LMFDB - Embedded newform 76.8.a.a.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:	$1$
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			1.2
Root			$-79.8251$ 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)

$f(q)$	$=$	$q-36.5155 q^{3} -266.507 q^{5} +1627.87 q^{7} -853.617 q^{9} +O(q^{10})$	$q-36.5155 q^{3} -266.507 q^{5} +1627.87 q^{7} -853.617 q^{9} +3312.50 q^{11} +6655.56 q^{13} +9731.63 q^{15} -30542.3 q^{17} -6859.00 q^{19} -59442.4 q^{21} -29491.6 q^{23} -7099.17 q^{25} +111030. q^{27} -119966. q^{29} +6726.33 q^{31} -120958. q^{33} -433838. q^{35} -297861. q^{37} -243031. q^{39} -111994. q^{41} -451608. q^{43} +227495. q^{45} -1.04264e6 q^{47} +1.82641e6 q^{49} +1.11527e6 q^{51} +24808.4 q^{53} -882803. q^{55} +250460. q^{57} -1.72955e6 q^{59} -1.66100e6 q^{61} -1.38958e6 q^{63} -1.77375e6 q^{65} +2.99362e6 q^{67} +1.07690e6 q^{69} -2.90204e6 q^{71} +4.36673e6 q^{73} +259230. q^{75} +5.39231e6 q^{77} +3.76244e6 q^{79} -2.18745e6 q^{81} -1.62199e6 q^{83} +8.13972e6 q^{85} +4.38064e6 q^{87} +2.25656e6 q^{89} +1.08344e7 q^{91} -245615. q^{93} +1.82797e6 q^{95} -1.60257e7 q^{97} -2.82760e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−36.5155	−0.780824	−0.390412	−	0.920640i	$-0.627667\pi$
$3$	−36.5155	−0.780824	−0.390412	+	0.920640i	$0.627667\pi$
$4$	0	0
$5$	−266.507	−0.953483	−0.476742	−	0.879043i	$-0.658182\pi$
$5$	−266.507	−0.953483	−0.476742	+	0.879043i	$0.658182\pi$
$6$	0	0
$7$	1627.87	1.79381	0.896904	−	0.442225i	$-0.145811\pi$
$7$	1627.87	1.79381	0.896904	+	0.442225i	$0.145811\pi$
$8$	0	0
$9$	−853.617	−0.390314
$10$	0	0
$11$	3312.50	0.750380	0.375190	−	0.926948i	$-0.377578\pi$
$11$	3312.50	0.750380	0.375190	+	0.926948i	$0.377578\pi$
$12$	0	0
$13$	6655.56	0.840201	0.420100	−	0.907478i	$-0.361995\pi$
$13$	6655.56	0.840201	0.420100	+	0.907478i	$0.361995\pi$
$14$	0	0
$15$	9731.63	0.744503
$16$	0	0
$17$	−30542.3	−1.50775	−0.753876	−	0.657016i	$-0.771818\pi$
$17$	−30542.3	−1.50775	−0.753876	+	0.657016i	$0.771818\pi$
$18$	0	0
$19$	−6859.00	−0.229416
$19$	−6859.00	−0.229416
$20$	0	0
$21$	−59442.4	−1.40065
$22$	0	0
$23$	−29491.6	−0.505417	−0.252709	−	0.967542i	$-0.581321\pi$
$23$	−29491.6	−0.505417	−0.252709	+	0.967542i	$0.581321\pi$
$24$	0	0
$25$	−7099.17	−0.0908693
$26$	0	0
$27$	111030.	1.08559
$28$	0	0
$29$	−119966.	−0.913412	−0.456706	−	0.889618i	$-0.650971\pi$
$29$	−119966.	−0.913412	−0.456706	+	0.889618i	$0.650971\pi$
$30$	0	0
$31$	6726.33	0.0405520	0.0202760	−	0.999794i	$-0.493546\pi$
$31$	6726.33	0.0405520	0.0202760	+	0.999794i	$0.493546\pi$
$32$	0	0
$33$	−120958.	−0.585914
$34$	0	0
$35$	−433838.	−1.71037
$36$	0	0
$37$	−297861.	−0.966735	−0.483367	−	0.875418i	$-0.660586\pi$
$37$	−297861.	−0.966735	−0.483367	+	0.875418i	$0.660586\pi$
$38$	0	0
$39$	−243031.	−0.656049
$40$	0	0
$41$	−111994.	−0.253777	−0.126889	−	0.991917i	$-0.540499\pi$
$41$	−111994.	−0.253777	−0.126889	+	0.991917i	$0.540499\pi$
$42$	0	0
$43$	−451608.	−0.866209	−0.433104	−	0.901344i	$-0.642582\pi$
$43$	−451608.	−0.866209	−0.433104	+	0.901344i	$0.642582\pi$
$44$	0	0
$45$	227495.	0.372158
$46$	0	0
$47$	−1.04264e6	−1.46485	−0.732426	−	0.680847i	$-0.761612\pi$
$47$	−1.04264e6	−1.46485	−0.732426	+	0.680847i	$0.761612\pi$
$48$	0	0
$49$	1.82641e6	2.21775
$50$	0	0
$51$	1.11527e6	1.17729
$52$	0	0
$53$	24808.4	0.0228894	0.0114447	−	0.999935i	$-0.496357\pi$
$53$	24808.4	0.0228894	0.0114447	+	0.999935i	$0.496357\pi$
$54$	0	0
$55$	−882803.	−0.715474
$56$	0	0
$57$	250460.	0.179133
$58$	0	0
$59$	−1.72955e6	−1.09635	−0.548177	−	0.836362i	$-0.684678\pi$
$59$	−1.72955e6	−1.09635	−0.548177	+	0.836362i	$0.684678\pi$
$60$	0	0
$61$	−1.66100e6	−0.936945	−0.468472	−	0.883478i	$-0.655195\pi$
$61$	−1.66100e6	−0.936945	−0.468472	+	0.883478i	$0.655195\pi$
$62$	0	0
$63$	−1.38958e6	−0.700149
$64$	0	0
$65$	−1.77375e6	−0.801118
$66$	0	0
$67$	2.99362e6	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$67$	2.99362e6	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$68$	0	0
$69$	1.07690e6	0.394642
$70$	0	0
$71$	−2.90204e6	−0.962274	−0.481137	−	0.876646i	$-0.659776\pi$
$71$	−2.90204e6	−0.962274	−0.481137	+	0.876646i	$0.659776\pi$
$72$	0	0
$73$	4.36673e6	1.31379	0.656895	−	0.753982i	$-0.271870\pi$
$73$	4.36673e6	1.31379	0.656895	+	0.753982i	$0.271870\pi$
$74$	0	0
$75$	259230.	0.0709529
$76$	0	0
$77$	5.39231e6	1.34604
$78$	0	0
$79$	3.76244e6	0.858568	0.429284	−	0.903170i	$-0.358766\pi$
$79$	3.76244e6	0.858568	0.429284	+	0.903170i	$0.358766\pi$
$80$	0	0
$81$	−2.18745e6	−0.457341
$82$	0	0
$83$	−1.62199e6	−0.311369	−0.155685	−	0.987807i	$-0.549758\pi$
$83$	−1.62199e6	−0.311369	−0.155685	+	0.987807i	$0.549758\pi$
$84$	0	0
$85$	8.13972e6	1.43762
$86$	0	0
$87$	4.38064e6	0.713214
$88$	0	0
$89$	2.25656e6	0.339298	0.169649	−	0.985505i	$-0.445737\pi$
$89$	2.25656e6	0.339298	0.169649	+	0.985505i	$0.445737\pi$
$90$	0	0
$91$	1.08344e7	1.50716
$92$	0	0
$93$	−245615.	−0.0316640
$94$	0	0
$95$	1.82797e6	0.218744
$96$	0	0
$97$	−1.60257e7	−1.78285	−0.891426	−	0.453166i	$-0.850294\pi$
$97$	−1.60257e7	−1.78285	−0.891426	+	0.453166i	$0.850294\pi$
$98$	0	0
$99$	−2.82760e6	−0.292884
$100$	0	0
$101$	1.91280e7	1.84733	0.923667	−	0.383195i	$-0.125176\pi$
$101$	1.91280e7	1.84733	0.923667	+	0.383195i	$0.125176\pi$
$102$	0	0
$103$	−1.76135e7	−1.58824	−0.794118	−	0.607764i	$-0.792067\pi$
$103$	−1.76135e7	−1.58824	−0.794118	+	0.607764i	$0.792067\pi$
$104$	0	0
$105$	1.58418e7	1.33549
$106$	0	0
$107$	9.05349e6	0.714451	0.357226	−	0.934018i	$-0.383723\pi$
$107$	9.05349e6	0.714451	0.357226	+	0.934018i	$0.383723\pi$
$108$	0	0
$109$	−1.44855e7	−1.07137	−0.535687	−	0.844417i	$-0.679947\pi$
$109$	−1.44855e7	−1.07137	−0.535687	+	0.844417i	$0.679947\pi$
$110$	0	0
$111$	1.08765e7	0.754849
$112$	0	0
$113$	−1.94862e7	−1.27043	−0.635217	−	0.772333i	$-0.719089\pi$
$113$	−1.94862e7	−1.27043	−0.635217	+	0.772333i	$0.719089\pi$
$114$	0	0
$115$	7.85970e6	0.481907
$116$	0	0
$117$	−5.68130e6	−0.327942
$118$	0	0
$119$	−4.97188e7	−2.70462
$120$	0	0
$121$	−8.51454e6	−0.436931
$122$	0	0
$123$	4.08953e6	0.198155
$124$	0	0
$125$	2.27128e7	1.04013
$126$	0	0
$127$	−3.95790e7	−1.71456	−0.857278	−	0.514853i	$-0.827846\pi$
$127$	−3.95790e7	−1.71456	−0.857278	+	0.514853i	$0.827846\pi$
$128$	0	0
$129$	1.64907e7	0.676356
$130$	0	0
$131$	3.22041e7	1.25159	0.625794	−	0.779988i	$-0.284775\pi$
$131$	3.22041e7	1.25159	0.625794	+	0.779988i	$0.284775\pi$
$132$	0	0
$133$	−1.11655e7	−0.411528
$134$	0	0
$135$	−2.95902e7	−1.03509
$136$	0	0
$137$	−2.71458e7	−0.901946	−0.450973	−	0.892538i	$-0.648923\pi$
$137$	−2.71458e7	−0.901946	−0.450973	+	0.892538i	$0.648923\pi$
$138$	0	0
$139$	−3.12951e7	−0.988381	−0.494191	−	0.869354i	$-0.664536\pi$
$139$	−3.12951e7	−0.988381	−0.494191	+	0.869354i	$0.664536\pi$
$140$	0	0
$141$	3.80727e7	1.14379
$142$	0	0
$143$	2.20465e7	0.630470
$144$	0	0
$145$	3.19719e7	0.870924
$146$	0	0
$147$	−6.66923e7	−1.73167
$148$	0	0
$149$	−2.76956e7	−0.685896	−0.342948	−	0.939354i	$-0.611426\pi$
$149$	−2.76956e7	−0.685896	−0.342948	+	0.939354i	$0.611426\pi$
$150$	0	0
$151$	2.59448e7	0.613240	0.306620	−	0.951832i	$-0.400802\pi$
$151$	2.59448e7	0.613240	0.306620	+	0.951832i	$0.400802\pi$
$152$	0	0
$153$	2.60714e7	0.588497
$154$	0	0
$155$	−1.79261e6	−0.0386656
$156$	0	0
$157$	6.65464e6	0.137238	0.0686192	−	0.997643i	$-0.478141\pi$
$157$	6.65464e6	0.137238	0.0686192	+	0.997643i	$0.478141\pi$
$158$	0	0
$159$	−905893.	−0.0178726
$160$	0	0
$161$	−4.80084e7	−0.906622
$162$	0	0
$163$	9.46484e6	0.171181	0.0855907	−	0.996330i	$-0.472722\pi$
$163$	9.46484e6	0.171181	0.0855907	+	0.996330i	$0.472722\pi$
$164$	0	0
$165$	3.22360e7	0.558659
$166$	0	0
$167$	−8.12016e7	−1.34914	−0.674570	−	0.738211i	$-0.735671\pi$
$167$	−8.12016e7	−1.34914	−0.674570	+	0.738211i	$0.735671\pi$
$168$	0	0
$169$	−1.84520e7	−0.294063
$170$	0	0
$171$	5.85496e6	0.0895442
$172$	0	0
$173$	−5.31944e7	−0.781097	−0.390548	−	0.920582i	$-0.627715\pi$
$173$	−5.31944e7	−0.781097	−0.390548	+	0.920582i	$0.627715\pi$
$174$	0	0
$175$	−1.15565e7	−0.163002
$176$	0	0
$177$	6.31554e7	0.856060
$178$	0	0
$179$	−5.79650e7	−0.755405	−0.377703	−	0.925927i	$-0.623286\pi$
$179$	−5.79650e7	−0.755405	−0.377703	+	0.925927i	$0.623286\pi$
$180$	0	0
$181$	1.25903e8	1.57820	0.789100	−	0.614265i	$-0.210547\pi$
$181$	1.25903e8	1.57820	0.789100	+	0.614265i	$0.210547\pi$
$182$	0	0
$183$	6.06521e7	0.731589
$184$	0	0
$185$	7.93819e7	0.921765
$186$	0	0
$187$	−1.01171e8	−1.13139
$188$	0	0
$189$	1.80742e8	1.94734
$190$	0	0
$191$	1.60824e7	0.167007	0.0835033	−	0.996508i	$-0.473389\pi$
$191$	1.60824e7	0.167007	0.0835033	+	0.996508i	$0.473389\pi$
$192$	0	0
$193$	1.34454e8	1.34625	0.673123	−	0.739530i	$-0.264952\pi$
$193$	1.34454e8	1.34625	0.673123	+	0.739530i	$0.264952\pi$
$194$	0	0
$195$	6.47695e7	0.625532
$196$	0	0
$197$	9.49248e7	0.884602	0.442301	−	0.896867i	$-0.354162\pi$
$197$	9.49248e7	0.884602	0.442301	+	0.896867i	$0.354162\pi$
$198$	0	0
$199$	−7.98652e7	−0.718409	−0.359205	−	0.933259i	$-0.616952\pi$
$199$	−7.98652e7	−0.718409	−0.359205	+	0.933259i	$0.616952\pi$
$200$	0	0
$201$	−1.09313e8	−0.949483
$202$	0	0
$203$	−1.95290e8	−1.63849
$204$	0	0
$205$	2.98472e7	0.241972
$206$	0	0
$207$	2.51745e7	0.197272
$208$	0	0
$209$	−2.27204e7	−0.172149
$210$	0	0
$211$	9.73927e7	0.713736	0.356868	−	0.934155i	$-0.383845\pi$
$211$	9.73927e7	0.713736	0.356868	+	0.934155i	$0.383845\pi$
$212$	0	0
$213$	1.05969e8	0.751366
$214$	0	0
$215$	1.20357e8	0.825916
$216$	0	0
$217$	1.09496e7	0.0727425
$218$	0	0
$219$	−1.59453e8	−1.02584
$220$	0	0
$221$	−2.03276e8	−1.26682
$222$	0	0
$223$	2.21179e8	1.33560	0.667800	−	0.744341i	$-0.267236\pi$
$223$	2.21179e8	1.33560	0.667800	+	0.744341i	$0.267236\pi$
$224$	0	0
$225$	6.05997e6	0.0354676
$226$	0	0
$227$	3.17166e8	1.79968	0.899842	−	0.436217i	$-0.143682\pi$
$227$	3.17166e8	1.79968	0.899842	+	0.436217i	$0.143682\pi$
$228$	0	0
$229$	1.24652e8	0.685920	0.342960	−	0.939350i	$-0.388570\pi$
$229$	1.24652e8	0.685920	0.342960	+	0.939350i	$0.388570\pi$
$230$	0	0
$231$	−1.96903e8	−1.05102
$232$	0	0
$233$	1.09237e8	0.565747	0.282874	−	0.959157i	$-0.408712\pi$
$233$	1.09237e8	0.565747	0.282874	+	0.959157i	$0.408712\pi$
$234$	0	0
$235$	2.77872e8	1.39671
$236$	0	0
$237$	−1.37387e8	−0.670390
$238$	0	0
$239$	5.73871e7	0.271908	0.135954	−	0.990715i	$-0.456590\pi$
$239$	5.73871e7	0.271908	0.135954	+	0.990715i	$0.456590\pi$
$240$	0	0
$241$	−1.03993e8	−0.478570	−0.239285	−	0.970949i	$-0.576913\pi$
$241$	−1.03993e8	−0.478570	−0.239285	+	0.970949i	$0.576913\pi$
$242$	0	0
$243$	−1.62946e8	−0.728488
$244$	0	0
$245$	−4.86751e8	−2.11459
$246$	0	0
$247$	−4.56505e7	−0.192755
$248$	0	0
$249$	5.92279e7	0.243124
$250$	0	0
$251$	2.83460e8	1.13145	0.565724	−	0.824595i	$-0.308597\pi$
$251$	2.83460e8	1.13145	0.565724	+	0.824595i	$0.308597\pi$
$252$	0	0
$253$	−9.76907e7	−0.379255
$254$	0	0
$255$	−2.97226e8	−1.12253
$256$	0	0
$257$	−4.16932e7	−0.153214	−0.0766072	−	0.997061i	$-0.524409\pi$
$257$	−4.16932e7	−0.153214	−0.0766072	+	0.997061i	$0.524409\pi$
$258$	0	0
$259$	−4.84878e8	−1.73414
$260$	0	0
$261$	1.02405e8	0.356518
$262$	0	0
$263$	−1.89687e8	−0.642972	−0.321486	−	0.946914i	$-0.604182\pi$
$263$	−1.89687e8	−0.642972	−0.321486	+	0.946914i	$0.604182\pi$
$264$	0	0
$265$	−6.61162e6	−0.0218246
$266$	0	0
$267$	−8.23994e7	−0.264932
$268$	0	0
$269$	−5.69865e8	−1.78500	−0.892501	−	0.451045i	$-0.851051\pi$
$269$	−5.69865e8	−1.78500	−0.892501	+	0.451045i	$0.851051\pi$
$270$	0	0
$271$	−4.73797e7	−0.144610	−0.0723052	−	0.997383i	$-0.523036\pi$
$271$	−4.73797e7	−0.144610	−0.0723052	+	0.997383i	$0.523036\pi$
$272$	0	0
$273$	−3.95623e8	−1.17683
$274$	0	0
$275$	−2.35160e7	−0.0681865
$276$	0	0
$277$	−5.59400e7	−0.158140	−0.0790702	−	0.996869i	$-0.525195\pi$
$277$	−5.59400e7	−0.158140	−0.0790702	+	0.996869i	$0.525195\pi$
$278$	0	0
$279$	−5.74171e6	−0.0158280
$280$	0	0
$281$	−4.53360e8	−1.21891	−0.609454	−	0.792821i	$-0.708611\pi$
$281$	−4.53360e8	−1.21891	−0.609454	+	0.792821i	$0.708611\pi$
$282$	0	0
$283$	4.64833e8	1.21911	0.609557	−	0.792743i	$-0.291347\pi$
$283$	4.64833e8	1.21911	0.609557	+	0.792743i	$0.291347\pi$
$284$	0	0
$285$	−6.67493e7	−0.170801
$286$	0	0
$287$	−1.82312e8	−0.455227
$288$	0	0
$289$	5.22492e8	1.27332
$290$	0	0
$291$	5.85186e8	1.39209
$292$	0	0
$293$	6.77277e8	1.57300	0.786502	−	0.617588i	$-0.211890\pi$
$293$	6.77277e8	1.57300	0.786502	+	0.617588i	$0.211890\pi$
$294$	0	0
$295$	4.60937e8	1.04536
$296$	0	0
$297$	3.67785e8	0.814605
$298$	0	0
$299$	−1.96283e8	−0.424652
$300$	0	0
$301$	−7.35159e8	−1.55381
$302$	0	0
$303$	−6.98470e8	−1.44244
$304$	0	0
$305$	4.42666e8	0.893361
$306$	0	0
$307$	8.20513e8	1.61846	0.809229	−	0.587494i	$-0.199885\pi$
$307$	8.20513e8	1.61846	0.809229	+	0.587494i	$0.199885\pi$
$308$	0	0
$309$	6.43165e8	1.24013
$310$	0	0
$311$	−7.38919e8	−1.39295	−0.696475	−	0.717581i	$-0.745249\pi$
$311$	−7.38919e8	−1.39295	−0.696475	+	0.717581i	$0.745249\pi$
$312$	0	0
$313$	1.29886e7	0.0239419	0.0119709	−	0.999928i	$-0.496189\pi$
$313$	1.29886e7	0.0239419	0.0119709	+	0.999928i	$0.496189\pi$
$314$	0	0
$315$	3.70331e8	0.667580
$316$	0	0
$317$	−9.28698e7	−0.163745	−0.0818723	−	0.996643i	$-0.526090\pi$
$317$	−9.28698e7	−0.163745	−0.0818723	+	0.996643i	$0.526090\pi$
$318$	0	0
$319$	−3.97388e8	−0.685406
$320$	0	0
$321$	−3.30593e8	−0.557861
$322$	0	0
$323$	2.09489e8	0.345902
$324$	0	0
$325$	−4.72490e7	−0.0763485
$326$	0	0
$327$	5.28946e8	0.836554
$328$	0	0
$329$	−1.69729e9	−2.62766
$330$	0	0
$331$	1.25428e7	0.0190107	0.00950534	−	0.999955i	$-0.496974\pi$
$331$	1.25428e7	0.0190107	0.00950534	+	0.999955i	$0.496974\pi$
$332$	0	0
$333$	2.54259e8	0.377330
$334$	0	0
$335$	−7.97819e8	−1.15944
$336$	0	0
$337$	−2.22454e8	−0.316619	−0.158309	−	0.987390i	$-0.550604\pi$
$337$	−2.22454e8	−0.316619	−0.158309	+	0.987390i	$0.550604\pi$
$338$	0	0
$339$	7.11548e8	0.991986
$340$	0	0
$341$	2.22809e7	0.0304294
$342$	0	0
$343$	1.63254e9	2.18441
$344$	0	0
$345$	−2.87001e8	−0.376285
$346$	0	0
$347$	−6.27729e8	−0.806527	−0.403264	−	0.915084i	$-0.632124\pi$
$347$	−6.27729e8	−0.806527	−0.403264	+	0.915084i	$0.632124\pi$
$348$	0	0
$349$	−4.40544e7	−0.0554754	−0.0277377	−	0.999615i	$-0.508830\pi$
$349$	−4.40544e7	−0.0554754	−0.0277377	+	0.999615i	$0.508830\pi$
$350$	0	0
$351$	7.38965e8	0.912114
$352$	0	0
$353$	8.73399e8	1.05682	0.528410	−	0.848989i	$-0.322788\pi$
$353$	8.73399e8	1.05682	0.528410	+	0.848989i	$0.322788\pi$
$354$	0	0
$355$	7.73412e8	0.917512
$356$	0	0
$357$	1.81551e9	2.11183
$358$	0	0
$359$	3.65022e8	0.416379	0.208189	−	0.978089i	$-0.433243\pi$
$359$	3.65022e8	0.416379	0.208189	+	0.978089i	$0.433243\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	3.10913e8	0.341166
$364$	0	0
$365$	−1.16376e9	−1.25268
$366$	0	0
$367$	1.74121e9	1.83873	0.919367	−	0.393401i	$-0.128701\pi$
$367$	1.74121e9	1.83873	0.919367	+	0.393401i	$0.128701\pi$
$368$	0	0
$369$	9.56002e7	0.0990528
$370$	0	0
$371$	4.03849e7	0.0410592
$372$	0	0
$373$	−1.36952e9	−1.36643	−0.683214	−	0.730218i	$-0.739419\pi$
$373$	−1.36952e9	−1.36643	−0.683214	+	0.730218i	$0.739419\pi$
$374$	0	0
$375$	−8.29370e8	−0.812155
$376$	0	0
$377$	−7.98445e8	−0.767450
$378$	0	0
$379$	−1.63943e9	−1.54688	−0.773439	−	0.633871i	$-0.781465\pi$
$379$	−1.63943e9	−1.54688	−0.773439	+	0.633871i	$0.781465\pi$
$380$	0	0
$381$	1.44525e9	1.33877
$382$	0	0
$383$	6.63452e8	0.603412	0.301706	−	0.953401i	$-0.402444\pi$
$383$	6.63452e8	0.603412	0.301706	+	0.953401i	$0.402444\pi$
$384$	0	0
$385$	−1.43709e9	−1.28342
$386$	0	0
$387$	3.85501e8	0.338094
$388$	0	0
$389$	1.82458e9	1.57159	0.785797	−	0.618485i	$-0.212253\pi$
$389$	1.82458e9	1.57159	0.785797	+	0.618485i	$0.212253\pi$
$390$	0	0
$391$	9.00739e8	0.762045
$392$	0	0
$393$	−1.17595e9	−0.977270
$394$	0	0
$395$	−1.00271e9	−0.818630
$396$	0	0
$397$	1.47702e9	1.18473	0.592367	−	0.805668i	$-0.298194\pi$
$397$	1.47702e9	1.18473	0.592367	+	0.805668i	$0.298194\pi$
$398$	0	0
$399$	4.07716e8	0.321331
$400$	0	0
$401$	1.63247e8	0.126427	0.0632134	−	0.998000i	$-0.479865\pi$
$401$	1.63247e8	0.126427	0.0632134	+	0.998000i	$0.479865\pi$
$402$	0	0
$403$	4.47675e7	0.0340718
$404$	0	0
$405$	5.82969e8	0.436067
$406$	0	0
$407$	−9.86663e8	−0.725418
$408$	0	0
$409$	−1.34540e9	−0.972343	−0.486171	−	0.873863i	$-0.661607\pi$
$409$	−1.34540e9	−0.972343	−0.486171	+	0.873863i	$0.661607\pi$
$410$	0	0
$411$	9.91242e8	0.704261
$412$	0	0
$413$	−2.81548e9	−1.96665
$414$	0	0
$415$	4.32272e8	0.296885
$416$	0	0
$417$	1.14276e9	0.771752
$418$	0	0
$419$	−1.22787e9	−0.815464	−0.407732	−	0.913102i	$-0.633680\pi$
$419$	−1.22787e9	−0.815464	−0.407732	+	0.913102i	$0.633680\pi$
$420$	0	0
$421$	−1.01531e9	−0.663149	−0.331574	−	0.943429i	$-0.607580\pi$
$421$	−1.01531e9	−0.663149	−0.331574	+	0.943429i	$0.607580\pi$
$422$	0	0
$423$	8.90019e8	0.571752
$424$	0	0
$425$	2.16825e8	0.137009
$426$	0	0
$427$	−2.70388e9	−1.68070
$428$	0	0
$429$	−8.05040e8	−0.492286
$430$	0	0
$431$	−4.35620e8	−0.262082	−0.131041	−	0.991377i	$-0.541832\pi$
$431$	−4.35620e8	−0.262082	−0.131041	+	0.991377i	$0.541832\pi$
$432$	0	0
$433$	2.31031e9	1.36761	0.683807	−	0.729663i	$-0.260323\pi$
$433$	2.31031e9	1.36761	0.683807	+	0.729663i	$0.260323\pi$
$434$	0	0
$435$	−1.16747e9	−0.680038
$436$	0	0
$437$	2.02283e8	0.115951
$438$	0	0
$439$	−1.38581e9	−0.781765	−0.390883	−	0.920441i	$-0.627830\pi$
$439$	−1.38581e9	−0.781765	−0.390883	+	0.920441i	$0.627830\pi$
$440$	0	0
$441$	−1.55906e9	−0.865618
$442$	0	0
$443$	9.50566e8	0.519481	0.259740	−	0.965679i	$-0.416363\pi$
$443$	9.50566e8	0.519481	0.259740	+	0.965679i	$0.416363\pi$
$444$	0	0
$445$	−6.01388e8	−0.323515
$446$	0	0
$447$	1.01132e9	0.535564
$448$	0	0
$449$	−2.53281e9	−1.32050	−0.660252	−	0.751044i	$-0.729551\pi$
$449$	−2.53281e9	−1.32050	−0.660252	+	0.751044i	$0.729551\pi$
$450$	0	0
$451$	−3.70981e8	−0.190429
$452$	0	0
$453$	−9.47387e8	−0.478833
$454$	0	0
$455$	−2.88743e9	−1.43705
$456$	0	0
$457$	−5.18819e8	−0.254278	−0.127139	−	0.991885i	$-0.540579\pi$
$457$	−5.18819e8	−0.254278	−0.127139	+	0.991885i	$0.540579\pi$
$458$	0	0
$459$	−3.39110e9	−1.63680
$460$	0	0
$461$	2.74067e9	1.30288	0.651438	−	0.758702i	$-0.274166\pi$
$461$	2.74067e9	1.30288	0.651438	+	0.758702i	$0.274166\pi$
$462$	0	0
$463$	1.96766e9	0.921331	0.460665	−	0.887574i	$-0.347611\pi$
$463$	1.96766e9	0.921331	0.460665	+	0.887574i	$0.347611\pi$
$464$	0	0
$465$	6.54581e7	0.0301911
$466$	0	0
$467$	−3.60124e9	−1.63622	−0.818112	−	0.575059i	$-0.804979\pi$
$467$	−3.60124e9	−1.63622	−0.818112	+	0.575059i	$0.804979\pi$
$468$	0	0
$469$	4.87321e9	2.18127
$470$	0	0
$471$	−2.42997e8	−0.107159
$472$	0	0
$473$	−1.49595e9	−0.649985
$474$	0	0
$475$	4.86932e7	0.0208469
$476$	0	0
$477$	−2.11769e7	−0.00893405
$478$	0	0
$479$	1.75192e9	0.728350	0.364175	−	0.931330i	$-0.381351\pi$
$479$	1.75192e9	0.728350	0.364175	+	0.931330i	$0.381351\pi$
$480$	0	0
$481$	−1.98243e9	−0.812251
$482$	0	0
$483$	1.75305e9	0.707912
$484$	0	0
$485$	4.27095e9	1.69992
$486$	0	0
$487$	3.79184e9	1.48764	0.743821	−	0.668379i	$-0.233011\pi$
$487$	3.79184e9	1.48764	0.743821	+	0.668379i	$0.233011\pi$
$488$	0	0
$489$	−3.45614e8	−0.133663
$490$	0	0
$491$	−1.11484e8	−0.0425039	−0.0212519	−	0.999774i	$-0.506765\pi$
$491$	−1.11484e8	−0.0425039	−0.0212519	+	0.999774i	$0.506765\pi$
$492$	0	0
$493$	3.66405e9	1.37720
$494$	0	0
$495$	7.53575e8	0.279260
$496$	0	0
$497$	−4.72413e9	−1.72613
$498$	0	0
$499$	−1.31431e9	−0.473527	−0.236764	−	0.971567i	$-0.576087\pi$
$499$	−1.31431e9	−0.473527	−0.236764	+	0.971567i	$0.576087\pi$
$500$	0	0
$501$	2.96512e9	1.05344
$502$	0	0
$503$	−4.40143e9	−1.54208	−0.771039	−	0.636788i	$-0.780263\pi$
$503$	−4.40143e9	−1.54208	−0.771039	+	0.636788i	$0.780263\pi$
$504$	0	0
$505$	−5.09775e9	−1.76140
$506$	0	0
$507$	6.73784e8	0.229611
$508$	0	0
$509$	−7.60902e8	−0.255751	−0.127875	−	0.991790i	$-0.540816\pi$
$509$	−7.60902e8	−0.255751	−0.127875	+	0.991790i	$0.540816\pi$
$510$	0	0
$511$	7.10845e9	2.35669
$512$	0	0
$513$	−7.61553e8	−0.249052
$514$	0	0
$515$	4.69411e9	1.51436
$516$	0	0
$517$	−3.45375e9	−1.09919
$518$	0	0
$519$	1.94242e9	0.609899
$520$	0	0
$521$	4.05257e9	1.25545	0.627724	−	0.778436i	$-0.283986\pi$
$521$	4.05257e9	1.25545	0.627724	+	0.778436i	$0.283986\pi$
$522$	0	0
$523$	1.45055e9	0.443379	0.221690	−	0.975117i	$-0.428843\pi$
$523$	1.45055e9	0.443379	0.221690	+	0.975117i	$0.428843\pi$
$524$	0	0
$525$	4.21992e8	0.127276
$526$	0	0
$527$	−2.05437e8	−0.0611424
$528$	0	0
$529$	−2.53507e9	−0.744553
$530$	0	0
$531$	1.47637e9	0.427923
$532$	0	0
$533$	−7.45385e8	−0.213224
$534$	0	0
$535$	−2.41282e9	−0.681218
$536$	0	0
$537$	2.11662e9	0.589838
$538$	0	0
$539$	6.04998e9	1.66415
$540$	0	0
$541$	9.90839e8	0.269037	0.134519	−	0.990911i	$-0.457051\pi$
$541$	9.90839e8	0.269037	0.134519	+	0.990911i	$0.457051\pi$
$542$	0	0
$543$	−4.59742e9	−1.23230
$544$	0	0
$545$	3.86049e9	1.02154
$546$	0	0
$547$	−1.99783e9	−0.521918	−0.260959	−	0.965350i	$-0.584039\pi$
$547$	−1.99783e9	−0.521918	−0.260959	+	0.965350i	$0.584039\pi$
$548$	0	0
$549$	1.41785e9	0.365703
$550$	0	0
$551$	8.22850e8	0.209551
$552$	0	0
$553$	6.12475e9	1.54011
$554$	0	0
$555$	−2.89867e9	−0.719736
$556$	0	0
$557$	4.06763e9	0.997351	0.498676	−	0.866789i	$-0.333820\pi$
$557$	4.06763e9	0.997351	0.498676	+	0.866789i	$0.333820\pi$
$558$	0	0
$559$	−3.00571e9	−0.727789
$560$	0	0
$561$	3.69432e9	0.883414
$562$	0	0
$563$	−6.98740e9	−1.65020	−0.825100	−	0.564987i	$-0.808881\pi$
$563$	−6.98740e9	−1.65020	−0.825100	+	0.564987i	$0.808881\pi$
$564$	0	0
$565$	5.19320e9	1.21134
$566$	0	0
$567$	−3.56087e9	−0.820381
$568$	0	0
$569$	−4.99115e7	−0.0113581	−0.00567907	−	0.999984i	$-0.501808\pi$
$569$	−4.99115e7	−0.0113581	−0.00567907	+	0.999984i	$0.501808\pi$
$570$	0	0
$571$	−3.32850e9	−0.748207	−0.374103	−	0.927387i	$-0.622049\pi$
$571$	−3.32850e9	−0.748207	−0.374103	+	0.927387i	$0.622049\pi$
$572$	0	0
$573$	−5.87256e8	−0.130403
$574$	0	0
$575$	2.09365e8	0.0459269
$576$	0	0
$577$	6.98807e9	1.51440	0.757202	−	0.653181i	$-0.226566\pi$
$577$	6.98807e9	1.51440	0.757202	+	0.653181i	$0.226566\pi$
$578$	0	0
$579$	−4.90967e9	−1.05118
$580$	0	0
$581$	−2.64039e9	−0.558536
$582$	0	0
$583$	8.21779e7	0.0171757
$584$	0	0
$585$	1.51411e9	0.312688
$586$	0	0
$587$	2.65909e8	0.0542624	0.0271312	−	0.999632i	$-0.491363\pi$
$587$	2.65909e8	0.0542624	0.0271312	+	0.999632i	$0.491363\pi$
$588$	0	0
$589$	−4.61359e7	−0.00930326
$590$	0	0
$591$	−3.46623e9	−0.690718
$592$	0	0
$593$	−6.30318e9	−1.24128	−0.620638	−	0.784098i	$-0.713126\pi$
$593$	−6.30318e9	−1.24128	−0.620638	+	0.784098i	$0.713126\pi$
$594$	0	0
$595$	1.32504e10	2.57881
$596$	0	0
$597$	2.91632e9	0.560951
$598$	0	0
$599$	−9.29186e9	−1.76648	−0.883240	−	0.468921i	$-0.844643\pi$
$599$	−9.29186e9	−1.76648	−0.883240	+	0.468921i	$0.844643\pi$
$600$	0	0
$601$	−6.64397e9	−1.24844	−0.624219	−	0.781249i	$-0.714583\pi$
$601$	−6.64397e9	−1.24844	−0.624219	+	0.781249i	$0.714583\pi$
$602$	0	0
$603$	−2.55540e9	−0.474623
$604$	0	0
$605$	2.26918e9	0.416606
$606$	0	0
$607$	−4.32378e9	−0.784700	−0.392350	−	0.919816i	$-0.628338\pi$
$607$	−4.32378e9	−0.784700	−0.392350	+	0.919816i	$0.628338\pi$
$608$	0	0
$609$	7.13110e9	1.27937
$610$	0	0
$611$	−6.93938e9	−1.23077
$612$	0	0
$613$	1.22685e9	0.215119	0.107559	−	0.994199i	$-0.465696\pi$
$613$	1.22685e9	0.215119	0.107559	+	0.994199i	$0.465696\pi$
$614$	0	0
$615$	−1.08989e9	−0.188938
$616$	0	0
$617$	7.41752e9	1.27134	0.635668	−	0.771963i	$-0.280725\pi$
$617$	7.41752e9	1.27134	0.635668	+	0.771963i	$0.280725\pi$
$618$	0	0
$619$	−8.99268e9	−1.52395	−0.761977	−	0.647605i	$-0.775771\pi$
$619$	−8.99268e9	−1.52395	−0.761977	+	0.647605i	$0.775771\pi$
$620$	0	0
$621$	−3.27444e9	−0.548676
$622$	0	0
$623$	3.67338e9	0.608636
$624$	0	0
$625$	−5.49850e9	−0.900873
$626$	0	0
$627$	8.29648e8	0.134418
$628$	0	0
$629$	9.09735e9	1.45760
$630$	0	0
$631$	−4.43901e9	−0.703370	−0.351685	−	0.936118i	$-0.614391\pi$
$631$	−4.43901e9	−0.703370	−0.351685	+	0.936118i	$0.614391\pi$
$632$	0	0
$633$	−3.55634e9	−0.557302
$634$	0	0
$635$	1.05481e10	1.63480
$636$	0	0
$637$	1.21558e10	1.86335
$638$	0	0
$639$	2.47723e9	0.375589
$640$	0	0
$641$	−9.77940e9	−1.46659	−0.733296	−	0.679910i	$-0.762019\pi$
$641$	−9.77940e9	−1.46659	−0.733296	+	0.679910i	$0.762019\pi$
$642$	0	0
$643$	−4.55347e9	−0.675467	−0.337734	−	0.941242i	$-0.609660\pi$
$643$	−4.55347e9	−0.675467	−0.337734	+	0.941242i	$0.609660\pi$
$644$	0	0
$645$	−4.39489e9	−0.644895
$646$	0	0
$647$	1.33205e10	1.93354	0.966772	−	0.255640i	$-0.0822861\pi$
$647$	1.33205e10	1.93354	0.966772	+	0.255640i	$0.0822861\pi$
$648$	0	0
$649$	−5.72913e9	−0.822682
$650$	0	0
$651$	−3.99829e8	−0.0567991
$652$	0	0
$653$	−6.99125e9	−0.982559	−0.491279	−	0.871002i	$-0.663471\pi$
$653$	−6.99125e9	−0.982559	−0.491279	+	0.871002i	$0.663471\pi$
$654$	0	0
$655$	−8.58261e9	−1.19337
$656$	0	0
$657$	−3.72751e9	−0.512791
$658$	0	0
$659$	6.12019e9	0.833040	0.416520	−	0.909127i	$-0.363250\pi$
$659$	6.12019e9	0.833040	0.416520	+	0.909127i	$0.363250\pi$
$660$	0	0
$661$	3.45719e9	0.465606	0.232803	−	0.972524i	$-0.425210\pi$
$661$	3.45719e9	0.465606	0.232803	+	0.972524i	$0.425210\pi$
$662$	0	0
$663$	7.42273e9	0.989160
$664$	0	0
$665$	2.97569e9	0.392385
$666$	0	0
$667$	3.53800e9	0.461655
$668$	0	0
$669$	−8.07646e9	−1.04287
$670$	0	0
$671$	−5.50204e9	−0.703064
$672$	0	0
$673$	−1.52296e10	−1.92591	−0.962953	−	0.269669i	$-0.913086\pi$
$673$	−1.52296e10	−1.92591	−0.962953	+	0.269669i	$0.913086\pi$
$674$	0	0
$675$	−7.88218e8	−0.0986469
$676$	0	0
$677$	−9.91870e9	−1.22855	−0.614277	−	0.789091i	$-0.710552\pi$
$677$	−9.91870e9	−1.22855	−0.614277	+	0.789091i	$0.710552\pi$
$678$	0	0
$679$	−2.60877e10	−3.19809
$680$	0	0
$681$	−1.15815e10	−1.40524
$682$	0	0
$683$	4.36189e9	0.523844	0.261922	−	0.965089i	$-0.415644\pi$
$683$	4.36189e9	0.523844	0.261922	+	0.965089i	$0.415644\pi$
$684$	0	0
$685$	7.23453e9	0.859990
$686$	0	0
$687$	−4.55172e9	−0.535583
$688$	0	0
$689$	1.65114e8	0.0192317
$690$	0	0
$691$	1.11399e8	0.0128443	0.00642213	−	0.999979i	$-0.497956\pi$
$691$	1.11399e8	0.0128443	0.00642213	+	0.999979i	$0.497956\pi$
$692$	0	0
$693$	−4.60296e9	−0.525377
$694$	0	0
$695$	8.34036e9	0.942405
$696$	0	0
$697$	3.42056e9	0.382633
$698$	0	0
$699$	−3.98883e9	−0.441749
$700$	0	0
$701$	4.01144e9	0.439833	0.219916	−	0.975519i	$-0.429422\pi$
$701$	4.01144e9	0.439833	0.219916	+	0.975519i	$0.429422\pi$
$702$	0	0
$703$	2.04303e9	0.221784
$704$	0	0
$705$	−1.01466e10	−1.09059
$706$	0	0
$707$	3.11379e10	3.31376
$708$	0	0
$709$	5.93278e9	0.625168	0.312584	−	0.949890i	$-0.398805\pi$
$709$	5.93278e9	0.625168	0.312584	+	0.949890i	$0.398805\pi$
$710$	0	0
$711$	−3.21168e9	−0.335111
$712$	0	0
$713$	−1.98370e8	−0.0204957
$714$	0	0
$715$	−5.87555e9	−0.601142
$716$	0	0
$717$	−2.09552e9	−0.212312
$718$	0	0
$719$	4.51400e9	0.452908	0.226454	−	0.974022i	$-0.427287\pi$
$719$	4.51400e9	0.452908	0.226454	+	0.974022i	$0.427287\pi$
$720$	0	0
$721$	−2.86724e10	−2.84899
$722$	0	0
$723$	3.79737e9	0.373679
$724$	0	0
$725$	8.51662e8	0.0830012
$726$	0	0
$727$	8.96487e9	0.865314	0.432657	−	0.901559i	$-0.357576\pi$
$727$	8.96487e9	0.865314	0.432657	+	0.901559i	$0.357576\pi$
$728$	0	0
$729$	1.07340e10	1.02616
$730$	0	0
$731$	1.37932e10	1.30603
$732$	0	0
$733$	2.02584e10	1.89994	0.949971	−	0.312338i	$-0.101112\pi$
$733$	2.02584e10	1.89994	0.949971	+	0.312338i	$0.101112\pi$
$734$	0	0
$735$	1.77740e10	1.65112
$736$	0	0
$737$	9.91634e9	0.912463
$738$	0	0
$739$	4.12238e9	0.375744	0.187872	−	0.982193i	$-0.439841\pi$
$739$	4.12238e9	0.375744	0.187872	+	0.982193i	$0.439841\pi$
$740$	0	0
$741$	1.66695e9	0.150508
$742$	0	0
$743$	2.33266e9	0.208636	0.104318	−	0.994544i	$-0.466734\pi$
$743$	2.33266e9	0.208636	0.104318	+	0.994544i	$0.466734\pi$
$744$	0	0
$745$	7.38106e9	0.653991
$746$	0	0
$747$	1.38456e9	0.121532
$748$	0	0
$749$	1.47379e10	1.28159
$750$	0	0
$751$	2.29418e10	1.97646	0.988230	−	0.152975i	$-0.0488855\pi$
$751$	2.29418e10	1.97646	0.988230	+	0.152975i	$0.0488855\pi$
$752$	0	0
$753$	−1.03507e10	−0.883461
$754$	0	0
$755$	−6.91446e9	−0.584715
$756$	0	0
$757$	−8.83607e9	−0.740327	−0.370164	−	0.928967i	$-0.620698\pi$
$757$	−8.83607e9	−0.740327	−0.370164	+	0.928967i	$0.620698\pi$
$758$	0	0
$759$	3.56723e9	0.296131
$760$	0	0
$761$	1.93655e10	1.59287	0.796437	−	0.604721i	$-0.206715\pi$
$761$	1.93655e10	1.59287	0.796437	+	0.604721i	$0.206715\pi$
$762$	0	0
$763$	−2.35805e10	−1.92184
$764$	0	0
$765$	−6.94821e9	−0.561123
$766$	0	0
$767$	−1.15111e10	−0.921158
$768$	0	0
$769$	−3.79471e9	−0.300910	−0.150455	−	0.988617i	$-0.548074\pi$
$769$	−3.79471e9	−0.300910	−0.150455	+	0.988617i	$0.548074\pi$
$770$	0	0
$771$	1.52245e9	0.119633
$772$	0	0
$773$	−2.83902e9	−0.221075	−0.110537	−	0.993872i	$-0.535257\pi$
$773$	−2.83902e9	−0.221075	−0.110537	+	0.993872i	$0.535257\pi$
$774$	0	0
$775$	−4.77513e7	−0.00368493
$776$	0	0
$777$	1.77056e10	1.35405
$778$	0	0
$779$	7.68169e8	0.0582205
$780$	0	0
$781$	−9.61298e9	−0.722070
$782$	0	0
$783$	−1.33198e10	−0.991592
$784$	0	0
$785$	−1.77351e9	−0.130855
$786$	0	0
$787$	1.30629e10	0.955274	0.477637	−	0.878557i	$-0.341493\pi$
$787$	1.30629e10	0.955274	0.477637	+	0.878557i	$0.341493\pi$
$788$	0	0
$789$	6.92651e9	0.502048
$790$	0	0
$791$	−3.17209e10	−2.27892
$792$	0	0
$793$	−1.10549e10	−0.787222
$794$	0	0
$795$	2.41427e8	0.0170412
$796$	0	0
$797$	−1.85425e10	−1.29737	−0.648687	−	0.761055i	$-0.724682\pi$
$797$	−1.85425e10	−1.29737	−0.648687	+	0.761055i	$0.724682\pi$
$798$	0	0
$799$	3.18447e10	2.20863
$800$	0	0
$801$	−1.92624e9	−0.132433
$802$	0	0
$803$	1.44648e10	0.985841
$804$	0	0
$805$	1.27945e10	0.864449
$806$	0	0
$807$	2.08089e10	1.39377
$808$	0	0
$809$	−3.36442e9	−0.223404	−0.111702	−	0.993742i	$-0.535630\pi$
$809$	−3.36442e9	−0.223404	−0.111702	+	0.993742i	$0.535630\pi$
$810$	0	0
$811$	−1.71841e10	−1.13124	−0.565618	−	0.824668i	$-0.691362\pi$
$811$	−1.71841e10	−1.13124	−0.565618	+	0.824668i	$0.691362\pi$
$812$	0	0
$813$	1.73009e9	0.112915
$814$	0	0
$815$	−2.52244e9	−0.163219
$816$	0	0
$817$	3.09758e9	0.198722
$818$	0	0
$819$	−9.24841e9	−0.588266
$820$	0	0
$821$	2.23026e10	1.40655	0.703274	−	0.710919i	$-0.251721\pi$
$821$	2.23026e10	1.40655	0.703274	+	0.710919i	$0.251721\pi$
$822$	0	0
$823$	2.62811e10	1.64340	0.821701	−	0.569918i	$-0.193025\pi$
$823$	2.62811e10	1.64340	0.821701	+	0.569918i	$0.193025\pi$
$824$	0	0
$825$	8.58697e8	0.0532416
$826$	0	0
$827$	−1.55677e10	−0.957093	−0.478546	−	0.878062i	$-0.658836\pi$
$827$	−1.55677e10	−0.957093	−0.478546	+	0.878062i	$0.658836\pi$
$828$	0	0
$829$	−8.16683e7	−0.00497866	−0.00248933	−	0.999997i	$-0.500792\pi$
$829$	−8.16683e7	−0.00497866	−0.00248933	+	0.999997i	$0.500792\pi$
$830$	0	0
$831$	2.04268e9	0.123480
$832$	0	0
$833$	−5.57827e10	−3.34382
$834$	0	0
$835$	2.16408e10	1.28638
$836$	0	0
$837$	7.46822e8	0.0440228
$838$	0	0
$839$	−1.44911e10	−0.847101	−0.423551	−	0.905872i	$-0.639216\pi$
$839$	−1.44911e10	−0.847101	−0.423551	+	0.905872i	$0.639216\pi$
$840$	0	0
$841$	−2.85792e9	−0.165678
$842$	0	0
$843$	1.65547e10	0.951753
$844$	0	0
$845$	4.91758e9	0.280384
$846$	0	0
$847$	−1.38605e10	−0.783770
$848$	0	0
$849$	−1.69736e10	−0.951913
$850$	0	0
$851$	8.78438e9	0.488605
$852$	0	0
$853$	−2.37032e10	−1.30763	−0.653815	−	0.756654i	$-0.726833\pi$
$853$	−2.37032e10	−1.30763	−0.653815	+	0.756654i	$0.726833\pi$
$854$	0	0
$855$	−1.56039e9	−0.0853789
$856$	0	0
$857$	1.75078e10	0.950163	0.475081	−	0.879942i	$-0.342419\pi$
$857$	1.75078e10	0.950163	0.475081	+	0.879942i	$0.342419\pi$
$858$	0	0
$859$	−2.17897e10	−1.17294	−0.586469	−	0.809972i	$-0.699482\pi$
$859$	−2.17897e10	−1.17294	−0.586469	+	0.809972i	$0.699482\pi$
$860$	0	0
$861$	6.65721e9	0.355452
$862$	0	0
$863$	−2.11606e10	−1.12070	−0.560351	−	0.828255i	$-0.689334\pi$
$863$	−2.11606e10	−1.12070	−0.560351	+	0.828255i	$0.689334\pi$
$864$	0	0
$865$	1.41767e10	0.744763
$866$	0	0
$867$	−1.90791e10	−0.994238
$868$	0	0
$869$	1.24631e10	0.644252
$870$	0	0
$871$	1.99242e10	1.02169
$872$	0	0
$873$	1.36798e10	0.695872
$874$	0	0
$875$	3.69735e10	1.86579
$876$	0	0
$877$	−1.68537e10	−0.843718	−0.421859	−	0.906661i	$-0.638622\pi$
$877$	−1.68537e10	−0.843718	−0.421859	+	0.906661i	$0.638622\pi$
$878$	0	0
$879$	−2.47311e10	−1.22824
$880$	0	0
$881$	7.63499e9	0.376178	0.188089	−	0.982152i	$-0.439771\pi$
$881$	7.63499e9	0.376178	0.188089	+	0.982152i	$0.439771\pi$
$882$	0	0
$883$	−3.53640e10	−1.72862	−0.864310	−	0.502960i	$-0.832244\pi$
$883$	−3.53640e10	−1.72862	−0.864310	+	0.502960i	$0.832244\pi$
$884$	0	0
$885$	−1.68313e10	−0.816239
$886$	0	0
$887$	5.18891e9	0.249657	0.124829	−	0.992178i	$-0.460162\pi$
$887$	5.18891e9	0.249657	0.124829	+	0.992178i	$0.460162\pi$
$888$	0	0
$889$	−6.44294e10	−3.07559
$890$	0	0
$891$	−7.24591e9	−0.343179
$892$	0	0
$893$	7.15150e9	0.336060
$894$	0	0
$895$	1.54481e10	0.720266
$896$	0	0
$897$	7.16737e9	0.331579
$898$	0	0
$899$	−8.06934e8	−0.0370407
$900$	0	0
$901$	−7.57707e8	−0.0345115
$902$	0	0
$903$	2.68447e10	1.21325
$904$	0	0
$905$	−3.35541e10	−1.50479
$906$	0	0
$907$	1.58922e10	0.707225	0.353612	−	0.935392i	$-0.384953\pi$
$907$	1.58922e10	0.707225	0.353612	+	0.935392i	$0.384953\pi$
$908$	0	0
$909$	−1.63280e10	−0.721041
$910$	0	0
$911$	−1.07719e10	−0.472041	−0.236020	−	0.971748i	$-0.575843\pi$
$911$	−1.07719e10	−0.472041	−0.236020	+	0.971748i	$0.575843\pi$
$912$	0	0
$913$	−5.37284e9	−0.233645
$914$	0	0
$915$	−1.61642e10	−0.697558
$916$	0	0
$917$	5.24240e10	2.24511
$918$	0	0
$919$	−2.44674e10	−1.03988	−0.519941	−	0.854202i	$-0.674046\pi$
$919$	−2.44674e10	−1.03988	−0.519941	+	0.854202i	$0.674046\pi$
$920$	0	0
$921$	−2.99615e10	−1.26373
$922$	0	0
$923$	−1.93147e10	−0.808503
$924$	0	0
$925$	2.11456e9	0.0878465
$926$	0	0
$927$	1.50352e10	0.619911
$928$	0	0
$929$	3.69154e10	1.51061	0.755306	−	0.655372i	$-0.227488\pi$
$929$	3.69154e10	1.51061	0.755306	+	0.655372i	$0.227488\pi$
$930$	0	0
$931$	−1.25274e10	−0.508786
$932$	0	0
$933$	2.69820e10	1.08765
$934$	0	0
$935$	2.69628e10	1.07876
$936$	0	0
$937$	3.71448e10	1.47506	0.737530	−	0.675314i	$-0.235992\pi$
$937$	3.71448e10	1.47506	0.737530	+	0.675314i	$0.235992\pi$
$938$	0	0
$939$	−4.74286e8	−0.0186944
$940$	0	0
$941$	1.56830e10	0.613573	0.306787	−	0.951778i	$-0.400746\pi$
$941$	1.56830e10	0.613573	0.306787	+	0.951778i	$0.400746\pi$
$942$	0	0
$943$	3.30289e9	0.128263
$944$	0	0
$945$	−4.81689e10	−1.85676
$946$	0	0
$947$	−2.19871e10	−0.841284	−0.420642	−	0.907227i	$-0.638195\pi$
$947$	−2.19871e10	−0.841284	−0.420642	+	0.907227i	$0.638195\pi$
$948$	0	0
$949$	2.90630e10	1.10385
$950$	0	0
$951$	3.39119e9	0.127856
$952$	0	0
$953$	3.88326e10	1.45335	0.726676	−	0.686980i	$-0.241064\pi$
$953$	3.88326e10	1.45335	0.726676	+	0.686980i	$0.241064\pi$
$954$	0	0
$955$	−4.28606e9	−0.159238
$956$	0	0
$957$	1.45108e10	0.535181
$958$	0	0
$959$	−4.41897e10	−1.61792
$960$	0	0
$961$	−2.74674e10	−0.998356
$962$	0	0
$963$	−7.72821e9	−0.278861
$964$	0	0
$965$	−3.58330e10	−1.28362
$966$	0	0
$967$	−2.05650e10	−0.731367	−0.365683	−	0.930739i	$-0.619165\pi$
$967$	−2.05650e10	−0.731367	−0.365683	+	0.930739i	$0.619165\pi$
$968$	0	0
$969$	−7.64962e9	−0.270089
$970$	0	0
$971$	2.17717e10	0.763178	0.381589	−	0.924332i	$-0.375377\pi$
$971$	2.17717e10	0.763178	0.381589	+	0.924332i	$0.375377\pi$
$972$	0	0
$973$	−5.09443e10	−1.77297
$974$	0	0
$975$	1.72532e9	0.0596147
$976$	0	0
$977$	1.55819e10	0.534551	0.267276	−	0.963620i	$-0.413877\pi$
$977$	1.55819e10	0.534551	0.267276	+	0.963620i	$0.413877\pi$
$978$	0	0
$979$	7.47485e9	0.254603
$980$	0	0
$981$	1.23651e10	0.418172
$982$	0	0
$983$	−2.07083e10	−0.695356	−0.347678	−	0.937614i	$-0.613030\pi$
$983$	−2.07083e10	−0.695356	−0.347678	+	0.937614i	$0.613030\pi$
$984$	0	0
$985$	−2.52981e10	−0.843453
$986$	0	0
$987$	6.19773e10	2.05174
$988$	0	0
$989$	1.33186e10	0.437797
$990$	0	0
$991$	−2.48904e10	−0.812407	−0.406204	−	0.913783i	$-0.633148\pi$
$991$	−2.48904e10	−0.812407	−0.406204	+	0.913783i	$0.633148\pi$
$992$	0	0
$993$	−4.58008e8	−0.0148440
$994$	0	0
$995$	2.12846e10	0.684992
$996$	0	0
$997$	−1.81625e10	−0.580421	−0.290211	−	0.956963i	$-0.593725\pi$
$997$	−1.81625e10	−0.580421	−0.290211	+	0.956963i	$0.593725\pi$
$998$	0	0
$999$	−3.30714e10	−1.04948

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.a.a.1.2	✓	5	1.1	even	1	trivial
304.8.a.g.1.4		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)

\(f(q)\)	\(=\)	\(q-36.5155 q^{3} -266.507 q^{5} +1627.87 q^{7} -853.617 q^{9} +O(q^{10})\)	\(q-36.5155 q^{3} -266.507 q^{5} +1627.87 q^{7} -853.617 q^{9} +3312.50 q^{11} +6655.56 q^{13} +9731.63 q^{15} -30542.3 q^{17} -6859.00 q^{19} -59442.4 q^{21} -29491.6 q^{23} -7099.17 q^{25} +111030. q^{27} -119966. q^{29} +6726.33 q^{31} -120958. q^{33} -433838. q^{35} -297861. q^{37} -243031. q^{39} -111994. q^{41} -451608. q^{43} +227495. q^{45} -1.04264e6 q^{47} +1.82641e6 q^{49} +1.11527e6 q^{51} +24808.4 q^{53} -882803. q^{55} +250460. q^{57} -1.72955e6 q^{59} -1.66100e6 q^{61} -1.38958e6 q^{63} -1.77375e6 q^{65} +2.99362e6 q^{67} +1.07690e6 q^{69} -2.90204e6 q^{71} +4.36673e6 q^{73} +259230. q^{75} +5.39231e6 q^{77} +3.76244e6 q^{79} -2.18745e6 q^{81} -1.62199e6 q^{83} +8.13972e6 q^{85} +4.38064e6 q^{87} +2.25656e6 q^{89} +1.08344e7 q^{91} -245615. q^{93} +1.82797e6 q^{95} -1.60257e7 q^{97} -2.82760e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 14 q^{3} - 280 q^{5} + 414 q^{7} + 3779 q^{9}+O(q^{10}) \) 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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