LMFDB - Embedded newform 76.8.a.b.1.2

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:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8376x^{4} + 135458x^{3} + 16275767x^{2} - 280013424x - 6276171312 $ x^6 - 2x^5 - 8376x^4 + 135458x^3 + 16275767x^2 - 280013424*x - 6276171312
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$49.5941$ 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)

$f(q)$	$=$	$q-42.5941 q^{3} +51.7645 q^{5} -1212.43 q^{7} -372.745 q^{9} +O(q^{10})$	$q-42.5941 q^{3} +51.7645 q^{5} -1212.43 q^{7} -372.745 q^{9} +2256.02 q^{11} +2287.47 q^{13} -2204.86 q^{15} -38018.6 q^{17} +6859.00 q^{19} +51642.5 q^{21} +68709.8 q^{23} -75445.4 q^{25} +109030. q^{27} +233248. q^{29} +312596. q^{31} -96093.2 q^{33} -62761.0 q^{35} +127969. q^{37} -97432.6 q^{39} -87683.8 q^{41} +939657. q^{43} -19295.0 q^{45} -975128. q^{47} +646455. q^{49} +1.61937e6 q^{51} -469035. q^{53} +116782. q^{55} -292153. q^{57} -130691. q^{59} +1.27306e6 q^{61} +451929. q^{63} +118410. q^{65} -2.74146e6 q^{67} -2.92663e6 q^{69} +3.10700e6 q^{71} -1.24592e6 q^{73} +3.21353e6 q^{75} -2.73528e6 q^{77} -2.65895e6 q^{79} -3.82884e6 q^{81} +6.59203e6 q^{83} -1.96801e6 q^{85} -9.93500e6 q^{87} -1.06005e7 q^{89} -2.77341e6 q^{91} -1.33147e7 q^{93} +355052. q^{95} +9.07223e6 q^{97} -840922. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9}+O(q^{10}) $ 6 * q + 40 * q^3 + 279 * q^5 - 1565 * q^7 + 3900 * q^9	$ 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9} + 7983 q^{11} + 1250 q^{13} + 26790 q^{15} + 21735 q^{17} + 41154 q^{19} - 86346 q^{21} + 100920 q^{23} + 373305 q^{25} + 534790 q^{27} - 58656 q^{29} + 403808 q^{31} + 916430 q^{33} + 463497 q^{35} + 808780 q^{37} + 758704 q^{39} + 556944 q^{41} + 1220735 q^{43} + 3234843 q^{45} + 1915305 q^{47} + 2045883 q^{49} + 908816 q^{51} + 511650 q^{53} + 1813341 q^{55} + 274360 q^{57} + 1300572 q^{59} + 565335 q^{61} - 7170325 q^{63} - 6195012 q^{65} - 45010 q^{67} - 7381528 q^{69} - 1424106 q^{71} - 11153825 q^{73} + 3941974 q^{75} - 17515425 q^{77} - 6392144 q^{79} + 6187530 q^{81} - 3164160 q^{83} - 19479255 q^{85} - 25999500 q^{87} - 14502678 q^{89} - 9736226 q^{91} - 18344300 q^{93} + 1913661 q^{95} - 21377010 q^{97} + 24032935 q^{99}+O(q^{100}) $ 6 * q + 40 * q^3 + 279 * q^5 - 1565 * q^7 + 3900 * q^9 + 7983 * q^11 + 1250 * q^13 + 26790 * q^15 + 21735 * q^17 + 41154 * q^19 - 86346 * q^21 + 100920 * q^23 + 373305 * q^25 + 534790 * q^27 - 58656 * q^29 + 403808 * q^31 + 916430 * q^33 + 463497 * q^35 + 808780 * q^37 + 758704 * q^39 + 556944 * q^41 + 1220735 * q^43 + 3234843 * q^45 + 1915305 * q^47 + 2045883 * q^49 + 908816 * q^51 + 511650 * q^53 + 1813341 * q^55 + 274360 * q^57 + 1300572 * q^59 + 565335 * q^61 - 7170325 * q^63 - 6195012 * q^65 - 45010 * q^67 - 7381528 * q^69 - 1424106 * q^71 - 11153825 * q^73 + 3941974 * q^75 - 17515425 * q^77 - 6392144 * q^79 + 6187530 * q^81 - 3164160 * q^83 - 19479255 * q^85 - 25999500 * q^87 - 14502678 * q^89 - 9736226 * q^91 - 18344300 * q^93 + 1913661 * q^95 - 21377010 * q^97 + 24032935 * 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
$3$	−42.5941	−0.910804	−0.455402	−	0.890286i	$-0.650504\pi$
$3$	−42.5941	−0.910804	−0.455402	+	0.890286i	$0.650504\pi$
$4$	0	0
$5$	51.7645	0.185198	0.0925991	−	0.995703i	$-0.470483\pi$
$5$	51.7645	0.185198	0.0925991	+	0.995703i	$0.470483\pi$
$6$	0	0
$7$	−1212.43	−1.33603	−0.668014	−	0.744149i	$-0.732855\pi$
$7$	−1212.43	−1.33603	−0.668014	+	0.744149i	$0.732855\pi$
$8$	0	0
$9$	−372.745	−0.170437
$10$	0	0
$11$	2256.02	0.511057	0.255528	−	0.966802i	$-0.417751\pi$
$11$	2256.02	0.511057	0.255528	+	0.966802i	$0.417751\pi$
$12$	0	0
$13$	2287.47	0.288771	0.144385	−	0.989522i	$-0.453879\pi$
$13$	2287.47	0.288771	0.144385	+	0.989522i	$0.453879\pi$
$14$	0	0
$15$	−2204.86	−0.168679
$16$	0	0
$17$	−38018.6	−1.87683	−0.938415	−	0.345511i	$-0.887706\pi$
$17$	−38018.6	−1.87683	−0.938415	+	0.345511i	$0.887706\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	51642.5	1.21686
$22$	0	0
$23$	68709.8	1.17753	0.588764	−	0.808305i	$-0.299615\pi$
$23$	68709.8	1.17753	0.588764	+	0.808305i	$0.299615\pi$
$24$	0	0
$25$	−75445.4	−0.965702
$26$	0	0
$27$	109030.	1.06604
$28$	0	0
$29$	233248.	1.77593	0.887965	−	0.459912i	$-0.152119\pi$
$29$	233248.	1.77593	0.887965	+	0.459912i	$0.152119\pi$
$30$	0	0
$31$	312596.	1.88459	0.942296	−	0.334781i	$-0.108662\pi$
$31$	312596.	1.88459	0.942296	+	0.334781i	$0.108662\pi$
$32$	0	0
$33$	−96093.2	−0.465472
$34$	0	0
$35$	−62761.0	−0.247430
$36$	0	0
$37$	127969.	0.415336	0.207668	−	0.978199i	$-0.433413\pi$
$37$	127969.	0.415336	0.207668	+	0.978199i	$0.433413\pi$
$38$	0	0
$39$	−97432.6	−0.263014
$40$	0	0
$41$	−87683.8	−0.198690	−0.0993450	−	0.995053i	$-0.531675\pi$
$41$	−87683.8	−0.198690	−0.0993450	+	0.995053i	$0.531675\pi$
$42$	0	0
$43$	939657.	1.80231	0.901156	−	0.433495i	$-0.142720\pi$
$43$	939657.	1.80231	0.901156	+	0.433495i	$0.142720\pi$
$44$	0	0
$45$	−19295.0	−0.0315646
$46$	0	0
$47$	−975128.	−1.37000	−0.684998	−	0.728545i	$-0.740197\pi$
$47$	−975128.	−1.37000	−0.684998	+	0.728545i	$0.740197\pi$
$48$	0	0
$49$	646455.	0.784968
$50$	0	0
$51$	1.61937e6	1.70942
$52$	0	0
$53$	−469035.	−0.432752	−0.216376	−	0.976310i	$-0.569424\pi$
$53$	−469035.	−0.432752	−0.216376	+	0.976310i	$0.569424\pi$
$54$	0	0
$55$	116782.	0.0946468
$56$	0	0
$57$	−292153.	−0.208953
$58$	0	0
$59$	−130691.	−0.0828445	−0.0414222	−	0.999142i	$-0.513189\pi$
$59$	−130691.	−0.0828445	−0.0414222	+	0.999142i	$0.513189\pi$
$60$	0	0
$61$	1.27306e6	0.718115	0.359057	−	0.933316i	$-0.383098\pi$
$61$	1.27306e6	0.718115	0.359057	+	0.933316i	$0.383098\pi$
$62$	0	0
$63$	451929.	0.227708
$64$	0	0
$65$	118410.	0.0534798
$66$	0	0
$67$	−2.74146e6	−1.11358	−0.556789	−	0.830654i	$-0.687967\pi$
$67$	−2.74146e6	−1.11358	−0.556789	+	0.830654i	$0.687967\pi$
$68$	0	0
$69$	−2.92663e6	−1.07250
$70$	0	0
$71$	3.10700e6	1.03024	0.515118	−	0.857120i	$-0.327748\pi$
$71$	3.10700e6	1.03024	0.515118	+	0.857120i	$0.327748\pi$
$72$	0	0
$73$	−1.24592e6	−0.374851	−0.187425	−	0.982279i	$-0.560014\pi$
$73$	−1.24592e6	−0.374851	−0.187425	+	0.982279i	$0.560014\pi$
$74$	0	0
$75$	3.21353e6	0.879565
$76$	0	0
$77$	−2.73528e6	−0.682786
$78$	0	0
$79$	−2.65895e6	−0.606757	−0.303379	−	0.952870i	$-0.598115\pi$
$79$	−2.65895e6	−0.606757	−0.303379	+	0.952870i	$0.598115\pi$
$80$	0	0
$81$	−3.82884e6	−0.800514
$82$	0	0
$83$	6.59203e6	1.26545	0.632726	−	0.774375i	$-0.281936\pi$
$83$	6.59203e6	1.26545	0.632726	+	0.774375i	$0.281936\pi$
$84$	0	0
$85$	−1.96801e6	−0.347585
$86$	0	0
$87$	−9.93500e6	−1.61752
$88$	0	0
$89$	−1.06005e7	−1.59390	−0.796951	−	0.604044i	$-0.793555\pi$
$89$	−1.06005e7	−1.59390	−0.796951	+	0.604044i	$0.793555\pi$
$90$	0	0
$91$	−2.77341e6	−0.385806
$92$	0	0
$93$	−1.33147e7	−1.71649
$94$	0	0
$95$	355052.	0.0424874
$96$	0	0
$97$	9.07223e6	1.00928	0.504641	−	0.863329i	$-0.331625\pi$
$97$	9.07223e6	1.00928	0.504641	+	0.863329i	$0.331625\pi$
$98$	0	0
$99$	−840922.	−0.0871029
$100$	0	0
$101$	3.84980e6	0.371803	0.185902	−	0.982568i	$-0.440479\pi$
$101$	3.84980e6	0.371803	0.185902	+	0.982568i	$0.440479\pi$
$102$	0	0
$103$	−8.09419e6	−0.729866	−0.364933	−	0.931034i	$-0.618908\pi$
$103$	−8.09419e6	−0.729866	−0.364933	+	0.931034i	$0.618908\pi$
$104$	0	0
$105$	2.67325e6	0.225360
$106$	0	0
$107$	−8.67188e6	−0.684337	−0.342168	−	0.939639i	$-0.611161\pi$
$107$	−8.67188e6	−0.684337	−0.342168	+	0.939639i	$0.611161\pi$
$108$	0	0
$109$	2.62891e7	1.94439	0.972194	−	0.234175i	$-0.0752390\pi$
$109$	2.62891e7	1.94439	0.972194	+	0.234175i	$0.0752390\pi$
$110$	0	0
$111$	−5.45073e6	−0.378289
$112$	0	0
$113$	1.14413e7	0.745934	0.372967	−	0.927845i	$-0.378340\pi$
$113$	1.14413e7	0.745934	0.372967	+	0.927845i	$0.378340\pi$
$114$	0	0
$115$	3.55672e6	0.218076
$116$	0	0
$117$	−852643.	−0.0492172
$118$	0	0
$119$	4.60951e7	2.50749
$120$	0	0
$121$	−1.43975e7	−0.738821
$122$	0	0
$123$	3.73481e6	0.180967
$124$	0	0
$125$	−7.94949e6	−0.364044
$126$	0	0
$127$	7.30945e6	0.316644	0.158322	−	0.987388i	$-0.449392\pi$
$127$	7.30945e6	0.316644	0.158322	+	0.987388i	$0.449392\pi$
$128$	0	0
$129$	−4.00238e7	−1.64155
$130$	0	0
$131$	3.45067e7	1.34108	0.670539	−	0.741874i	$-0.266063\pi$
$131$	3.45067e7	1.34108	0.670539	+	0.741874i	$0.266063\pi$
$132$	0	0
$133$	−8.31609e6	−0.306506
$134$	0	0
$135$	5.64388e6	0.197428
$136$	0	0
$137$	3.26058e7	1.08336	0.541680	−	0.840585i	$-0.317788\pi$
$137$	3.26058e7	1.08336	0.541680	+	0.840585i	$0.317788\pi$
$138$	0	0
$139$	−5.22975e6	−0.165169	−0.0825846	−	0.996584i	$-0.526317\pi$
$139$	−5.22975e6	−0.165169	−0.0825846	+	0.996584i	$0.526317\pi$
$140$	0	0
$141$	4.15347e7	1.24780
$142$	0	0
$143$	5.16058e6	0.147578
$144$	0	0
$145$	1.20740e7	0.328899
$146$	0	0
$147$	−2.75352e7	−0.714952
$148$	0	0
$149$	−2.33523e7	−0.578332	−0.289166	−	0.957279i	$-0.593378\pi$
$149$	−2.33523e7	−0.578332	−0.289166	+	0.957279i	$0.593378\pi$
$150$	0	0
$151$	4.91085e7	1.16075	0.580373	−	0.814351i	$-0.302907\pi$
$151$	4.91085e7	1.16075	0.580373	+	0.814351i	$0.302907\pi$
$152$	0	0
$153$	1.41713e7	0.319881
$154$	0	0
$155$	1.61814e7	0.349023
$156$	0	0
$157$	−6.77966e7	−1.39817	−0.699084	−	0.715039i	$-0.746409\pi$
$157$	−6.77966e7	−1.39817	−0.699084	+	0.715039i	$0.746409\pi$
$158$	0	0
$159$	1.99781e7	0.394152
$160$	0	0
$161$	−8.33061e7	−1.57321
$162$	0	0
$163$	4.72620e7	0.854782	0.427391	−	0.904067i	$-0.359433\pi$
$163$	4.72620e7	0.854782	0.427391	+	0.904067i	$0.359433\pi$
$164$	0	0
$165$	−4.97421e6	−0.0862046
$166$	0	0
$167$	2.91407e7	0.484164	0.242082	−	0.970256i	$-0.422170\pi$
$167$	2.91407e7	0.484164	0.242082	+	0.970256i	$0.422170\pi$
$168$	0	0
$169$	−5.75160e7	−0.916611
$170$	0	0
$171$	−2.55666e6	−0.0391009
$172$	0	0
$173$	997984.	0.0146542	0.00732710	−	0.999973i	$-0.497668\pi$
$173$	997984.	0.0146542	0.00732710	+	0.999973i	$0.497668\pi$
$174$	0	0
$175$	9.14727e7	1.29020
$176$	0	0
$177$	5.56666e6	0.0754551
$178$	0	0
$179$	3.20947e7	0.418261	0.209131	−	0.977888i	$-0.432937\pi$
$179$	3.20947e7	0.418261	0.209131	+	0.977888i	$0.432937\pi$
$180$	0	0
$181$	−2.00517e6	−0.0251348	−0.0125674	−	0.999921i	$-0.504000\pi$
$181$	−2.00517e6	−0.0251348	−0.0125674	+	0.999921i	$0.504000\pi$
$182$	0	0
$183$	−5.42247e7	−0.654061
$184$	0	0
$185$	6.62426e6	0.0769195
$186$	0	0
$187$	−8.57708e7	−0.959166
$188$	0	0
$189$	−1.32192e8	−1.42426
$190$	0	0
$191$	1.12003e8	1.16309	0.581547	−	0.813513i	$-0.302448\pi$
$191$	1.12003e8	1.16309	0.581547	+	0.813513i	$0.302448\pi$
$192$	0	0
$193$	−1.55506e7	−0.155703	−0.0778516	−	0.996965i	$-0.524806\pi$
$193$	−1.55506e7	−0.155703	−0.0778516	+	0.996965i	$0.524806\pi$
$194$	0	0
$195$	−5.04355e6	−0.0487096
$196$	0	0
$197$	−1.06637e8	−0.993747	−0.496874	−	0.867823i	$-0.665519\pi$
$197$	−1.06637e8	−0.993747	−0.496874	+	0.867823i	$0.665519\pi$
$198$	0	0
$199$	1.43722e8	1.29281	0.646407	−	0.762992i	$-0.276271\pi$
$199$	1.43722e8	1.29281	0.646407	+	0.762992i	$0.276271\pi$
$200$	0	0
$201$	1.16770e8	1.01425
$202$	0	0
$203$	−2.82798e8	−2.37269
$204$	0	0
$205$	−4.53890e6	−0.0367970
$206$	0	0
$207$	−2.56112e7	−0.200694
$208$	0	0
$209$	1.54741e7	0.117244
$210$	0	0
$211$	−2.19907e8	−1.61157	−0.805786	−	0.592206i	$-0.798257\pi$
$211$	−2.19907e8	−1.61157	−0.805786	+	0.592206i	$0.798257\pi$
$212$	0	0
$213$	−1.32340e8	−0.938342
$214$	0	0
$215$	4.86409e7	0.333785
$216$	0	0
$217$	−3.79002e8	−2.51787
$218$	0	0
$219$	5.30686e7	0.341416
$220$	0	0
$221$	−8.69663e7	−0.541974
$222$	0	0
$223$	−1.58758e8	−0.958671	−0.479335	−	0.877632i	$-0.659122\pi$
$223$	−1.58758e8	−0.958671	−0.479335	+	0.877632i	$0.659122\pi$
$224$	0	0
$225$	2.81219e7	0.164591
$226$	0	0
$227$	2.41294e8	1.36916	0.684581	−	0.728936i	$-0.259985\pi$
$227$	2.41294e8	1.36916	0.684581	+	0.728936i	$0.259985\pi$
$228$	0	0
$229$	2.27161e8	1.25000	0.625000	−	0.780624i	$-0.285099\pi$
$229$	2.27161e8	1.25000	0.625000	+	0.780624i	$0.285099\pi$
$230$	0	0
$231$	1.16507e8	0.621884
$232$	0	0
$233$	8.65053e7	0.448020	0.224010	−	0.974587i	$-0.428085\pi$
$233$	8.65053e7	0.448020	0.224010	+	0.974587i	$0.428085\pi$
$234$	0	0
$235$	−5.04770e7	−0.253721
$236$	0	0
$237$	1.13255e8	0.552637
$238$	0	0
$239$	2.10306e8	0.996457	0.498229	−	0.867046i	$-0.333984\pi$
$239$	2.10306e8	0.996457	0.498229	+	0.867046i	$0.333984\pi$
$240$	0	0
$241$	1.44188e8	0.663546	0.331773	−	0.943359i	$-0.392353\pi$
$241$	1.44188e8	0.663546	0.331773	+	0.943359i	$0.392353\pi$
$242$	0	0
$243$	−7.53628e7	−0.336927
$244$	0	0
$245$	3.34634e7	0.145375
$246$	0	0
$247$	1.56897e7	0.0662486
$248$	0	0
$249$	−2.80781e8	−1.15258
$250$	0	0
$251$	1.60301e8	0.639849	0.319924	−	0.947443i	$-0.396342\pi$
$251$	1.60301e8	0.639849	0.319924	+	0.947443i	$0.396342\pi$
$252$	0	0
$253$	1.55011e8	0.601783
$254$	0	0
$255$	8.38256e7	0.316582
$256$	0	0
$257$	5.16177e8	1.89685	0.948425	−	0.317001i	$-0.102676\pi$
$257$	5.16177e8	1.89685	0.948425	+	0.317001i	$0.102676\pi$
$258$	0	0
$259$	−1.55154e8	−0.554900
$260$	0	0
$261$	−8.69422e7	−0.302684
$262$	0	0
$263$	2.35146e8	0.797064	0.398532	−	0.917154i	$-0.369520\pi$
$263$	2.35146e8	0.797064	0.398532	+	0.917154i	$0.369520\pi$
$264$	0	0
$265$	−2.42793e7	−0.0801449
$266$	0	0
$267$	4.51519e8	1.45173
$268$	0	0
$269$	6.76843e7	0.212009	0.106005	−	0.994366i	$-0.466194\pi$
$269$	6.76843e7	0.212009	0.106005	+	0.994366i	$0.466194\pi$
$270$	0	0
$271$	−5.60223e8	−1.70989	−0.854946	−	0.518718i	$-0.826410\pi$
$271$	−5.60223e8	−1.70989	−0.854946	+	0.518718i	$0.826410\pi$
$272$	0	0
$273$	1.18131e8	0.351393
$274$	0	0
$275$	−1.70207e8	−0.493528
$276$	0	0
$277$	2.85315e8	0.806576	0.403288	−	0.915073i	$-0.367867\pi$
$277$	2.85315e8	0.806576	0.403288	+	0.915073i	$0.367867\pi$
$278$	0	0
$279$	−1.16519e8	−0.321204
$280$	0	0
$281$	−1.26172e8	−0.339228	−0.169614	−	0.985511i	$-0.554252\pi$
$281$	−1.26172e8	−0.339228	−0.169614	+	0.985511i	$0.554252\pi$
$282$	0	0
$283$	−8.51458e7	−0.223311	−0.111656	−	0.993747i	$-0.535615\pi$
$283$	−8.51458e7	−0.223311	−0.111656	+	0.993747i	$0.535615\pi$
$284$	0	0
$285$	−1.51231e7	−0.0386977
$286$	0	0
$287$	1.06311e8	0.265455
$288$	0	0
$289$	1.03507e9	2.52249
$290$	0	0
$291$	−3.86423e8	−0.919258
$292$	0	0
$293$	6.28053e8	1.45868	0.729339	−	0.684153i	$-0.239828\pi$
$293$	6.28053e8	1.45868	0.729339	+	0.684153i	$0.239828\pi$
$294$	0	0
$295$	−6.76515e6	−0.0153426
$296$	0	0
$297$	2.45974e8	0.544806
$298$	0	0
$299$	1.57171e8	0.340036
$300$	0	0
$301$	−1.13927e9	−2.40794
$302$	0	0
$303$	−1.63979e8	−0.338640
$304$	0	0
$305$	6.58992e7	0.132994
$306$	0	0
$307$	−4.45350e8	−0.878451	−0.439225	−	0.898377i	$-0.644747\pi$
$307$	−4.45350e8	−0.878451	−0.439225	+	0.898377i	$0.644747\pi$
$308$	0	0
$309$	3.44764e8	0.664764
$310$	0	0
$311$	−2.75339e8	−0.519046	−0.259523	−	0.965737i	$-0.583565\pi$
$311$	−2.75339e8	−0.519046	−0.259523	+	0.965737i	$0.583565\pi$
$312$	0	0
$313$	−7.78975e8	−1.43588	−0.717940	−	0.696105i	$-0.754915\pi$
$313$	−7.78975e8	−1.43588	−0.717940	+	0.696105i	$0.754915\pi$
$314$	0	0
$315$	2.33939e7	0.0421711
$316$	0	0
$317$	−4.33779e8	−0.764824	−0.382412	−	0.923992i	$-0.624907\pi$
$317$	−4.33779e8	−0.764824	−0.382412	+	0.923992i	$0.624907\pi$
$318$	0	0
$319$	5.26214e8	0.907601
$320$	0	0
$321$	3.69370e8	0.623296
$322$	0	0
$323$	−2.60770e8	−0.430574
$324$	0	0
$325$	−1.72579e8	−0.278867
$326$	0	0
$327$	−1.11976e9	−1.77096
$328$	0	0
$329$	1.18228e9	1.83035
$330$	0	0
$331$	−5.41971e8	−0.821444	−0.410722	−	0.911761i	$-0.634723\pi$
$331$	−5.41971e8	−0.821444	−0.410722	+	0.911761i	$0.634723\pi$
$332$	0	0
$333$	−4.76999e7	−0.0707885
$334$	0	0
$335$	−1.41910e8	−0.206233
$336$	0	0
$337$	−2.81248e8	−0.400299	−0.200150	−	0.979765i	$-0.564143\pi$
$337$	−2.81248e8	−0.400299	−0.200150	+	0.979765i	$0.564143\pi$
$338$	0	0
$339$	−4.87331e8	−0.679399
$340$	0	0
$341$	7.05223e8	0.963133
$342$	0	0
$343$	2.14707e8	0.287288
$344$	0	0
$345$	−1.51495e8	−0.198624
$346$	0	0
$347$	8.21034e8	1.05489	0.527446	−	0.849589i	$-0.323150\pi$
$347$	8.21034e8	1.05489	0.527446	+	0.849589i	$0.323150\pi$
$348$	0	0
$349$	6.68221e8	0.841455	0.420727	−	0.907187i	$-0.361775\pi$
$349$	6.68221e8	0.841455	0.420727	+	0.907187i	$0.361775\pi$
$350$	0	0
$351$	2.49403e8	0.307841
$352$	0	0
$353$	−5.84297e8	−0.707004	−0.353502	−	0.935434i	$-0.615009\pi$
$353$	−5.84297e8	−0.707004	−0.353502	+	0.935434i	$0.615009\pi$
$354$	0	0
$355$	1.60832e8	0.190798
$356$	0	0
$357$	−1.96338e9	−2.28384
$358$	0	0
$359$	−1.10866e9	−1.26464	−0.632322	−	0.774706i	$-0.717898\pi$
$359$	−1.10866e9	−1.26464	−0.632322	+	0.774706i	$0.717898\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	6.13249e8	0.672921
$364$	0	0
$365$	−6.44941e7	−0.0694217
$366$	0	0
$367$	2.54022e7	0.0268250	0.0134125	−	0.999910i	$-0.495731\pi$
$367$	2.54022e7	0.0268250	0.0134125	+	0.999910i	$0.495731\pi$
$368$	0	0
$369$	3.26837e7	0.0338641
$370$	0	0
$371$	5.68674e8	0.578169
$372$	0	0
$373$	−1.61195e9	−1.60831	−0.804157	−	0.594417i	$-0.797383\pi$
$373$	−1.61195e9	−1.60831	−0.804157	+	0.594417i	$0.797383\pi$
$374$	0	0
$375$	3.38601e8	0.331573
$376$	0	0
$377$	5.33548e8	0.512837
$378$	0	0
$379$	8.72992e8	0.823707	0.411854	−	0.911250i	$-0.364882\pi$
$379$	8.72992e8	0.823707	0.411854	+	0.911250i	$0.364882\pi$
$380$	0	0
$381$	−3.11339e8	−0.288401
$382$	0	0
$383$	−1.15484e9	−1.05033	−0.525167	−	0.850999i	$-0.675997\pi$
$383$	−1.15484e9	−1.05033	−0.525167	+	0.850999i	$0.675997\pi$
$384$	0	0
$385$	−1.41590e8	−0.126451
$386$	0	0
$387$	−3.50253e8	−0.307180
$388$	0	0
$389$	2.27717e8	0.196143	0.0980713	−	0.995179i	$-0.468733\pi$
$389$	2.27717e8	0.196143	0.0980713	+	0.995179i	$0.468733\pi$
$390$	0	0
$391$	−2.61225e9	−2.21002
$392$	0	0
$393$	−1.46978e9	−1.22146
$394$	0	0
$395$	−1.37639e8	−0.112370
$396$	0	0
$397$	1.53175e9	1.22863	0.614316	−	0.789060i	$-0.289432\pi$
$397$	1.53175e9	1.22863	0.614316	+	0.789060i	$0.289432\pi$
$398$	0	0
$399$	3.54216e8	0.279166
$400$	0	0
$401$	1.85275e9	1.43487	0.717433	−	0.696627i	$-0.245317\pi$
$401$	1.85275e9	1.43487	0.717433	+	0.696627i	$0.245317\pi$
$402$	0	0
$403$	7.15053e8	0.544215
$404$	0	0
$405$	−1.98198e8	−0.148254
$406$	0	0
$407$	2.88701e8	0.212260
$408$	0	0
$409$	−5.95251e8	−0.430198	−0.215099	−	0.976592i	$-0.569007\pi$
$409$	−5.95251e8	−0.430198	−0.215099	+	0.976592i	$0.569007\pi$
$410$	0	0
$411$	−1.38881e9	−0.986728
$412$	0	0
$413$	1.58454e8	0.110682
$414$	0	0
$415$	3.41233e8	0.234360
$416$	0	0
$417$	2.22756e8	0.150437
$418$	0	0
$419$	2.07152e9	1.37575	0.687876	−	0.725828i	$-0.258543\pi$
$419$	2.07152e9	1.37575	0.687876	+	0.725828i	$0.258543\pi$
$420$	0	0
$421$	1.71895e9	1.12273	0.561366	−	0.827568i	$-0.310276\pi$
$421$	1.71895e9	1.12273	0.561366	+	0.827568i	$0.310276\pi$
$422$	0	0
$423$	3.63474e8	0.233498
$424$	0	0
$425$	2.86833e9	1.81246
$426$	0	0
$427$	−1.54350e9	−0.959420
$428$	0	0
$429$	−2.19810e8	−0.134415
$430$	0	0
$431$	2.41978e9	1.45581	0.727905	−	0.685678i	$-0.240494\pi$
$431$	2.41978e9	1.45581	0.727905	+	0.685678i	$0.240494\pi$
$432$	0	0
$433$	2.12996e9	1.26085	0.630425	−	0.776250i	$-0.282881\pi$
$433$	2.12996e9	1.26085	0.630425	+	0.776250i	$0.282881\pi$
$434$	0	0
$435$	−5.14280e8	−0.299562
$436$	0	0
$437$	4.71280e8	0.270143
$438$	0	0
$439$	−6.87761e8	−0.387982	−0.193991	−	0.981003i	$-0.562143\pi$
$439$	−6.87761e8	−0.387982	−0.193991	+	0.981003i	$0.562143\pi$
$440$	0	0
$441$	−2.40963e8	−0.133788
$442$	0	0
$443$	1.14951e9	0.628202	0.314101	−	0.949389i	$-0.398297\pi$
$443$	1.14951e9	0.628202	0.314101	+	0.949389i	$0.398297\pi$
$444$	0	0
$445$	−5.48730e8	−0.295188
$446$	0	0
$447$	9.94668e8	0.526747
$448$	0	0
$449$	3.08059e9	1.60610	0.803048	−	0.595915i	$-0.203210\pi$
$449$	3.08059e9	1.60610	0.803048	+	0.595915i	$0.203210\pi$
$450$	0	0
$451$	−1.97817e8	−0.101542
$452$	0	0
$453$	−2.09173e9	−1.05721
$454$	0	0
$455$	−1.43564e8	−0.0714505
$456$	0	0
$457$	−7.75524e8	−0.380092	−0.190046	−	0.981775i	$-0.560864\pi$
$457$	−7.75524e8	−0.380092	−0.190046	+	0.981775i	$0.560864\pi$
$458$	0	0
$459$	−4.14517e9	−2.00077
$460$	0	0
$461$	3.95073e8	0.187812	0.0939061	−	0.995581i	$-0.470065\pi$
$461$	3.95073e8	0.187812	0.0939061	+	0.995581i	$0.470065\pi$
$462$	0	0
$463$	8.45981e8	0.396120	0.198060	−	0.980190i	$-0.436536\pi$
$463$	8.45981e8	0.396120	0.198060	+	0.980190i	$0.436536\pi$
$464$	0	0
$465$	−6.89230e8	−0.317891
$466$	0	0
$467$	9.45610e8	0.429638	0.214819	−	0.976654i	$-0.431084\pi$
$467$	9.45610e8	0.429638	0.214819	+	0.976654i	$0.431084\pi$
$468$	0	0
$469$	3.32385e9	1.48777
$470$	0	0
$471$	2.88773e9	1.27346
$472$	0	0
$473$	2.11989e9	0.921084
$474$	0	0
$475$	−5.17480e8	−0.221547
$476$	0	0
$477$	1.74830e8	0.0737569
$478$	0	0
$479$	−1.47508e9	−0.613257	−0.306628	−	0.951829i	$-0.599201\pi$
$479$	−1.47508e9	−0.613257	−0.306628	+	0.951829i	$0.599201\pi$
$480$	0	0
$481$	2.92726e8	0.119937
$482$	0	0
$483$	3.54835e9	1.43288
$484$	0	0
$485$	4.69619e8	0.186917
$486$	0	0
$487$	−2.65521e9	−1.04171	−0.520855	−	0.853645i	$-0.674387\pi$
$487$	−2.65521e9	−1.04171	−0.520855	+	0.853645i	$0.674387\pi$
$488$	0	0
$489$	−2.01308e9	−0.778538
$490$	0	0
$491$	−1.96164e9	−0.747882	−0.373941	−	0.927452i	$-0.621994\pi$
$491$	−1.96164e9	−0.747882	−0.373941	+	0.927452i	$0.621994\pi$
$492$	0	0
$493$	−8.86778e9	−3.33312
$494$	0	0
$495$	−4.35299e7	−0.0161313
$496$	0	0
$497$	−3.76703e9	−1.37642
$498$	0	0
$499$	1.95682e9	0.705016	0.352508	−	0.935809i	$-0.385329\pi$
$499$	1.95682e9	0.705016	0.352508	+	0.935809i	$0.385329\pi$
$500$	0	0
$501$	−1.24122e9	−0.440978
$502$	0	0
$503$	4.41795e9	1.54786	0.773931	−	0.633270i	$-0.218288\pi$
$503$	4.41795e9	1.54786	0.773931	+	0.633270i	$0.218288\pi$
$504$	0	0
$505$	1.99283e8	0.0688573
$506$	0	0
$507$	2.44984e9	0.834853
$508$	0	0
$509$	−1.90035e9	−0.638734	−0.319367	−	0.947631i	$-0.603470\pi$
$509$	−1.90035e9	−0.638734	−0.319367	+	0.947631i	$0.603470\pi$
$510$	0	0
$511$	1.51059e9	0.500811
$512$	0	0
$513$	7.47837e8	0.244566
$514$	0	0
$515$	−4.18991e8	−0.135170
$516$	0	0
$517$	−2.19991e9	−0.700146
$518$	0	0
$519$	−4.25082e7	−0.0133471
$520$	0	0
$521$	−4.21546e8	−0.130591	−0.0652955	−	0.997866i	$-0.520799\pi$
$521$	−4.21546e8	−0.130591	−0.0652955	+	0.997866i	$0.520799\pi$
$522$	0	0
$523$	−2.47415e8	−0.0756258	−0.0378129	−	0.999285i	$-0.512039\pi$
$523$	−2.47415e8	−0.0756258	−0.0378129	+	0.999285i	$0.512039\pi$
$524$	0	0
$525$	−3.89619e9	−1.17512
$526$	0	0
$527$	−1.18845e10	−3.53706
$528$	0	0
$529$	1.31620e9	0.386570
$530$	0	0
$531$	4.87144e7	0.0141197
$532$	0	0
$533$	−2.00574e8	−0.0573759
$534$	0	0
$535$	−4.48895e8	−0.126738
$536$	0	0
$537$	−1.36704e9	−0.380954
$538$	0	0
$539$	1.45842e9	0.401163
$540$	0	0
$541$	−2.03800e9	−0.553368	−0.276684	−	0.960961i	$-0.589235\pi$
$541$	−2.03800e9	−0.553368	−0.276684	+	0.960961i	$0.589235\pi$
$542$	0	0
$543$	8.54082e7	0.0228929
$544$	0	0
$545$	1.36084e9	0.360097
$546$	0	0
$547$	1.81698e9	0.474672	0.237336	−	0.971428i	$-0.423726\pi$
$547$	1.81698e9	0.474672	0.237336	+	0.971428i	$0.423726\pi$
$548$	0	0
$549$	−4.74526e8	−0.122393
$550$	0	0
$551$	1.59985e9	0.407426
$552$	0	0
$553$	3.22380e9	0.810644
$554$	0	0
$555$	−2.82154e8	−0.0700585
$556$	0	0
$557$	3.39447e8	0.0832297	0.0416149	−	0.999134i	$-0.486750\pi$
$557$	3.39447e8	0.0832297	0.0416149	+	0.999134i	$0.486750\pi$
$558$	0	0
$559$	2.14944e9	0.520455
$560$	0	0
$561$	3.65333e9	0.873612
$562$	0	0
$563$	1.15614e9	0.273044	0.136522	−	0.990637i	$-0.456408\pi$
$563$	1.15614e9	0.273044	0.136522	+	0.990637i	$0.456408\pi$
$564$	0	0
$565$	5.92252e8	0.138146
$566$	0	0
$567$	4.64221e9	1.06951
$568$	0	0
$569$	−1.81463e9	−0.412947	−0.206473	−	0.978452i	$-0.566199\pi$
$569$	−1.81463e9	−0.412947	−0.206473	+	0.978452i	$0.566199\pi$
$570$	0	0
$571$	4.22286e9	0.949250	0.474625	−	0.880188i	$-0.342584\pi$
$571$	4.22286e9	0.949250	0.474625	+	0.880188i	$0.342584\pi$
$572$	0	0
$573$	−4.77068e9	−1.05935
$574$	0	0
$575$	−5.18384e9	−1.13714
$576$	0	0
$577$	4.62442e8	0.100217	0.0501086	−	0.998744i	$-0.484043\pi$
$577$	4.62442e8	0.100217	0.0501086	+	0.998744i	$0.484043\pi$
$578$	0	0
$579$	6.62365e8	0.141815
$580$	0	0
$581$	−7.99241e9	−1.69068
$582$	0	0
$583$	−1.05815e9	−0.221161
$584$	0	0
$585$	−4.41366e7	−0.00911493
$586$	0	0
$587$	3.80196e9	0.775843	0.387921	−	0.921692i	$-0.373193\pi$
$587$	3.80196e9	0.775843	0.387921	+	0.921692i	$0.373193\pi$
$588$	0	0
$589$	2.14409e9	0.432355
$590$	0	0
$591$	4.54210e9	0.905109
$592$	0	0
$593$	−9.18168e9	−1.80813	−0.904067	−	0.427390i	$-0.859433\pi$
$593$	−9.18168e9	−1.80813	−0.904067	+	0.427390i	$0.859433\pi$
$594$	0	0
$595$	2.38609e9	0.464384
$596$	0	0
$597$	−6.12169e9	−1.17750
$598$	0	0
$599$	2.82098e9	0.536298	0.268149	−	0.963377i	$-0.413588\pi$
$599$	2.82098e9	0.536298	0.268149	+	0.963377i	$0.413588\pi$
$600$	0	0
$601$	−5.83093e9	−1.09566	−0.547831	−	0.836589i	$-0.684547\pi$
$601$	−5.83093e9	−1.09566	−0.547831	+	0.836589i	$0.684547\pi$
$602$	0	0
$603$	1.02187e9	0.189795
$604$	0	0
$605$	−7.45281e8	−0.136828
$606$	0	0
$607$	−4.93096e9	−0.894892	−0.447446	−	0.894311i	$-0.647666\pi$
$607$	−4.93096e9	−0.894892	−0.447446	+	0.894311i	$0.647666\pi$
$608$	0	0
$609$	1.20455e10	2.16105
$610$	0	0
$611$	−2.23058e9	−0.395615
$612$	0	0
$613$	−5.15868e8	−0.0904538	−0.0452269	−	0.998977i	$-0.514401\pi$
$613$	−5.15868e8	−0.0904538	−0.0452269	+	0.998977i	$0.514401\pi$
$614$	0	0
$615$	1.93330e8	0.0335149
$616$	0	0
$617$	−4.60228e8	−0.0788815	−0.0394408	−	0.999222i	$-0.512558\pi$
$617$	−4.60228e8	−0.0788815	−0.0394408	+	0.999222i	$0.512558\pi$
$618$	0	0
$619$	−9.84343e9	−1.66813	−0.834064	−	0.551668i	$-0.813992\pi$
$619$	−9.84343e9	−1.66813	−0.834064	+	0.551668i	$0.813992\pi$
$620$	0	0
$621$	7.49142e9	1.25529
$622$	0	0
$623$	1.28524e10	2.12950
$624$	0	0
$625$	5.48267e9	0.898281
$626$	0	0
$627$	−6.59103e8	−0.106787
$628$	0	0
$629$	−4.86521e9	−0.779515
$630$	0	0
$631$	6.19888e7	0.00982223	0.00491112	−	0.999988i	$-0.498437\pi$
$631$	6.19888e7	0.00982223	0.00491112	+	0.999988i	$0.498437\pi$
$632$	0	0
$633$	9.36672e9	1.46783
$634$	0	0
$635$	3.78370e8	0.0586420
$636$	0	0
$637$	1.47875e9	0.226676
$638$	0	0
$639$	−1.15812e9	−0.175590
$640$	0	0
$641$	4.01306e8	0.0601828	0.0300914	−	0.999547i	$-0.490420\pi$
$641$	4.01306e8	0.0601828	0.0300914	+	0.999547i	$0.490420\pi$
$642$	0	0
$643$	−1.13622e10	−1.68548	−0.842740	−	0.538321i	$-0.819059\pi$
$643$	−1.13622e10	−1.68548	−0.842740	+	0.538321i	$0.819059\pi$
$644$	0	0
$645$	−2.07181e9	−0.304012
$646$	0	0
$647$	1.36393e9	0.197982	0.0989911	−	0.995088i	$-0.468438\pi$
$647$	1.36393e9	0.197982	0.0989911	+	0.995088i	$0.468438\pi$
$648$	0	0
$649$	−2.94842e8	−0.0423382
$650$	0	0
$651$	1.61432e10	2.29328
$652$	0	0
$653$	8.97787e9	1.26176	0.630881	−	0.775880i	$-0.282694\pi$
$653$	8.97787e9	1.26176	0.630881	+	0.775880i	$0.282694\pi$
$654$	0	0
$655$	1.78622e9	0.248365
$656$	0	0
$657$	4.64409e8	0.0638884
$658$	0	0
$659$	2.35102e9	0.320006	0.160003	−	0.987117i	$-0.448850\pi$
$659$	2.35102e9	0.320006	0.160003	+	0.987117i	$0.448850\pi$
$660$	0	0
$661$	−6.65204e9	−0.895879	−0.447940	−	0.894064i	$-0.647842\pi$
$661$	−6.65204e9	−0.895879	−0.447940	+	0.894064i	$0.647842\pi$
$662$	0	0
$663$	3.70425e9	0.493632
$664$	0	0
$665$	−4.30478e8	−0.0567643
$666$	0	0
$667$	1.60264e10	2.09121
$668$	0	0
$669$	6.76216e9	0.873161
$670$	0	0
$671$	2.87205e9	0.366997
$672$	0	0
$673$	1.50577e10	1.90417	0.952084	−	0.305838i	$-0.0989366\pi$
$673$	1.50577e10	1.90417	0.952084	+	0.305838i	$0.0989366\pi$
$674$	0	0
$675$	−8.22581e9	−1.02947
$676$	0	0
$677$	−1.45878e10	−1.80688	−0.903440	−	0.428715i	$-0.858966\pi$
$677$	−1.45878e10	−1.80688	−0.903440	+	0.428715i	$0.858966\pi$
$678$	0	0
$679$	−1.09995e10	−1.34843
$680$	0	0
$681$	−1.02777e10	−1.24704
$682$	0	0
$683$	6.04295e9	0.725732	0.362866	−	0.931841i	$-0.381798\pi$
$683$	6.04295e9	0.725732	0.362866	+	0.931841i	$0.381798\pi$
$684$	0	0
$685$	1.68782e9	0.200636
$686$	0	0
$687$	−9.67573e9	−1.13851
$688$	0	0
$689$	−1.07290e9	−0.124966
$690$	0	0
$691$	1.02900e10	1.18643	0.593217	−	0.805042i	$-0.297858\pi$
$691$	1.02900e10	1.18643	0.593217	+	0.805042i	$0.297858\pi$
$692$	0	0
$693$	1.01956e9	0.116372
$694$	0	0
$695$	−2.70715e8	−0.0305890
$696$	0	0
$697$	3.33361e9	0.372907
$698$	0	0
$699$	−3.68461e9	−0.408058
$700$	0	0
$701$	−6.26930e9	−0.687394	−0.343697	−	0.939081i	$-0.611679\pi$
$701$	−6.26930e9	−0.687394	−0.343697	+	0.939081i	$0.611679\pi$
$702$	0	0
$703$	8.77741e8	0.0952846
$704$	0	0
$705$	2.15002e9	0.231090
$706$	0	0
$707$	−4.66763e9	−0.496739
$708$	0	0
$709$	8.02429e9	0.845560	0.422780	−	0.906232i	$-0.361054\pi$
$709$	8.02429e9	0.845560	0.422780	+	0.906232i	$0.361054\pi$
$710$	0	0
$711$	9.91110e8	0.103414
$712$	0	0
$713$	2.14784e10	2.21916
$714$	0	0
$715$	2.67135e8	0.0273312
$716$	0	0
$717$	−8.95778e9	−0.907577
$718$	0	0
$719$	−3.73707e9	−0.374956	−0.187478	−	0.982269i	$-0.560031\pi$
$719$	−3.73707e9	−0.374956	−0.187478	+	0.982269i	$0.560031\pi$
$720$	0	0
$721$	9.81368e9	0.975120
$722$	0	0
$723$	−6.14157e9	−0.604360
$724$	0	0
$725$	−1.75975e10	−1.71502
$726$	0	0
$727$	3.32395e9	0.320837	0.160418	−	0.987049i	$-0.448716\pi$
$727$	3.32395e9	0.320837	0.160418	+	0.987049i	$0.448716\pi$
$728$	0	0
$729$	1.15837e10	1.10739
$730$	0	0
$731$	−3.57244e10	−3.38263
$732$	0	0
$733$	4.35160e9	0.408117	0.204059	−	0.978959i	$-0.434587\pi$
$733$	4.35160e9	0.408117	0.204059	+	0.978959i	$0.434587\pi$
$734$	0	0
$735$	−1.42534e9	−0.132408
$736$	0	0
$737$	−6.18481e9	−0.569102
$738$	0	0
$739$	1.10497e9	0.100715	0.0503575	−	0.998731i	$-0.483964\pi$
$739$	1.10497e9	0.100715	0.0503575	+	0.998731i	$0.483964\pi$
$740$	0	0
$741$	−6.68290e8	−0.0603395
$742$	0	0
$743$	−8.24924e9	−0.737824	−0.368912	−	0.929464i	$-0.620270\pi$
$743$	−8.24924e9	−0.737824	−0.368912	+	0.929464i	$0.620270\pi$
$744$	0	0
$745$	−1.20882e9	−0.107106
$746$	0	0
$747$	−2.45715e9	−0.215680
$748$	0	0
$749$	1.05141e10	0.914292
$750$	0	0
$751$	1.34895e10	1.16213	0.581067	−	0.813856i	$-0.302635\pi$
$751$	1.34895e10	1.16213	0.581067	+	0.813856i	$0.302635\pi$
$752$	0	0
$753$	−6.82786e9	−0.582777
$754$	0	0
$755$	2.54207e9	0.214968
$756$	0	0
$757$	−1.71461e10	−1.43658	−0.718288	−	0.695745i	$-0.755074\pi$
$757$	−1.71461e10	−1.43658	−0.718288	+	0.695745i	$0.755074\pi$
$758$	0	0
$759$	−6.60254e9	−0.548106
$760$	0	0
$761$	8.50603e9	0.699650	0.349825	−	0.936815i	$-0.386241\pi$
$761$	8.50603e9	0.699650	0.349825	+	0.936815i	$0.386241\pi$
$762$	0	0
$763$	−3.18738e10	−2.59776
$764$	0	0
$765$	7.33567e8	0.0592413
$766$	0	0
$767$	−2.98952e8	−0.0239231
$768$	0	0
$769$	4.00986e9	0.317971	0.158985	−	0.987281i	$-0.449178\pi$
$769$	4.00986e9	0.317971	0.158985	+	0.987281i	$0.449178\pi$
$770$	0	0
$771$	−2.19861e10	−1.72766
$772$	0	0
$773$	−4.29948e9	−0.334802	−0.167401	−	0.985889i	$-0.553537\pi$
$773$	−4.29948e9	−0.334802	−0.167401	+	0.985889i	$0.553537\pi$
$774$	0	0
$775$	−2.35839e10	−1.81995
$776$	0	0
$777$	6.60865e9	0.505405
$778$	0	0
$779$	−6.01423e8	−0.0455826
$780$	0	0
$781$	7.00945e9	0.526509
$782$	0	0
$783$	2.54311e10	1.89321
$784$	0	0
$785$	−3.50946e9	−0.258938
$786$	0	0
$787$	−2.08312e10	−1.52336	−0.761680	−	0.647954i	$-0.775625\pi$
$787$	−2.08312e10	−1.52336	−0.761680	+	0.647954i	$0.775625\pi$
$788$	0	0
$789$	−1.00158e10	−0.725969
$790$	0	0
$791$	−1.38718e10	−0.996588
$792$	0	0
$793$	2.91208e9	0.207371
$794$	0	0
$795$	1.03416e9	0.0729963
$796$	0	0
$797$	9.84013e9	0.688489	0.344244	−	0.938880i	$-0.388135\pi$
$797$	9.84013e9	0.688489	0.344244	+	0.938880i	$0.388135\pi$
$798$	0	0
$799$	3.70730e10	2.57125
$800$	0	0
$801$	3.95129e9	0.271660
$802$	0	0
$803$	−2.81081e9	−0.191570
$804$	0	0
$805$	−4.31230e9	−0.291355
$806$	0	0
$807$	−2.88295e9	−0.193099
$808$	0	0
$809$	−2.11413e10	−1.40382	−0.701910	−	0.712265i	$-0.747669\pi$
$809$	−2.11413e10	−1.40382	−0.701910	+	0.712265i	$0.747669\pi$
$810$	0	0
$811$	−8.99755e9	−0.592313	−0.296156	−	0.955139i	$-0.595705\pi$
$811$	−8.99755e9	−0.592313	−0.296156	+	0.955139i	$0.595705\pi$
$812$	0	0
$813$	2.38622e10	1.55738
$814$	0	0
$815$	2.44649e9	0.158304
$816$	0	0
$817$	6.44511e9	0.413479
$818$	0	0
$819$	1.03377e9	0.0657555
$820$	0	0
$821$	−1.50430e10	−0.948708	−0.474354	−	0.880334i	$-0.657318\pi$
$821$	−1.50430e10	−0.948708	−0.474354	+	0.880334i	$0.657318\pi$
$822$	0	0
$823$	2.86488e10	1.79146	0.895729	−	0.444600i	$-0.146654\pi$
$823$	2.86488e10	1.79146	0.895729	+	0.444600i	$0.146654\pi$
$824$	0	0
$825$	7.24979e9	0.449507
$826$	0	0
$827$	1.90228e10	1.16951	0.584755	−	0.811210i	$-0.301191\pi$
$827$	1.90228e10	1.16951	0.584755	+	0.811210i	$0.301191\pi$
$828$	0	0
$829$	−7.54277e9	−0.459822	−0.229911	−	0.973212i	$-0.573844\pi$
$829$	−7.54277e9	−0.459822	−0.229911	+	0.973212i	$0.573844\pi$
$830$	0	0
$831$	−1.21527e10	−0.734632
$832$	0	0
$833$	−2.45773e10	−1.47325
$834$	0	0
$835$	1.50845e9	0.0896662
$836$	0	0
$837$	3.40823e10	2.00905
$838$	0	0
$839$	9.01446e9	0.526954	0.263477	−	0.964666i	$-0.415131\pi$
$839$	9.01446e9	0.526954	0.263477	+	0.964666i	$0.415131\pi$
$840$	0	0
$841$	3.71549e10	2.15392
$842$	0	0
$843$	5.37419e9	0.308970
$844$	0	0
$845$	−2.97729e9	−0.169755
$846$	0	0
$847$	1.74561e10	0.987085
$848$	0	0
$849$	3.62671e9	0.203393
$850$	0	0
$851$	8.79273e9	0.489069
$852$	0	0
$853$	5.68194e9	0.313455	0.156728	−	0.987642i	$-0.449906\pi$
$853$	5.68194e9	0.313455	0.156728	+	0.987642i	$0.449906\pi$
$854$	0	0
$855$	−1.32344e8	−0.00724141
$856$	0	0
$857$	−2.26368e10	−1.22852	−0.614261	−	0.789103i	$-0.710546\pi$
$857$	−2.26368e10	−1.22852	−0.614261	+	0.789103i	$0.710546\pi$
$858$	0	0
$859$	1.38340e10	0.744685	0.372343	−	0.928095i	$-0.378555\pi$
$859$	1.38340e10	0.744685	0.372343	+	0.928095i	$0.378555\pi$
$860$	0	0
$861$	−4.52821e9	−0.241777
$862$	0	0
$863$	−3.03741e10	−1.60867	−0.804334	−	0.594178i	$-0.797477\pi$
$863$	−3.03741e10	−1.60867	−0.804334	+	0.594178i	$0.797477\pi$
$864$	0	0
$865$	5.16601e7	0.00271393
$866$	0	0
$867$	−4.40880e10	−2.29749
$868$	0	0
$869$	−5.99865e9	−0.310087
$870$	0	0
$871$	−6.27101e9	−0.321569
$872$	0	0
$873$	−3.38163e9	−0.172019
$874$	0	0
$875$	9.63824e9	0.486373
$876$	0	0
$877$	1.83593e9	0.0919088	0.0459544	−	0.998944i	$-0.485367\pi$
$877$	1.83593e9	0.0919088	0.0459544	+	0.998944i	$0.485367\pi$
$878$	0	0
$879$	−2.67513e10	−1.32857
$880$	0	0
$881$	2.50289e10	1.23318	0.616589	−	0.787285i	$-0.288514\pi$
$881$	2.50289e10	1.23318	0.616589	+	0.787285i	$0.288514\pi$
$882$	0	0
$883$	3.10246e10	1.51651	0.758253	−	0.651961i	$-0.226053\pi$
$883$	3.10246e10	1.51651	0.758253	+	0.651961i	$0.226053\pi$
$884$	0	0
$885$	2.88155e8	0.0139741
$886$	0	0
$887$	−5.19460e9	−0.249931	−0.124965	−	0.992161i	$-0.539882\pi$
$887$	−5.19460e9	−0.249931	−0.124965	+	0.992161i	$0.539882\pi$
$888$	0	0
$889$	−8.86223e9	−0.423045
$890$	0	0
$891$	−8.63794e9	−0.409108
$892$	0	0
$893$	−6.68840e9	−0.314299
$894$	0	0
$895$	1.66136e9	0.0774612
$896$	0	0
$897$	−6.69457e9	−0.309706
$898$	0	0
$899$	7.29125e10	3.34690
$900$	0	0
$901$	1.78320e10	0.812202
$902$	0	0
$903$	4.85263e10	2.19316
$904$	0	0
$905$	−1.03796e8	−0.00465492
$906$	0	0
$907$	−3.33724e10	−1.48512	−0.742560	−	0.669780i	$-0.766389\pi$
$907$	−3.33724e10	−1.48512	−0.742560	+	0.669780i	$0.766389\pi$
$908$	0	0
$909$	−1.43499e9	−0.0633690
$910$	0	0
$911$	2.71857e8	0.0119131	0.00595656	−	0.999982i	$-0.498104\pi$
$911$	2.71857e8	0.0119131	0.00595656	+	0.999982i	$0.498104\pi$
$912$	0	0
$913$	1.48718e10	0.646718
$914$	0	0
$915$	−2.80691e9	−0.121131
$916$	0	0
$917$	−4.18371e10	−1.79172
$918$	0	0
$919$	−1.31476e10	−0.558783	−0.279391	−	0.960177i	$-0.590133\pi$
$919$	−1.31476e10	−0.558783	−0.279391	+	0.960177i	$0.590133\pi$
$920$	0	0
$921$	1.89693e10	0.800096
$922$	0	0
$923$	7.10715e9	0.297502
$924$	0	0
$925$	−9.65469e9	−0.401091
$926$	0	0
$927$	3.01707e9	0.124396
$928$	0	0
$929$	2.49695e10	1.02177	0.510886	−	0.859648i	$-0.329317\pi$
$929$	2.49695e10	1.02177	0.510886	+	0.859648i	$0.329317\pi$
$930$	0	0
$931$	4.43404e9	0.180084
$932$	0	0
$933$	1.17278e10	0.472749
$934$	0	0
$935$	−4.43988e9	−0.177636
$936$	0	0
$937$	−3.99459e10	−1.58629	−0.793146	−	0.609031i	$-0.791558\pi$
$937$	−3.99459e10	−1.58629	−0.793146	+	0.609031i	$0.791558\pi$
$938$	0	0
$939$	3.31797e10	1.30781
$940$	0	0
$941$	2.08944e10	0.817459	0.408729	−	0.912656i	$-0.365972\pi$
$941$	2.08944e10	0.817459	0.408729	+	0.912656i	$0.365972\pi$
$942$	0	0
$943$	−6.02473e9	−0.233963
$944$	0	0
$945$	−6.84283e9	−0.263770
$946$	0	0
$947$	−1.29152e10	−0.494169	−0.247084	−	0.968994i	$-0.579472\pi$
$947$	−1.29152e10	−0.494169	−0.247084	+	0.968994i	$0.579472\pi$
$948$	0	0
$949$	−2.84999e9	−0.108246
$950$	0	0
$951$	1.84764e10	0.696605
$952$	0	0
$953$	−2.87193e10	−1.07485	−0.537427	−	0.843310i	$-0.680604\pi$
$953$	−2.87193e10	−1.07485	−0.537427	+	0.843310i	$0.680604\pi$
$954$	0	0
$955$	5.79780e9	0.215403
$956$	0	0
$957$	−2.24136e10	−0.826646
$958$	0	0
$959$	−3.95324e10	−1.44740
$960$	0	0
$961$	7.02035e10	2.55169
$962$	0	0
$963$	3.23240e9	0.116636
$964$	0	0
$965$	−8.04971e8	−0.0288360
$966$	0	0
$967$	−1.06364e10	−0.378270	−0.189135	−	0.981951i	$-0.560568\pi$
$967$	−1.06364e10	−0.378270	−0.189135	+	0.981951i	$0.560568\pi$
$968$	0	0
$969$	1.11072e10	0.392169
$970$	0	0
$971$	3.09069e10	1.08340	0.541699	−	0.840572i	$-0.317781\pi$
$971$	3.09069e10	1.08340	0.541699	+	0.840572i	$0.317781\pi$
$972$	0	0
$973$	6.34073e9	0.220671
$974$	0	0
$975$	7.35085e9	0.253993
$976$	0	0
$977$	1.35513e10	0.464890	0.232445	−	0.972610i	$-0.425327\pi$
$977$	1.35513e10	0.464890	0.232445	+	0.972610i	$0.425327\pi$
$978$	0	0
$979$	−2.39150e10	−0.814574
$980$	0	0
$981$	−9.79914e9	−0.331395
$982$	0	0
$983$	−4.99042e10	−1.67571	−0.837857	−	0.545890i	$-0.816192\pi$
$983$	−4.99042e10	−1.67571	−0.837857	+	0.545890i	$0.816192\pi$
$984$	0	0
$985$	−5.52001e9	−0.184040
$986$	0	0
$987$	−5.03581e10	−1.66709
$988$	0	0
$989$	6.45636e10	2.12227
$990$	0	0
$991$	−2.95439e10	−0.964295	−0.482147	−	0.876090i	$-0.660143\pi$
$991$	−2.95439e10	−0.964295	−0.482147	+	0.876090i	$0.660143\pi$
$992$	0	0
$993$	2.30847e10	0.748174
$994$	0	0
$995$	7.43967e9	0.239427
$996$	0	0
$997$	−4.15419e10	−1.32756	−0.663778	−	0.747929i	$-0.731048\pi$
$997$	−4.15419e10	−1.32756	−0.663778	+	0.747929i	$0.731048\pi$
$998$	0	0
$999$	1.39525e10	0.442764

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.a.b.1.2	✓	6
4.3	odd	2		304.8.a.i.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.a.b.1.2	✓	6	1.1	even	1	trivial
304.8.a.i.1.5		6	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)

\(f(q)\)	\(=\)	\(q-42.5941 q^{3} +51.7645 q^{5} -1212.43 q^{7} -372.745 q^{9} +O(q^{10})\)	\(q-42.5941 q^{3} +51.7645 q^{5} -1212.43 q^{7} -372.745 q^{9} +2256.02 q^{11} +2287.47 q^{13} -2204.86 q^{15} -38018.6 q^{17} +6859.00 q^{19} +51642.5 q^{21} +68709.8 q^{23} -75445.4 q^{25} +109030. q^{27} +233248. q^{29} +312596. q^{31} -96093.2 q^{33} -62761.0 q^{35} +127969. q^{37} -97432.6 q^{39} -87683.8 q^{41} +939657. q^{43} -19295.0 q^{45} -975128. q^{47} +646455. q^{49} +1.61937e6 q^{51} -469035. q^{53} +116782. q^{55} -292153. q^{57} -130691. q^{59} +1.27306e6 q^{61} +451929. q^{63} +118410. q^{65} -2.74146e6 q^{67} -2.92663e6 q^{69} +3.10700e6 q^{71} -1.24592e6 q^{73} +3.21353e6 q^{75} -2.73528e6 q^{77} -2.65895e6 q^{79} -3.82884e6 q^{81} +6.59203e6 q^{83} -1.96801e6 q^{85} -9.93500e6 q^{87} -1.06005e7 q^{89} -2.77341e6 q^{91} -1.33147e7 q^{93} +355052. q^{95} +9.07223e6 q^{97} -840922. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9}+O(q^{10}) \) 6 * q + 40 * q^3 + 279 * q^5 - 1565 * q^7 + 3900 * q^9	\( 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9} + 7983 q^{11} + 1250 q^{13} + 26790 q^{15} + 21735 q^{17} + 41154 q^{19} - 86346 q^{21} + 100920 q^{23} + 373305 q^{25} + 534790 q^{27} - 58656 q^{29} + 403808 q^{31} + 916430 q^{33} + 463497 q^{35} + 808780 q^{37} + 758704 q^{39} + 556944 q^{41} + 1220735 q^{43} + 3234843 q^{45} + 1915305 q^{47} + 2045883 q^{49} + 908816 q^{51} + 511650 q^{53} + 1813341 q^{55} + 274360 q^{57} + 1300572 q^{59} + 565335 q^{61} - 7170325 q^{63} - 6195012 q^{65} - 45010 q^{67} - 7381528 q^{69} - 1424106 q^{71} - 11153825 q^{73} + 3941974 q^{75} - 17515425 q^{77} - 6392144 q^{79} + 6187530 q^{81} - 3164160 q^{83} - 19479255 q^{85} - 25999500 q^{87} - 14502678 q^{89} - 9736226 q^{91} - 18344300 q^{93} + 1913661 q^{95} - 21377010 q^{97} + 24032935 q^{99}+O(q^{100}) \) 6 * q + 40 * q^3 + 279 * q^5 - 1565 * q^7 + 3900 * q^9 + 7983 * q^11 + 1250 * q^13 + 26790 * q^15 + 21735 * q^17 + 41154 * q^19 - 86346 * q^21 + 100920 * q^23 + 373305 * q^25 + 534790 * q^27 - 58656 * q^29 + 403808 * q^31 + 916430 * q^33 + 463497 * q^35 + 808780 * q^37 + 758704 * q^39 + 556944 * q^41 + 1220735 * q^43 + 3234843 * q^45 + 1915305 * q^47 + 2045883 * q^49 + 908816 * q^51 + 511650 * q^53 + 1813341 * q^55 + 274360 * q^57 + 1300572 * q^59 + 565335 * q^61 - 7170325 * q^63 - 6195012 * q^65 - 45010 * q^67 - 7381528 * q^69 - 1424106 * q^71 - 11153825 * q^73 + 3941974 * q^75 - 17515425 * q^77 - 6392144 * q^79 + 6187530 * q^81 - 3164160 * q^83 - 19479255 * q^85 - 25999500 * q^87 - 14502678 * q^89 - 9736226 * q^91 - 18344300 * q^93 + 1913661 * q^95 - 21377010 * q^97 + 24032935 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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