LMFDB - Embedded newform 76.8.a.b.1.3

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			1.3
Root			$44.4724$ 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)

$f(q)$	$=$	$q-37.4724 q^{3} +534.866 q^{5} +1051.36 q^{7} -782.818 q^{9} +O(q^{10})$	$q-37.4724 q^{3} +534.866 q^{5} +1051.36 q^{7} -782.818 q^{9} +295.636 q^{11} -5972.92 q^{13} -20042.7 q^{15} +8602.67 q^{17} +6859.00 q^{19} -39397.1 q^{21} +71079.5 q^{23} +207956. q^{25} +111286. q^{27} -134576. q^{29} +123349. q^{31} -11078.2 q^{33} +562338. q^{35} -16358.8 q^{37} +223820. q^{39} -559477. q^{41} +80276.3 q^{43} -418702. q^{45} +786873. q^{47} +281821. q^{49} -322363. q^{51} +1.46372e6 q^{53} +158126. q^{55} -257023. q^{57} +1.32299e6 q^{59} +2.75642e6 q^{61} -823026. q^{63} -3.19471e6 q^{65} +2.31966e6 q^{67} -2.66352e6 q^{69} -2.33227e6 q^{71} -795003. q^{73} -7.79262e6 q^{75} +310821. q^{77} +2.75154e6 q^{79} -2.45814e6 q^{81} -9.02658e6 q^{83} +4.60127e6 q^{85} +5.04288e6 q^{87} +6.58315e6 q^{89} -6.27971e6 q^{91} -4.62218e6 q^{93} +3.66864e6 q^{95} -1.49695e7 q^{97} -231429. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−37.4724	−0.801286	−0.400643	−	0.916234i	$-0.631213\pi$
$3$	−37.4724	−0.801286	−0.400643	+	0.916234i	$0.631213\pi$
$4$	0	0
$5$	534.866	1.91359	0.956797	−	0.290758i	$-0.0939073\pi$
$5$	534.866	1.91359	0.956797	+	0.290758i	$0.0939073\pi$
$6$	0	0
$7$	1051.36	1.15854	0.579268	−	0.815137i	$-0.303338\pi$
$7$	1051.36	1.15854	0.579268	+	0.815137i	$0.303338\pi$
$8$	0	0
$9$	−782.818	−0.357941
$10$	0	0
$11$	295.636	0.0669704	0.0334852	−	0.999439i	$-0.489339\pi$
$11$	295.636	0.0669704	0.0334852	+	0.999439i	$0.489339\pi$
$12$	0	0
$13$	−5972.92	−0.754024	−0.377012	−	0.926208i	$-0.623048\pi$
$13$	−5972.92	−0.754024	−0.377012	+	0.926208i	$0.623048\pi$
$14$	0	0
$15$	−20042.7	−1.53333
$16$	0	0
$17$	8602.67	0.424680	0.212340	−	0.977196i	$-0.431892\pi$
$17$	8602.67	0.424680	0.212340	+	0.977196i	$0.431892\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	−39397.1	−0.928318
$22$	0	0
$23$	71079.5	1.21814	0.609070	−	0.793116i	$-0.291543\pi$
$23$	71079.5	1.21814	0.609070	+	0.793116i	$0.291543\pi$
$24$	0	0
$25$	207956.	2.66184
$26$	0	0
$27$	111286.	1.08810
$28$	0	0
$29$	−134576.	−1.02465	−0.512323	−	0.858793i	$-0.671215\pi$
$29$	−134576.	−1.02465	−0.512323	+	0.858793i	$0.671215\pi$
$30$	0	0
$31$	123349.	0.743651	0.371826	−	0.928303i	$-0.378732\pi$
$31$	123349.	0.743651	0.371826	+	0.928303i	$0.378732\pi$
$32$	0	0
$33$	−11078.2	−0.0536624
$34$	0	0
$35$	562338.	2.21697
$36$	0	0
$37$	−16358.8	−0.0530940	−0.0265470	−	0.999648i	$-0.508451\pi$
$37$	−16358.8	−0.0530940	−0.0265470	+	0.999648i	$0.508451\pi$
$38$	0	0
$39$	223820.	0.604189
$40$	0	0
$41$	−559477.	−1.26777	−0.633883	−	0.773429i	$-0.718540\pi$
$41$	−559477.	−1.26777	−0.633883	+	0.773429i	$0.718540\pi$
$42$	0	0
$43$	80276.3	0.153974	0.0769871	−	0.997032i	$-0.475470\pi$
$43$	80276.3	0.153974	0.0769871	+	0.997032i	$0.475470\pi$
$44$	0	0
$45$	−418702.	−0.684954
$46$	0	0
$47$	786873.	1.10551	0.552755	−	0.833344i	$-0.313577\pi$
$47$	786873.	1.10551	0.552755	+	0.833344i	$0.313577\pi$
$48$	0	0
$49$	281821.	0.342206
$50$	0	0
$51$	−322363.	−0.340290
$52$	0	0
$53$	1.46372e6	1.35050	0.675249	−	0.737590i	$-0.264036\pi$
$53$	1.46372e6	1.35050	0.675249	+	0.737590i	$0.264036\pi$
$54$	0	0
$55$	158126.	0.128154
$56$	0	0
$57$	−257023.	−0.183828
$58$	0	0
$59$	1.32299e6	0.838638	0.419319	−	0.907839i	$-0.362269\pi$
$59$	1.32299e6	0.838638	0.419319	+	0.907839i	$0.362269\pi$
$60$	0	0
$61$	2.75642e6	1.55486	0.777430	−	0.628970i	$-0.216523\pi$
$61$	2.75642e6	1.55486	0.777430	+	0.628970i	$0.216523\pi$
$62$	0	0
$63$	−823026.	−0.414688
$64$	0	0
$65$	−3.19471e6	−1.44290
$66$	0	0
$67$	2.31966e6	0.942240	0.471120	−	0.882069i	$-0.343850\pi$
$67$	2.31966e6	0.942240	0.471120	+	0.882069i	$0.343850\pi$
$68$	0	0
$69$	−2.66352e6	−0.976078
$70$	0	0
$71$	−2.33227e6	−0.773349	−0.386674	−	0.922216i	$-0.626376\pi$
$71$	−2.33227e6	−0.773349	−0.386674	+	0.922216i	$0.626376\pi$
$72$	0	0
$73$	−795003.	−0.239188	−0.119594	−	0.992823i	$-0.538159\pi$
$73$	−795003.	−0.239188	−0.119594	+	0.992823i	$0.538159\pi$
$74$	0	0
$75$	−7.79262e6	−2.13289
$76$	0	0
$77$	310821.	0.0775876
$78$	0	0
$79$	2.75154e6	0.627887	0.313944	−	0.949442i	$-0.398350\pi$
$79$	2.75154e6	0.627887	0.313944	+	0.949442i	$0.398350\pi$
$80$	0	0
$81$	−2.45814e6	−0.513937
$82$	0	0
$83$	−9.02658e6	−1.73281	−0.866403	−	0.499345i	$-0.833574\pi$
$83$	−9.02658e6	−1.73281	−0.866403	+	0.499345i	$0.833574\pi$
$84$	0	0
$85$	4.60127e6	0.812665
$86$	0	0
$87$	5.04288e6	0.821035
$88$	0	0
$89$	6.58315e6	0.989848	0.494924	−	0.868936i	$-0.335196\pi$
$89$	6.58315e6	0.989848	0.494924	+	0.868936i	$0.335196\pi$
$90$	0	0
$91$	−6.27971e6	−0.873564
$92$	0	0
$93$	−4.62218e6	−0.595877
$94$	0	0
$95$	3.66864e6	0.439008
$96$	0	0
$97$	−1.49695e7	−1.66535	−0.832676	−	0.553761i	$-0.813192\pi$
$97$	−1.49695e7	−1.66535	−0.832676	+	0.553761i	$0.813192\pi$
$98$	0	0
$99$	−231429.	−0.0239715
$100$	0	0
$101$	−1.08324e7	−1.04617	−0.523085	−	0.852281i	$-0.675219\pi$
$101$	−1.08324e7	−1.04617	−0.523085	+	0.852281i	$0.675219\pi$
$102$	0	0
$103$	2.05785e7	1.85560	0.927798	−	0.373083i	$-0.121699\pi$
$103$	2.05785e7	1.85560	0.927798	+	0.373083i	$0.121699\pi$
$104$	0	0
$105$	−2.10722e7	−1.77642
$106$	0	0
$107$	−1.41665e7	−1.11794	−0.558972	−	0.829186i	$-0.688804\pi$
$107$	−1.41665e7	−1.11794	−0.558972	+	0.829186i	$0.688804\pi$
$108$	0	0
$109$	6.85578e6	0.507066	0.253533	−	0.967327i	$-0.418407\pi$
$109$	6.85578e6	0.507066	0.253533	+	0.967327i	$0.418407\pi$
$110$	0	0
$111$	613003.	0.0425434
$112$	0	0
$113$	−1.68743e7	−1.10015	−0.550075	−	0.835115i	$-0.685401\pi$
$113$	−1.68743e7	−1.10015	−0.550075	+	0.835115i	$0.685401\pi$
$114$	0	0
$115$	3.80180e7	2.33102
$116$	0	0
$117$	4.67571e6	0.269896
$118$	0	0
$119$	9.04453e6	0.492007
$120$	0	0
$121$	−1.93998e7	−0.995515
$122$	0	0
$123$	2.09650e7	1.01584
$124$	0	0
$125$	6.94423e7	3.18009
$126$	0	0
$127$	−5.87849e6	−0.254655	−0.127328	−	0.991861i	$-0.540640\pi$
$127$	−5.87849e6	−0.254655	−0.127328	+	0.991861i	$0.540640\pi$
$128$	0	0
$129$	−3.00815e6	−0.123377
$130$	0	0
$131$	−7.59672e6	−0.295241	−0.147621	−	0.989044i	$-0.547161\pi$
$131$	−7.59672e6	−0.295241	−0.147621	+	0.989044i	$0.547161\pi$
$132$	0	0
$133$	7.21130e6	0.265786
$134$	0	0
$135$	5.95232e7	2.08218
$136$	0	0
$137$	5.05849e7	1.68073	0.840366	−	0.542019i	$-0.182340\pi$
$137$	5.05849e7	1.68073	0.840366	+	0.542019i	$0.182340\pi$
$138$	0	0
$139$	−9.90580e6	−0.312851	−0.156426	−	0.987690i	$-0.549997\pi$
$139$	−9.90580e6	−0.312851	−0.156426	+	0.987690i	$0.549997\pi$
$140$	0	0
$141$	−2.94861e7	−0.885829
$142$	0	0
$143$	−1.76581e6	−0.0504973
$144$	0	0
$145$	−7.19800e7	−1.96076
$146$	0	0
$147$	−1.05605e7	−0.274204
$148$	0	0
$149$	−3.20460e7	−0.793638	−0.396819	−	0.917897i	$-0.629886\pi$
$149$	−3.20460e7	−0.793638	−0.396819	+	0.917897i	$0.629886\pi$
$150$	0	0
$151$	−6.65362e7	−1.57267	−0.786337	−	0.617798i	$-0.788025\pi$
$151$	−6.65362e7	−1.57267	−0.786337	+	0.617798i	$0.788025\pi$
$152$	0	0
$153$	−6.73432e6	−0.152011
$154$	0	0
$155$	6.59751e7	1.42305
$156$	0	0
$157$	4.68675e7	0.966547	0.483274	−	0.875469i	$-0.339448\pi$
$157$	4.68675e7	0.966547	0.483274	+	0.875469i	$0.339448\pi$
$158$	0	0
$159$	−5.48493e7	−1.08213
$160$	0	0
$161$	7.47304e7	1.41126
$162$	0	0
$163$	−1.67212e7	−0.302420	−0.151210	−	0.988502i	$-0.548317\pi$
$163$	−1.67212e7	−0.302420	−0.151210	+	0.988502i	$0.548317\pi$
$164$	0	0
$165$	−5.92535e6	−0.102688
$166$	0	0
$167$	4.29076e7	0.712896	0.356448	−	0.934315i	$-0.383988\pi$
$167$	4.29076e7	0.712896	0.356448	+	0.934315i	$0.383988\pi$
$168$	0	0
$169$	−2.70727e7	−0.431447
$170$	0	0
$171$	−5.36935e6	−0.0821174
$172$	0	0
$173$	9.83994e6	0.144488	0.0722439	−	0.997387i	$-0.476984\pi$
$173$	9.83994e6	0.144488	0.0722439	+	0.997387i	$0.476984\pi$
$174$	0	0
$175$	2.18637e8	3.08384
$176$	0	0
$177$	−4.95757e7	−0.671989
$178$	0	0
$179$	1.83927e7	0.239696	0.119848	−	0.992792i	$-0.461759\pi$
$179$	1.83927e7	0.239696	0.119848	+	0.992792i	$0.461759\pi$
$180$	0	0
$181$	−1.39402e8	−1.74741	−0.873703	−	0.486459i	$-0.838288\pi$
$181$	−1.39402e8	−1.74741	−0.873703	+	0.486459i	$0.838288\pi$
$182$	0	0
$183$	−1.03290e8	−1.24589
$184$	0	0
$185$	−8.74975e6	−0.101600
$186$	0	0
$187$	2.54326e6	0.0284410
$188$	0	0
$189$	1.17002e8	1.26060
$190$	0	0
$191$	1.29534e8	1.34514	0.672568	−	0.740035i	$-0.265191\pi$
$191$	1.29534e8	1.34514	0.672568	+	0.740035i	$0.265191\pi$
$192$	0	0
$193$	−1.61263e8	−1.61467	−0.807334	−	0.590094i	$-0.799091\pi$
$193$	−1.61263e8	−1.61467	−0.807334	+	0.590094i	$0.799091\pi$
$194$	0	0
$195$	1.19714e8	1.15617
$196$	0	0
$197$	−1.94001e8	−1.80789	−0.903945	−	0.427649i	$-0.859342\pi$
$197$	−1.94001e8	−1.80789	−0.903945	+	0.427649i	$0.859342\pi$
$198$	0	0
$199$	−1.08522e8	−0.976188	−0.488094	−	0.872791i	$-0.662308\pi$
$199$	−1.08522e8	−0.976188	−0.488094	+	0.872791i	$0.662308\pi$
$200$	0	0
$201$	−8.69231e7	−0.755003
$202$	0	0
$203$	−1.41488e8	−1.18709
$204$	0	0
$205$	−2.99245e8	−2.42599
$206$	0	0
$207$	−5.56423e7	−0.436023
$208$	0	0
$209$	2.02777e6	0.0153641
$210$	0	0
$211$	−1.29901e8	−0.951969	−0.475985	−	0.879454i	$-0.657908\pi$
$211$	−1.29901e8	−0.951969	−0.475985	+	0.879454i	$0.657908\pi$
$212$	0	0
$213$	8.73960e7	0.619673
$214$	0	0
$215$	4.29371e7	0.294644
$216$	0	0
$217$	1.29684e8	0.861547
$218$	0	0
$219$	2.97907e7	0.191658
$220$	0	0
$221$	−5.13831e7	−0.320219
$222$	0	0
$223$	2.62253e8	1.58363	0.791813	−	0.610764i	$-0.209137\pi$
$223$	2.62253e8	1.58363	0.791813	+	0.610764i	$0.209137\pi$
$224$	0	0
$225$	−1.62792e8	−0.952783
$226$	0	0
$227$	−1.00523e8	−0.570395	−0.285198	−	0.958469i	$-0.592059\pi$
$227$	−1.00523e8	−0.570395	−0.285198	+	0.958469i	$0.592059\pi$
$228$	0	0
$229$	5.80242e7	0.319290	0.159645	−	0.987174i	$-0.448965\pi$
$229$	5.80242e7	0.319290	0.159645	+	0.987174i	$0.448965\pi$
$230$	0	0
$231$	−1.16472e7	−0.0621698
$232$	0	0
$233$	−8.68438e7	−0.449773	−0.224886	−	0.974385i	$-0.572201\pi$
$233$	−8.68438e7	−0.449773	−0.224886	+	0.974385i	$0.572201\pi$
$234$	0	0
$235$	4.20872e8	2.11550
$236$	0	0
$237$	−1.03107e8	−0.503117
$238$	0	0
$239$	3.27755e8	1.55294	0.776472	−	0.630151i	$-0.217007\pi$
$239$	3.27755e8	1.55294	0.776472	+	0.630151i	$0.217007\pi$
$240$	0	0
$241$	−1.06019e8	−0.487890	−0.243945	−	0.969789i	$-0.578442\pi$
$241$	−1.06019e8	−0.487890	−0.243945	+	0.969789i	$0.578442\pi$
$242$	0	0
$243$	−1.51270e8	−0.676289
$244$	0	0
$245$	1.50736e8	0.654843
$246$	0	0
$247$	−4.09683e7	−0.172985
$248$	0	0
$249$	3.38248e8	1.38847
$250$	0	0
$251$	−1.26777e8	−0.506037	−0.253019	−	0.967461i	$-0.581423\pi$
$251$	−1.26777e8	−0.506037	−0.253019	+	0.967461i	$0.581423\pi$
$252$	0	0
$253$	2.10137e7	0.0815793
$254$	0	0
$255$	−1.72421e8	−0.651177
$256$	0	0
$257$	4.69212e8	1.72426	0.862131	−	0.506685i	$-0.169129\pi$
$257$	4.69212e8	1.72426	0.862131	+	0.506685i	$0.169129\pi$
$258$	0	0
$259$	−1.71990e7	−0.0615113
$260$	0	0
$261$	1.05348e8	0.366764
$262$	0	0
$263$	−4.76939e8	−1.61666	−0.808329	−	0.588731i	$-0.799628\pi$
$263$	−4.76939e8	−1.61666	−0.808329	+	0.588731i	$0.799628\pi$
$264$	0	0
$265$	7.82896e8	2.58430
$266$	0	0
$267$	−2.46687e8	−0.793151
$268$	0	0
$269$	2.76028e8	0.864609	0.432305	−	0.901728i	$-0.357700\pi$
$269$	2.76028e8	0.864609	0.432305	+	0.901728i	$0.357700\pi$
$270$	0	0
$271$	1.35400e8	0.413263	0.206632	−	0.978419i	$-0.433750\pi$
$271$	1.35400e8	0.413263	0.206632	+	0.978419i	$0.433750\pi$
$272$	0	0
$273$	2.35316e8	0.699974
$274$	0	0
$275$	6.14793e7	0.178264
$276$	0	0
$277$	−2.17285e8	−0.614258	−0.307129	−	0.951668i	$-0.599368\pi$
$277$	−2.17285e8	−0.614258	−0.307129	+	0.951668i	$0.599368\pi$
$278$	0	0
$279$	−9.65597e7	−0.266184
$280$	0	0
$281$	3.98290e8	1.07085	0.535423	−	0.844584i	$-0.320152\pi$
$281$	3.98290e8	1.07085	0.535423	+	0.844584i	$0.320152\pi$
$282$	0	0
$283$	−5.48396e8	−1.43828	−0.719138	−	0.694868i	$-0.755463\pi$
$283$	−5.48396e8	−1.43828	−0.719138	+	0.694868i	$0.755463\pi$
$284$	0	0
$285$	−1.37473e8	−0.351771
$286$	0	0
$287$	−5.88214e8	−1.46875
$288$	0	0
$289$	−3.36333e8	−0.819647
$290$	0	0
$291$	5.60943e8	1.33442
$292$	0	0
$293$	7.46332e7	0.173339	0.0866693	−	0.996237i	$-0.472378\pi$
$293$	7.46332e7	0.173339	0.0866693	+	0.996237i	$0.472378\pi$
$294$	0	0
$295$	7.07622e8	1.60481
$296$	0	0
$297$	3.29002e7	0.0728704
$298$	0	0
$299$	−4.24553e8	−0.918507
$300$	0	0
$301$	8.43996e7	0.178385
$302$	0	0
$303$	4.05918e8	0.838280
$304$	0	0
$305$	1.47432e9	2.97537
$306$	0	0
$307$	1.33882e8	0.264082	0.132041	−	0.991244i	$-0.457847\pi$
$307$	1.33882e8	0.264082	0.132041	+	0.991244i	$0.457847\pi$
$308$	0	0
$309$	−7.71126e8	−1.48686
$310$	0	0
$311$	−2.10338e8	−0.396512	−0.198256	−	0.980150i	$-0.563528\pi$
$311$	−2.10338e8	−0.396512	−0.198256	+	0.980150i	$0.563528\pi$
$312$	0	0
$313$	−1.04231e9	−1.92128	−0.960639	−	0.277800i	$-0.910395\pi$
$313$	−1.04231e9	−1.92128	−0.960639	+	0.277800i	$0.910395\pi$
$314$	0	0
$315$	−4.40208e8	−0.793544
$316$	0	0
$317$	7.60268e7	0.134048	0.0670238	−	0.997751i	$-0.478650\pi$
$317$	7.60268e7	0.134048	0.0670238	+	0.997751i	$0.478650\pi$
$318$	0	0
$319$	−3.97855e7	−0.0686210
$320$	0	0
$321$	5.30854e8	0.895793
$322$	0	0
$323$	5.90057e7	0.0974283
$324$	0	0
$325$	−1.24211e9	−2.00709
$326$	0	0
$327$	−2.56903e8	−0.406305
$328$	0	0
$329$	8.27290e8	1.28077
$330$	0	0
$331$	−1.17391e9	−1.77925	−0.889625	−	0.456693i	$-0.849034\pi$
$331$	−1.17391e9	−1.77925	−0.889625	+	0.456693i	$0.849034\pi$
$332$	0	0
$333$	1.28060e7	0.0190045
$334$	0	0
$335$	1.24070e9	1.80306
$336$	0	0
$337$	−7.26345e8	−1.03380	−0.516902	−	0.856045i	$-0.672915\pi$
$337$	−7.26345e8	−1.03380	−0.516902	+	0.856045i	$0.672915\pi$
$338$	0	0
$339$	6.32322e8	0.881535
$340$	0	0
$341$	3.64664e7	0.0498026
$342$	0	0
$343$	−5.69546e8	−0.762078
$344$	0	0
$345$	−1.42463e9	−1.86782
$346$	0	0
$347$	−7.76723e8	−0.997960	−0.498980	−	0.866614i	$-0.666292\pi$
$347$	−7.76723e8	−0.997960	−0.498980	+	0.866614i	$0.666292\pi$
$348$	0	0
$349$	5.01723e8	0.631793	0.315897	−	0.948794i	$-0.397695\pi$
$349$	5.01723e8	0.631793	0.315897	+	0.948794i	$0.397695\pi$
$350$	0	0
$351$	−6.64704e8	−0.820453
$352$	0	0
$353$	2.62792e8	0.317981	0.158990	−	0.987280i	$-0.449176\pi$
$353$	2.62792e8	0.317981	0.158990	+	0.987280i	$0.449176\pi$
$354$	0	0
$355$	−1.24745e9	−1.47988
$356$	0	0
$357$	−3.38920e8	−0.394238
$358$	0	0
$359$	9.71367e8	1.10803	0.554017	−	0.832506i	$-0.313094\pi$
$359$	9.71367e8	1.10803	0.554017	+	0.832506i	$0.313094\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	7.26956e8	0.797692
$364$	0	0
$365$	−4.25220e8	−0.457708
$366$	0	0
$367$	5.96031e8	0.629416	0.314708	−	0.949189i	$-0.398093\pi$
$367$	5.96031e8	0.629416	0.314708	+	0.949189i	$0.398093\pi$
$368$	0	0
$369$	4.37969e8	0.453786
$370$	0	0
$371$	1.53891e9	1.56460
$372$	0	0
$373$	2.55628e8	0.255052	0.127526	−	0.991835i	$-0.459296\pi$
$373$	2.55628e8	0.255052	0.127526	+	0.991835i	$0.459296\pi$
$374$	0	0
$375$	−2.60217e9	−2.54816
$376$	0	0
$377$	8.03812e8	0.772609
$378$	0	0
$379$	−8.10096e8	−0.764362	−0.382181	−	0.924088i	$-0.624827\pi$
$379$	−8.10096e8	−0.764362	−0.382181	+	0.924088i	$0.624827\pi$
$380$	0	0
$381$	2.20281e8	0.204052
$382$	0	0
$383$	−2.28109e8	−0.207466	−0.103733	−	0.994605i	$-0.533079\pi$
$383$	−2.28109e8	−0.207466	−0.103733	+	0.994605i	$0.533079\pi$
$384$	0	0
$385$	1.66247e8	0.148471
$386$	0	0
$387$	−6.28417e7	−0.0551137
$388$	0	0
$389$	1.18396e9	1.01980	0.509898	−	0.860235i	$-0.329683\pi$
$389$	1.18396e9	1.01980	0.509898	+	0.860235i	$0.329683\pi$
$390$	0	0
$391$	6.11474e8	0.517320
$392$	0	0
$393$	2.84668e8	0.236572
$394$	0	0
$395$	1.47171e9	1.20152
$396$	0	0
$397$	−9.15418e8	−0.734264	−0.367132	−	0.930169i	$-0.619660\pi$
$397$	−9.15418e8	−0.734264	−0.367132	+	0.930169i	$0.619660\pi$
$398$	0	0
$399$	−2.70225e8	−0.212971
$400$	0	0
$401$	−1.59847e9	−1.23794	−0.618971	−	0.785414i	$-0.712450\pi$
$401$	−1.59847e9	−1.23794	−0.618971	+	0.785414i	$0.712450\pi$
$402$	0	0
$403$	−7.36754e8	−0.560731
$404$	0	0
$405$	−1.31478e9	−0.983466
$406$	0	0
$407$	−4.83625e6	−0.00355572
$408$	0	0
$409$	7.87777e8	0.569340	0.284670	−	0.958626i	$-0.408116\pi$
$409$	7.87777e8	0.569340	0.284670	+	0.958626i	$0.408116\pi$
$410$	0	0
$411$	−1.89554e9	−1.34675
$412$	0	0
$413$	1.39094e9	0.971593
$414$	0	0
$415$	−4.82801e9	−3.31589
$416$	0	0
$417$	3.71194e8	0.250683
$418$	0	0
$419$	9.26948e8	0.615611	0.307806	−	0.951449i	$-0.400405\pi$
$419$	9.26948e8	0.615611	0.307806	+	0.951449i	$0.400405\pi$
$420$	0	0
$421$	−4.95207e8	−0.323444	−0.161722	−	0.986836i	$-0.551705\pi$
$421$	−4.95207e8	−0.323444	−0.161722	+	0.986836i	$0.551705\pi$
$422$	0	0
$423$	−6.15979e8	−0.395708
$424$	0	0
$425$	1.78898e9	1.13043
$426$	0	0
$427$	2.89800e9	1.80136
$428$	0	0
$429$	6.61692e7	0.0404628
$430$	0	0
$431$	2.06186e9	1.24048	0.620240	−	0.784412i	$-0.287035\pi$
$431$	2.06186e9	1.24048	0.620240	+	0.784412i	$0.287035\pi$
$432$	0	0
$433$	1.48205e9	0.877315	0.438658	−	0.898654i	$-0.355454\pi$
$433$	1.48205e9	0.877315	0.438658	+	0.898654i	$0.355454\pi$
$434$	0	0
$435$	2.69727e9	1.57113
$436$	0	0
$437$	4.87535e8	0.279460
$438$	0	0
$439$	2.70017e9	1.52323	0.761615	−	0.648029i	$-0.224407\pi$
$439$	2.70017e9	1.52323	0.761615	+	0.648029i	$0.224407\pi$
$440$	0	0
$441$	−2.20615e8	−0.122490
$442$	0	0
$443$	4.93525e8	0.269710	0.134855	−	0.990865i	$-0.456943\pi$
$443$	4.93525e8	0.269710	0.134855	+	0.990865i	$0.456943\pi$
$444$	0	0
$445$	3.52110e9	1.89417
$446$	0	0
$447$	1.20084e9	0.635931
$448$	0	0
$449$	−1.53120e9	−0.798308	−0.399154	−	0.916884i	$-0.630696\pi$
$449$	−1.53120e9	−0.798308	−0.399154	+	0.916884i	$0.630696\pi$
$450$	0	0
$451$	−1.65402e8	−0.0849027
$452$	0	0
$453$	2.49327e9	1.26016
$454$	0	0
$455$	−3.35880e9	−1.67165
$456$	0	0
$457$	1.64400e9	0.805742	0.402871	−	0.915257i	$-0.368012\pi$
$457$	1.64400e9	0.805742	0.402871	+	0.915257i	$0.368012\pi$
$458$	0	0
$459$	9.57359e8	0.462094
$460$	0	0
$461$	3.58510e9	1.70431	0.852154	−	0.523291i	$-0.175296\pi$
$461$	3.58510e9	1.70431	0.852154	+	0.523291i	$0.175296\pi$
$462$	0	0
$463$	5.30000e8	0.248166	0.124083	−	0.992272i	$-0.460401\pi$
$463$	5.30000e8	0.248166	0.124083	+	0.992272i	$0.460401\pi$
$464$	0	0
$465$	−2.47225e9	−1.14027
$466$	0	0
$467$	−2.38209e9	−1.08230	−0.541151	−	0.840925i	$-0.682011\pi$
$467$	−2.38209e9	−1.08230	−0.541151	+	0.840925i	$0.682011\pi$
$468$	0	0
$469$	2.43880e9	1.09162
$470$	0	0
$471$	−1.75624e9	−0.774480
$472$	0	0
$473$	2.37326e7	0.0103117
$474$	0	0
$475$	1.42637e9	0.610668
$476$	0	0
$477$	−1.14583e9	−0.483399
$478$	0	0
$479$	2.57694e9	1.07135	0.535673	−	0.844425i	$-0.320058\pi$
$479$	2.57694e9	1.07135	0.535673	+	0.844425i	$0.320058\pi$
$480$	0	0
$481$	9.77098e7	0.0400341
$482$	0	0
$483$	−2.80033e9	−1.13082
$484$	0	0
$485$	−8.00666e9	−3.18680
$486$	0	0
$487$	−1.20964e9	−0.474573	−0.237286	−	0.971440i	$-0.576258\pi$
$487$	−1.20964e9	−0.474573	−0.237286	+	0.971440i	$0.576258\pi$
$488$	0	0
$489$	6.26583e8	0.242324
$490$	0	0
$491$	−1.68870e9	−0.643824	−0.321912	−	0.946770i	$-0.604326\pi$
$491$	−1.68870e9	−0.643824	−0.321912	+	0.946770i	$0.604326\pi$
$492$	0	0
$493$	−1.15771e9	−0.435147
$494$	0	0
$495$	−1.23783e8	−0.0458717
$496$	0	0
$497$	−2.45207e9	−0.895952
$498$	0	0
$499$	2.36842e9	0.853310	0.426655	−	0.904415i	$-0.359692\pi$
$499$	2.36842e9	0.853310	0.426655	+	0.904415i	$0.359692\pi$
$500$	0	0
$501$	−1.60785e9	−0.571234
$502$	0	0
$503$	−2.37989e9	−0.833814	−0.416907	−	0.908949i	$-0.636886\pi$
$503$	−2.37989e9	−0.833814	−0.416907	+	0.908949i	$0.636886\pi$
$504$	0	0
$505$	−5.79390e9	−2.00194
$506$	0	0
$507$	1.01448e9	0.345713
$508$	0	0
$509$	−4.99063e9	−1.67743	−0.838713	−	0.544574i	$-0.816692\pi$
$509$	−4.99063e9	−1.67743	−0.838713	+	0.544574i	$0.816692\pi$
$510$	0	0
$511$	−8.35837e8	−0.277108
$512$	0	0
$513$	7.63312e8	0.249627
$514$	0	0
$515$	1.10067e10	3.55086
$516$	0	0
$517$	2.32628e8	0.0740364
$518$	0	0
$519$	−3.68726e8	−0.115776
$520$	0	0
$521$	4.55457e9	1.41096	0.705481	−	0.708729i	$-0.250731\pi$
$521$	4.55457e9	1.41096	0.705481	+	0.708729i	$0.250731\pi$
$522$	0	0
$523$	5.64306e6	0.00172488	0.000862439	−	1.00000i	$-0.499725\pi$
$523$	5.64306e6	0.00172488	0.000862439		1.00000i	$0.499725\pi$
$524$	0	0
$525$	−8.19288e9	−2.47103
$526$	0	0
$527$	1.06113e9	0.315814
$528$	0	0
$529$	1.64748e9	0.483865
$530$	0	0
$531$	−1.03566e9	−0.300183
$532$	0	0
$533$	3.34171e9	0.955926
$534$	0	0
$535$	−7.57719e9	−2.13929
$536$	0	0
$537$	−6.89221e8	−0.192065
$538$	0	0
$539$	8.33164e7	0.0229176
$540$	0	0
$541$	7.40309e8	0.201012	0.100506	−	0.994936i	$-0.467954\pi$
$541$	7.40309e8	0.201012	0.100506	+	0.994936i	$0.467954\pi$
$542$	0	0
$543$	5.22373e9	1.40017
$544$	0	0
$545$	3.66692e9	0.970318
$546$	0	0
$547$	4.43452e9	1.15849	0.579243	−	0.815155i	$-0.303348\pi$
$547$	4.43452e9	1.15849	0.579243	+	0.815155i	$0.303348\pi$
$548$	0	0
$549$	−2.15778e9	−0.556549
$550$	0	0
$551$	−9.23056e8	−0.235070
$552$	0	0
$553$	2.89287e9	0.727430
$554$	0	0
$555$	3.27874e8	0.0814108
$556$	0	0
$557$	6.87397e8	0.168544	0.0842722	−	0.996443i	$-0.473143\pi$
$557$	6.87397e8	0.168544	0.0842722	+	0.996443i	$0.473143\pi$
$558$	0	0
$559$	−4.79484e8	−0.116100
$560$	0	0
$561$	−9.53020e7	−0.0227894
$562$	0	0
$563$	−8.03254e9	−1.89703	−0.948513	−	0.316738i	$-0.897412\pi$
$563$	−8.03254e9	−1.89703	−0.948513	+	0.316738i	$0.897412\pi$
$564$	0	0
$565$	−9.02550e9	−2.10524
$566$	0	0
$567$	−2.58440e9	−0.595414
$568$	0	0
$569$	−2.86266e9	−0.651444	−0.325722	−	0.945466i	$-0.605607\pi$
$569$	−2.86266e9	−0.651444	−0.325722	+	0.945466i	$0.605607\pi$
$570$	0	0
$571$	−5.07196e9	−1.14012	−0.570059	−	0.821604i	$-0.693080\pi$
$571$	−5.07196e9	−1.14012	−0.570059	+	0.821604i	$0.693080\pi$
$572$	0	0
$573$	−4.85394e9	−1.07784
$574$	0	0
$575$	1.47814e10	3.24249
$576$	0	0
$577$	−2.62615e9	−0.569121	−0.284561	−	0.958658i	$-0.591848\pi$
$577$	−2.62615e9	−0.569121	−0.284561	+	0.958658i	$0.591848\pi$
$578$	0	0
$579$	6.04291e9	1.29381
$580$	0	0
$581$	−9.49022e9	−2.00752
$582$	0	0
$583$	4.32730e8	0.0904434
$584$	0	0
$585$	2.50088e9	0.516472
$586$	0	0
$587$	−6.80829e9	−1.38933	−0.694664	−	0.719334i	$-0.744447\pi$
$587$	−6.80829e9	−1.38933	−0.694664	+	0.719334i	$0.744447\pi$
$588$	0	0
$589$	8.46050e8	0.170605
$590$	0	0
$591$	7.26968e9	1.44864
$592$	0	0
$593$	1.35760e9	0.267350	0.133675	−	0.991025i	$-0.457322\pi$
$593$	1.35760e9	0.267350	0.133675	+	0.991025i	$0.457322\pi$
$594$	0	0
$595$	4.83761e9	0.941502
$596$	0	0
$597$	4.06660e9	0.782206
$598$	0	0
$599$	2.74137e9	0.521164	0.260582	−	0.965452i	$-0.416086\pi$
$599$	2.74137e9	0.521164	0.260582	+	0.965452i	$0.416086\pi$
$600$	0	0
$601$	7.08436e8	0.133119	0.0665595	−	0.997782i	$-0.478798\pi$
$601$	7.08436e8	0.133119	0.0665595	+	0.997782i	$0.478798\pi$
$602$	0	0
$603$	−1.81587e9	−0.337267
$604$	0	0
$605$	−1.03763e10	−1.90501
$606$	0	0
$607$	−7.89344e9	−1.43254	−0.716268	−	0.697825i	$-0.754151\pi$
$607$	−7.89344e9	−1.43254	−0.716268	+	0.697825i	$0.754151\pi$
$608$	0	0
$609$	5.30190e9	0.951198
$610$	0	0
$611$	−4.69994e9	−0.833581
$612$	0	0
$613$	−6.19208e9	−1.08574	−0.542869	−	0.839817i	$-0.682662\pi$
$613$	−6.19208e9	−1.08574	−0.542869	+	0.839817i	$0.682662\pi$
$614$	0	0
$615$	1.12134e10	1.94391
$616$	0	0
$617$	−5.15487e9	−0.883527	−0.441763	−	0.897132i	$-0.645647\pi$
$617$	−5.15487e9	−0.883527	−0.441763	+	0.897132i	$0.645647\pi$
$618$	0	0
$619$	−5.41867e9	−0.918280	−0.459140	−	0.888364i	$-0.651842\pi$
$619$	−5.41867e9	−0.918280	−0.459140	+	0.888364i	$0.651842\pi$
$620$	0	0
$621$	7.91018e9	1.32546
$622$	0	0
$623$	6.92128e9	1.14678
$624$	0	0
$625$	2.08957e10	3.42355
$626$	0	0
$627$	−7.59853e7	−0.0123110
$628$	0	0
$629$	−1.40729e8	−0.0225480
$630$	0	0
$631$	2.19937e9	0.348494	0.174247	−	0.984702i	$-0.444251\pi$
$631$	2.19937e9	0.348494	0.174247	+	0.984702i	$0.444251\pi$
$632$	0	0
$633$	4.86769e9	0.762799
$634$	0	0
$635$	−3.14420e9	−0.487307
$636$	0	0
$637$	−1.68330e9	−0.258031
$638$	0	0
$639$	1.82575e9	0.276814
$640$	0	0
$641$	1.11320e10	1.66944	0.834721	−	0.550673i	$-0.185629\pi$
$641$	1.11320e10	1.66944	0.834721	+	0.550673i	$0.185629\pi$
$642$	0	0
$643$	7.18274e8	0.106550	0.0532748	−	0.998580i	$-0.483034\pi$
$643$	7.18274e8	0.106550	0.0532748	+	0.998580i	$0.483034\pi$
$644$	0	0
$645$	−1.60896e9	−0.236094
$646$	0	0
$647$	1.53093e9	0.222224	0.111112	−	0.993808i	$-0.464559\pi$
$647$	1.53093e9	0.222224	0.111112	+	0.993808i	$0.464559\pi$
$648$	0	0
$649$	3.91124e8	0.0561639
$650$	0	0
$651$	−4.85959e9	−0.690345
$652$	0	0
$653$	2.82808e9	0.397463	0.198731	−	0.980054i	$-0.436318\pi$
$653$	2.82808e9	0.397463	0.198731	+	0.980054i	$0.436318\pi$
$654$	0	0
$655$	−4.06323e9	−0.564971
$656$	0	0
$657$	6.22343e8	0.0856152
$658$	0	0
$659$	4.64597e9	0.632378	0.316189	−	0.948696i	$-0.397597\pi$
$659$	4.64597e9	0.632378	0.316189	+	0.948696i	$0.397597\pi$
$660$	0	0
$661$	−1.11820e10	−1.50596	−0.752981	−	0.658043i	$-0.771385\pi$
$661$	−1.11820e10	−1.50596	−0.752981	+	0.658043i	$0.771385\pi$
$662$	0	0
$663$	1.92545e9	0.256587
$664$	0	0
$665$	3.85708e9	0.508607
$666$	0	0
$667$	−9.56559e9	−1.24816
$668$	0	0
$669$	−9.82724e9	−1.26894
$670$	0	0
$671$	8.14897e8	0.104130
$672$	0	0
$673$	1.17454e10	1.48531	0.742654	−	0.669675i	$-0.233567\pi$
$673$	1.17454e10	1.48531	0.742654	+	0.669675i	$0.233567\pi$
$674$	0	0
$675$	2.31427e10	2.89635
$676$	0	0
$677$	−4.66475e9	−0.577788	−0.288894	−	0.957361i	$-0.593287\pi$
$677$	−4.66475e9	−0.577788	−0.288894	+	0.957361i	$0.593287\pi$
$678$	0	0
$679$	−1.57384e10	−1.92937
$680$	0	0
$681$	3.76685e9	0.457049
$682$	0	0
$683$	−4.53766e8	−0.0544953	−0.0272477	−	0.999629i	$-0.508674\pi$
$683$	−4.53766e8	−0.0544953	−0.0272477	+	0.999629i	$0.508674\pi$
$684$	0	0
$685$	2.70561e10	3.21624
$686$	0	0
$687$	−2.17431e9	−0.255842
$688$	0	0
$689$	−8.74272e9	−1.01831
$690$	0	0
$691$	8.62215e9	0.994129	0.497064	−	0.867714i	$-0.334411\pi$
$691$	8.62215e9	0.994129	0.497064	+	0.867714i	$0.334411\pi$
$692$	0	0
$693$	−2.43316e8	−0.0277718
$694$	0	0
$695$	−5.29827e9	−0.598670
$696$	0	0
$697$	−4.81300e9	−0.538395
$698$	0	0
$699$	3.25425e9	0.360396
$700$	0	0
$701$	5.32283e9	0.583619	0.291810	−	0.956476i	$-0.405743\pi$
$701$	5.32283e9	0.583619	0.291810	+	0.956476i	$0.405743\pi$
$702$	0	0
$703$	−1.12205e8	−0.0121806
$704$	0	0
$705$	−1.57711e10	−1.69512
$706$	0	0
$707$	−1.13888e10	−1.21202
$708$	0	0
$709$	2.91261e9	0.306917	0.153458	−	0.988155i	$-0.450959\pi$
$709$	2.91261e9	0.306917	0.153458	+	0.988155i	$0.450959\pi$
$710$	0	0
$711$	−2.15396e9	−0.224747
$712$	0	0
$713$	8.76758e9	0.905871
$714$	0	0
$715$	−9.44472e8	−0.0966313
$716$	0	0
$717$	−1.22818e10	−1.24435
$718$	0	0
$719$	−6.09958e9	−0.611997	−0.305998	−	0.952032i	$-0.598990\pi$
$719$	−6.09958e9	−0.611997	−0.305998	+	0.952032i	$0.598990\pi$
$720$	0	0
$721$	2.16355e10	2.14978
$722$	0	0
$723$	3.97277e9	0.390939
$724$	0	0
$725$	−2.79859e10	−2.72745
$726$	0	0
$727$	1.05056e9	0.101403	0.0507013	−	0.998714i	$-0.483854\pi$
$727$	1.05056e9	0.101403	0.0507013	+	0.998714i	$0.483854\pi$
$728$	0	0
$729$	1.10444e10	1.05584
$730$	0	0
$731$	6.90591e8	0.0653898
$732$	0	0
$733$	5.70042e9	0.534617	0.267309	−	0.963611i	$-0.413866\pi$
$733$	5.70042e9	0.534617	0.267309	+	0.963611i	$0.413866\pi$
$734$	0	0
$735$	−5.64846e9	−0.524716
$736$	0	0
$737$	6.85773e8	0.0631022
$738$	0	0
$739$	2.96646e9	0.270385	0.135193	−	0.990819i	$-0.456835\pi$
$739$	2.96646e9	0.270385	0.135193	+	0.990819i	$0.456835\pi$
$740$	0	0
$741$	1.53518e9	0.138610
$742$	0	0
$743$	2.86275e9	0.256048	0.128024	−	0.991771i	$-0.459137\pi$
$743$	2.86275e9	0.256048	0.128024	+	0.991771i	$0.459137\pi$
$744$	0	0
$745$	−1.71403e10	−1.51870
$746$	0	0
$747$	7.06617e9	0.620243
$748$	0	0
$749$	−1.48942e10	−1.29518
$750$	0	0
$751$	1.32221e10	1.13910	0.569548	−	0.821958i	$-0.307118\pi$
$751$	1.32221e10	1.13910	0.569548	+	0.821958i	$0.307118\pi$
$752$	0	0
$753$	4.75064e9	0.405480
$754$	0	0
$755$	−3.55879e10	−3.00946
$756$	0	0
$757$	−9.63101e9	−0.806931	−0.403465	−	0.914995i	$-0.632194\pi$
$757$	−9.63101e9	−0.806931	−0.403465	+	0.914995i	$0.632194\pi$
$758$	0	0
$759$	−7.87433e8	−0.0653683
$760$	0	0
$761$	−7.17923e8	−0.0590516	−0.0295258	−	0.999564i	$-0.509400\pi$
$761$	−7.17923e8	−0.0590516	−0.0295258	+	0.999564i	$0.509400\pi$
$762$	0	0
$763$	7.20792e9	0.587454
$764$	0	0
$765$	−3.60196e9	−0.290886
$766$	0	0
$767$	−7.90212e9	−0.632354
$768$	0	0
$769$	−1.17288e10	−0.930064	−0.465032	−	0.885294i	$-0.653957\pi$
$769$	−1.17288e10	−0.930064	−0.465032	+	0.885294i	$0.653957\pi$
$770$	0	0
$771$	−1.75825e10	−1.38163
$772$	0	0
$773$	6.52830e8	0.0508361	0.0254180	−	0.999677i	$-0.491908\pi$
$773$	6.52830e8	0.0508361	0.0254180	+	0.999677i	$0.491908\pi$
$774$	0	0
$775$	2.56512e10	1.97948
$776$	0	0
$777$	6.44489e8	0.0492881
$778$	0	0
$779$	−3.83745e9	−0.290845
$780$	0	0
$781$	−6.89504e8	−0.0517915
$782$	0	0
$783$	−1.49764e10	−1.11492
$784$	0	0
$785$	2.50678e10	1.84958
$786$	0	0
$787$	−1.04425e10	−0.763648	−0.381824	−	0.924235i	$-0.624704\pi$
$787$	−1.04425e10	−0.763648	−0.381824	+	0.924235i	$0.624704\pi$
$788$	0	0
$789$	1.78721e10	1.29540
$790$	0	0
$791$	−1.77410e10	−1.27456
$792$	0	0
$793$	−1.64639e10	−1.17240
$794$	0	0
$795$	−2.93370e10	−2.07077
$796$	0	0
$797$	−2.15545e10	−1.50811	−0.754057	−	0.656809i	$-0.771906\pi$
$797$	−2.15545e10	−1.50811	−0.754057	+	0.656809i	$0.771906\pi$
$798$	0	0
$799$	6.76921e9	0.469488
$800$	0	0
$801$	−5.15341e9	−0.354308
$802$	0	0
$803$	−2.35031e8	−0.0160185
$804$	0	0
$805$	3.99707e10	2.70058
$806$	0	0
$807$	−1.03434e10	−0.692799
$808$	0	0
$809$	2.57297e10	1.70850	0.854249	−	0.519864i	$-0.174017\pi$
$809$	2.57297e10	1.70850	0.854249	+	0.519864i	$0.174017\pi$
$810$	0	0
$811$	−2.39614e10	−1.57739	−0.788694	−	0.614786i	$-0.789242\pi$
$811$	−2.39614e10	−1.57739	−0.788694	+	0.614786i	$0.789242\pi$
$812$	0	0
$813$	−5.07377e9	−0.331142
$814$	0	0
$815$	−8.94358e9	−0.578708
$816$	0	0
$817$	5.50615e8	0.0353241
$818$	0	0
$819$	4.91587e9	0.312685
$820$	0	0
$821$	2.05187e10	1.29405	0.647023	−	0.762471i	$-0.276014\pi$
$821$	2.05187e10	1.29405	0.647023	+	0.762471i	$0.276014\pi$
$822$	0	0
$823$	2.42320e10	1.51527	0.757633	−	0.652680i	$-0.226356\pi$
$823$	2.42320e10	1.51527	0.757633	+	0.652680i	$0.226356\pi$
$824$	0	0
$825$	−2.30378e9	−0.142841
$826$	0	0
$827$	1.04858e10	0.644663	0.322332	−	0.946627i	$-0.395533\pi$
$827$	1.04858e10	0.644663	0.322332	+	0.946627i	$0.395533\pi$
$828$	0	0
$829$	9.37341e9	0.571422	0.285711	−	0.958316i	$-0.407770\pi$
$829$	9.37341e9	0.571422	0.285711	+	0.958316i	$0.407770\pi$
$830$	0	0
$831$	8.14220e9	0.492196
$832$	0	0
$833$	2.42441e9	0.145328
$834$	0	0
$835$	2.29498e10	1.36419
$836$	0	0
$837$	1.37270e10	0.809166
$838$	0	0
$839$	−2.02569e10	−1.18415	−0.592073	−	0.805884i	$-0.701690\pi$
$839$	−2.02569e10	−1.18415	−0.592073	+	0.805884i	$0.701690\pi$
$840$	0	0
$841$	8.60791e8	0.0499013
$842$	0	0
$843$	−1.49249e10	−0.858054
$844$	0	0
$845$	−1.44803e10	−0.825615
$846$	0	0
$847$	−2.03962e10	−1.15334
$848$	0	0
$849$	2.05497e10	1.15247
$850$	0	0
$851$	−1.16278e9	−0.0646759
$852$	0	0
$853$	1.69670e10	0.936014	0.468007	−	0.883725i	$-0.344972\pi$
$853$	1.69670e10	0.936014	0.468007	+	0.883725i	$0.344972\pi$
$854$	0	0
$855$	−2.87188e9	−0.157139
$856$	0	0
$857$	−1.35498e10	−0.735358	−0.367679	−	0.929953i	$-0.619847\pi$
$857$	−1.35498e10	−0.735358	−0.367679	+	0.929953i	$0.619847\pi$
$858$	0	0
$859$	2.71673e10	1.46241	0.731206	−	0.682157i	$-0.238958\pi$
$859$	2.71673e10	1.46241	0.731206	+	0.682157i	$0.238958\pi$
$860$	0	0
$861$	2.20418e10	1.17689
$862$	0	0
$863$	1.28684e10	0.681535	0.340768	−	0.940148i	$-0.389313\pi$
$863$	1.28684e10	0.681535	0.340768	+	0.940148i	$0.389313\pi$
$864$	0	0
$865$	5.26305e9	0.276491
$866$	0	0
$867$	1.26032e10	0.656771
$868$	0	0
$869$	8.13455e8	0.0420498
$870$	0	0
$871$	−1.38551e10	−0.710472
$872$	0	0
$873$	1.17184e10	0.596098
$874$	0	0
$875$	7.30090e10	3.68424
$876$	0	0
$877$	−1.97165e9	−0.0987032	−0.0493516	−	0.998781i	$-0.515715\pi$
$877$	−1.97165e9	−0.0987032	−0.0493516	+	0.998781i	$0.515715\pi$
$878$	0	0
$879$	−2.79668e9	−0.138894
$880$	0	0
$881$	−1.68548e10	−0.830439	−0.415219	−	0.909721i	$-0.636295\pi$
$881$	−1.68548e10	−0.830439	−0.415219	+	0.909721i	$0.636295\pi$
$882$	0	0
$883$	−2.12610e10	−1.03925	−0.519625	−	0.854394i	$-0.673928\pi$
$883$	−2.12610e10	−1.03925	−0.519625	+	0.854394i	$0.673928\pi$
$884$	0	0
$885$	−2.65163e10	−1.28591
$886$	0	0
$887$	5.52895e9	0.266017	0.133009	−	0.991115i	$-0.457536\pi$
$887$	5.52895e9	0.266017	0.133009	+	0.991115i	$0.457536\pi$
$888$	0	0
$889$	−6.18042e9	−0.295027
$890$	0	0
$891$	−7.26715e8	−0.0344185
$892$	0	0
$893$	5.39717e9	0.253621
$894$	0	0
$895$	9.83765e9	0.458681
$896$	0	0
$897$	1.59090e10	0.735986
$898$	0	0
$899$	−1.65998e10	−0.761980
$900$	0	0
$901$	1.25919e10	0.573530
$902$	0	0
$903$	−3.16266e9	−0.142937
$904$	0	0
$905$	−7.45613e10	−3.34383
$906$	0	0
$907$	−2.17690e10	−0.968752	−0.484376	−	0.874860i	$-0.660953\pi$
$907$	−2.17690e10	−0.968752	−0.484376	+	0.874860i	$0.660953\pi$
$908$	0	0
$909$	8.47983e9	0.374467
$910$	0	0
$911$	1.28234e10	0.561939	0.280969	−	0.959717i	$-0.409344\pi$
$911$	1.28234e10	0.561939	0.280969	+	0.959717i	$0.409344\pi$
$912$	0	0
$913$	−2.66858e9	−0.116047
$914$	0	0
$915$	−5.52461e10	−2.38412
$916$	0	0
$917$	−7.98691e9	−0.342047
$918$	0	0
$919$	4.13325e10	1.75666	0.878329	−	0.478056i	$-0.158658\pi$
$919$	4.13325e10	1.75666	0.878329	+	0.478056i	$0.158658\pi$
$920$	0	0
$921$	−5.01689e9	−0.211605
$922$	0	0
$923$	1.39305e10	0.583124
$924$	0	0
$925$	−3.40191e9	−0.141328
$926$	0	0
$927$	−1.61092e10	−0.664195
$928$	0	0
$929$	9.79812e9	0.400948	0.200474	−	0.979699i	$-0.435752\pi$
$929$	9.79812e9	0.400948	0.200474	+	0.979699i	$0.435752\pi$
$930$	0	0
$931$	1.93301e9	0.0785074
$932$	0	0
$933$	7.88186e9	0.317719
$934$	0	0
$935$	1.36030e9	0.0544245
$936$	0	0
$937$	−3.98497e10	−1.58248	−0.791238	−	0.611509i	$-0.790563\pi$
$937$	−3.98497e10	−1.58248	−0.791238	+	0.611509i	$0.790563\pi$
$938$	0	0
$939$	3.90577e10	1.53949
$940$	0	0
$941$	1.81033e10	0.708261	0.354131	−	0.935196i	$-0.384777\pi$
$941$	1.81033e10	0.708261	0.354131	+	0.935196i	$0.384777\pi$
$942$	0	0
$943$	−3.97674e10	−1.54432
$944$	0	0
$945$	6.25805e10	2.41228
$946$	0	0
$947$	4.04379e9	0.154726	0.0773631	−	0.997003i	$-0.475350\pi$
$947$	4.04379e9	0.154726	0.0773631	+	0.997003i	$0.475350\pi$
$948$	0	0
$949$	4.74849e9	0.180353
$950$	0	0
$951$	−2.84891e9	−0.107410
$952$	0	0
$953$	4.73322e10	1.77146	0.885731	−	0.464199i	$-0.153658\pi$
$953$	4.73322e10	1.77146	0.885731	+	0.464199i	$0.153658\pi$
$954$	0	0
$955$	6.92831e10	2.57404
$956$	0	0
$957$	1.49086e9	0.0549850
$958$	0	0
$959$	5.31831e10	1.94719
$960$	0	0
$961$	−1.22977e10	−0.446983
$962$	0	0
$963$	1.10898e10	0.400159
$964$	0	0
$965$	−8.62539e10	−3.08982
$966$	0	0
$967$	9.97179e9	0.354634	0.177317	−	0.984154i	$-0.443258\pi$
$967$	9.97179e9	0.354634	0.177317	+	0.984154i	$0.443258\pi$
$968$	0	0
$969$	−2.21109e9	−0.0780679
$970$	0	0
$971$	4.21331e10	1.47692	0.738459	−	0.674298i	$-0.235554\pi$
$971$	4.21331e10	1.47692	0.738459	+	0.674298i	$0.235554\pi$
$972$	0	0
$973$	−1.04146e10	−0.362449
$974$	0	0
$975$	4.65448e10	1.60825
$976$	0	0
$977$	1.45583e10	0.499435	0.249717	−	0.968319i	$-0.419662\pi$
$977$	1.45583e10	0.499435	0.249717	+	0.968319i	$0.419662\pi$
$978$	0	0
$979$	1.94622e9	0.0662905
$980$	0	0
$981$	−5.36683e9	−0.181500
$982$	0	0
$983$	1.56642e10	0.525983	0.262992	−	0.964798i	$-0.415291\pi$
$983$	1.56642e10	0.525983	0.262992	+	0.964798i	$0.415291\pi$
$984$	0	0
$985$	−1.03764e11	−3.45957
$986$	0	0
$987$	−3.10005e10	−1.02626
$988$	0	0
$989$	5.70600e9	0.187562
$990$	0	0
$991$	−5.33740e10	−1.74210	−0.871048	−	0.491198i	$-0.836559\pi$
$991$	−5.33740e10	−1.74210	−0.871048	+	0.491198i	$0.836559\pi$
$992$	0	0
$993$	4.39892e10	1.42569
$994$	0	0
$995$	−5.80449e10	−1.86803
$996$	0	0
$997$	2.54290e10	0.812637	0.406318	−	0.913732i	$-0.366812\pi$
$997$	2.54290e10	0.812637	0.406318	+	0.913732i	$0.366812\pi$
$998$	0	0
$999$	−1.82051e9	−0.0577715

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.a.b.1.3	✓	6
4.3	odd	2		304.8.a.i.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.a.b.1.3	✓	6	1.1	even	1	trivial
304.8.a.i.1.4		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)

\(f(q)\)	\(=\)	\(q-37.4724 q^{3} +534.866 q^{5} +1051.36 q^{7} -782.818 q^{9} +O(q^{10})\)	\(q-37.4724 q^{3} +534.866 q^{5} +1051.36 q^{7} -782.818 q^{9} +295.636 q^{11} -5972.92 q^{13} -20042.7 q^{15} +8602.67 q^{17} +6859.00 q^{19} -39397.1 q^{21} +71079.5 q^{23} +207956. q^{25} +111286. q^{27} -134576. q^{29} +123349. q^{31} -11078.2 q^{33} +562338. q^{35} -16358.8 q^{37} +223820. q^{39} -559477. q^{41} +80276.3 q^{43} -418702. q^{45} +786873. q^{47} +281821. q^{49} -322363. q^{51} +1.46372e6 q^{53} +158126. q^{55} -257023. q^{57} +1.32299e6 q^{59} +2.75642e6 q^{61} -823026. q^{63} -3.19471e6 q^{65} +2.31966e6 q^{67} -2.66352e6 q^{69} -2.33227e6 q^{71} -795003. q^{73} -7.79262e6 q^{75} +310821. q^{77} +2.75154e6 q^{79} -2.45814e6 q^{81} -9.02658e6 q^{83} +4.60127e6 q^{85} +5.04288e6 q^{87} +6.58315e6 q^{89} -6.27971e6 q^{91} -4.62218e6 q^{93} +3.66864e6 q^{95} -1.49695e7 q^{97} -231429. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 40 q^{3} + 279 q^{5} - 1565 q^{7} + 3900 q^{9}+O(q^{10}) \) 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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