LMFDB - Embedded newform 171.10.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,10,Mod(1,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("171.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	171.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:	$88.0711279840$
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} - 3x^{5} - 2124x^{4} - 384x^{3} + 1071312x^{2} + 1260144x - 135644992 $ x^6 - 3x^5 - 2124x^4 - 384x^3 + 1071312x^2 + 1260144*x - 135644992
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-15.8741$ of defining polynomial
Character	$\chi$	$=$	171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+21.8741 q^{2} -33.5233 q^{4} +698.718 q^{5} +5865.75 q^{7} -11932.8 q^{8} +15283.8 q^{10} +69006.8 q^{11} -99409.7 q^{13} +128308. q^{14} -243856. q^{16} +357590. q^{17} -130321. q^{19} -23423.3 q^{20} +1.50946e6 q^{22} +620216. q^{23} -1.46492e6 q^{25} -2.17450e6 q^{26} -196639. q^{28} -1.49302e6 q^{29} +4.48863e6 q^{31} +775473. q^{32} +7.82197e6 q^{34} +4.09851e6 q^{35} +1.88749e7 q^{37} -2.85066e6 q^{38} -8.33769e6 q^{40} -1.29370e7 q^{41} -1.66989e7 q^{43} -2.31333e6 q^{44} +1.35667e7 q^{46} +3.70039e7 q^{47} -5.94653e6 q^{49} -3.20438e7 q^{50} +3.33254e6 q^{52} +3.57930e7 q^{53} +4.82163e7 q^{55} -6.99951e7 q^{56} -3.26584e7 q^{58} +4.81006e7 q^{59} +5.33864e7 q^{61} +9.81847e7 q^{62} +1.41817e8 q^{64} -6.94594e7 q^{65} +2.23693e8 q^{67} -1.19876e7 q^{68} +8.96513e7 q^{70} +7.22238e7 q^{71} +1.56596e8 q^{73} +4.12871e8 q^{74} +4.36879e6 q^{76} +4.04777e8 q^{77} -2.65933e7 q^{79} -1.70387e8 q^{80} -2.82985e8 q^{82} +7.06750e8 q^{83} +2.49855e8 q^{85} -3.65273e8 q^{86} -8.23447e8 q^{88} -1.88064e8 q^{89} -5.83113e8 q^{91} -2.07917e7 q^{92} +8.09428e8 q^{94} -9.10577e7 q^{95} +1.15689e9 q^{97} -1.30075e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 33 q^{2} + 1365 q^{4} + 3612 q^{5} + 4085 q^{7} + 23511 q^{8} - 93884 q^{10} + 69312 q^{11} - 191747 q^{13} + 644691 q^{14} + 13905 q^{16} + 288195 q^{17} - 781926 q^{19} + 1551444 q^{20} + 2409710 q^{22}+ \cdots + 4245664590 q^{98}+O(q^{100}) $ 6 * q + 33 * q^2 + 1365 * q^4 + 3612 * q^5 + 4085 * q^7 + 23511 * q^8 - 93884 * q^10 + 69312 * q^11 - 191747 * q^13 + 644691 * q^14 + 13905 * q^16 + 288195 * q^17 - 781926 * q^19 + 1551444 * q^20 + 2409710 * q^22 + 50697 * q^23 + 7126888 * q^25 - 4394073 * q^26 + 19295623 * q^28 + 7178667 * q^29 - 4310716 * q^31 + 5388975 * q^32 + 18049 * q^34 + 4166490 * q^35 - 15687040 * q^37 - 4300593 * q^38 + 33173148 * q^40 + 25134306 * q^41 - 75118674 * q^43 + 94242102 * q^44 - 85549087 * q^46 + 83731938 * q^47 + 21010019 * q^49 - 141211941 * q^50 + 147001043 * q^52 + 138019203 * q^53 - 1518258 * q^55 + 182020629 * q^56 + 14051197 * q^58 + 7809915 * q^59 + 191946566 * q^61 - 564818736 * q^62 + 167338593 * q^64 + 146177172 * q^65 - 16109787 * q^67 - 561722511 * q^68 + 692634996 * q^70 + 264469698 * q^71 - 287572857 * q^73 - 376807098 * q^74 - 177888165 * q^76 + 1619915436 * q^77 - 86534002 * q^79 + 1300596708 * q^80 + 1582686496 * q^82 + 359214570 * q^83 - 648092718 * q^85 + 1061832924 * q^86 + 779241042 * q^88 + 2263866306 * q^89 + 1172864023 * q^91 + 373904061 * q^92 - 864022488 * q^94 - 470719452 * q^95 + 2705893460 * q^97 + 4245664590 * q^98

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 $

$2$	21.8741	0.966708	0.483354	−	0.875425i	$-0.339418\pi$
$2$	21.8741	0.966708	0.483354	+	0.875425i	$0.339418\pi$
$3$	0	0
$3$	0	0
$4$	−33.5233	−0.0654752
$5$	698.718	0.499962	0.249981	−	0.968251i	$-0.419576\pi$
$5$	698.718	0.499962	0.249981	+	0.968251i	$0.419576\pi$
$6$	0	0
$7$	5865.75	0.923385	0.461692	−	0.887040i	$-0.347242\pi$
$7$	5865.75	0.923385	0.461692	+	0.887040i	$0.347242\pi$
$8$	−11932.8	−1.03000
$9$	0	0
$10$	15283.8	0.483318
$11$	69006.8	1.42110	0.710550	−	0.703646i	$-0.248446\pi$
$11$	69006.8	1.42110	0.710550	+	0.703646i	$0.248446\pi$
$12$	0	0
$13$	−99409.7	−0.965348	−0.482674	−	0.875800i	$-0.660334\pi$
$13$	−99409.7	−0.965348	−0.482674	+	0.875800i	$0.660334\pi$
$14$	128308.	0.892644
$15$	0	0
$16$	−243856.	−0.930238
$17$	357590.	1.03840	0.519201	−	0.854652i	$-0.326230\pi$
$17$	357590.	1.03840	0.519201	+	0.854652i	$0.326230\pi$
$18$	0	0
$19$	−130321.	−0.229416
$19$	−130321.	−0.229416
$20$	−23423.3	−0.0327351
$21$	0	0
$22$	1.50946e6	1.37379
$23$	620216.	0.462134	0.231067	−	0.972938i	$-0.425778\pi$
$23$	620216.	0.462134	0.231067	+	0.972938i	$0.425778\pi$
$24$	0	0
$25$	−1.46492e6	−0.750038
$26$	−2.17450e6	−0.933210
$27$	0	0
$28$	−196639.	−0.0604588
$29$	−1.49302e6	−0.391989	−0.195994	−	0.980605i	$-0.562793\pi$
$29$	−1.49302e6	−0.391989	−0.195994	+	0.980605i	$0.562793\pi$
$30$	0	0
$31$	4.48863e6	0.872943	0.436471	−	0.899718i	$-0.356228\pi$
$31$	4.48863e6	0.872943	0.436471	+	0.899718i	$0.356228\pi$
$32$	775473.	0.130735
$33$	0	0
$34$	7.82197e6	1.00383
$35$	4.09851e6	0.461657
$36$	0	0
$37$	1.88749e7	1.65568	0.827840	−	0.560964i	$-0.189569\pi$
$37$	1.88749e7	1.65568	0.827840	+	0.560964i	$0.189569\pi$
$38$	−2.85066e6	−0.221778
$39$	0	0
$40$	−8.33769e6	−0.514963
$41$	−1.29370e7	−0.715000	−0.357500	−	0.933913i	$-0.616371\pi$
$41$	−1.29370e7	−0.715000	−0.357500	+	0.933913i	$0.616371\pi$
$42$	0	0
$43$	−1.66989e7	−0.744868	−0.372434	−	0.928059i	$-0.621477\pi$
$43$	−1.66989e7	−0.744868	−0.372434	+	0.928059i	$0.621477\pi$
$44$	−2.31333e6	−0.0930468
$45$	0	0
$46$	1.35667e7	0.446749
$47$	3.70039e7	1.10613	0.553067	−	0.833137i	$-0.313457\pi$
$47$	3.70039e7	1.10613	0.553067	+	0.833137i	$0.313457\pi$
$48$	0	0
$49$	−5.94653e6	−0.147361
$50$	−3.20438e7	−0.725068
$51$	0	0
$52$	3.33254e6	0.0632063
$53$	3.57930e7	0.623098	0.311549	−	0.950230i	$-0.399152\pi$
$53$	3.57930e7	0.623098	0.311549	+	0.950230i	$0.399152\pi$
$54$	0	0
$55$	4.82163e7	0.710496
$56$	−6.99951e7	−0.951090
$57$	0	0
$58$	−3.26584e7	−0.378939
$59$	4.81006e7	0.516793	0.258397	−	0.966039i	$-0.416806\pi$
$59$	4.81006e7	0.516793	0.258397	+	0.966039i	$0.416806\pi$
$60$	0	0
$61$	5.33864e7	0.493681	0.246841	−	0.969056i	$-0.420608\pi$
$61$	5.33864e7	0.493681	0.246841	+	0.969056i	$0.420608\pi$
$62$	9.81847e7	0.843881
$63$	0	0
$64$	1.41817e8	1.05662
$65$	−6.94594e7	−0.482637
$66$	0	0
$67$	2.23693e8	1.35617	0.678086	−	0.734982i	$-0.262810\pi$
$67$	2.23693e8	1.35617	0.678086	+	0.734982i	$0.262810\pi$
$68$	−1.19876e7	−0.0679896
$69$	0	0
$70$	8.96513e7	0.446288
$71$	7.22238e7	0.337301	0.168650	−	0.985676i	$-0.446059\pi$
$71$	7.22238e7	0.337301	0.168650	+	0.985676i	$0.446059\pi$
$72$	0	0
$73$	1.56596e8	0.645398	0.322699	−	0.946502i	$-0.395410\pi$
$73$	1.56596e8	0.645398	0.322699	+	0.946502i	$0.395410\pi$
$74$	4.12871e8	1.60056
$75$	0	0
$76$	4.36879e6	0.0150210
$77$	4.04777e8	1.31222
$78$	0	0
$79$	−2.65933e7	−0.0768159	−0.0384079	−	0.999262i	$-0.512229\pi$
$79$	−2.65933e7	−0.0768159	−0.0384079	+	0.999262i	$0.512229\pi$
$80$	−1.70387e8	−0.465084
$81$	0	0
$82$	−2.82985e8	−0.691196
$83$	7.06750e8	1.63461	0.817306	−	0.576204i	$-0.195467\pi$
$83$	7.06750e8	1.63461	0.817306	+	0.576204i	$0.195467\pi$
$84$	0	0
$85$	2.49855e8	0.519162
$86$	−3.65273e8	−0.720070
$87$	0	0
$88$	−8.23447e8	−1.46374
$89$	−1.88064e8	−0.317724	−0.158862	−	0.987301i	$-0.550782\pi$
$89$	−1.88064e8	−0.317724	−0.158862	+	0.987301i	$0.550782\pi$
$90$	0	0
$91$	−5.83113e8	−0.891387
$92$	−2.07917e7	−0.0302583
$93$	0	0
$94$	8.09428e8	1.06931
$95$	−9.10577e7	−0.114699
$96$	0	0
$97$	1.15689e9	1.32685	0.663423	−	0.748244i	$-0.269103\pi$
$97$	1.15689e9	1.32685	0.663423	+	0.748244i	$0.269103\pi$
$98$	−1.30075e8	−0.142455
$99$	0	0
$100$	4.91089e7	0.0491089
$101$	−2.53441e8	−0.242343	−0.121172	−	0.992632i	$-0.538665\pi$
$101$	−2.53441e8	−0.242343	−0.121172	+	0.992632i	$0.538665\pi$
$102$	0	0
$103$	1.60503e9	1.40513	0.702563	−	0.711622i	$-0.252039\pi$
$103$	1.60503e9	1.40513	0.702563	+	0.711622i	$0.252039\pi$
$104$	1.18624e9	0.994312
$105$	0	0
$106$	7.82939e8	0.602354
$107$	2.15646e9	1.59043	0.795214	−	0.606329i	$-0.207359\pi$
$107$	2.15646e9	1.59043	0.795214	+	0.606329i	$0.207359\pi$
$108$	0	0
$109$	−2.41905e9	−1.64144	−0.820721	−	0.571330i	$-0.806428\pi$
$109$	−2.41905e9	−1.64144	−0.820721	+	0.571330i	$0.806428\pi$
$110$	1.05469e9	0.686843
$111$	0	0
$112$	−1.43040e9	−0.858967
$113$	−1.54624e9	−0.892120	−0.446060	−	0.895003i	$-0.647173\pi$
$113$	−1.54624e9	−0.892120	−0.446060	+	0.895003i	$0.647173\pi$
$114$	0	0
$115$	4.33357e8	0.231050
$116$	5.00508e7	0.0256655
$117$	0	0
$118$	1.05216e9	0.499588
$119$	2.09754e9	0.958845
$120$	0	0
$121$	2.40399e9	1.01953
$122$	1.16778e9	0.477246
$123$	0	0
$124$	−1.50474e8	−0.0571561
$125$	−2.38825e9	−0.874953
$126$	0	0
$127$	−9.47766e8	−0.323284	−0.161642	−	0.986849i	$-0.551679\pi$
$127$	−9.47766e8	−0.323284	−0.161642	+	0.986849i	$0.551679\pi$
$128$	2.70508e9	0.890709
$129$	0	0
$130$	−1.51936e9	−0.466569
$131$	−6.59678e9	−1.95709	−0.978546	−	0.206028i	$-0.933946\pi$
$131$	−6.59678e9	−1.95709	−0.978546	+	0.206028i	$0.933946\pi$
$132$	0	0
$133$	−7.64431e8	−0.211839
$134$	4.89308e9	1.31102
$135$	0	0
$136$	−4.26707e9	−1.06956
$137$	7.31940e9	1.77514	0.887570	−	0.460672i	$-0.152392\pi$
$137$	7.31940e9	1.77514	0.887570	+	0.460672i	$0.152392\pi$
$138$	0	0
$139$	−2.60855e9	−0.592697	−0.296348	−	0.955080i	$-0.595769\pi$
$139$	−2.60855e9	−0.592697	−0.296348	+	0.955080i	$0.595769\pi$
$140$	−1.37396e8	−0.0302271
$141$	0	0
$142$	1.57983e9	0.326072
$143$	−6.85995e9	−1.37186
$144$	0	0
$145$	−1.04320e9	−0.195980
$146$	3.42540e9	0.623911
$147$	0	0
$148$	−6.32748e8	−0.108406
$149$	−6.36482e9	−1.05791	−0.528954	−	0.848650i	$-0.677416\pi$
$149$	−6.36482e9	−1.05791	−0.528954	+	0.848650i	$0.677416\pi$
$150$	0	0
$151$	6.80337e9	1.06495	0.532473	−	0.846447i	$-0.321263\pi$
$151$	6.80337e9	1.06495	0.532473	+	0.846447i	$0.321263\pi$
$152$	1.55510e9	0.236299
$153$	0	0
$154$	8.85414e9	1.26854
$155$	3.13629e9	0.436438
$156$	0	0
$157$	−9.76564e8	−0.128278	−0.0641390	−	0.997941i	$-0.520430\pi$
$157$	−9.76564e8	−0.128278	−0.0641390	+	0.997941i	$0.520430\pi$
$158$	−5.81706e8	−0.0742585
$159$	0	0
$160$	5.41838e8	0.0653626
$161$	3.63804e9	0.426728
$162$	0	0
$163$	1.11013e10	1.23177	0.615884	−	0.787837i	$-0.288799\pi$
$163$	1.11013e10	1.23177	0.615884	+	0.787837i	$0.288799\pi$
$164$	4.33691e8	0.0468148
$165$	0	0
$166$	1.54595e10	1.58019
$167$	1.38400e10	1.37693	0.688465	−	0.725269i	$-0.258285\pi$
$167$	1.38400e10	1.37693	0.688465	+	0.725269i	$0.258285\pi$
$168$	0	0
$169$	−7.22205e8	−0.0681036
$170$	5.46535e9	0.501878
$171$	0	0
$172$	5.59801e8	0.0487704
$173$	1.57166e10	1.33399	0.666993	−	0.745064i	$-0.267581\pi$
$173$	1.57166e10	1.33399	0.666993	+	0.745064i	$0.267581\pi$
$174$	0	0
$175$	−8.59285e9	−0.692574
$176$	−1.68277e10	−1.32196
$177$	0	0
$178$	−4.11372e9	−0.307146
$179$	1.45874e10	1.06203	0.531017	−	0.847361i	$-0.321810\pi$
$179$	1.45874e10	1.06203	0.531017	+	0.847361i	$0.321810\pi$
$180$	0	0
$181$	−2.66724e10	−1.84718	−0.923588	−	0.383388i	$-0.874757\pi$
$181$	−2.66724e10	−1.84718	−0.923588	+	0.383388i	$0.874757\pi$
$182$	−1.27551e10	−0.861712
$183$	0	0
$184$	−7.40094e9	−0.476000
$185$	1.31882e10	0.827778
$186$	0	0
$187$	2.46762e10	1.47567
$188$	−1.24049e9	−0.0724243
$189$	0	0
$190$	−1.99181e9	−0.110881
$191$	−2.05405e10	−1.11676	−0.558382	−	0.829584i	$-0.688578\pi$
$191$	−2.05405e10	−1.11676	−0.558382	+	0.829584i	$0.688578\pi$
$192$	0	0
$193$	−1.71562e10	−0.890049	−0.445024	−	0.895518i	$-0.646805\pi$
$193$	−1.71562e10	−0.890049	−0.445024	+	0.895518i	$0.646805\pi$
$194$	2.53060e10	1.28267
$195$	0	0
$196$	1.99347e8	0.00964846
$197$	−2.45099e9	−0.115943	−0.0579713	−	0.998318i	$-0.518463\pi$
$197$	−2.45099e9	−0.115943	−0.0579713	+	0.998318i	$0.518463\pi$
$198$	0	0
$199$	−9.88469e9	−0.446811	−0.223406	−	0.974726i	$-0.571717\pi$
$199$	−9.88469e9	−0.446811	−0.223406	+	0.974726i	$0.571717\pi$
$200$	1.74806e10	0.772542
$201$	0	0
$202$	−5.54380e9	−0.234275
$203$	−8.75767e9	−0.361957
$204$	0	0
$205$	−9.03932e9	−0.357473
$206$	3.51086e10	1.35835
$207$	0	0
$208$	2.42417e10	0.898003
$209$	−8.99304e9	−0.326023
$210$	0	0
$211$	3.09600e10	1.07530	0.537650	−	0.843168i	$-0.319312\pi$
$211$	3.09600e10	1.07530	0.537650	+	0.843168i	$0.319312\pi$
$212$	−1.19990e9	−0.0407974
$213$	0	0
$214$	4.71706e10	1.53748
$215$	−1.16678e10	−0.372406
$216$	0	0
$217$	2.63292e10	0.806062
$218$	−5.29145e10	−1.58679
$219$	0	0
$220$	−1.61637e9	−0.0465199
$221$	−3.55480e10	−1.00242
$222$	0	0
$223$	1.02056e10	0.276353	0.138177	−	0.990408i	$-0.455876\pi$
$223$	1.02056e10	0.276353	0.138177	+	0.990408i	$0.455876\pi$
$224$	4.54874e9	0.120719
$225$	0	0
$226$	−3.38226e10	−0.862420
$227$	−7.93357e8	−0.0198313	−0.00991567	−	0.999951i	$-0.503156\pi$
$227$	−7.93357e8	−0.0198313	−0.00991567	+	0.999951i	$0.503156\pi$
$228$	0	0
$229$	−6.64052e10	−1.59567	−0.797833	−	0.602878i	$-0.794021\pi$
$229$	−6.64052e10	−1.59567	−0.797833	+	0.602878i	$0.794021\pi$
$230$	9.47929e9	0.223357
$231$	0	0
$232$	1.78159e10	0.403750
$233$	−7.11080e10	−1.58058	−0.790290	−	0.612732i	$-0.790070\pi$
$233$	−7.11080e10	−1.58058	−0.790290	+	0.612732i	$0.790070\pi$
$234$	0	0
$235$	2.58553e10	0.553025
$236$	−1.61249e9	−0.0338371
$237$	0	0
$238$	4.58818e10	0.926923
$239$	3.97648e10	0.788331	0.394166	−	0.919039i	$-0.371034\pi$
$239$	3.97648e10	0.788331	0.394166	+	0.919039i	$0.371034\pi$
$240$	0	0
$241$	−7.89042e10	−1.50669	−0.753344	−	0.657626i	$-0.771561\pi$
$241$	−7.89042e10	−1.50669	−0.753344	+	0.657626i	$0.771561\pi$
$242$	5.25852e10	0.985585
$243$	0	0
$244$	−1.78969e9	−0.0323239
$245$	−4.15495e9	−0.0736747
$246$	0	0
$247$	1.29552e10	0.221466
$248$	−5.35620e10	−0.899134
$249$	0	0
$250$	−5.22408e10	−0.845824
$251$	5.62203e10	0.894049	0.447025	−	0.894522i	$-0.352484\pi$
$251$	5.62203e10	0.894049	0.447025	+	0.894522i	$0.352484\pi$
$252$	0	0
$253$	4.27991e10	0.656739
$254$	−2.07315e10	−0.312522
$255$	0	0
$256$	−1.34391e10	−0.195565
$257$	−7.57735e10	−1.08347	−0.541737	−	0.840548i	$-0.682233\pi$
$257$	−7.57735e10	−1.08347	−0.541737	+	0.840548i	$0.682233\pi$
$258$	0	0
$259$	1.10715e11	1.52883
$260$	2.32851e9	0.0316008
$261$	0	0
$262$	−1.44299e11	−1.89194
$263$	−6.82969e10	−0.880237	−0.440119	−	0.897940i	$-0.645064\pi$
$263$	−6.82969e10	−0.880237	−0.440119	+	0.897940i	$0.645064\pi$
$264$	0	0
$265$	2.50092e10	0.311525
$266$	−1.67212e10	−0.204786
$267$	0	0
$268$	−7.49891e9	−0.0887956
$269$	5.77957e10	0.672993	0.336496	−	0.941685i	$-0.390758\pi$
$269$	5.77957e10	0.672993	0.336496	+	0.941685i	$0.390758\pi$
$270$	0	0
$271$	−4.08135e10	−0.459665	−0.229833	−	0.973230i	$-0.573818\pi$
$271$	−4.08135e10	−0.459665	−0.229833	+	0.973230i	$0.573818\pi$
$272$	−8.72006e10	−0.965961
$273$	0	0
$274$	1.60105e11	1.71604
$275$	−1.01089e11	−1.06588
$276$	0	0
$277$	−1.75586e11	−1.79197	−0.895985	−	0.444084i	$-0.853529\pi$
$277$	−1.75586e11	−1.79197	−0.895985	+	0.444084i	$0.853529\pi$
$278$	−5.70597e10	−0.572965
$279$	0	0
$280$	−4.89069e10	−0.475509
$281$	7.04649e10	0.674209	0.337105	−	0.941467i	$-0.390552\pi$
$281$	7.04649e10	0.674209	0.337105	+	0.941467i	$0.390552\pi$
$282$	0	0
$283$	−2.04828e11	−1.89824	−0.949119	−	0.314919i	$-0.898023\pi$
$283$	−2.04828e11	−1.89824	−0.949119	+	0.314919i	$0.898023\pi$
$284$	−2.42118e9	−0.0220848
$285$	0	0
$286$	−1.50055e11	−1.32618
$287$	−7.58852e10	−0.660220
$288$	0	0
$289$	9.28297e9	0.0782793
$290$	−2.28190e10	−0.189455
$291$	0	0
$292$	−5.24961e9	−0.0422575
$293$	−2.01249e11	−1.59525	−0.797626	−	0.603152i	$-0.793911\pi$
$293$	−2.01249e11	−1.59525	−0.797626	+	0.603152i	$0.793911\pi$
$294$	0	0
$295$	3.36088e10	0.258377
$296$	−2.25231e11	−1.70536
$297$	0	0
$298$	−1.39225e11	−1.02269
$299$	−6.16555e10	−0.446120
$300$	0	0
$301$	−9.79515e10	−0.687800
$302$	1.48818e11	1.02949
$303$	0	0
$304$	3.17796e10	0.213411
$305$	3.73021e10	0.246822
$306$	0	0
$307$	7.21281e10	0.463428	0.231714	−	0.972784i	$-0.425567\pi$
$307$	7.21281e10	0.463428	0.231714	+	0.972784i	$0.425567\pi$
$308$	−1.35695e10	−0.0859180
$309$	0	0
$310$	6.86035e10	0.421909
$311$	8.09818e10	0.490869	0.245434	−	0.969413i	$-0.421069\pi$
$311$	8.09818e10	0.490869	0.245434	+	0.969413i	$0.421069\pi$
$312$	0	0
$313$	1.25838e10	0.0741073	0.0370536	−	0.999313i	$-0.488203\pi$
$313$	1.25838e10	0.0741073	0.0370536	+	0.999313i	$0.488203\pi$
$314$	−2.13615e10	−0.124007
$315$	0	0
$316$	8.91496e8	0.00502953
$317$	1.28488e11	0.714654	0.357327	−	0.933979i	$-0.383688\pi$
$317$	1.28488e11	0.714654	0.357327	+	0.933979i	$0.383688\pi$
$318$	0	0
$319$	−1.03028e11	−0.557056
$320$	9.90903e10	0.528270
$321$	0	0
$322$	7.95788e10	0.412521
$323$	−4.66015e10	−0.238226
$324$	0	0
$325$	1.45627e11	0.724047
$326$	2.42831e11	1.19076
$327$	0	0
$328$	1.54375e11	0.736453
$329$	2.17056e11	1.02139
$330$	0	0
$331$	2.59970e10	0.119041	0.0595206	−	0.998227i	$-0.481043\pi$
$331$	2.59970e10	0.119041	0.0595206	+	0.998227i	$0.481043\pi$
$332$	−2.36926e10	−0.107026
$333$	0	0
$334$	3.02738e11	1.33109
$335$	1.56298e11	0.678035
$336$	0	0
$337$	−3.50929e11	−1.48212	−0.741061	−	0.671437i	$-0.765677\pi$
$337$	−3.50929e11	−1.48212	−0.741061	+	0.671437i	$0.765677\pi$
$338$	−1.57976e10	−0.0658364
$339$	0	0
$340$	−8.37596e9	−0.0339922
$341$	3.09746e11	1.24054
$342$	0	0
$343$	−2.71585e11	−1.05946
$344$	1.99265e11	0.767217
$345$	0	0
$346$	3.43787e11	1.28957
$347$	−4.08443e11	−1.51234	−0.756169	−	0.654376i	$-0.772931\pi$
$347$	−4.08443e11	−1.51234	−0.756169	+	0.654376i	$0.772931\pi$
$348$	0	0
$349$	−6.73337e9	−0.0242951	−0.0121475	−	0.999926i	$-0.503867\pi$
$349$	−6.73337e9	−0.0242951	−0.0121475	+	0.999926i	$0.503867\pi$
$350$	−1.87961e11	−0.669517
$351$	0	0
$352$	5.35129e10	0.185788
$353$	1.85813e11	0.636929	0.318464	−	0.947935i	$-0.396833\pi$
$353$	1.85813e11	0.636929	0.318464	+	0.947935i	$0.396833\pi$
$354$	0	0
$355$	5.04641e10	0.168638
$356$	6.30451e9	0.0208030
$357$	0	0
$358$	3.19086e11	1.02668
$359$	−4.19699e11	−1.33356	−0.666780	−	0.745254i	$-0.732328\pi$
$359$	−4.19699e11	−1.33356	−0.666780	+	0.745254i	$0.732328\pi$
$360$	0	0
$361$	1.69836e10	0.0526316
$362$	−5.83434e11	−1.78568
$363$	0	0
$364$	1.95479e10	0.0583637
$365$	1.09416e11	0.322674
$366$	0	0
$367$	−2.93032e11	−0.843174	−0.421587	−	0.906788i	$-0.638527\pi$
$367$	−2.93032e11	−0.843174	−0.421587	+	0.906788i	$0.638527\pi$
$368$	−1.51244e11	−0.429895
$369$	0	0
$370$	2.88481e11	0.800219
$371$	2.09953e11	0.575359
$372$	0	0
$373$	5.79223e11	1.54937	0.774687	−	0.632345i	$-0.217907\pi$
$373$	5.79223e11	1.54937	0.774687	+	0.632345i	$0.217907\pi$
$374$	5.39769e11	1.42655
$375$	0	0
$376$	−4.41562e11	−1.13932
$377$	1.48420e11	0.378406
$378$	0	0
$379$	2.86017e11	0.712058	0.356029	−	0.934475i	$-0.384130\pi$
$379$	2.86017e11	0.712058	0.356029	+	0.934475i	$0.384130\pi$
$380$	3.05255e9	0.00750995
$381$	0	0
$382$	−4.49306e11	−1.07958
$383$	8.00783e11	1.90160	0.950802	−	0.309798i	$-0.100262\pi$
$383$	8.00783e11	1.90160	0.950802	+	0.309798i	$0.100262\pi$
$384$	0	0
$385$	2.82825e11	0.656062
$386$	−3.75277e11	−0.860418
$387$	0	0
$388$	−3.87829e10	−0.0868755
$389$	5.45637e11	1.20818	0.604089	−	0.796917i	$-0.293537\pi$
$389$	5.45637e11	1.20818	0.604089	+	0.796917i	$0.293537\pi$
$390$	0	0
$391$	2.21783e11	0.479881
$392$	7.09590e10	0.151782
$393$	0	0
$394$	−5.36132e10	−0.112083
$395$	−1.85813e10	−0.0384050
$396$	0	0
$397$	4.38015e11	0.884977	0.442488	−	0.896774i	$-0.354096\pi$
$397$	4.38015e11	0.884977	0.442488	+	0.896774i	$0.354096\pi$
$398$	−2.16219e11	−0.431936
$399$	0	0
$400$	3.57229e11	0.697714
$401$	−4.41087e11	−0.851872	−0.425936	−	0.904753i	$-0.640055\pi$
$401$	−4.41087e11	−0.851872	−0.425936	+	0.904753i	$0.640055\pi$
$402$	0	0
$403$	−4.46213e11	−0.842694
$404$	8.49618e9	0.0158675
$405$	0	0
$406$	−1.91566e11	−0.349906
$407$	1.30250e12	2.35289
$408$	0	0
$409$	6.33909e11	1.12014	0.560069	−	0.828446i	$-0.310774\pi$
$409$	6.33909e11	1.12014	0.560069	+	0.828446i	$0.310774\pi$
$410$	−1.97727e11	−0.345572
$411$	0	0
$412$	−5.38058e10	−0.0920008
$413$	2.82147e11	0.477199
$414$	0	0
$415$	4.93819e11	0.817244
$416$	−7.70896e10	−0.126205
$417$	0	0
$418$	−1.96715e11	−0.315169
$419$	4.50874e10	0.0714648	0.0357324	−	0.999361i	$-0.488624\pi$
$419$	4.50874e10	0.0714648	0.0357324	+	0.999361i	$0.488624\pi$
$420$	0	0
$421$	2.38622e11	0.370205	0.185102	−	0.982719i	$-0.440738\pi$
$421$	2.38622e11	0.370205	0.185102	+	0.982719i	$0.440738\pi$
$422$	6.77222e11	1.03950
$423$	0	0
$424$	−4.27112e11	−0.641793
$425$	−5.23840e11	−0.778841
$426$	0	0
$427$	3.13152e11	0.455858
$428$	−7.22915e10	−0.104133
$429$	0	0
$430$	−2.55223e11	−0.360008
$431$	4.50145e10	0.0628355	0.0314177	−	0.999506i	$-0.489998\pi$
$431$	4.50145e10	0.0628355	0.0314177	+	0.999506i	$0.489998\pi$
$432$	0	0
$433$	7.10784e11	0.971722	0.485861	−	0.874036i	$-0.338506\pi$
$433$	7.10784e11	0.971722	0.485861	+	0.874036i	$0.338506\pi$
$434$	5.75927e11	0.779227
$435$	0	0
$436$	8.10945e10	0.107474
$437$	−8.08272e10	−0.106021
$438$	0	0
$439$	−3.18777e10	−0.0409635	−0.0204817	−	0.999790i	$-0.506520\pi$
$439$	−3.18777e10	−0.0409635	−0.0204817	+	0.999790i	$0.506520\pi$
$440$	−5.75357e11	−0.731814
$441$	0	0
$442$	−7.77580e11	−0.969047
$443$	7.05739e11	0.870617	0.435308	−	0.900281i	$-0.356639\pi$
$443$	7.05739e11	0.870617	0.435308	+	0.900281i	$0.356639\pi$
$444$	0	0
$445$	−1.31403e11	−0.158850
$446$	2.23237e11	0.267153
$447$	0	0
$448$	8.31865e11	0.975667
$449$	9.97531e10	0.115829	0.0579146	−	0.998322i	$-0.481555\pi$
$449$	9.97531e10	0.115829	0.0579146	+	0.998322i	$0.481555\pi$
$450$	0	0
$451$	−8.92741e11	−1.01609
$452$	5.18350e10	0.0584117
$453$	0	0
$454$	−1.73540e10	−0.0191711
$455$	−4.07432e11	−0.445660
$456$	0	0
$457$	4.00903e11	0.429949	0.214974	−	0.976620i	$-0.431033\pi$
$457$	4.00903e11	0.429949	0.214974	+	0.976620i	$0.431033\pi$
$458$	−1.45255e12	−1.54254
$459$	0	0
$460$	−1.45275e10	−0.0151280
$461$	9.84581e10	0.101531	0.0507653	−	0.998711i	$-0.483834\pi$
$461$	9.84581e10	0.101531	0.0507653	+	0.998711i	$0.483834\pi$
$462$	0	0
$463$	−4.87218e11	−0.492730	−0.246365	−	0.969177i	$-0.579236\pi$
$463$	−4.87218e11	−0.492730	−0.246365	+	0.969177i	$0.579236\pi$
$464$	3.64082e11	0.364643
$465$	0	0
$466$	−1.55542e12	−1.52796
$467$	1.77701e12	1.72888	0.864440	−	0.502737i	$-0.167673\pi$
$467$	1.77701e12	1.72888	0.864440	+	0.502737i	$0.167673\pi$
$468$	0	0
$469$	1.31213e12	1.25227
$470$	5.65562e11	0.534614
$471$	0	0
$472$	−5.73977e11	−0.532299
$473$	−1.15234e12	−1.05853
$474$	0	0
$475$	1.90910e11	0.172070
$476$	−7.03163e10	−0.0627805
$477$	0	0
$478$	8.69821e11	0.762086
$479$	−8.74626e11	−0.759124	−0.379562	−	0.925166i	$-0.623925\pi$
$479$	−8.74626e11	−0.759124	−0.379562	+	0.925166i	$0.623925\pi$
$480$	0	0
$481$	−1.87635e12	−1.59831
$482$	−1.72596e12	−1.45653
$483$	0	0
$484$	−8.05897e10	−0.0667537
$485$	8.08343e11	0.663373
$486$	0	0
$487$	−1.12352e12	−0.905112	−0.452556	−	0.891736i	$-0.649488\pi$
$487$	−1.12352e12	−0.905112	−0.452556	+	0.891736i	$0.649488\pi$
$488$	−6.37052e11	−0.508494
$489$	0	0
$490$	−9.08858e10	−0.0712220
$491$	−6.54532e11	−0.508234	−0.254117	−	0.967173i	$-0.581785\pi$
$491$	−6.54532e11	−0.508234	−0.254117	+	0.967173i	$0.581785\pi$
$492$	0	0
$493$	−5.33889e11	−0.407042
$494$	2.83383e11	0.214093
$495$	0	0
$496$	−1.09458e12	−0.812045
$497$	4.23647e11	0.311459
$498$	0	0
$499$	−1.45315e12	−1.04920	−0.524600	−	0.851349i	$-0.675785\pi$
$499$	−1.45315e12	−1.04920	−0.524600	+	0.851349i	$0.675785\pi$
$500$	8.00620e10	0.0572877
$501$	0	0
$502$	1.22977e12	0.864285
$503$	−1.41787e12	−0.987602	−0.493801	−	0.869575i	$-0.664393\pi$
$503$	−1.41787e12	−0.987602	−0.493801	+	0.869575i	$0.664393\pi$
$504$	0	0
$505$	−1.77084e11	−0.121162
$506$	9.36193e11	0.634875
$507$	0	0
$508$	3.17722e10	0.0211671
$509$	8.90571e11	0.588083	0.294042	−	0.955793i	$-0.405000\pi$
$509$	8.90571e11	0.588083	0.294042	+	0.955793i	$0.405000\pi$
$510$	0	0
$511$	9.18553e11	0.595951
$512$	−1.67897e12	−1.07976
$513$	0	0
$514$	−1.65748e12	−1.04740
$515$	1.12146e12	0.702509
$516$	0	0
$517$	2.55352e12	1.57193
$518$	2.42180e12	1.47793
$519$	0	0
$520$	8.28848e11	0.497118
$521$	−9.36152e11	−0.556643	−0.278322	−	0.960488i	$-0.589778\pi$
$521$	−9.36152e11	−0.556643	−0.278322	+	0.960488i	$0.589778\pi$
$522$	0	0
$523$	−1.01211e12	−0.591519	−0.295759	−	0.955262i	$-0.595573\pi$
$523$	−1.01211e12	−0.591519	−0.295759	+	0.955262i	$0.595573\pi$
$524$	2.21146e11	0.128141
$525$	0	0
$526$	−1.49393e12	−0.850933
$527$	1.60509e12	0.906466
$528$	0	0
$529$	−1.41648e12	−0.786432
$530$	5.47054e11	0.301154
$531$	0	0
$532$	2.56262e10	0.0138702
$533$	1.28606e12	0.690224
$534$	0	0
$535$	1.50676e12	0.795153
$536$	−2.66929e12	−1.39686
$537$	0	0
$538$	1.26423e12	0.650588
$539$	−4.10351e11	−0.209414
$540$	0	0
$541$	−1.24268e12	−0.623692	−0.311846	−	0.950133i	$-0.600947\pi$
$541$	−1.24268e12	−0.623692	−0.311846	+	0.950133i	$0.600947\pi$
$542$	−8.92758e11	−0.444362
$543$	0	0
$544$	2.77302e11	0.135756
$545$	−1.69023e12	−0.820659
$546$	0	0
$547$	3.21870e12	1.53722	0.768612	−	0.639715i	$-0.220948\pi$
$547$	3.21870e12	1.53722	0.768612	+	0.639715i	$0.220948\pi$
$548$	−2.45370e11	−0.116228
$549$	0	0
$550$	−2.21124e12	−1.03039
$551$	1.94571e11	0.0899284
$552$	0	0
$553$	−1.55990e11	−0.0709306
$554$	−3.84079e12	−1.73231
$555$	0	0
$556$	8.74472e10	0.0388069
$557$	−3.17464e12	−1.39748	−0.698741	−	0.715375i	$-0.746256\pi$
$557$	−3.17464e12	−1.39748	−0.698741	+	0.715375i	$0.746256\pi$
$558$	0	0
$559$	1.66003e12	0.719057
$560$	−9.99447e11	−0.429451
$561$	0	0
$562$	1.54136e12	0.651763
$563$	−1.80639e12	−0.757746	−0.378873	−	0.925449i	$-0.623688\pi$
$563$	−1.80639e12	−0.757746	−0.378873	+	0.925449i	$0.623688\pi$
$564$	0	0
$565$	−1.08039e12	−0.446026
$566$	−4.48043e12	−1.83504
$567$	0	0
$568$	−8.61834e11	−0.347421
$569$	1.89764e12	0.758940	0.379470	−	0.925204i	$-0.376106\pi$
$569$	1.89764e12	0.758940	0.379470	+	0.925204i	$0.376106\pi$
$570$	0	0
$571$	4.80173e12	1.89032	0.945160	−	0.326608i	$-0.105906\pi$
$571$	4.80173e12	1.89032	0.945160	+	0.326608i	$0.105906\pi$
$572$	2.29968e11	0.0898225
$573$	0	0
$574$	−1.65992e12	−0.638240
$575$	−9.08566e11	−0.346618
$576$	0	0
$577$	2.07557e12	0.779555	0.389777	−	0.920909i	$-0.372552\pi$
$577$	2.07557e12	0.779555	0.389777	+	0.920909i	$0.372552\pi$
$578$	2.03057e11	0.0756732
$579$	0	0
$580$	3.49714e10	0.0128318
$581$	4.14562e12	1.50938
$582$	0	0
$583$	2.46996e12	0.885485
$584$	−1.86863e12	−0.664762
$585$	0	0
$586$	−4.40214e12	−1.54214
$587$	3.37346e12	1.17275	0.586373	−	0.810041i	$-0.300555\pi$
$587$	3.37346e12	1.17275	0.586373	+	0.810041i	$0.300555\pi$
$588$	0	0
$589$	−5.84962e11	−0.200267
$590$	7.35163e11	0.249775
$591$	0	0
$592$	−4.60276e12	−1.54018
$593$	−1.13559e12	−0.377115	−0.188558	−	0.982062i	$-0.560381\pi$
$593$	−1.13559e12	−0.377115	−0.188558	+	0.982062i	$0.560381\pi$
$594$	0	0
$595$	1.46559e12	0.479386
$596$	2.13370e11	0.0692667
$597$	0	0
$598$	−1.34866e12	−0.431268
$599$	1.09523e12	0.347603	0.173801	−	0.984781i	$-0.444395\pi$
$599$	1.09523e12	0.347603	0.173801	+	0.984781i	$0.444395\pi$
$600$	0	0
$601$	3.69611e12	1.15561	0.577803	−	0.816176i	$-0.303910\pi$
$601$	3.69611e12	1.15561	0.577803	+	0.816176i	$0.303910\pi$
$602$	−2.14260e12	−0.664902
$603$	0	0
$604$	−2.28071e11	−0.0697276
$605$	1.67971e12	0.509725
$606$	0	0
$607$	2.37375e12	0.709720	0.354860	−	0.934920i	$-0.384529\pi$
$607$	2.37375e12	0.709720	0.354860	+	0.934920i	$0.384529\pi$
$608$	−1.01060e11	−0.0299927
$609$	0	0
$610$	8.15950e11	0.238605
$611$	−3.67855e12	−1.06780
$612$	0	0
$613$	−1.26316e12	−0.361314	−0.180657	−	0.983546i	$-0.557822\pi$
$613$	−1.26316e12	−0.361314	−0.180657	+	0.983546i	$0.557822\pi$
$614$	1.57774e12	0.447999
$615$	0	0
$616$	−4.83014e12	−1.35159
$617$	2.28805e12	0.635599	0.317799	−	0.948158i	$-0.397056\pi$
$617$	2.28805e12	0.635599	0.317799	+	0.948158i	$0.397056\pi$
$618$	0	0
$619$	−4.26483e12	−1.16760	−0.583799	−	0.811898i	$-0.698434\pi$
$619$	−4.26483e12	−1.16760	−0.583799	+	0.811898i	$0.698434\pi$
$620$	−1.05139e11	−0.0285759
$621$	0	0
$622$	1.77140e12	0.474527
$623$	−1.10313e12	−0.293381
$624$	0	0
$625$	1.19245e12	0.312595
$626$	2.75259e11	0.0716401
$627$	0	0
$628$	3.27376e10	0.00839902
$629$	6.74948e12	1.71926
$630$	0	0
$631$	4.18289e12	1.05038	0.525188	−	0.850986i	$-0.323995\pi$
$631$	4.18289e12	1.05038	0.525188	+	0.850986i	$0.323995\pi$
$632$	3.17334e11	0.0791206
$633$	0	0
$634$	2.81056e12	0.690862
$635$	−6.62222e11	−0.161630
$636$	0	0
$637$	5.91143e11	0.142254
$638$	−2.25365e12	−0.538510
$639$	0	0
$640$	1.89009e12	0.445321
$641$	−1.75683e12	−0.411026	−0.205513	−	0.978654i	$-0.565886\pi$
$641$	−1.75683e12	−0.411026	−0.205513	+	0.978654i	$0.565886\pi$
$642$	0	0
$643$	−4.78899e12	−1.10483	−0.552414	−	0.833570i	$-0.686293\pi$
$643$	−4.78899e12	−1.10483	−0.552414	+	0.833570i	$0.686293\pi$
$644$	−1.21959e11	−0.0279401
$645$	0	0
$646$	−1.01937e12	−0.230295
$647$	−5.52406e12	−1.23934	−0.619668	−	0.784864i	$-0.712733\pi$
$647$	−5.52406e12	−1.23934	−0.619668	+	0.784864i	$0.712733\pi$
$648$	0	0
$649$	3.31927e12	0.734415
$650$	3.18546e12	0.699943
$651$	0	0
$652$	−3.72151e11	−0.0806502
$653$	4.76047e12	1.02457	0.512283	−	0.858816i	$-0.328800\pi$
$653$	4.76047e12	1.02457	0.512283	+	0.858816i	$0.328800\pi$
$654$	0	0
$655$	−4.60929e12	−0.978472
$656$	3.15477e12	0.665120
$657$	0	0
$658$	4.74791e12	0.987383
$659$	1.13341e12	0.234100	0.117050	−	0.993126i	$-0.462656\pi$
$659$	1.13341e12	0.234100	0.117050	+	0.993126i	$0.462656\pi$
$660$	0	0
$661$	−4.58129e12	−0.933428	−0.466714	−	0.884408i	$-0.654562\pi$
$661$	−4.58129e12	−0.933428	−0.466714	+	0.884408i	$0.654562\pi$
$662$	5.68662e11	0.115078
$663$	0	0
$664$	−8.43354e12	−1.68366
$665$	−5.34122e11	−0.105911
$666$	0	0
$667$	−9.25994e11	−0.181151
$668$	−4.63962e11	−0.0901548
$669$	0	0
$670$	3.41888e12	0.655462
$671$	3.68403e12	0.701571
$672$	0	0
$673$	−7.86344e12	−1.47756	−0.738780	−	0.673947i	$-0.764598\pi$
$673$	−7.86344e12	−1.47756	−0.738780	+	0.673947i	$0.764598\pi$
$674$	−7.67625e12	−1.43278
$675$	0	0
$676$	2.42107e10	0.00445910
$677$	5.46353e12	0.999595	0.499798	−	0.866142i	$-0.333408\pi$
$677$	5.46353e12	0.999595	0.499798	+	0.866142i	$0.333408\pi$
$678$	0	0
$679$	6.78606e12	1.22519
$680$	−2.98148e12	−0.534739
$681$	0	0
$682$	6.77541e12	1.19924
$683$	1.78774e12	0.314349	0.157174	−	0.987571i	$-0.449762\pi$
$683$	1.78774e12	0.314349	0.157174	+	0.987571i	$0.449762\pi$
$684$	0	0
$685$	5.11420e12	0.887503
$686$	−5.94069e12	−1.02418
$687$	0	0
$688$	4.07213e12	0.692904
$689$	−3.55817e12	−0.601506
$690$	0	0
$691$	−2.11550e12	−0.352989	−0.176495	−	0.984302i	$-0.556476\pi$
$691$	−2.11550e12	−0.352989	−0.176495	+	0.984302i	$0.556476\pi$
$692$	−5.26872e11	−0.0873429
$693$	0	0
$694$	−8.93433e12	−1.46199
$695$	−1.82264e12	−0.296326
$696$	0	0
$697$	−4.62614e12	−0.742458
$698$	−1.47287e11	−0.0234863
$699$	0	0
$700$	2.88061e11	0.0453464
$701$	−4.18078e12	−0.653923	−0.326962	−	0.945038i	$-0.606025\pi$
$701$	−4.18078e12	−0.653923	−0.326962	+	0.945038i	$0.606025\pi$
$702$	0	0
$703$	−2.45979e12	−0.379839
$704$	9.78635e12	1.50156
$705$	0	0
$706$	4.06450e12	0.615725
$707$	−1.48662e12	−0.223776
$708$	0	0
$709$	−6.70507e12	−0.996541	−0.498271	−	0.867022i	$-0.666031\pi$
$709$	−6.70507e12	−0.996541	−0.498271	+	0.867022i	$0.666031\pi$
$710$	1.10386e12	0.163023
$711$	0	0
$712$	2.24413e12	0.327257
$713$	2.78392e12	0.403417
$714$	0	0
$715$	−4.79317e12	−0.685876
$716$	−4.89017e11	−0.0695369
$717$	0	0
$718$	−9.18054e12	−1.28916
$719$	−2.80366e12	−0.391241	−0.195621	−	0.980680i	$-0.562672\pi$
$719$	−2.80366e12	−0.391241	−0.195621	+	0.980680i	$0.562672\pi$
$720$	0	0
$721$	9.41470e12	1.29747
$722$	3.71500e11	0.0508794
$723$	0	0
$724$	8.94146e11	0.120944
$725$	2.18715e12	0.294007
$726$	0	0
$727$	9.92301e12	1.31746	0.658732	−	0.752378i	$-0.271093\pi$
$727$	9.92301e12	1.31746	0.658732	+	0.752378i	$0.271093\pi$
$728$	6.95819e12	0.918132
$729$	0	0
$730$	2.39339e12	0.311932
$731$	−5.97136e12	−0.773473
$732$	0	0
$733$	−2.54793e12	−0.326001	−0.163000	−	0.986626i	$-0.552117\pi$
$733$	−2.54793e12	−0.326001	−0.163000	+	0.986626i	$0.552117\pi$
$734$	−6.40981e12	−0.815103
$735$	0	0
$736$	4.80961e11	0.0604171
$737$	1.54363e13	1.92726
$738$	0	0
$739$	−1.44078e13	−1.77704	−0.888520	−	0.458839i	$-0.848266\pi$
$739$	−1.44078e13	−1.77704	−0.888520	+	0.458839i	$0.848266\pi$
$740$	−4.42113e11	−0.0541989
$741$	0	0
$742$	4.59253e12	0.556204
$743$	−2.72437e12	−0.327957	−0.163978	−	0.986464i	$-0.552433\pi$
$743$	−2.72437e12	−0.327957	−0.163978	+	0.986464i	$0.552433\pi$
$744$	0	0
$745$	−4.44722e12	−0.528914
$746$	1.26700e13	1.49779
$747$	0	0
$748$	−8.27226e11	−0.0966200
$749$	1.26492e13	1.46858
$750$	0	0
$751$	1.59091e13	1.82501	0.912504	−	0.409067i	$-0.134146\pi$
$751$	1.59091e13	1.82501	0.912504	+	0.409067i	$0.134146\pi$
$752$	−9.02364e12	−1.02897
$753$	0	0
$754$	3.24657e12	0.365808
$755$	4.75364e12	0.532433
$756$	0	0
$757$	1.65456e12	0.183126	0.0915631	−	0.995799i	$-0.470814\pi$
$757$	1.65456e12	0.183126	0.0915631	+	0.995799i	$0.470814\pi$
$758$	6.25637e12	0.688352
$759$	0	0
$760$	1.08658e12	0.118141
$761$	−2.67031e12	−0.288622	−0.144311	−	0.989532i	$-0.546097\pi$
$761$	−2.67031e12	−0.288622	−0.144311	+	0.989532i	$0.546097\pi$
$762$	0	0
$763$	−1.41895e13	−1.51568
$764$	6.88586e11	0.0731203
$765$	0	0
$766$	1.75164e13	1.83830
$767$	−4.78167e12	−0.498885
$768$	0	0
$769$	−2.16207e12	−0.222947	−0.111473	−	0.993767i	$-0.535557\pi$
$769$	−2.16207e12	−0.222947	−0.111473	+	0.993767i	$0.535557\pi$
$770$	6.18655e12	0.634220
$771$	0	0
$772$	5.75133e11	0.0582761
$773$	−1.95566e12	−0.197009	−0.0985044	−	0.995137i	$-0.531406\pi$
$773$	−1.95566e12	−0.197009	−0.0985044	+	0.995137i	$0.531406\pi$
$774$	0	0
$775$	−6.57547e12	−0.654740
$776$	−1.38050e13	−1.36666
$777$	0	0
$778$	1.19353e13	1.16796
$779$	1.68596e12	0.164032
$780$	0	0
$781$	4.98393e12	0.479339
$782$	4.85131e12	0.463905
$783$	0	0
$784$	1.45010e12	0.137080
$785$	−6.82343e11	−0.0641341
$786$	0	0
$787$	7.57561e12	0.703933	0.351966	−	0.936013i	$-0.385513\pi$
$787$	7.57561e12	0.703933	0.351966	+	0.936013i	$0.385513\pi$
$788$	8.21652e10	0.00759137
$789$	0	0
$790$	−4.06448e11	−0.0371265
$791$	−9.06985e12	−0.823770
$792$	0	0
$793$	−5.30713e12	−0.476574
$794$	9.58119e12	0.855514
$795$	0	0
$796$	3.31367e11	0.0292550
$797$	5.83097e12	0.511892	0.255946	−	0.966691i	$-0.417613\pi$
$797$	5.83097e12	0.511892	0.255946	+	0.966691i	$0.417613\pi$
$798$	0	0
$799$	1.32323e13	1.14861
$800$	−1.13600e12	−0.0980562
$801$	0	0
$802$	−9.64838e12	−0.823511
$803$	1.08062e13	0.917175
$804$	0	0
$805$	2.54196e12	0.213348
$806$	−9.76052e12	−0.814639
$807$	0	0
$808$	3.02427e12	0.249614
$809$	−6.52428e12	−0.535506	−0.267753	−	0.963488i	$-0.586281\pi$
$809$	−6.52428e12	−0.535506	−0.267753	+	0.963488i	$0.586281\pi$
$810$	0	0
$811$	−1.79836e12	−0.145977	−0.0729883	−	0.997333i	$-0.523254\pi$
$811$	−1.79836e12	−0.145977	−0.0729883	+	0.997333i	$0.523254\pi$
$812$	2.93586e11	0.0236992
$813$	0	0
$814$	2.84909e13	2.27456
$815$	7.75667e12	0.615837
$816$	0	0
$817$	2.17622e12	0.170884
$818$	1.38662e13	1.08285
$819$	0	0
$820$	3.03028e11	0.0234056
$821$	1.75567e13	1.34865	0.674324	−	0.738436i	$-0.264435\pi$
$821$	1.75567e13	1.34865	0.674324	+	0.738436i	$0.264435\pi$
$822$	0	0
$823$	−1.95759e12	−0.148738	−0.0743692	−	0.997231i	$-0.523694\pi$
$823$	−1.95759e12	−0.148738	−0.0743692	+	0.997231i	$0.523694\pi$
$824$	−1.91525e13	−1.44728
$825$	0	0
$826$	6.17171e12	0.461312
$827$	−1.90700e13	−1.41767	−0.708835	−	0.705374i	$-0.750779\pi$
$827$	−1.90700e13	−1.41767	−0.708835	+	0.705374i	$0.750779\pi$
$828$	0	0
$829$	−2.30481e13	−1.69489	−0.847443	−	0.530886i	$-0.821859\pi$
$829$	−2.30481e13	−1.69489	−0.847443	+	0.530886i	$0.821859\pi$
$830$	1.08019e13	0.790036
$831$	0	0
$832$	−1.40980e13	−1.02001
$833$	−2.12642e12	−0.153020
$834$	0	0
$835$	9.67026e12	0.688413
$836$	3.01476e11	0.0213464
$837$	0	0
$838$	9.86248e11	0.0690857
$839$	2.80134e12	0.195181	0.0975905	−	0.995227i	$-0.468886\pi$
$839$	2.80134e12	0.195181	0.0975905	+	0.995227i	$0.468886\pi$
$840$	0	0
$841$	−1.22780e13	−0.846345
$842$	5.21965e12	0.357880
$843$	0	0
$844$	−1.03788e12	−0.0704054
$845$	−5.04618e11	−0.0340492
$846$	0	0
$847$	1.41012e13	0.941415
$848$	−8.72834e12	−0.579629
$849$	0	0
$850$	−1.14585e13	−0.752912
$851$	1.17065e13	0.765146
$852$	0	0
$853$	−9.10919e12	−0.589128	−0.294564	−	0.955632i	$-0.595174\pi$
$853$	−9.10919e12	−0.589128	−0.294564	+	0.955632i	$0.595174\pi$
$854$	6.84992e12	0.440682
$855$	0	0
$856$	−2.57327e13	−1.63815
$857$	−1.60042e13	−1.01349	−0.506747	−	0.862095i	$-0.669152\pi$
$857$	−1.60042e13	−1.01349	−0.506747	+	0.862095i	$0.669152\pi$
$858$	0	0
$859$	1.88776e13	1.18298	0.591491	−	0.806312i	$-0.298539\pi$
$859$	1.88776e13	1.18298	0.591491	+	0.806312i	$0.298539\pi$
$860$	3.91144e11	0.0243833
$861$	0	0
$862$	9.84653e11	0.0607436
$863$	2.95424e12	0.181299	0.0906497	−	0.995883i	$-0.471106\pi$
$863$	2.95424e12	0.181299	0.0906497	+	0.995883i	$0.471106\pi$
$864$	0	0
$865$	1.09815e13	0.666942
$866$	1.55478e13	0.939372
$867$	0	0
$868$	−8.82641e11	−0.0527771
$869$	−1.83512e12	−0.109163
$870$	0	0
$871$	−2.22372e13	−1.30918
$872$	2.88661e13	1.69069
$873$	0	0
$874$	−1.76802e12	−0.102491
$875$	−1.40089e13	−0.807918
$876$	0	0
$877$	−1.95296e13	−1.11480	−0.557398	−	0.830246i	$-0.688200\pi$
$877$	−1.95296e13	−1.11480	−0.557398	+	0.830246i	$0.688200\pi$
$878$	−6.97297e11	−0.0395997
$879$	0	0
$880$	−1.17579e13	−0.660931
$881$	−1.59453e13	−0.891745	−0.445872	−	0.895096i	$-0.647107\pi$
$881$	−1.59453e13	−0.891745	−0.445872	+	0.895096i	$0.647107\pi$
$882$	0	0
$883$	3.41632e13	1.89119	0.945597	−	0.325341i	$-0.105479\pi$
$883$	3.41632e13	1.89119	0.945597	+	0.325341i	$0.105479\pi$
$884$	1.19168e12	0.0656336
$885$	0	0
$886$	1.54374e13	0.841633
$887$	−1.65254e13	−0.896388	−0.448194	−	0.893936i	$-0.647933\pi$
$887$	−1.65254e13	−0.896388	−0.448194	+	0.893936i	$0.647933\pi$
$888$	0	0
$889$	−5.55937e12	−0.298516
$890$	−2.87433e12	−0.153561
$891$	0	0
$892$	−3.42124e11	−0.0180943
$893$	−4.82239e12	−0.253764
$894$	0	0
$895$	1.01925e13	0.530977
$896$	1.58674e13	0.822467
$897$	0	0
$898$	2.18201e12	0.111973
$899$	−6.70160e12	−0.342184
$900$	0	0
$901$	1.27992e13	0.647026
$902$	−1.95279e13	−0.982260
$903$	0	0
$904$	1.84510e13	0.918887
$905$	−1.86365e13	−0.923518
$906$	0	0
$907$	−3.06612e13	−1.50438	−0.752188	−	0.658949i	$-0.771001\pi$
$907$	−3.06612e13	−1.50438	−0.752188	+	0.658949i	$0.771001\pi$
$908$	2.65959e10	0.00129846
$909$	0	0
$910$	−8.91221e12	−0.430823
$911$	1.70337e13	0.819365	0.409683	−	0.912228i	$-0.365639\pi$
$911$	1.70337e13	0.819365	0.409683	+	0.912228i	$0.365639\pi$
$912$	0	0
$913$	4.87706e13	2.32295
$914$	8.76940e12	0.415635
$915$	0	0
$916$	2.22612e12	0.104477
$917$	−3.86951e13	−1.80715
$918$	0	0
$919$	−1.92109e13	−0.888441	−0.444221	−	0.895917i	$-0.646519\pi$
$919$	−1.92109e13	−0.888441	−0.444221	+	0.895917i	$0.646519\pi$
$920$	−5.17117e12	−0.237982
$921$	0	0
$922$	2.15368e12	0.0981505
$923$	−7.17974e12	−0.325613
$924$	0	0
$925$	−2.76502e13	−1.24182
$926$	−1.06575e13	−0.476326
$927$	0	0
$928$	−1.15780e12	−0.0512467
$929$	−6.24519e12	−0.275090	−0.137545	−	0.990496i	$-0.543921\pi$
$929$	−6.24519e12	−0.275090	−0.137545	+	0.990496i	$0.543921\pi$
$930$	0	0
$931$	7.74958e11	0.0338068
$932$	2.38377e12	0.103489
$933$	0	0
$934$	3.88706e13	1.67132
$935$	1.72417e13	0.737781
$936$	0	0
$937$	1.80123e13	0.763379	0.381689	−	0.924291i	$-0.375342\pi$
$937$	1.80123e13	0.763379	0.381689	+	0.924291i	$0.375342\pi$
$938$	2.87016e13	1.21058
$939$	0	0
$940$	−8.66756e11	−0.0362094
$941$	2.60871e13	1.08461	0.542304	−	0.840182i	$-0.317552\pi$
$941$	2.60871e13	1.08461	0.542304	+	0.840182i	$0.317552\pi$
$942$	0	0
$943$	−8.02374e12	−0.330426
$944$	−1.17296e13	−0.480741
$945$	0	0
$946$	−2.52063e13	−1.02329
$947$	−1.24739e13	−0.503998	−0.251999	−	0.967728i	$-0.581088\pi$
$947$	−1.24739e13	−0.503998	−0.251999	+	0.967728i	$0.581088\pi$
$948$	0	0
$949$	−1.55672e13	−0.623033
$950$	4.17598e12	0.166342
$951$	0	0
$952$	−2.50296e13	−0.987614
$953$	−3.64825e12	−0.143274	−0.0716369	−	0.997431i	$-0.522822\pi$
$953$	−3.64825e12	−0.143274	−0.0716369	+	0.997431i	$0.522822\pi$
$954$	0	0
$955$	−1.43520e13	−0.558340
$956$	−1.33305e12	−0.0516161
$957$	0	0
$958$	−1.91317e13	−0.733851
$959$	4.29338e13	1.63914
$960$	0	0
$961$	−6.29185e12	−0.237971
$962$	−4.10434e13	−1.54510
$963$	0	0
$964$	2.64513e12	0.0986507
$965$	−1.19874e13	−0.444991
$966$	0	0
$967$	−4.99854e13	−1.83833	−0.919165	−	0.393872i	$-0.871135\pi$
$967$	−4.99854e13	−1.83833	−0.919165	+	0.393872i	$0.871135\pi$
$968$	−2.86864e13	−1.05012
$969$	0	0
$970$	1.76818e13	0.641288
$971$	−2.19803e13	−0.793501	−0.396750	−	0.917927i	$-0.629862\pi$
$971$	−2.19803e13	−0.793501	−0.396750	+	0.917927i	$0.629862\pi$
$972$	0	0
$973$	−1.53011e13	−0.547287
$974$	−2.45761e13	−0.874979
$975$	0	0
$976$	−1.30186e13	−0.459241
$977$	−1.75824e13	−0.617379	−0.308690	−	0.951163i	$-0.599890\pi$
$977$	−1.75824e13	−0.617379	−0.308690	+	0.951163i	$0.599890\pi$
$978$	0	0
$979$	−1.29777e13	−0.451517
$980$	1.39288e11	0.00482386
$981$	0	0
$982$	−1.43173e13	−0.491314
$983$	−7.28755e12	−0.248938	−0.124469	−	0.992224i	$-0.539723\pi$
$983$	−7.28755e12	−0.248938	−0.124469	+	0.992224i	$0.539723\pi$
$984$	0	0
$985$	−1.71255e12	−0.0579669
$986$	−1.16783e13	−0.393491
$987$	0	0
$988$	−4.34300e11	−0.0145005
$989$	−1.03569e13	−0.344229
$990$	0	0
$991$	−1.09639e13	−0.361107	−0.180553	−	0.983565i	$-0.557789\pi$
$991$	−1.09639e13	−0.361107	−0.180553	+	0.983565i	$0.557789\pi$
$992$	3.48081e12	0.114124
$993$	0	0
$994$	9.26690e12	0.301090
$995$	−6.90661e12	−0.223389
$996$	0	0
$997$	2.25491e12	0.0722770	0.0361385	−	0.999347i	$-0.488494\pi$
$997$	2.25491e12	0.0722770	0.0361385	+	0.999347i	$0.488494\pi$
$998$	−3.17864e13	−1.01427
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.10.a.c.1.4		6
3.2	odd	2		19.10.a.a.1.3	✓	6
12.11	even	2		304.10.a.f.1.2		6
57.56	even	2		361.10.a.b.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.10.a.a.1.3	✓	6	3.2	odd	2
171.10.a.c.1.4		6	1.1	even	1	trivial
304.10.a.f.1.2		6	12.11	even	2
361.10.a.b.1.4		6	57.56	even	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:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)

\(f(q)\)	\(=\)	\(q+21.8741 q^{2} -33.5233 q^{4} +698.718 q^{5} +5865.75 q^{7} -11932.8 q^{8} +15283.8 q^{10} +69006.8 q^{11} -99409.7 q^{13} +128308. q^{14} -243856. q^{16} +357590. q^{17} -130321. q^{19} -23423.3 q^{20} +1.50946e6 q^{22} +620216. q^{23} -1.46492e6 q^{25} -2.17450e6 q^{26} -196639. q^{28} -1.49302e6 q^{29} +4.48863e6 q^{31} +775473. q^{32} +7.82197e6 q^{34} +4.09851e6 q^{35} +1.88749e7 q^{37} -2.85066e6 q^{38} -8.33769e6 q^{40} -1.29370e7 q^{41} -1.66989e7 q^{43} -2.31333e6 q^{44} +1.35667e7 q^{46} +3.70039e7 q^{47} -5.94653e6 q^{49} -3.20438e7 q^{50} +3.33254e6 q^{52} +3.57930e7 q^{53} +4.82163e7 q^{55} -6.99951e7 q^{56} -3.26584e7 q^{58} +4.81006e7 q^{59} +5.33864e7 q^{61} +9.81847e7 q^{62} +1.41817e8 q^{64} -6.94594e7 q^{65} +2.23693e8 q^{67} -1.19876e7 q^{68} +8.96513e7 q^{70} +7.22238e7 q^{71} +1.56596e8 q^{73} +4.12871e8 q^{74} +4.36879e6 q^{76} +4.04777e8 q^{77} -2.65933e7 q^{79} -1.70387e8 q^{80} -2.82985e8 q^{82} +7.06750e8 q^{83} +2.49855e8 q^{85} -3.65273e8 q^{86} -8.23447e8 q^{88} -1.88064e8 q^{89} -5.83113e8 q^{91} -2.07917e7 q^{92} +8.09428e8 q^{94} -9.10577e7 q^{95} +1.15689e9 q^{97} -1.30075e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 33 q^{2} + 1365 q^{4} + 3612 q^{5} + 4085 q^{7} + 23511 q^{8} - 93884 q^{10} + 69312 q^{11} - 191747 q^{13} + 644691 q^{14} + 13905 q^{16} + 288195 q^{17} - 781926 q^{19} + 1551444 q^{20} + 2409710 q^{22}+ \cdots + 4245664590 q^{98}+O(q^{100}) \) 6 * q + 33 * q^2 + 1365 * q^4 + 3612 * q^5 + 4085 * q^7 + 23511 * q^8 - 93884 * q^10 + 69312 * q^11 - 191747 * q^13 + 644691 * q^14 + 13905 * q^16 + 288195 * q^17 - 781926 * q^19 + 1551444 * q^20 + 2409710 * q^22 + 50697 * q^23 + 7126888 * q^25 - 4394073 * q^26 + 19295623 * q^28 + 7178667 * q^29 - 4310716 * q^31 + 5388975 * q^32 + 18049 * q^34 + 4166490 * q^35 - 15687040 * q^37 - 4300593 * q^38 + 33173148 * q^40 + 25134306 * q^41 - 75118674 * q^43 + 94242102 * q^44 - 85549087 * q^46 + 83731938 * q^47 + 21010019 * q^49 - 141211941 * q^50 + 147001043 * q^52 + 138019203 * q^53 - 1518258 * q^55 + 182020629 * q^56 + 14051197 * q^58 + 7809915 * q^59 + 191946566 * q^61 - 564818736 * q^62 + 167338593 * q^64 + 146177172 * q^65 - 16109787 * q^67 - 561722511 * q^68 + 692634996 * q^70 + 264469698 * q^71 - 287572857 * q^73 - 376807098 * q^74 - 177888165 * q^76 + 1619915436 * q^77 - 86534002 * q^79 + 1300596708 * q^80 + 1582686496 * q^82 + 359214570 * q^83 - 648092718 * q^85 + 1061832924 * q^86 + 779241042 * q^88 + 2263866306 * q^89 + 1172864023 * q^91 + 373904061 * q^92 - 864022488 * q^94 - 470719452 * q^95 + 2705893460 * q^97 + 4245664590 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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