LMFDB - Embedded newform 76.8.a.a.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(1,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("76.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.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:	$23.7412619368$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5014x^{3} + 113222x^{2} - 625803x + 567036 $ x^5 - 2x^4 - 5014x^3 + 113222x^2 - 625803x + 567036
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{2}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$6.88490$ of defining polynomial
Character	$\chi$	$=$	76.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.4201 q^{3} -3.35619 q^{5} -173.262 q^{7} -1540.82 q^{9} +O(q^{10})$	$q+25.4201 q^{3} -3.35619 q^{5} -173.262 q^{7} -1540.82 q^{9} +3804.09 q^{11} -14439.0 q^{13} -85.3148 q^{15} +3292.85 q^{17} -6859.00 q^{19} -4404.35 q^{21} +643.599 q^{23} -78113.7 q^{25} -94761.6 q^{27} +93767.3 q^{29} -103884. q^{31} +96700.4 q^{33} +581.501 q^{35} -322176. q^{37} -367042. q^{39} -843931. q^{41} -289803. q^{43} +5171.27 q^{45} +772392. q^{47} -793523. q^{49} +83704.8 q^{51} -596172. q^{53} -12767.2 q^{55} -174357. q^{57} -2.76656e6 q^{59} +2.39723e6 q^{61} +266965. q^{63} +48460.1 q^{65} +2.71365e6 q^{67} +16360.4 q^{69} +5.03316e6 q^{71} +3.65348e6 q^{73} -1.98566e6 q^{75} -659104. q^{77} -4.72959e6 q^{79} +960912. q^{81} +2.28839e6 q^{83} -11051.4 q^{85} +2.38358e6 q^{87} +8.80023e6 q^{89} +2.50174e6 q^{91} -2.64074e6 q^{93} +23020.1 q^{95} +1.21813e7 q^{97} -5.86140e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 14 q^{3} - 280 q^{5} + 414 q^{7} + 3779 q^{9}+O(q^{10}) $ 5 * q - 14 * q^3 - 280 * q^5 + 414 * q^7 + 3779 * q^9	$ 5 q - 14 q^{3} - 280 q^{5} + 414 q^{7} + 3779 q^{9} - 2662 q^{11} - 602 q^{13} - 20800 q^{15} - 27366 q^{17} - 34295 q^{19} - 59964 q^{21} - 67096 q^{23} - 109115 q^{25} - 178778 q^{27} - 372398 q^{29} - 271372 q^{31} - 792700 q^{33} - 608250 q^{35} - 562630 q^{37} - 963904 q^{39} - 956714 q^{41} - 827362 q^{43} - 1165100 q^{45} - 1812982 q^{47} - 862031 q^{49} - 2458254 q^{51} + 486998 q^{53} + 467930 q^{55} + 96026 q^{57} - 367182 q^{59} + 1879732 q^{61} - 1007274 q^{63} + 1790920 q^{65} - 1046394 q^{67} + 7261712 q^{69} - 4664572 q^{71} + 4224942 q^{73} + 8194850 q^{75} + 8611110 q^{77} + 9574024 q^{79} + 11351813 q^{81} + 11754804 q^{83} + 18711750 q^{85} + 3801472 q^{87} + 2782542 q^{89} + 7385214 q^{91} + 29535004 q^{93} + 1920520 q^{95} + 1291574 q^{97} - 9760310 q^{99}+O(q^{100}) $ 5 * q - 14 * q^3 - 280 * q^5 + 414 * q^7 + 3779 * q^9 - 2662 * q^11 - 602 * q^13 - 20800 * q^15 - 27366 * q^17 - 34295 * q^19 - 59964 * q^21 - 67096 * q^23 - 109115 * q^25 - 178778 * q^27 - 372398 * q^29 - 271372 * q^31 - 792700 * q^33 - 608250 * q^35 - 562630 * q^37 - 963904 * q^39 - 956714 * q^41 - 827362 * q^43 - 1165100 * q^45 - 1812982 * q^47 - 862031 * q^49 - 2458254 * q^51 + 486998 * q^53 + 467930 * q^55 + 96026 * q^57 - 367182 * q^59 + 1879732 * q^61 - 1007274 * q^63 + 1790920 * q^65 - 1046394 * q^67 + 7261712 * q^69 - 4664572 * q^71 + 4224942 * q^73 + 8194850 * q^75 + 8611110 * q^77 + 9574024 * q^79 + 11351813 * q^81 + 11754804 * q^83 + 18711750 * q^85 + 3801472 * q^87 + 2782542 * q^89 + 7385214 * q^91 + 29535004 * q^93 + 1920520 * q^95 + 1291574 * q^97 - 9760310 * 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
$3$	25.4201	0.543568	0.271784	−	0.962358i	$-0.412386\pi$
$3$	25.4201	0.543568	0.271784	+	0.962358i	$0.412386\pi$
$4$	0	0
$5$	−3.35619	−0.0120075	−0.00600374	−	0.999982i	$-0.501911\pi$
$5$	−3.35619	−0.0120075	−0.00600374	+	0.999982i	$0.501911\pi$
$6$	0	0
$7$	−173.262	−0.190924	−0.0954620	−	0.995433i	$-0.530433\pi$
$7$	−173.262	−0.190924	−0.0954620	+	0.995433i	$0.530433\pi$
$8$	0	0
$9$	−1540.82	−0.704534
$10$	0	0
$11$	3804.09	0.861740	0.430870	−	0.902414i	$-0.358207\pi$
$11$	3804.09	0.861740	0.430870	+	0.902414i	$0.358207\pi$
$12$	0	0
$13$	−14439.0	−1.82279	−0.911394	−	0.411536i	$-0.864993\pi$
$13$	−14439.0	−1.82279	−0.911394	+	0.411536i	$0.864993\pi$
$14$	0	0
$15$	−85.3148	−0.00652687
$16$	0	0
$17$	3292.85	0.162555	0.0812776	−	0.996691i	$-0.474100\pi$
$17$	3292.85	0.162555	0.0812776	+	0.996691i	$0.474100\pi$
$18$	0	0
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	0	0
$21$	−4404.35	−0.103780
$22$	0	0
$23$	643.599	0.0110298	0.00551491	−	0.999985i	$-0.498245\pi$
$23$	643.599	0.0110298	0.00551491	+	0.999985i	$0.498245\pi$
$24$	0	0
$25$	−78113.7	−0.999856
$26$	0	0
$27$	−94761.6	−0.926530
$28$	0	0
$29$	93767.3	0.713935	0.356967	−	0.934117i	$-0.383811\pi$
$29$	93767.3	0.713935	0.356967	+	0.934117i	$0.383811\pi$
$30$	0	0
$31$	−103884.	−0.626299	−0.313150	−	0.949704i	$-0.601384\pi$
$31$	−103884.	−0.626299	−0.313150	+	0.949704i	$0.601384\pi$
$32$	0	0
$33$	96700.4	0.468414
$34$	0	0
$35$	581.501	0.00229251
$36$	0	0
$37$	−322176.	−1.04565	−0.522825	−	0.852440i	$-0.675122\pi$
$37$	−322176.	−1.04565	−0.522825	+	0.852440i	$0.675122\pi$
$38$	0	0
$39$	−367042.	−0.990808
$40$	0	0
$41$	−843931.	−1.91233	−0.956166	−	0.292824i	$-0.905405\pi$
$41$	−843931.	−1.91233	−0.956166	+	0.292824i	$0.905405\pi$
$42$	0	0
$43$	−289803.	−0.555858	−0.277929	−	0.960602i	$-0.589648\pi$
$43$	−289803.	−0.555858	−0.277929	+	0.960602i	$0.589648\pi$
$44$	0	0
$45$	5171.27	0.00845968
$46$	0	0
$47$	772392.	1.08516	0.542582	−	0.840003i	$-0.317447\pi$
$47$	772392.	1.08516	0.542582	+	0.840003i	$0.317447\pi$
$48$	0	0
$49$	−793523.	−0.963548
$50$	0	0
$51$	83704.8	0.0883598
$52$	0	0
$53$	−596172.	−0.550055	−0.275028	−	0.961436i	$-0.588687\pi$
$53$	−596172.	−0.550055	−0.275028	+	0.961436i	$0.588687\pi$
$54$	0	0
$55$	−12767.2	−0.0103473
$56$	0	0
$57$	−174357.	−0.124703
$58$	0	0
$59$	−2.76656e6	−1.75371	−0.876856	−	0.480753i	$-0.840363\pi$
$59$	−2.76656e6	−1.75371	−0.876856	+	0.480753i	$0.840363\pi$
$60$	0	0
$61$	2.39723e6	1.35224	0.676121	−	0.736791i	$-0.263660\pi$
$61$	2.39723e6	1.35224	0.676121	+	0.736791i	$0.263660\pi$
$62$	0	0
$63$	266965.	0.134513
$64$	0	0
$65$	48460.1	0.0218871
$66$	0	0
$67$	2.71365e6	1.10228	0.551140	−	0.834413i	$-0.314193\pi$
$67$	2.71365e6	1.10228	0.551140	+	0.834413i	$0.314193\pi$
$68$	0	0
$69$	16360.4	0.00599545
$70$	0	0
$71$	5.03316e6	1.66893	0.834463	−	0.551065i	$-0.185778\pi$
$71$	5.03316e6	1.66893	0.834463	+	0.551065i	$0.185778\pi$
$72$	0	0
$73$	3.65348e6	1.09920	0.549600	−	0.835428i	$-0.314780\pi$
$73$	3.65348e6	1.09920	0.549600	+	0.835428i	$0.314780\pi$
$74$	0	0
$75$	−1.98566e6	−0.543489
$76$	0	0
$77$	−659104.	−0.164527
$78$	0	0
$79$	−4.72959e6	−1.07927	−0.539633	−	0.841900i	$-0.681437\pi$
$79$	−4.72959e6	−1.07927	−0.539633	+	0.841900i	$0.681437\pi$
$80$	0	0
$81$	960912.	0.200903
$82$	0	0
$83$	2.28839e6	0.439296	0.219648	−	0.975579i	$-0.429509\pi$
$83$	2.28839e6	0.439296	0.219648	+	0.975579i	$0.429509\pi$
$84$	0	0
$85$	−11051.4	−0.00195188
$86$	0	0
$87$	2.38358e6	0.388072
$88$	0	0
$89$	8.80023e6	1.32321	0.661605	−	0.749852i	$-0.269876\pi$
$89$	8.80023e6	1.32321	0.661605	+	0.749852i	$0.269876\pi$
$90$	0	0
$91$	2.50174e6	0.348014
$92$	0	0
$93$	−2.64074e6	−0.340436
$94$	0	0
$95$	23020.1	0.00275470
$96$	0	0
$97$	1.21813e7	1.35516	0.677581	−	0.735448i	$-0.263028\pi$
$97$	1.21813e7	1.35516	0.677581	+	0.735448i	$0.263028\pi$
$98$	0	0
$99$	−5.86140e6	−0.607125
$100$	0	0
$101$	2.27068e6	0.219296	0.109648	−	0.993970i	$-0.465028\pi$
$101$	2.27068e6	0.219296	0.109648	+	0.993970i	$0.465028\pi$
$102$	0	0
$103$	−4.11736e6	−0.371269	−0.185634	−	0.982619i	$-0.559434\pi$
$103$	−4.11736e6	−0.371269	−0.185634	+	0.982619i	$0.559434\pi$
$104$	0	0
$105$	14781.8	0.00124614
$106$	0	0
$107$	−2.18710e6	−0.172594	−0.0862970	−	0.996269i	$-0.527503\pi$
$107$	−2.18710e6	−0.172594	−0.0862970	+	0.996269i	$0.527503\pi$
$108$	0	0
$109$	−6.58258e6	−0.486860	−0.243430	−	0.969919i	$-0.578273\pi$
$109$	−6.58258e6	−0.486860	−0.243430	+	0.969919i	$0.578273\pi$
$110$	0	0
$111$	−8.18975e6	−0.568382
$112$	0	0
$113$	1.01191e7	0.659732	0.329866	−	0.944028i	$-0.392996\pi$
$113$	1.01191e7	0.659732	0.329866	+	0.944028i	$0.392996\pi$
$114$	0	0
$115$	−2160.04	−0.000132440 0
$116$	0	0
$117$	2.22479e7	1.28422
$118$	0	0
$119$	−570527.	−0.0310357
$120$	0	0
$121$	−5.01609e6	−0.257405
$122$	0	0
$123$	−2.14528e7	−1.03948
$124$	0	0
$125$	524367.	0.0240132
$126$	0	0
$127$	−5.27049e6	−0.228317	−0.114158	−	0.993463i	$-0.536417\pi$
$127$	−5.27049e6	−0.228317	−0.114158	+	0.993463i	$0.536417\pi$
$128$	0	0
$129$	−7.36684e6	−0.302146
$130$	0	0
$131$	−3.23909e7	−1.25885	−0.629425	−	0.777061i	$-0.716710\pi$
$131$	−3.23909e7	−1.25885	−0.629425	+	0.777061i	$0.716710\pi$
$132$	0	0
$133$	1.18840e6	0.0438010
$134$	0	0
$135$	318038.	0.0111253
$136$	0	0
$137$	−3.71836e7	−1.23546	−0.617730	−	0.786390i	$-0.711948\pi$
$137$	−3.71836e7	−1.23546	−0.617730	+	0.786390i	$0.711948\pi$
$138$	0	0
$139$	1.43249e7	0.452418	0.226209	−	0.974079i	$-0.427367\pi$
$139$	1.43249e7	0.452418	0.226209	+	0.974079i	$0.427367\pi$
$140$	0	0
$141$	1.96343e7	0.589860
$142$	0	0
$143$	−5.49273e7	−1.57077
$144$	0	0
$145$	−314701.	−0.00857255
$146$	0	0
$147$	−2.01715e7	−0.523753
$148$	0	0
$149$	4.39404e6	0.108821	0.0544104	−	0.998519i	$-0.482672\pi$
$149$	4.39404e6	0.108821	0.0544104	+	0.998519i	$0.482672\pi$
$150$	0	0
$151$	−2.48731e7	−0.587908	−0.293954	−	0.955819i	$-0.594971\pi$
$151$	−2.48731e7	−0.587908	−0.293954	+	0.955819i	$0.594971\pi$
$152$	0	0
$153$	−5.07368e6	−0.114526
$154$	0	0
$155$	348654.	0.00752027
$156$	0	0
$157$	4.07527e6	0.0840441	0.0420221	−	0.999117i	$-0.486620\pi$
$157$	4.07527e6	0.0840441	0.0420221	+	0.999117i	$0.486620\pi$
$158$	0	0
$159$	−1.51548e7	−0.298992
$160$	0	0
$161$	−111511.	−0.00210586
$162$	0	0
$163$	−7.57086e7	−1.36927	−0.684635	−	0.728887i	$-0.740038\pi$
$163$	−7.57086e7	−1.36927	−0.684635	+	0.728887i	$0.740038\pi$
$164$	0	0
$165$	−324545.	−0.00562446
$166$	0	0
$167$	3.19156e7	0.530268	0.265134	−	0.964212i	$-0.414584\pi$
$167$	3.19156e7	0.530268	0.265134	+	0.964212i	$0.414584\pi$
$168$	0	0
$169$	1.45737e8	2.32255
$170$	0	0
$171$	1.05685e7	0.161631
$172$	0	0
$173$	6.07975e7	0.892738	0.446369	−	0.894849i	$-0.352717\pi$
$173$	6.07975e7	0.892738	0.446369	+	0.894849i	$0.352717\pi$
$174$	0	0
$175$	1.35342e7	0.190896
$176$	0	0
$177$	−7.03264e7	−0.953261
$178$	0	0
$179$	1.12224e7	0.146252	0.0731259	−	0.997323i	$-0.476703\pi$
$179$	1.12224e7	0.146252	0.0731259	+	0.997323i	$0.476703\pi$
$180$	0	0
$181$	5.70223e7	0.714775	0.357388	−	0.933956i	$-0.383668\pi$
$181$	5.70223e7	0.714775	0.357388	+	0.933956i	$0.383668\pi$
$182$	0	0
$183$	6.09378e7	0.735035
$184$	0	0
$185$	1.08128e6	0.0125556
$186$	0	0
$187$	1.25263e7	0.140080
$188$	0	0
$189$	1.64186e7	0.176897
$190$	0	0
$191$	−3.66566e7	−0.380658	−0.190329	−	0.981720i	$-0.560956\pi$
$191$	−3.66566e7	−0.380658	−0.190329	+	0.981720i	$0.560956\pi$
$192$	0	0
$193$	−1.06650e8	−1.06785	−0.533924	−	0.845533i	$-0.679283\pi$
$193$	−1.06650e8	−1.06785	−0.533924	+	0.845533i	$0.679283\pi$
$194$	0	0
$195$	1.23186e6	0.0118971
$196$	0	0
$197$	1.20480e8	1.12275	0.561375	−	0.827561i	$-0.310272\pi$
$197$	1.20480e8	1.12275	0.561375	+	0.827561i	$0.310272\pi$
$198$	0	0
$199$	2.12531e8	1.91177	0.955885	−	0.293741i	$-0.0949002\pi$
$199$	2.12531e8	1.91177	0.955885	+	0.293741i	$0.0949002\pi$
$200$	0	0
$201$	6.89813e7	0.599164
$202$	0	0
$203$	−1.62463e7	−0.136307
$204$	0	0
$205$	2.83239e6	0.0229623
$206$	0	0
$207$	−991668.	−0.00777088
$208$	0	0
$209$	−2.60922e7	−0.197697
$210$	0	0
$211$	−1.23530e8	−0.905279	−0.452640	−	0.891693i	$-0.649518\pi$
$211$	−1.23530e8	−0.905279	−0.452640	+	0.891693i	$0.649518\pi$
$212$	0	0
$213$	1.27944e8	0.907174
$214$	0	0
$215$	972635.	0.00667445
$216$	0	0
$217$	1.79991e7	0.119576
$218$	0	0
$219$	9.28719e7	0.597489
$220$	0	0
$221$	−4.75456e7	−0.296304
$222$	0	0
$223$	1.75568e7	0.106018	0.0530088	−	0.998594i	$-0.483119\pi$
$223$	1.75568e7	0.106018	0.0530088	+	0.998594i	$0.483119\pi$
$224$	0	0
$225$	1.20359e8	0.704433
$226$	0	0
$227$	−2.25659e8	−1.28045	−0.640225	−	0.768187i	$-0.721159\pi$
$227$	−2.25659e8	−1.28045	−0.640225	+	0.768187i	$0.721159\pi$
$228$	0	0
$229$	−2.22000e8	−1.22160	−0.610800	−	0.791785i	$-0.709152\pi$
$229$	−2.22000e8	−1.22160	−0.610800	+	0.791785i	$0.709152\pi$
$230$	0	0
$231$	−1.67545e7	−0.0894314
$232$	0	0
$233$	1.03946e8	0.538347	0.269174	−	0.963092i	$-0.413249\pi$
$233$	1.03946e8	0.538347	0.269174	+	0.963092i	$0.413249\pi$
$234$	0	0
$235$	−2.59229e6	−0.0130301
$236$	0	0
$237$	−1.20227e8	−0.586654
$238$	0	0
$239$	−2.75718e8	−1.30639	−0.653195	−	0.757189i	$-0.726572\pi$
$239$	−2.75718e8	−1.30639	−0.653195	+	0.757189i	$0.726572\pi$
$240$	0	0
$241$	3.22608e8	1.48462	0.742311	−	0.670055i	$-0.233730\pi$
$241$	3.22608e8	1.48462	0.742311	+	0.670055i	$0.233730\pi$
$242$	0	0
$243$	2.31670e8	1.03573
$244$	0	0
$245$	2.66322e6	0.0115698
$246$	0	0
$247$	9.90372e7	0.418176
$248$	0	0
$249$	5.81713e7	0.238787
$250$	0	0
$251$	−3.37531e8	−1.34727	−0.673637	−	0.739062i	$-0.735269\pi$
$251$	−3.37531e8	−1.34727	−0.673637	+	0.739062i	$0.735269\pi$
$252$	0	0
$253$	2.44831e6	0.00950482
$254$	0	0
$255$	−280929.	−0.00106098
$256$	0	0
$257$	−3.32504e8	−1.22189	−0.610944	−	0.791674i	$-0.709210\pi$
$257$	−3.32504e8	−1.22189	−0.610944	+	0.791674i	$0.709210\pi$
$258$	0	0
$259$	5.58208e7	0.199640
$260$	0	0
$261$	−1.44478e8	−0.502992
$262$	0	0
$263$	−7.63765e7	−0.258890	−0.129445	−	0.991587i	$-0.541320\pi$
$263$	−7.63765e7	−0.258890	−0.129445	+	0.991587i	$0.541320\pi$
$264$	0	0
$265$	2.00087e6	0.00660477
$266$	0	0
$267$	2.23703e8	0.719254
$268$	0	0
$269$	−3.10285e8	−0.971914	−0.485957	−	0.873983i	$-0.661529\pi$
$269$	−3.10285e8	−0.971914	−0.485957	+	0.873983i	$0.661529\pi$
$270$	0	0
$271$	−1.79448e8	−0.547704	−0.273852	−	0.961772i	$-0.588298\pi$
$271$	−1.79448e8	−0.547704	−0.273852	+	0.961772i	$0.588298\pi$
$272$	0	0
$273$	6.35945e7	0.189169
$274$	0	0
$275$	−2.97151e8	−0.861615
$276$	0	0
$277$	2.27108e8	0.642027	0.321014	−	0.947075i	$-0.395976\pi$
$277$	2.27108e8	0.642027	0.321014	+	0.947075i	$0.395976\pi$
$278$	0	0
$279$	1.60066e8	0.441249
$280$	0	0
$281$	−2.49532e8	−0.670895	−0.335448	−	0.942059i	$-0.608888\pi$
$281$	−2.49532e8	−0.670895	−0.335448	+	0.942059i	$0.608888\pi$
$282$	0	0
$283$	−4.15917e8	−1.09082	−0.545412	−	0.838168i	$-0.683627\pi$
$283$	−4.15917e8	−1.09082	−0.545412	+	0.838168i	$0.683627\pi$
$284$	0	0
$285$	585174.	0.00149737
$286$	0	0
$287$	1.46221e8	0.365110
$288$	0	0
$289$	−3.99496e8	−0.973576
$290$	0	0
$291$	3.09649e8	0.736622
$292$	0	0
$293$	−3.24502e8	−0.753670	−0.376835	−	0.926281i	$-0.622988\pi$
$293$	−3.24502e8	−0.753670	−0.376835	+	0.926281i	$0.622988\pi$
$294$	0	0
$295$	9.28511e6	0.0210577
$296$	0	0
$297$	−3.60481e8	−0.798427
$298$	0	0
$299$	−9.29294e6	−0.0201050
$300$	0	0
$301$	5.02119e7	0.106127
$302$	0	0
$303$	5.77211e7	0.119202
$304$	0	0
$305$	−8.04555e6	−0.0162370
$306$	0	0
$307$	5.69052e8	1.12245	0.561226	−	0.827663i	$-0.310330\pi$
$307$	5.69052e8	1.12245	0.561226	+	0.827663i	$0.310330\pi$
$308$	0	0
$309$	−1.04664e8	−0.201810
$310$	0	0
$311$	−1.05324e8	−0.198549	−0.0992744	−	0.995060i	$-0.531652\pi$
$311$	−1.05324e8	−0.198549	−0.0992744	+	0.995060i	$0.531652\pi$
$312$	0	0
$313$	−4.13697e8	−0.762566	−0.381283	−	0.924458i	$-0.624518\pi$
$313$	−4.13697e8	−0.762566	−0.381283	+	0.924458i	$0.624518\pi$
$314$	0	0
$315$	−895986.	−0.00161516
$316$	0	0
$317$	3.97209e8	0.700344	0.350172	−	0.936685i	$-0.386123\pi$
$317$	3.97209e8	0.700344	0.350172	+	0.936685i	$0.386123\pi$
$318$	0	0
$319$	3.56699e8	0.615226
$320$	0	0
$321$	−5.55964e7	−0.0938165
$322$	0	0
$323$	−2.25857e7	−0.0372927
$324$	0	0
$325$	1.12789e9	1.82252
$326$	0	0
$327$	−1.67330e8	−0.264641
$328$	0	0
$329$	−1.33826e8	−0.207184
$330$	0	0
$331$	5.36235e8	0.812750	0.406375	−	0.913706i	$-0.366793\pi$
$331$	5.36235e8	0.812750	0.406375	+	0.913706i	$0.366793\pi$
$332$	0	0
$333$	4.96413e8	0.736697
$334$	0	0
$335$	−9.10752e6	−0.0132356
$336$	0	0
$337$	−3.86284e8	−0.549796	−0.274898	−	0.961473i	$-0.588644\pi$
$337$	−3.86284e8	−0.549796	−0.274898	+	0.961473i	$0.588644\pi$
$338$	0	0
$339$	2.57229e8	0.358609
$340$	0	0
$341$	−3.95183e8	−0.539707
$342$	0	0
$343$	2.80176e8	0.374888
$344$	0	0
$345$	−54908.6	−7.19902e−5 0
$346$	0	0
$347$	7.18308e8	0.922906	0.461453	−	0.887165i	$-0.347328\pi$
$347$	7.18308e8	0.922906	0.461453	+	0.887165i	$0.347328\pi$
$348$	0	0
$349$	2.29357e8	0.288817	0.144409	−	0.989518i	$-0.453872\pi$
$349$	2.29357e8	0.288817	0.144409	+	0.989518i	$0.453872\pi$
$350$	0	0
$351$	1.36826e9	1.68887
$352$	0	0
$353$	−1.26980e9	−1.53647	−0.768237	−	0.640166i	$-0.778866\pi$
$353$	−1.26980e9	−1.53647	−0.768237	+	0.640166i	$0.778866\pi$
$354$	0	0
$355$	−1.68923e7	−0.0200396
$356$	0	0
$357$	−1.45029e7	−0.0168700
$358$	0	0
$359$	1.43972e9	1.64228	0.821142	−	0.570724i	$-0.193337\pi$
$359$	1.43972e9	1.64228	0.821142	+	0.570724i	$0.193337\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	−1.27510e8	−0.139917
$364$	0	0
$365$	−1.22618e7	−0.0131986
$366$	0	0
$367$	8.07332e8	0.852552	0.426276	−	0.904593i	$-0.359825\pi$
$367$	8.07332e8	0.852552	0.426276	+	0.904593i	$0.359825\pi$
$368$	0	0
$369$	1.30034e9	1.34730
$370$	0	0
$371$	1.03294e8	0.105019
$372$	0	0
$373$	2.62122e8	0.261530	0.130765	−	0.991413i	$-0.458257\pi$
$373$	2.62122e8	0.261530	0.130765	+	0.991413i	$0.458257\pi$
$374$	0	0
$375$	1.33295e7	0.0130528
$376$	0	0
$377$	−1.35391e9	−1.30135
$378$	0	0
$379$	−3.34919e8	−0.316011	−0.158006	−	0.987438i	$-0.550506\pi$
$379$	−3.34919e8	−0.316011	−0.158006	+	0.987438i	$0.550506\pi$
$380$	0	0
$381$	−1.33977e8	−0.124106
$382$	0	0
$383$	−2.17105e9	−1.97457	−0.987287	−	0.158951i	$-0.949189\pi$
$383$	−2.17105e9	−1.97457	−0.987287	+	0.158951i	$0.949189\pi$
$384$	0	0
$385$	2.21208e6	0.00197555
$386$	0	0
$387$	4.46534e8	0.391621
$388$	0	0
$389$	−5.23430e8	−0.450853	−0.225426	−	0.974260i	$-0.572378\pi$
$389$	−5.23430e8	−0.450853	−0.225426	+	0.974260i	$0.572378\pi$
$390$	0	0
$391$	2.11928e6	0.00179295
$392$	0	0
$393$	−8.23382e8	−0.684270
$394$	0	0
$395$	1.58734e7	0.0129593
$396$	0	0
$397$	8.46818e8	0.679240	0.339620	−	0.940563i	$-0.389702\pi$
$397$	8.46818e8	0.679240	0.339620	+	0.940563i	$0.389702\pi$
$398$	0	0
$399$	3.02094e7	0.0238088
$400$	0	0
$401$	−1.56110e9	−1.20900	−0.604499	−	0.796606i	$-0.706627\pi$
$401$	−1.56110e9	−1.20900	−0.604499	+	0.796606i	$0.706627\pi$
$402$	0	0
$403$	1.49998e9	1.14161
$404$	0	0
$405$	−3.22501e6	−0.00241234
$406$	0	0
$407$	−1.22558e9	−0.901078
$408$	0	0
$409$	7.63724e8	0.551957	0.275978	−	0.961164i	$-0.410998\pi$
$409$	7.63724e8	0.551957	0.275978	+	0.961164i	$0.410998\pi$
$410$	0	0
$411$	−9.45211e8	−0.671556
$412$	0	0
$413$	4.79340e8	0.334826
$414$	0	0
$415$	−7.68029e6	−0.00527484
$416$	0	0
$417$	3.64141e8	0.245920
$418$	0	0
$419$	1.80707e9	1.20013	0.600063	−	0.799953i	$-0.295142\pi$
$419$	1.80707e9	1.20013	0.600063	+	0.799953i	$0.295142\pi$
$420$	0	0
$421$	−1.17005e9	−0.764215	−0.382108	−	0.924118i	$-0.624802\pi$
$421$	−1.17005e9	−0.764215	−0.382108	+	0.924118i	$0.624802\pi$
$422$	0	0
$423$	−1.19011e9	−0.764535
$424$	0	0
$425$	−2.57217e8	−0.162532
$426$	0	0
$427$	−4.15348e8	−0.258175
$428$	0	0
$429$	−1.39626e9	−0.853818
$430$	0	0
$431$	−3.33701e8	−0.200764	−0.100382	−	0.994949i	$-0.532007\pi$
$431$	−3.33701e8	−0.200764	−0.100382	+	0.994949i	$0.532007\pi$
$432$	0	0
$433$	−1.18240e9	−0.699934	−0.349967	−	0.936762i	$-0.613807\pi$
$433$	−1.18240e9	−0.699934	−0.349967	+	0.936762i	$0.613807\pi$
$434$	0	0
$435$	−7.99974e6	−0.00465976
$436$	0	0
$437$	−4.41445e6	−0.00253041
$438$	0	0
$439$	1.36067e8	0.0767585	0.0383792	−	0.999263i	$-0.487781\pi$
$439$	1.36067e8	0.0767585	0.0383792	+	0.999263i	$0.487781\pi$
$440$	0	0
$441$	1.22267e9	0.678853
$442$	0	0
$443$	−2.42005e9	−1.32255	−0.661274	−	0.750144i	$-0.729984\pi$
$443$	−2.42005e9	−1.32255	−0.661274	+	0.750144i	$0.729984\pi$
$444$	0	0
$445$	−2.95352e7	−0.0158884
$446$	0	0
$447$	1.11697e8	0.0591515
$448$	0	0
$449$	−1.03162e9	−0.537845	−0.268923	−	0.963162i	$-0.586668\pi$
$449$	−1.03162e9	−0.537845	−0.268923	+	0.963162i	$0.586668\pi$
$450$	0	0
$451$	−3.21039e9	−1.64793
$452$	0	0
$453$	−6.32276e8	−0.319568
$454$	0	0
$455$	−8.39630e6	−0.00417877
$456$	0	0
$457$	2.98845e8	0.146467	0.0732335	−	0.997315i	$-0.476668\pi$
$457$	2.98845e8	0.146467	0.0732335	+	0.997315i	$0.476668\pi$
$458$	0	0
$459$	−3.12036e8	−0.150612
$460$	0	0
$461$	2.56152e9	1.21771	0.608857	−	0.793280i	$-0.291628\pi$
$461$	2.56152e9	1.21771	0.608857	+	0.793280i	$0.291628\pi$
$462$	0	0
$463$	3.45788e9	1.61911	0.809556	−	0.587042i	$-0.199708\pi$
$463$	3.45788e9	1.61911	0.809556	+	0.587042i	$0.199708\pi$
$464$	0	0
$465$	8.86283e6	0.00408778
$466$	0	0
$467$	2.11651e9	0.961638	0.480819	−	0.876820i	$-0.340339\pi$
$467$	2.11651e9	0.961638	0.480819	+	0.876820i	$0.340339\pi$
$468$	0	0
$469$	−4.70173e8	−0.210452
$470$	0	0
$471$	1.03594e8	0.0456837
$472$	0	0
$473$	−1.10244e9	−0.479004
$474$	0	0
$475$	5.35782e8	0.229383
$476$	0	0
$477$	9.18592e8	0.387533
$478$	0	0
$479$	−3.94044e9	−1.63822	−0.819108	−	0.573640i	$-0.805531\pi$
$479$	−3.94044e9	−1.63822	−0.819108	+	0.573640i	$0.805531\pi$
$480$	0	0
$481$	4.65190e9	1.90600
$482$	0	0
$483$	−2.83463e6	−0.00114467
$484$	0	0
$485$	−4.08827e7	−0.0162721
$486$	0	0
$487$	−4.70134e9	−1.84446	−0.922231	−	0.386639i	$-0.873636\pi$
$487$	−4.70134e9	−1.84446	−0.922231	+	0.386639i	$0.873636\pi$
$488$	0	0
$489$	−1.92452e9	−0.744290
$490$	0	0
$491$	−3.24351e9	−1.23660	−0.618302	−	0.785941i	$-0.712179\pi$
$491$	−3.24351e9	−1.23660	−0.618302	+	0.785941i	$0.712179\pi$
$492$	0	0
$493$	3.08762e8	0.116054
$494$	0	0
$495$	1.96720e7	0.00729004
$496$	0	0
$497$	−8.72057e8	−0.318638
$498$	0	0
$499$	3.31230e8	0.119338	0.0596689	−	0.998218i	$-0.480996\pi$
$499$	3.31230e8	0.119338	0.0596689	+	0.998218i	$0.480996\pi$
$500$	0	0
$501$	8.11298e8	0.288236
$502$	0	0
$503$	−3.42162e9	−1.19879	−0.599396	−	0.800453i	$-0.704593\pi$
$503$	−3.42162e9	−1.19879	−0.599396	+	0.800453i	$0.704593\pi$
$504$	0	0
$505$	−7.62084e6	−0.00263320
$506$	0	0
$507$	3.70465e9	1.26246
$508$	0	0
$509$	4.79856e9	1.61287	0.806435	−	0.591323i	$-0.201394\pi$
$509$	4.79856e9	1.61287	0.806435	+	0.591323i	$0.201394\pi$
$510$	0	0
$511$	−6.33009e8	−0.209864
$512$	0	0
$513$	6.49970e8	0.212560
$514$	0	0
$515$	1.38186e7	0.00445800
$516$	0	0
$517$	2.93825e9	0.935128
$518$	0	0
$519$	1.54548e9	0.485263
$520$	0	0
$521$	−3.01575e9	−0.934250	−0.467125	−	0.884191i	$-0.654710\pi$
$521$	−3.01575e9	−0.934250	−0.467125	+	0.884191i	$0.654710\pi$
$522$	0	0
$523$	5.76903e9	1.76338	0.881692	−	0.471826i	$-0.156405\pi$
$523$	5.76903e9	1.76338	0.881692	+	0.471826i	$0.156405\pi$
$524$	0	0
$525$	3.44040e8	0.103765
$526$	0	0
$527$	−3.42074e8	−0.101808
$528$	0	0
$529$	−3.40441e9	−0.999878
$530$	0	0
$531$	4.26276e9	1.23555
$532$	0	0
$533$	1.21855e10	3.48578
$534$	0	0
$535$	7.34033e6	0.00207242
$536$	0	0
$537$	2.85276e8	0.0794978
$538$	0	0
$539$	−3.01863e9	−0.830327
$540$	0	0
$541$	−3.72538e8	−0.101153	−0.0505767	−	0.998720i	$-0.516106\pi$
$541$	−3.72538e8	−0.101153	−0.0505767	+	0.998720i	$0.516106\pi$
$542$	0	0
$543$	1.44951e9	0.388529
$544$	0	0
$545$	2.20924e7	0.00584595
$546$	0	0
$547$	−5.36797e9	−1.40234	−0.701172	−	0.712992i	$-0.747339\pi$
$547$	−5.36797e9	−1.40234	−0.701172	+	0.712992i	$0.747339\pi$
$548$	0	0
$549$	−3.69368e9	−0.952701
$550$	0	0
$551$	−6.43150e8	−0.163788
$552$	0	0
$553$	8.19459e8	0.206058
$554$	0	0
$555$	2.74864e7	0.00682483
$556$	0	0
$557$	6.93092e9	1.69941	0.849705	−	0.527259i	$-0.176780\pi$
$557$	6.93092e9	1.69941	0.849705	+	0.527259i	$0.176780\pi$
$558$	0	0
$559$	4.18447e9	1.01321
$560$	0	0
$561$	3.18420e8	0.0761431
$562$	0	0
$563$	−1.24931e9	−0.295047	−0.147523	−	0.989059i	$-0.547130\pi$
$563$	−1.24931e9	−0.295047	−0.147523	+	0.989059i	$0.547130\pi$
$564$	0	0
$565$	−3.39617e7	−0.00792172
$566$	0	0
$567$	−1.66490e8	−0.0383572
$568$	0	0
$569$	−3.91417e9	−0.890732	−0.445366	−	0.895349i	$-0.646926\pi$
$569$	−3.91417e9	−0.890732	−0.445366	+	0.895349i	$0.646926\pi$
$570$	0	0
$571$	4.12496e9	0.927242	0.463621	−	0.886034i	$-0.346550\pi$
$571$	4.12496e9	0.927242	0.463621	+	0.886034i	$0.346550\pi$
$572$	0	0
$573$	−9.31816e8	−0.206914
$574$	0	0
$575$	−5.02739e7	−0.0110282
$576$	0	0
$577$	2.84776e9	0.617147	0.308574	−	0.951200i	$-0.400148\pi$
$577$	2.84776e9	0.617147	0.308574	+	0.951200i	$0.400148\pi$
$578$	0	0
$579$	−2.71105e9	−0.580447
$580$	0	0
$581$	−3.96492e8	−0.0838722
$582$	0	0
$583$	−2.26789e9	−0.474004
$584$	0	0
$585$	−7.46681e7	−0.0154202
$586$	0	0
$587$	7.85805e9	1.60355	0.801773	−	0.597629i	$-0.203890\pi$
$587$	7.85805e9	1.60355	0.801773	+	0.597629i	$0.203890\pi$
$588$	0	0
$589$	7.12539e8	0.143683
$590$	0	0
$591$	3.06262e9	0.610291
$592$	0	0
$593$	−2.49384e9	−0.491109	−0.245554	−	0.969383i	$-0.578970\pi$
$593$	−2.49384e9	−0.491109	−0.245554	+	0.969383i	$0.578970\pi$
$594$	0	0
$595$	1.91480e6	0.000372660 0
$596$	0	0
$597$	5.40256e9	1.03918
$598$	0	0
$599$	1.77144e9	0.336769	0.168385	−	0.985721i	$-0.446145\pi$
$599$	1.77144e9	0.336769	0.168385	+	0.985721i	$0.446145\pi$
$600$	0	0
$601$	−6.53447e9	−1.22786	−0.613931	−	0.789360i	$-0.710413\pi$
$601$	−6.53447e9	−1.22786	−0.613931	+	0.789360i	$0.710413\pi$
$602$	0	0
$603$	−4.18124e9	−0.776594
$604$	0	0
$605$	1.68350e7	0.00309078
$606$	0	0
$607$	−6.98740e9	−1.26811	−0.634053	−	0.773290i	$-0.718610\pi$
$607$	−6.98740e9	−1.26811	−0.634053	+	0.773290i	$0.718610\pi$
$608$	0	0
$609$	−4.12984e8	−0.0740922
$610$	0	0
$611$	−1.11526e10	−1.97802
$612$	0	0
$613$	−4.48052e9	−0.785627	−0.392814	−	0.919618i	$-0.628498\pi$
$613$	−4.48052e9	−0.785627	−0.392814	+	0.919618i	$0.628498\pi$
$614$	0	0
$615$	7.19998e7	0.0124816
$616$	0	0
$617$	3.28136e9	0.562413	0.281207	−	0.959647i	$-0.409265\pi$
$617$	3.28136e9	0.562413	0.281207	+	0.959647i	$0.409265\pi$
$618$	0	0
$619$	−1.03952e10	−1.76163	−0.880817	−	0.473457i	$-0.843006\pi$
$619$	−1.03952e10	−1.76163	−0.880817	+	0.473457i	$0.843006\pi$
$620$	0	0
$621$	−6.09885e7	−0.0102194
$622$	0	0
$623$	−1.52475e9	−0.252633
$624$	0	0
$625$	6.10088e9	0.999567
$626$	0	0
$627$	−6.63268e8	−0.107461
$628$	0	0
$629$	−1.06088e9	−0.169976
$630$	0	0
$631$	3.86959e9	0.613143	0.306572	−	0.951848i	$-0.400818\pi$
$631$	3.86959e9	0.613143	0.306572	+	0.951848i	$0.400818\pi$
$632$	0	0
$633$	−3.14014e9	−0.492081
$634$	0	0
$635$	1.76888e7	0.00274151
$636$	0	0
$637$	1.14577e10	1.75634
$638$	0	0
$639$	−7.75518e9	−1.17581
$640$	0	0
$641$	−7.14267e9	−1.07117	−0.535584	−	0.844482i	$-0.679908\pi$
$641$	−7.14267e9	−1.07117	−0.535584	+	0.844482i	$0.679908\pi$
$642$	0	0
$643$	−9.59951e9	−1.42400	−0.712001	−	0.702178i	$-0.752211\pi$
$643$	−9.59951e9	−1.42400	−0.712001	+	0.702178i	$0.752211\pi$
$644$	0	0
$645$	2.47245e7	0.00362801
$646$	0	0
$647$	9.05075e9	1.31377	0.656885	−	0.753991i	$-0.271874\pi$
$647$	9.05075e9	1.31377	0.656885	+	0.753991i	$0.271874\pi$
$648$	0	0
$649$	−1.05242e10	−1.51124
$650$	0	0
$651$	4.57541e8	0.0649974
$652$	0	0
$653$	2.68998e9	0.378053	0.189026	−	0.981972i	$-0.439467\pi$
$653$	2.68998e9	0.378053	0.189026	+	0.981972i	$0.439467\pi$
$654$	0	0
$655$	1.08710e8	0.0151156
$656$	0	0
$657$	−5.62934e9	−0.774424
$658$	0	0
$659$	2.21462e9	0.301439	0.150720	−	0.988577i	$-0.451841\pi$
$659$	2.21462e9	0.301439	0.150720	+	0.988577i	$0.451841\pi$
$660$	0	0
$661$	−2.65391e9	−0.357422	−0.178711	−	0.983902i	$-0.557193\pi$
$661$	−2.65391e9	−0.357422	−0.178711	+	0.983902i	$0.557193\pi$
$662$	0	0
$663$	−1.20861e9	−0.161061
$664$	0	0
$665$	−3.98851e6	−0.000525939 0
$666$	0	0
$667$	6.03486e7	0.00787457
$668$	0	0
$669$	4.46296e8	0.0576278
$670$	0	0
$671$	9.11925e9	1.16528
$672$	0	0
$673$	−5.88089e9	−0.743688	−0.371844	−	0.928295i	$-0.621274\pi$
$673$	−5.88089e9	−0.743688	−0.371844	+	0.928295i	$0.621274\pi$
$674$	0	0
$675$	7.40218e9	0.926396
$676$	0	0
$677$	−7.08195e9	−0.877188	−0.438594	−	0.898685i	$-0.644523\pi$
$677$	−7.08195e9	−0.877188	−0.438594	+	0.898685i	$0.644523\pi$
$678$	0	0
$679$	−2.11055e9	−0.258733
$680$	0	0
$681$	−5.73629e9	−0.696011
$682$	0	0
$683$	3.66860e8	0.0440583	0.0220292	−	0.999757i	$-0.492987\pi$
$683$	3.66860e8	0.0440583	0.0220292	+	0.999757i	$0.492987\pi$
$684$	0	0
$685$	1.24795e8	0.0148348
$686$	0	0
$687$	−5.64327e9	−0.664022
$688$	0	0
$689$	8.60814e9	1.00263
$690$	0	0
$691$	3.15284e9	0.363520	0.181760	−	0.983343i	$-0.441821\pi$
$691$	3.15284e9	0.363520	0.181760	+	0.983343i	$0.441821\pi$
$692$	0	0
$693$	1.01556e9	0.115915
$694$	0	0
$695$	−4.80771e7	−0.00543239
$696$	0	0
$697$	−2.77894e9	−0.310860
$698$	0	0
$699$	2.64232e9	0.292628
$700$	0	0
$701$	8.59305e9	0.942181	0.471091	−	0.882085i	$-0.343860\pi$
$701$	8.59305e9	0.942181	0.471091	+	0.882085i	$0.343860\pi$
$702$	0	0
$703$	2.20980e9	0.239889
$704$	0	0
$705$	−6.58965e7	−0.00708272
$706$	0	0
$707$	−3.93423e8	−0.0418690
$708$	0	0
$709$	7.95777e9	0.838551	0.419275	−	0.907859i	$-0.362284\pi$
$709$	7.95777e9	0.838551	0.419275	+	0.907859i	$0.362284\pi$
$710$	0	0
$711$	7.28743e9	0.760380
$712$	0	0
$713$	−6.68596e7	−0.00690797
$714$	0	0
$715$	1.84346e8	0.0188610
$716$	0	0
$717$	−7.00880e9	−0.710112
$718$	0	0
$719$	−1.47897e10	−1.48391	−0.741955	−	0.670450i	$-0.766101\pi$
$719$	−1.47897e10	−1.48391	−0.741955	+	0.670450i	$0.766101\pi$
$720$	0	0
$721$	7.13382e8	0.0708841
$722$	0	0
$723$	8.20075e9	0.806993
$724$	0	0
$725$	−7.32452e9	−0.713832
$726$	0	0
$727$	8.97168e9	0.865971	0.432986	−	0.901401i	$-0.357460\pi$
$727$	8.97168e9	0.865971	0.432986	+	0.901401i	$0.357460\pi$
$728$	0	0
$729$	3.78757e9	0.362088
$730$	0	0
$731$	−9.54279e8	−0.0903576
$732$	0	0
$733$	−1.73669e9	−0.162876	−0.0814382	−	0.996678i	$-0.525951\pi$
$733$	−1.73669e9	−0.162876	−0.0814382	+	0.996678i	$0.525951\pi$
$734$	0	0
$735$	6.76993e7	0.00628896
$736$	0	0
$737$	1.03230e10	0.949878
$738$	0	0
$739$	−2.95911e9	−0.269716	−0.134858	−	0.990865i	$-0.543058\pi$
$739$	−2.95911e9	−0.269716	−0.134858	+	0.990865i	$0.543058\pi$
$740$	0	0
$741$	2.51754e9	0.227307
$742$	0	0
$743$	2.67191e9	0.238979	0.119490	−	0.992835i	$-0.461874\pi$
$743$	2.67191e9	0.238979	0.119490	+	0.992835i	$0.461874\pi$
$744$	0	0
$745$	−1.47472e7	−0.00130666
$746$	0	0
$747$	−3.52600e9	−0.309499
$748$	0	0
$749$	3.78942e8	0.0329523
$750$	0	0
$751$	−1.12390e10	−0.968248	−0.484124	−	0.874999i	$-0.660862\pi$
$751$	−1.12390e10	−0.968248	−0.484124	+	0.874999i	$0.660862\pi$
$752$	0	0
$753$	−8.58009e9	−0.732335
$754$	0	0
$755$	8.34787e7	0.00705930
$756$	0	0
$757$	6.77994e9	0.568055	0.284027	−	0.958816i	$-0.408329\pi$
$757$	6.77994e9	0.568055	0.284027	+	0.958816i	$0.408329\pi$
$758$	0	0
$759$	6.22363e7	0.00516651
$760$	0	0
$761$	1.53202e10	1.26014	0.630068	−	0.776540i	$-0.283027\pi$
$761$	1.53202e10	1.26014	0.630068	+	0.776540i	$0.283027\pi$
$762$	0	0
$763$	1.14051e9	0.0929532
$764$	0	0
$765$	1.70282e7	0.00137517
$766$	0	0
$767$	3.99464e10	3.19664
$768$	0	0
$769$	−1.94253e10	−1.54037	−0.770187	−	0.637818i	$-0.779837\pi$
$769$	−1.94253e10	−1.54037	−0.770187	+	0.637818i	$0.779837\pi$
$770$	0	0
$771$	−8.45230e9	−0.664178
$772$	0	0
$773$	2.09447e10	1.63097	0.815487	−	0.578776i	$-0.196469\pi$
$773$	2.09447e10	1.63097	0.815487	+	0.578776i	$0.196469\pi$
$774$	0	0
$775$	8.11475e9	0.626209
$776$	0	0
$777$	1.41897e9	0.108518
$778$	0	0
$779$	5.78852e9	0.438719
$780$	0	0
$781$	1.91466e10	1.43818
$782$	0	0
$783$	−8.88554e9	−0.661482
$784$	0	0
$785$	−1.36774e7	−0.00100916
$786$	0	0
$787$	3.02490e9	0.221207	0.110604	−	0.993865i	$-0.464722\pi$
$787$	3.02490e9	0.221207	0.110604	+	0.993865i	$0.464722\pi$
$788$	0	0
$789$	−1.94150e9	−0.140724
$790$	0	0
$791$	−1.75326e9	−0.125959
$792$	0	0
$793$	−3.46136e10	−2.46485
$794$	0	0
$795$	5.08623e7	0.00359014
$796$	0	0
$797$	4.28296e9	0.299667	0.149834	−	0.988711i	$-0.452126\pi$
$797$	4.28296e9	0.299667	0.149834	+	0.988711i	$0.452126\pi$
$798$	0	0
$799$	2.54337e9	0.176399
$800$	0	0
$801$	−1.35595e10	−0.932247
$802$	0	0
$803$	1.38981e10	0.947224
$804$	0	0
$805$	374253.	2.52860e−5 0
$806$	0	0
$807$	−7.88749e9	−0.528301
$808$	0	0
$809$	−4.60253e9	−0.305617	−0.152808	−	0.988256i	$-0.548832\pi$
$809$	−4.60253e9	−0.305617	−0.152808	+	0.988256i	$0.548832\pi$
$810$	0	0
$811$	2.38895e10	1.57266	0.786328	−	0.617810i	$-0.211980\pi$
$811$	2.38895e10	1.57266	0.786328	+	0.617810i	$0.211980\pi$
$812$	0	0
$813$	−4.56159e9	−0.297714
$814$	0	0
$815$	2.54093e8	0.0164415
$816$	0	0
$817$	1.98776e9	0.127522
$818$	0	0
$819$	−3.85472e9	−0.245188
$820$	0	0
$821$	−4.67646e9	−0.294928	−0.147464	−	0.989067i	$-0.547111\pi$
$821$	−4.67646e9	−0.294928	−0.147464	+	0.989067i	$0.547111\pi$
$822$	0	0
$823$	3.03215e10	1.89605	0.948027	−	0.318189i	$-0.103075\pi$
$823$	3.03215e10	1.89605	0.948027	+	0.318189i	$0.103075\pi$
$824$	0	0
$825$	−7.55363e9	−0.468346
$826$	0	0
$827$	2.53758e10	1.56009	0.780047	−	0.625721i	$-0.215195\pi$
$827$	2.53758e10	1.56009	0.780047	+	0.625721i	$0.215195\pi$
$828$	0	0
$829$	−2.75451e10	−1.67920	−0.839601	−	0.543204i	$-0.817211\pi$
$829$	−2.75451e10	−1.67920	−0.839601	+	0.543204i	$0.817211\pi$
$830$	0	0
$831$	5.77312e9	0.348985
$832$	0	0
$833$	−2.61296e9	−0.156630
$834$	0	0
$835$	−1.07115e8	−0.00636717
$836$	0	0
$837$	9.84420e9	0.580285
$838$	0	0
$839$	−1.46274e10	−0.855065	−0.427533	−	0.904000i	$-0.640617\pi$
$839$	−1.46274e10	−0.855065	−0.427533	+	0.904000i	$0.640617\pi$
$840$	0	0
$841$	−8.45756e9	−0.490297
$842$	0	0
$843$	−6.34314e9	−0.364677
$844$	0	0
$845$	−4.89120e8	−0.0278880
$846$	0	0
$847$	8.69099e8	0.0491448
$848$	0	0
$849$	−1.05727e10	−0.592936
$850$	0	0
$851$	−2.07352e8	−0.0115333
$852$	0	0
$853$	1.16171e10	0.640880	0.320440	−	0.947269i	$-0.396169\pi$
$853$	1.16171e10	0.640880	0.320440	+	0.947269i	$0.396169\pi$
$854$	0	0
$855$	−3.54698e7	−0.00194078
$856$	0	0
$857$	1.76278e10	0.956678	0.478339	−	0.878175i	$-0.341239\pi$
$857$	1.76278e10	0.956678	0.478339	+	0.878175i	$0.341239\pi$
$858$	0	0
$859$	−1.92669e10	−1.03714	−0.518569	−	0.855036i	$-0.673535\pi$
$859$	−1.92669e10	−1.03714	−0.518569	+	0.855036i	$0.673535\pi$
$860$	0	0
$861$	3.71697e9	0.198462
$862$	0	0
$863$	2.06979e10	1.09620	0.548098	−	0.836414i	$-0.315352\pi$
$863$	2.06979e10	1.09620	0.548098	+	0.836414i	$0.315352\pi$
$864$	0	0
$865$	−2.04048e8	−0.0107195
$866$	0	0
$867$	−1.01552e10	−0.529204
$868$	0	0
$869$	−1.79918e10	−0.930046
$870$	0	0
$871$	−3.91824e10	−2.00922
$872$	0	0
$873$	−1.87691e10	−0.954758
$874$	0	0
$875$	−9.08529e7	−0.00458470
$876$	0	0
$877$	−2.31739e10	−1.16011	−0.580056	−	0.814576i	$-0.696969\pi$
$877$	−2.31739e10	−1.16011	−0.580056	+	0.814576i	$0.696969\pi$
$878$	0	0
$879$	−8.24889e9	−0.409670
$880$	0	0
$881$	−1.29374e10	−0.637426	−0.318713	−	0.947851i	$-0.603251\pi$
$881$	−1.29374e10	−0.637426	−0.318713	+	0.947851i	$0.603251\pi$
$882$	0	0
$883$	8.34205e9	0.407765	0.203883	−	0.978995i	$-0.434644\pi$
$883$	8.34205e9	0.407765	0.203883	+	0.978995i	$0.434644\pi$
$884$	0	0
$885$	2.36029e8	0.0114463
$886$	0	0
$887$	−5.97628e9	−0.287540	−0.143770	−	0.989611i	$-0.545923\pi$
$887$	−5.97628e9	−0.287540	−0.143770	+	0.989611i	$0.545923\pi$
$888$	0	0
$889$	9.13176e8	0.0435912
$890$	0	0
$891$	3.65539e9	0.173126
$892$	0	0
$893$	−5.29783e9	−0.248954
$894$	0	0
$895$	−3.76646e7	−0.00175612
$896$	0	0
$897$	−2.36228e8	−0.0109284
$898$	0	0
$899$	−9.74091e9	−0.447137
$900$	0	0
$901$	−1.96311e9	−0.0894144
$902$	0	0
$903$	1.27639e9	0.0576870
$904$	0	0
$905$	−1.91378e8	−0.00858264
$906$	0	0
$907$	5.54811e8	0.0246899	0.0123450	−	0.999924i	$-0.496070\pi$
$907$	5.54811e8	0.0246899	0.0123450	+	0.999924i	$0.496070\pi$
$908$	0	0
$909$	−3.49871e9	−0.154502
$910$	0	0
$911$	−2.42935e10	−1.06458	−0.532288	−	0.846564i	$-0.678668\pi$
$911$	−2.42935e10	−1.06458	−0.532288	+	0.846564i	$0.678668\pi$
$912$	0	0
$913$	8.70525e9	0.378559
$914$	0	0
$915$	−2.04519e8	−0.00882591
$916$	0	0
$917$	5.61212e9	0.240345
$918$	0	0
$919$	6.29454e9	0.267522	0.133761	−	0.991014i	$-0.457295\pi$
$919$	6.29454e9	0.267522	0.133761	+	0.991014i	$0.457295\pi$
$920$	0	0
$921$	1.44654e10	0.610128
$922$	0	0
$923$	−7.26739e10	−3.04210
$924$	0	0
$925$	2.51663e10	1.04550
$926$	0	0
$927$	6.34409e9	0.261572
$928$	0	0
$929$	8.78717e9	0.359579	0.179789	−	0.983705i	$-0.442458\pi$
$929$	8.78717e9	0.359579	0.179789	+	0.983705i	$0.442458\pi$
$930$	0	0
$931$	5.44278e9	0.221053
$932$	0	0
$933$	−2.67736e9	−0.107925
$934$	0	0
$935$	−4.20407e7	−0.00168201
$936$	0	0
$937$	−4.18401e9	−0.166152	−0.0830758	−	0.996543i	$-0.526474\pi$
$937$	−4.18401e9	−0.166152	−0.0830758	+	0.996543i	$0.526474\pi$
$938$	0	0
$939$	−1.05162e10	−0.414506
$940$	0	0
$941$	2.92628e10	1.14486	0.572429	−	0.819954i	$-0.306001\pi$
$941$	2.92628e10	1.14486	0.572429	+	0.819954i	$0.306001\pi$
$942$	0	0
$943$	−5.43153e8	−0.0210927
$944$	0	0
$945$	−5.51040e7	−0.00212408
$946$	0	0
$947$	3.35848e10	1.28504	0.642521	−	0.766268i	$-0.277889\pi$
$947$	3.35848e10	1.28504	0.642521	+	0.766268i	$0.277889\pi$
$948$	0	0
$949$	−5.27526e10	−2.00361
$950$	0	0
$951$	1.00971e10	0.380684
$952$	0	0
$953$	−3.10142e10	−1.16074	−0.580370	−	0.814353i	$-0.697092\pi$
$953$	−3.10142e10	−1.16074	−0.580370	+	0.814353i	$0.697092\pi$
$954$	0	0
$955$	1.23027e8	0.00457075
$956$	0	0
$957$	9.06734e9	0.334417
$958$	0	0
$959$	6.44250e9	0.235879
$960$	0	0
$961$	−1.67208e10	−0.607749
$962$	0	0
$963$	3.36992e9	0.121598
$964$	0	0
$965$	3.57937e8	0.0128222
$966$	0	0
$967$	3.04256e10	1.08205	0.541025	−	0.841007i	$-0.318037\pi$
$967$	3.04256e10	1.08205	0.541025	+	0.841007i	$0.318037\pi$
$968$	0	0
$969$	−5.74131e8	−0.0202711
$970$	0	0
$971$	1.46525e10	0.513623	0.256811	−	0.966462i	$-0.417328\pi$
$971$	1.46525e10	0.513623	0.256811	+	0.966462i	$0.417328\pi$
$972$	0	0
$973$	−2.48196e9	−0.0863774
$974$	0	0
$975$	2.86710e10	0.990665
$976$	0	0
$977$	5.10251e10	1.75046	0.875232	−	0.483704i	$-0.160709\pi$
$977$	5.10251e10	1.75046	0.875232	+	0.483704i	$0.160709\pi$
$978$	0	0
$979$	3.34768e10	1.14026
$980$	0	0
$981$	1.01426e10	0.343009
$982$	0	0
$983$	−1.29089e10	−0.433463	−0.216732	−	0.976231i	$-0.569540\pi$
$983$	−1.29089e10	−0.433463	−0.216732	+	0.976231i	$0.569540\pi$
$984$	0	0
$985$	−4.04354e8	−0.0134814
$986$	0	0
$987$	−3.40188e9	−0.112618
$988$	0	0
$989$	−1.86517e8	−0.00613100
$990$	0	0
$991$	−4.14437e10	−1.35270	−0.676348	−	0.736582i	$-0.736439\pi$
$991$	−4.14437e10	−1.35270	−0.676348	+	0.736582i	$0.736439\pi$
$992$	0	0
$993$	1.36312e10	0.441785
$994$	0	0
$995$	−7.13293e8	−0.0229555
$996$	0	0
$997$	−5.44529e9	−0.174015	−0.0870077	−	0.996208i	$-0.527730\pi$
$997$	−5.44529e9	−0.174015	−0.0870077	+	0.996208i	$0.527730\pi$
$998$	0	0
$999$	3.05299e10	0.968826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.a.a.1.4	✓	5
4.3	odd	2		304.8.a.g.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.a.a.1.4	✓	5	1.1	even	1	trivial
304.8.a.g.1.2		5	4.3	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:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.a (trivial)

\(f(q)\)	\(=\)	\(q+25.4201 q^{3} -3.35619 q^{5} -173.262 q^{7} -1540.82 q^{9} +O(q^{10})\)	\(q+25.4201 q^{3} -3.35619 q^{5} -173.262 q^{7} -1540.82 q^{9} +3804.09 q^{11} -14439.0 q^{13} -85.3148 q^{15} +3292.85 q^{17} -6859.00 q^{19} -4404.35 q^{21} +643.599 q^{23} -78113.7 q^{25} -94761.6 q^{27} +93767.3 q^{29} -103884. q^{31} +96700.4 q^{33} +581.501 q^{35} -322176. q^{37} -367042. q^{39} -843931. q^{41} -289803. q^{43} +5171.27 q^{45} +772392. q^{47} -793523. q^{49} +83704.8 q^{51} -596172. q^{53} -12767.2 q^{55} -174357. q^{57} -2.76656e6 q^{59} +2.39723e6 q^{61} +266965. q^{63} +48460.1 q^{65} +2.71365e6 q^{67} +16360.4 q^{69} +5.03316e6 q^{71} +3.65348e6 q^{73} -1.98566e6 q^{75} -659104. q^{77} -4.72959e6 q^{79} +960912. q^{81} +2.28839e6 q^{83} -11051.4 q^{85} +2.38358e6 q^{87} +8.80023e6 q^{89} +2.50174e6 q^{91} -2.64074e6 q^{93} +23020.1 q^{95} +1.21813e7 q^{97} -5.86140e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 14 q^{3} - 280 q^{5} + 414 q^{7} + 3779 q^{9}+O(q^{10}) \) 5 * q - 14 * q^3 - 280 * q^5 + 414 * q^7 + 3779 * q^9	\( 5 q - 14 q^{3} - 280 q^{5} + 414 q^{7} + 3779 q^{9} - 2662 q^{11} - 602 q^{13} - 20800 q^{15} - 27366 q^{17} - 34295 q^{19} - 59964 q^{21} - 67096 q^{23} - 109115 q^{25} - 178778 q^{27} - 372398 q^{29} - 271372 q^{31} - 792700 q^{33} - 608250 q^{35} - 562630 q^{37} - 963904 q^{39} - 956714 q^{41} - 827362 q^{43} - 1165100 q^{45} - 1812982 q^{47} - 862031 q^{49} - 2458254 q^{51} + 486998 q^{53} + 467930 q^{55} + 96026 q^{57} - 367182 q^{59} + 1879732 q^{61} - 1007274 q^{63} + 1790920 q^{65} - 1046394 q^{67} + 7261712 q^{69} - 4664572 q^{71} + 4224942 q^{73} + 8194850 q^{75} + 8611110 q^{77} + 9574024 q^{79} + 11351813 q^{81} + 11754804 q^{83} + 18711750 q^{85} + 3801472 q^{87} + 2782542 q^{89} + 7385214 q^{91} + 29535004 q^{93} + 1920520 q^{95} + 1291574 q^{97} - 9760310 q^{99}+O(q^{100}) \) 5 * q - 14 * q^3 - 280 * q^5 + 414 * q^7 + 3779 * q^9 - 2662 * q^11 - 602 * q^13 - 20800 * q^15 - 27366 * q^17 - 34295 * q^19 - 59964 * q^21 - 67096 * q^23 - 109115 * q^25 - 178778 * q^27 - 372398 * q^29 - 271372 * q^31 - 792700 * q^33 - 608250 * q^35 - 562630 * q^37 - 963904 * q^39 - 956714 * q^41 - 827362 * q^43 - 1165100 * q^45 - 1812982 * q^47 - 862031 * q^49 - 2458254 * q^51 + 486998 * q^53 + 467930 * q^55 + 96026 * q^57 - 367182 * q^59 + 1879732 * q^61 - 1007274 * q^63 + 1790920 * q^65 - 1046394 * q^67 + 7261712 * q^69 - 4664572 * q^71 + 4224942 * q^73 + 8194850 * q^75 + 8611110 * q^77 + 9574024 * q^79 + 11351813 * q^81 + 11754804 * q^83 + 18711750 * q^85 + 3801472 * q^87 + 2782542 * q^89 + 7385214 * q^91 + 29535004 * q^93 + 1920520 * q^95 + 1291574 * q^97 - 9760310 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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