$	0	0
$765$	1.79577	0.0649262
$766$	0	0
$767$	−4.96068	−0.179120
$768$	0	0
$769$	0.211623	0.00763130	0.00381565	−	0.999993i	$-0.498785\pi$
$769$	0.211623	0.00763130	0.00381565	+	0.999993i	$0.498785\pi$
$770$	0	0
$771$	47.5525	1.71256
$772$	0	0
$773$	−15.2808	−0.549613	−0.274807	−	0.961500i	$-0.588614\pi$
$773$	−15.2808	−0.549613	−0.274807	+	0.961500i	$0.588614\pi$
$774$	0	0
$775$	−40.3856	−1.45069
$776$	0	0
$777$	−6.63048	−0.237867
$778$	0	0
$779$	43.0644	1.54294
$780$	0	0
$781$	−47.2022	−1.68903
$782$	0	0
$783$	8.15804	0.291544
$784$	0	0
$785$	−21.9144	−0.782157
$786$	0	0
$787$	41.3904	1.47541	0.737705	−	0.675123i	$-0.235910\pi$
$787$	41.3904	1.47541	0.737705	+	0.675123i	$0.235910\pi$
$788$	0	0
$789$	−21.3624	−0.760522
$790$	0	0
$791$	−5.43461	−0.193232
$792$	0	0
$793$	−8.94721	−0.317725
$794$	0	0
$795$	4.33288	0.153672
$796$	0	0
$797$	1.67212	0.0592294	0.0296147	−	0.999561i	$-0.490572\pi$
$797$	1.67212	0.0592294	0.0296147	+	0.999561i	$0.490572\pi$
$798$	0	0
$799$	−0.232649	−0.00823052
$800$	0	0
$801$	−39.3415	−1.39006
$802$	0	0
$803$	−52.0306	−1.83612
$804$	0	0
$805$	−3.46162	−0.122006
$806$	0	0
$807$	45.4426	1.59966
$808$	0	0
$809$	36.6250	1.28767	0.643833	−	0.765166i	$-0.277343\pi$
$809$	36.6250	1.28767	0.643833	+	0.765166i	$0.277343\pi$
$810$	0	0
$811$	32.4359	1.13898	0.569489	−	0.821999i	$-0.307141\pi$
$811$	32.4359	1.13898	0.569489	+	0.821999i	$0.307141\pi$
$812$	0	0
$813$	37.1972	1.30456
$814$	0	0
$815$	45.8783	1.60705
$816$	0	0
$817$	25.4899	0.891780
$818$	0	0
$819$	2.86792	0.100213
$820$	0	0
$821$	49.2576	1.71910	0.859551	−	0.511049i	$-0.170743\pi$
$821$	49.2576	1.71910	0.859551	+	0.511049i	$0.170743\pi$
$822$	0	0
$823$	−12.2443	−0.426810	−0.213405	−	0.976964i	$-0.568455\pi$
$823$	−12.2443	−0.426810	−0.213405	+	0.976964i	$0.568455\pi$
$824$	0	0
$825$	−66.8190	−2.32634
$826$	0	0
$827$	24.0204	0.835273	0.417636	−	0.908614i	$-0.362859\pi$
$827$	24.0204	0.835273	0.417636	+	0.908614i	$0.362859\pi$
$828$	0	0
$829$	−1.14959	−0.0399269	−0.0199634	−	0.999801i	$-0.506355\pi$
$829$	−1.14959	−0.0399269	−0.0199634	+	0.999801i	$0.506355\pi$
$830$	0	0
$831$	−53.6906	−1.86251
$832$	0	0
$833$	1.25539	0.0434966
$834$	0	0
$835$	1.55875	0.0539429
$836$	0	0
$837$	−7.19465	−0.248683
$838$	0	0
$839$	36.9364	1.27518	0.637592	−	0.770374i	$-0.279930\pi$
$839$	36.9364	1.27518	0.637592	+	0.770374i	$0.279930\pi$
$840$	0	0
$841$	54.1630	1.86769
$842$	0	0
$843$	−71.8680	−2.47526
$844$	0	0
$845$	−3.16568	−0.108903
$846$	0	0
$847$	−22.4058	−0.769872
$848$	0	0
$849$	10.2302	0.351100
$850$	0	0
$851$	2.55717	0.0876586
$852$	0	0
$853$	2.42115	0.0828985	0.0414493	−	0.999141i	$-0.486803\pi$
$853$	2.42115	0.0828985	0.0414493	+	0.999141i	$0.486803\pi$
$854$	0	0
$855$	50.0877	1.71296
$856$	0	0
$857$	−30.2965	−1.03491	−0.517455	−	0.855710i	$-0.673121\pi$
$857$	−30.2965	−1.03491	−0.517455	+	0.855710i	$0.673121\pi$
$858$	0	0
$859$	3.26790	0.111499	0.0557496	−	0.998445i	$-0.482245\pi$
$859$	3.26790	0.111499	0.0557496	+	0.998445i	$0.482245\pi$
$860$	0	0
$861$	−18.5095	−0.630804
$862$	0	0
$863$	−38.8796	−1.32348	−0.661738	−	0.749735i	$-0.730181\pi$
$863$	−38.8796	−1.32348	−0.661738	+	0.749735i	$0.730181\pi$
$864$	0	0
$865$	21.9124	0.745044
$866$	0	0
$867$	40.2000	1.36526
$868$	0	0
$869$	−23.8107	−0.807723
$870$	0	0
$871$	7.26955	0.246319
$872$	0	0
$873$	−41.3484	−1.39943
$874$	0	0
$875$	0.0746069	0.00252217
$876$	0	0
$877$	0.673607	0.0227461	0.0113731	−	0.999935i	$-0.496380\pi$
$877$	0.673607	0.0227461	0.0113731	+	0.999935i	$0.496380\pi$
$878$	0	0
$879$	61.9505	2.08954
$880$	0	0
$881$	8.16669	0.275143	0.137571	−	0.990492i	$-0.456070\pi$
$881$	8.16669	0.275143	0.137571	+	0.990492i	$0.456070\pi$
$882$	0	0
$883$	15.7235	0.529137	0.264568	−	0.964367i	$-0.414771\pi$
$883$	15.7235	0.529137	0.264568	+	0.964367i	$0.414771\pi$
$884$	0	0
$885$	37.2377	1.25173
$886$	0	0
$887$	27.0798	0.909251	0.454625	−	0.890683i	$-0.349773\pi$
$887$	27.0798	0.909251	0.454625	+	0.890683i	$0.349773\pi$
$888$	0	0
$889$	−4.00486	−0.134319
$890$	0	0
$891$	−56.0571	−1.87798
$892$	0	0
$893$	−6.48904	−0.217148
$894$	0	0
$895$	76.4312	2.55481
$896$	0	0
$897$	−2.37123	−0.0791731
$898$	0	0
$899$	−73.3421	−2.44610
$900$	0	0
$901$	−0.124843	−0.00415913
$902$	0	0
$903$	−10.9558	−0.364587
$904$	0	0
$905$	−46.3004	−1.53908
$906$	0	0
$907$	−9.71580	−0.322608	−0.161304	−	0.986905i	$-0.551570\pi$
$907$	−9.71580	−0.322608	−0.161304	+	0.986905i	$0.551570\pi$
$908$	0	0
$909$	12.3576	0.409875
$910$	0	0
$911$	3.68558	0.122109	0.0610543	−	0.998134i	$-0.480554\pi$
$911$	3.68558	0.122109	0.0610543	+	0.998134i	$0.480554\pi$
$912$	0	0
$913$	37.8592	1.25296
$914$	0	0
$915$	67.1628	2.22034
$916$	0	0
$917$	10.9002	0.359956
$918$	0	0
$919$	1.97858	0.0652673	0.0326337	−	0.999467i	$-0.489611\pi$
$919$	1.97858	0.0652673	0.0326337	+	0.999467i	$0.489611\pi$
$920$	0	0
$921$	−35.9552	−1.18476
$922$	0	0
$923$	8.41151	0.276868
$924$	0	0
$925$	−12.8410	−0.422208
$926$	0	0
$927$	28.7431	0.944047
$928$	0	0
$929$	−0.206658	−0.00678023	−0.00339012	−	0.999994i	$-0.501079\pi$
$929$	−0.206658	−0.00678023	−0.00339012	+	0.999994i	$0.501079\pi$
$930$	0	0
$931$	35.0153	1.14758
$932$	0	0
$933$	−18.1476	−0.594124
$934$	0	0
$935$	3.84224	0.125655
$936$	0	0
$937$	36.0425	1.17746	0.588728	−	0.808331i	$-0.299629\pi$
$937$	36.0425	1.17746	0.588728	+	0.808331i	$0.299629\pi$
$938$	0	0
$939$	−39.4077	−1.28602
$940$	0	0
$941$	−15.5431	−0.506690	−0.253345	−	0.967376i	$-0.581531\pi$
$941$	−15.5431	−0.506690	−0.253345	+	0.967376i	$0.581531\pi$
$942$	0	0
$943$	7.13855	0.232463
$944$	0	0
$945$	3.09671	0.100736
$946$	0	0
$947$	2.87566	0.0934464	0.0467232	−	0.998908i	$-0.485122\pi$
$947$	2.87566	0.0934464	0.0467232	+	0.998908i	$0.485122\pi$
$948$	0	0
$949$	9.27194	0.300980
$950$	0	0
$951$	49.9569	1.61996
$952$	0	0
$953$	30.3101	0.981841	0.490921	−	0.871204i	$-0.336661\pi$
$953$	30.3101	0.981841	0.490921	+	0.871204i	$0.336661\pi$
$954$	0	0
$955$	−68.8653	−2.22843
$956$	0	0
$957$	−121.346	−3.92257
$958$	0	0
$959$	1.13402	0.0366193
$960$	0	0
$961$	33.6811	1.08649
$962$	0	0
$963$	−39.9523	−1.28745
$964$	0	0
$965$	−17.1796	−0.553030
$966$	0	0
$967$	−19.2278	−0.618323	−0.309162	−	0.951010i	$-0.600048\pi$
$967$	−19.2278	−0.618323	−0.309162	+	0.951010i	$0.600048\pi$
$968$	0	0
$969$	−3.09394	−0.0993916
$970$	0	0
$971$	−26.9172	−0.863815	−0.431907	−	0.901918i	$-0.642159\pi$
$971$	−26.9172	−0.863815	−0.431907	+	0.901918i	$0.642159\pi$
$972$	0	0
$973$	−15.3832	−0.493163
$974$	0	0
$975$	11.9073	0.381337
$976$	0	0
$977$	33.5682	1.07394	0.536971	−	0.843601i	$-0.319569\pi$
$977$	33.5682	1.07394	0.536971	+	0.843601i	$0.319569\pi$
$978$	0	0
$979$	−84.1754	−2.69026
$980$	0	0
$981$	1.73676	0.0554505
$982$	0	0
$983$	−7.98670	−0.254736	−0.127368	−	0.991856i	$-0.540653\pi$
$983$	−7.98670	−0.254736	−0.127368	+	0.991856i	$0.540653\pi$
$984$	0	0
$985$	−60.7145	−1.93452
$986$	0	0
$987$	2.78906	0.0887767
$988$	0	0
$989$	4.22532	0.134357
$990$	0	0
$991$	32.4610	1.03116	0.515578	−	0.856842i	$-0.327577\pi$
$991$	32.4610	1.03116	0.515578	+	0.856842i	$0.327577\pi$
$992$	0	0
$993$	53.8227	1.70801
$994$	0	0
$995$	−47.5232	−1.50659
$996$	0	0
$997$	−57.7905	−1.83024	−0.915121	−	0.403178i	$-0.867905\pi$
$997$	−57.7905	−1.83024	−0.915121	+	0.403178i	$0.867905\pi$
$998$	0	0
$999$	−2.28760	−0.0723765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4784.2.a.bh.1.3		10
4.3	odd	2		299.2.a.g.1.9	✓	10
12.11	even	2		2691.2.a.bc.1.2		10
20.19	odd	2		7475.2.a.w.1.2		10
52.51	odd	2		3887.2.a.p.1.2		10
92.91	even	2		6877.2.a.o.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
299.2.a.g.1.9	✓	10	4.3	odd	2
2691.2.a.bc.1.2		10	12.11	even	2
3887.2.a.p.1.2		10	52.51	odd	2
4784.2.a.bh.1.3		10	1.1	even	1	trivial
6877.2.a.o.1.9		10	92.91	even	2
7475.2.a.w.1.2		10	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4784 = 2^{4} \cdot 13 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4784.a (trivial)

\(f(q)\)	\(=\)	\(q-2.37123 q^{3} -3.16568 q^{5} -1.09348 q^{7} +2.62273 q^{9} +O(q^{10})\)	\(q-2.37123 q^{3} -3.16568 q^{5} -1.09348 q^{7} +2.62273 q^{9} +5.61162 q^{11} -1.00000 q^{13} +7.50657 q^{15} -0.216286 q^{17} -6.03266 q^{19} +2.59290 q^{21} -1.00000 q^{23} +5.02155 q^{25} +0.894583 q^{27} +9.11937 q^{29} -8.04245 q^{31} -13.3064 q^{33} +3.46162 q^{35} -2.55717 q^{37} +2.37123 q^{39} -7.13855 q^{41} -4.22532 q^{43} -8.30275 q^{45} +1.07565 q^{47} -5.80429 q^{49} +0.512864 q^{51} +0.577213 q^{53} -17.7646 q^{55} +14.3048 q^{57} +4.96068 q^{59} +8.94721 q^{61} -2.86792 q^{63} +3.16568 q^{65} -7.26955 q^{67} +2.37123 q^{69} -8.41151 q^{71} -9.27194 q^{73} -11.9073 q^{75} -6.13621 q^{77} -4.24311 q^{79} -9.98947 q^{81} +6.74657 q^{83} +0.684694 q^{85} -21.6241 q^{87} -15.0002 q^{89} +1.09348 q^{91} +19.0705 q^{93} +19.0975 q^{95} -15.7654 q^{97} +14.7178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 3 q^{3} + 3 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) \) 10 * q - 3 * q^3 + 3 * q^5 + 2 * q^7 + 17 * q^9	\( 10 q - 3 q^{3} + 3 q^{5} + 2 q^{7} + 17 q^{9} - 3 q^{11} - 10 q^{13} - 2 q^{15} - 3 q^{17} - 2 q^{19} + 21 q^{21} - 10 q^{23} + 33 q^{25} - 6 q^{27} + 17 q^{29} - 5 q^{31} - 23 q^{33} - 3 q^{35} + 16 q^{37} + 3 q^{39} - 16 q^{41} + 9 q^{43} + 32 q^{45} + 11 q^{47} + 40 q^{49} + 31 q^{51} + 8 q^{53} + 14 q^{55} - 35 q^{57} - 2 q^{59} + 48 q^{61} + 15 q^{63} - 3 q^{65} + 6 q^{67} + 3 q^{69} - 24 q^{71} - 33 q^{73} + 22 q^{75} + 15 q^{77} - 17 q^{79} + 30 q^{81} + 21 q^{83} + 58 q^{85} - 23 q^{87} - 16 q^{89} - 2 q^{91} + 15 q^{93} + 27 q^{95} - 40 q^{97} + 17 q^{99}+O(q^{100}) \) 10 * q - 3 * q^3 + 3 * q^5 + 2 * q^7 + 17 * q^9 - 3 * q^11 - 10 * q^13 - 2 * q^15 - 3 * q^17 - 2 * q^19 + 21 * q^21 - 10 * q^23 + 33 * q^25 - 6 * q^27 + 17 * q^29 - 5 * q^31 - 23 * q^33 - 3 * q^35 + 16 * q^37 + 3 * q^39 - 16 * q^41 + 9 * q^43 + 32 * q^45 + 11 * q^47 + 40 * q^49 + 31 * q^51 + 8 * q^53 + 14 * q^55 - 35 * q^57 - 2 * q^59 + 48 * q^61 + 15 * q^63 - 3 * q^65 + 6 * q^67 + 3 * q^69 - 24 * q^71 - 33 * q^73 + 22 * q^75 + 15 * q^77 - 17 * q^79 + 30 * q^81 + 21 * q^83 + 58 * q^85 - 23 * q^87 - 16 * q^89 - 2 * q^91 + 15 * q^93 + 27 * q^95 - 40 * q^97 + 17 * q^99

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