LMFDB - Embedded newform 176.10.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,10,Mod(1,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("176.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 176 = 2^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	176.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:	$90.6463071648$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{889}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 222 $ x^2 - x - 222
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 22)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$15.4081$ of defining polynomial
Character	$\chi$	$=$	176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+144.672 q^{3} -1706.58 q^{5} +8664.66 q^{7} +1247.12 q^{9} +O(q^{10})$	$q+144.672 q^{3} -1706.58 q^{5} +8664.66 q^{7} +1247.12 q^{9} -14641.0 q^{11} +67852.1 q^{13} -246895. q^{15} +249646. q^{17} +293189. q^{19} +1.25354e6 q^{21} -1.10315e6 q^{23} +959294. q^{25} -2.66716e6 q^{27} -5.23603e6 q^{29} +8.38378e6 q^{31} -2.11815e6 q^{33} -1.47869e7 q^{35} +7.34600e6 q^{37} +9.81632e6 q^{39} +1.01256e7 q^{41} +2.56347e7 q^{43} -2.12831e6 q^{45} -5.88993e7 q^{47} +3.47227e7 q^{49} +3.61168e7 q^{51} +8.68723e7 q^{53} +2.49861e7 q^{55} +4.24164e7 q^{57} +1.82302e8 q^{59} +1.22104e8 q^{61} +1.08059e7 q^{63} -1.15795e8 q^{65} +6.80532e7 q^{67} -1.59596e8 q^{69} +1.67349e8 q^{71} -1.51799e8 q^{73} +1.38783e8 q^{75} -1.26859e8 q^{77} -2.95735e8 q^{79} -4.10412e8 q^{81} -4.61094e8 q^{83} -4.26040e8 q^{85} -7.57509e8 q^{87} +4.63623e8 q^{89} +5.87915e8 q^{91} +1.21290e9 q^{93} -5.00351e8 q^{95} +1.06866e8 q^{97} -1.82591e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 21 q^{3} - 521 q^{5} + 7490 q^{7} - 3141 q^{9}+O(q^{10}) $ 2 * q + 21 * q^3 - 521 * q^5 + 7490 * q^7 - 3141 * q^9	$ 2 q + 21 q^{3} - 521 q^{5} + 7490 q^{7} - 3141 q^{9} - 29282 q^{11} + 150314 q^{13} - 393519 q^{15} + 690472 q^{17} - 511212 q^{19} + 1398810 q^{21} - 874751 q^{23} + 411771 q^{25} + 309771 q^{27} - 2951058 q^{29} + 5818705 q^{31} - 307461 q^{33} - 16179590 q^{35} + 2658905 q^{37} - 381948 q^{39} + 13427994 q^{41} + 17820762 q^{43} - 7330788 q^{45} - 56044104 q^{47} - 4251114 q^{49} - 18401250 q^{51} + 96842752 q^{53} + 7627961 q^{55} + 141898680 q^{57} + 119136183 q^{59} - 90424326 q^{61} + 15960420 q^{63} - 18029712 q^{65} + 295944891 q^{67} - 187843215 q^{69} + 322953267 q^{71} - 255975514 q^{73} + 206496864 q^{75} - 109661090 q^{77} + 889658 q^{79} - 692205750 q^{81} + 277699042 q^{83} + 96595042 q^{85} - 1040096232 q^{87} + 1363672217 q^{89} + 491050280 q^{91} + 1530132009 q^{93} - 1454033872 q^{95} + 1398434043 q^{97} + 45987381 q^{99}+O(q^{100}) $ 2 * q + 21 * q^3 - 521 * q^5 + 7490 * q^7 - 3141 * q^9 - 29282 * q^11 + 150314 * q^13 - 393519 * q^15 + 690472 * q^17 - 511212 * q^19 + 1398810 * q^21 - 874751 * q^23 + 411771 * q^25 + 309771 * q^27 - 2951058 * q^29 + 5818705 * q^31 - 307461 * q^33 - 16179590 * q^35 + 2658905 * q^37 - 381948 * q^39 + 13427994 * q^41 + 17820762 * q^43 - 7330788 * q^45 - 56044104 * q^47 - 4251114 * q^49 - 18401250 * q^51 + 96842752 * q^53 + 7627961 * q^55 + 141898680 * q^57 + 119136183 * q^59 - 90424326 * q^61 + 15960420 * q^63 - 18029712 * q^65 + 295944891 * q^67 - 187843215 * q^69 + 322953267 * q^71 - 255975514 * q^73 + 206496864 * q^75 - 109661090 * q^77 + 889658 * q^79 - 692205750 * q^81 + 277699042 * q^83 + 96595042 * q^85 - 1040096232 * q^87 + 1363672217 * q^89 + 491050280 * q^91 + 1530132009 * q^93 - 1454033872 * q^95 + 1398434043 * q^97 + 45987381 * 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
$3$	144.672	1.03119	0.515597	−	0.856831i	$-0.327570\pi$
$3$	144.672	1.03119	0.515597	+	0.856831i	$0.327570\pi$
$4$	0	0
$5$	−1706.58	−1.22113	−0.610565	−	0.791966i	$-0.709058\pi$
$5$	−1706.58	−1.22113	−0.610565	+	0.791966i	$0.709058\pi$
$6$	0	0
$7$	8664.66	1.36399	0.681993	−	0.731358i	$-0.261113\pi$
$7$	8664.66	1.36399	0.681993	+	0.731358i	$0.261113\pi$
$8$	0	0
$9$	1247.12	0.0633603
$10$	0	0
$11$	−14641.0	−0.301511
$11$	−14641.0	−0.301511
$12$	0	0
$13$	67852.1	0.658898	0.329449	−	0.944173i	$-0.393137\pi$
$13$	67852.1	0.658898	0.329449	+	0.944173i	$0.393137\pi$
$14$	0	0
$15$	−246895.	−1.25922
$16$	0	0
$17$	249646.	0.724943	0.362471	−	0.931995i	$-0.381933\pi$
$17$	249646.	0.724943	0.362471	+	0.931995i	$0.381933\pi$
$18$	0	0
$19$	293189.	0.516127	0.258064	−	0.966128i	$-0.416916\pi$
$19$	293189.	0.516127	0.258064	+	0.966128i	$0.416916\pi$
$20$	0	0
$21$	1.25354e6	1.40653
$22$	0	0
$23$	−1.10315e6	−0.821979	−0.410990	−	0.911640i	$-0.634817\pi$
$23$	−1.10315e6	−0.821979	−0.410990	+	0.911640i	$0.634817\pi$
$24$	0	0
$25$	959294.	0.491158
$26$	0	0
$27$	−2.66716e6	−0.965857
$28$	0	0
$29$	−5.23603e6	−1.37471	−0.687354	−	0.726322i	$-0.741228\pi$
$29$	−5.23603e6	−1.37471	−0.687354	+	0.726322i	$0.741228\pi$
$30$	0	0
$31$	8.38378e6	1.63047	0.815234	−	0.579131i	$-0.196608\pi$
$31$	8.38378e6	1.63047	0.815234	+	0.579131i	$0.196608\pi$
$32$	0	0
$33$	−2.11815e6	−0.310917
$34$	0	0
$35$	−1.47869e7	−1.66561
$36$	0	0
$37$	7.34600e6	0.644382	0.322191	−	0.946675i	$-0.395581\pi$
$37$	7.34600e6	0.644382	0.322191	+	0.946675i	$0.395581\pi$
$38$	0	0
$39$	9.81632e6	0.679451
$40$	0	0
$41$	1.01256e7	0.559623	0.279811	−	0.960055i	$-0.409728\pi$
$41$	1.01256e7	0.559623	0.279811	+	0.960055i	$0.409728\pi$
$42$	0	0
$43$	2.56347e7	1.14346	0.571730	−	0.820442i	$-0.306273\pi$
$43$	2.56347e7	1.14346	0.571730	+	0.820442i	$0.306273\pi$
$44$	0	0
$45$	−2.12831e6	−0.0773712
$46$	0	0
$47$	−5.88993e7	−1.76064	−0.880318	−	0.474385i	$-0.842671\pi$
$47$	−5.88993e7	−1.76064	−0.880318	+	0.474385i	$0.842671\pi$
$48$	0	0
$49$	3.47227e7	0.860460
$50$	0	0
$51$	3.61168e7	0.747556
$52$	0	0
$53$	8.68723e7	1.51231	0.756153	−	0.654395i	$-0.227076\pi$
$53$	8.68723e7	1.51231	0.756153	+	0.654395i	$0.227076\pi$
$54$	0	0
$55$	2.49861e7	0.368185
$56$	0	0
$57$	4.24164e7	0.532227
$58$	0	0
$59$	1.82302e8	1.95865	0.979326	−	0.202289i	$-0.0648379\pi$
$59$	1.82302e8	1.95865	0.979326	+	0.202289i	$0.0648379\pi$
$60$	0	0
$61$	1.22104e8	1.12914	0.564569	−	0.825386i	$-0.309042\pi$
$61$	1.22104e8	1.12914	0.564569	+	0.825386i	$0.309042\pi$
$62$	0	0
$63$	1.08059e7	0.0864227
$64$	0	0
$65$	−1.15795e8	−0.804600
$66$	0	0
$67$	6.80532e7	0.412584	0.206292	−	0.978491i	$-0.433860\pi$
$67$	6.80532e7	0.412584	0.206292	+	0.978491i	$0.433860\pi$
$68$	0	0
$69$	−1.59596e8	−0.847620
$70$	0	0
$71$	1.67349e8	0.781555	0.390777	−	0.920485i	$-0.372206\pi$
$71$	1.67349e8	0.781555	0.390777	+	0.920485i	$0.372206\pi$
$72$	0	0
$73$	−1.51799e8	−0.625626	−0.312813	−	0.949815i	$-0.601271\pi$
$73$	−1.51799e8	−0.625626	−0.312813	+	0.949815i	$0.601271\pi$
$74$	0	0
$75$	1.38783e8	0.506479
$76$	0	0
$77$	−1.26859e8	−0.411258
$78$	0	0
$79$	−2.95735e8	−0.854241	−0.427120	−	0.904195i	$-0.640472\pi$
$79$	−2.95735e8	−0.854241	−0.427120	+	0.904195i	$0.640472\pi$
$80$	0	0
$81$	−4.10412e8	−1.05935
$82$	0	0
$83$	−4.61094e8	−1.06644	−0.533222	−	0.845975i	$-0.679019\pi$
$83$	−4.61094e8	−1.06644	−0.533222	+	0.845975i	$0.679019\pi$
$84$	0	0
$85$	−4.26040e8	−0.885249
$86$	0	0
$87$	−7.57509e8	−1.41759
$88$	0	0
$89$	4.63623e8	0.783267	0.391634	−	0.920121i	$-0.371910\pi$
$89$	4.63623e8	0.783267	0.391634	+	0.920121i	$0.371910\pi$
$90$	0	0
$91$	5.87915e8	0.898728
$92$	0	0
$93$	1.21290e9	1.68133
$94$	0	0
$95$	−5.00351e8	−0.630258
$96$	0	0
$97$	1.06866e8	0.122565	0.0612825	−	0.998120i	$-0.480481\pi$
$97$	1.06866e8	0.122565	0.0612825	+	0.998120i	$0.480481\pi$
$98$	0	0
$99$	−1.82591e7	−0.0191039
$100$	0	0
$101$	1.38561e9	1.32494	0.662470	−	0.749088i	$-0.269508\pi$
$101$	1.38561e9	1.32494	0.662470	+	0.749088i	$0.269508\pi$
$102$	0	0
$103$	1.45242e9	1.27152	0.635760	−	0.771886i	$-0.280687\pi$
$103$	1.45242e9	1.27152	0.635760	+	0.771886i	$0.280687\pi$
$104$	0	0
$105$	−2.13926e9	−1.71756
$106$	0	0
$107$	−1.44377e9	−1.06481	−0.532405	−	0.846490i	$-0.678712\pi$
$107$	−1.44377e9	−1.06481	−0.532405	+	0.846490i	$0.678712\pi$
$108$	0	0
$109$	2.58675e9	1.75524	0.877618	−	0.479361i	$-0.159132\pi$
$109$	2.58675e9	1.75524	0.877618	+	0.479361i	$0.159132\pi$
$110$	0	0
$111$	1.06276e9	0.664482
$112$	0	0
$113$	1.13601e9	0.655433	0.327717	−	0.944776i	$-0.393721\pi$
$113$	1.13601e9	0.655433	0.327717	+	0.944776i	$0.393721\pi$
$114$	0	0
$115$	1.88262e9	1.00374
$116$	0	0
$117$	8.46198e7	0.0417480
$118$	0	0
$119$	2.16309e9	0.988812
$120$	0	0
$121$	2.14359e8	0.0909091
$122$	0	0
$123$	1.46490e9	0.577079
$124$	0	0
$125$	1.69605e9	0.621362
$126$	0	0
$127$	3.27848e9	1.11829	0.559147	−	0.829068i	$-0.311129\pi$
$127$	3.27848e9	1.11829	0.559147	+	0.829068i	$0.311129\pi$
$128$	0	0
$129$	3.70864e9	1.17913
$130$	0	0
$131$	−1.81056e9	−0.537146	−0.268573	−	0.963259i	$-0.586552\pi$
$131$	−1.81056e9	−0.537146	−0.268573	+	0.963259i	$0.586552\pi$
$132$	0	0
$133$	2.54038e9	0.703991
$134$	0	0
$135$	4.55173e9	1.17944
$136$	0	0
$137$	−3.19317e9	−0.774425	−0.387213	−	0.921990i	$-0.626562\pi$
$137$	−3.19317e9	−0.774425	−0.387213	+	0.921990i	$0.626562\pi$
$138$	0	0
$139$	8.30571e8	0.188717	0.0943583	−	0.995538i	$-0.469920\pi$
$139$	8.30571e8	0.188717	0.0943583	+	0.995538i	$0.469920\pi$
$140$	0	0
$141$	−8.52110e9	−1.81556
$142$	0	0
$143$	−9.93422e8	−0.198665
$144$	0	0
$145$	8.93570e9	1.67870
$146$	0	0
$147$	5.02341e9	0.887301
$148$	0	0
$149$	8.83876e9	1.46911	0.734553	−	0.678551i	$-0.237392\pi$
$149$	8.83876e9	1.46911	0.734553	+	0.678551i	$0.237392\pi$
$150$	0	0
$151$	6.05461e9	0.947742	0.473871	−	0.880594i	$-0.342856\pi$
$151$	6.05461e9	0.947742	0.473871	+	0.880594i	$0.342856\pi$
$152$	0	0
$153$	3.11338e8	0.0459326
$154$	0	0
$155$	−1.43076e10	−1.99101
$156$	0	0
$157$	−3.21421e9	−0.422208	−0.211104	−	0.977464i	$-0.567706\pi$
$157$	−3.21421e9	−0.422208	−0.211104	+	0.977464i	$0.567706\pi$
$158$	0	0
$159$	1.25680e10	1.55948
$160$	0	0
$161$	−9.55845e9	−1.12117
$162$	0	0
$163$	−3.35516e9	−0.372279	−0.186140	−	0.982523i	$-0.559598\pi$
$163$	−3.35516e9	−0.372279	−0.186140	+	0.982523i	$0.559598\pi$
$164$	0	0
$165$	3.61479e9	0.379670
$166$	0	0
$167$	−1.13205e10	−1.12627	−0.563136	−	0.826364i	$-0.690405\pi$
$167$	−1.13205e10	−1.12627	−0.563136	+	0.826364i	$0.690405\pi$
$168$	0	0
$169$	−6.00060e9	−0.565854
$170$	0	0
$171$	3.65643e8	0.0327020
$172$	0	0
$173$	1.24081e10	1.05317	0.526586	−	0.850122i	$-0.323472\pi$
$173$	1.24081e10	1.05317	0.526586	+	0.850122i	$0.323472\pi$
$174$	0	0
$175$	8.31195e9	0.669934
$176$	0	0
$177$	2.63741e10	2.01975
$178$	0	0
$179$	1.78548e10	1.29992	0.649960	−	0.759968i	$-0.274785\pi$
$179$	1.78548e10	1.29992	0.649960	+	0.759968i	$0.274785\pi$
$180$	0	0
$181$	2.69794e10	1.86844	0.934220	−	0.356698i	$-0.116098\pi$
$181$	2.69794e10	1.86844	0.934220	+	0.356698i	$0.116098\pi$
$182$	0	0
$183$	1.76652e10	1.16436
$184$	0	0
$185$	−1.25365e10	−0.786874
$186$	0	0
$187$	−3.65506e9	−0.218578
$188$	0	0
$189$	−2.31101e10	−1.31742
$190$	0	0
$191$	1.88704e10	1.02596	0.512980	−	0.858400i	$-0.328541\pi$
$191$	1.88704e10	1.02596	0.512980	+	0.858400i	$0.328541\pi$
$192$	0	0
$193$	−1.63769e10	−0.849618	−0.424809	−	0.905283i	$-0.639659\pi$
$193$	−1.63769e10	−0.849618	−0.424809	+	0.905283i	$0.639659\pi$
$194$	0	0
$195$	−1.67524e10	−0.829698
$196$	0	0
$197$	−2.41377e10	−1.14182	−0.570909	−	0.821013i	$-0.693409\pi$
$197$	−2.41377e10	−1.14182	−0.570909	+	0.821013i	$0.693409\pi$
$198$	0	0
$199$	6.98474e7	0.00315727	0.00157863	−	0.999999i	$-0.499498\pi$
$199$	6.98474e7	0.00315727	0.00157863	+	0.999999i	$0.499498\pi$
$200$	0	0
$201$	9.84543e9	0.425454
$202$	0	0
$203$	−4.53684e10	−1.87508
$204$	0	0
$205$	−1.72802e10	−0.683372
$206$	0	0
$207$	−1.37577e9	−0.0520809
$208$	0	0
$209$	−4.29258e9	−0.155618
$210$	0	0
$211$	−3.34436e10	−1.16156	−0.580780	−	0.814061i	$-0.697252\pi$
$211$	−3.34436e10	−1.16156	−0.580780	+	0.814061i	$0.697252\pi$
$212$	0	0
$213$	2.42107e10	0.805935
$214$	0	0
$215$	−4.37478e10	−1.39631
$216$	0	0
$217$	7.26426e10	2.22394
$218$	0	0
$219$	−2.19611e10	−0.645141
$220$	0	0
$221$	1.69390e10	0.477663
$222$	0	0
$223$	1.74786e10	0.473299	0.236650	−	0.971595i	$-0.423951\pi$
$223$	1.74786e10	0.473299	0.236650	+	0.971595i	$0.423951\pi$
$224$	0	0
$225$	1.19636e9	0.0311200
$226$	0	0
$227$	1.99552e10	0.498816	0.249408	−	0.968398i	$-0.419764\pi$
$227$	1.99552e10	0.498816	0.249408	+	0.968398i	$0.419764\pi$
$228$	0	0
$229$	−7.45741e10	−1.79196	−0.895980	−	0.444095i	$-0.853525\pi$
$229$	−7.45741e10	−1.79196	−0.895980	+	0.444095i	$0.853525\pi$
$230$	0	0
$231$	−1.83530e10	−0.424086
$232$	0	0
$233$	−4.84331e10	−1.07657	−0.538283	−	0.842764i	$-0.680927\pi$
$233$	−4.84331e10	−1.07657	−0.538283	+	0.842764i	$0.680927\pi$
$234$	0	0
$235$	1.00516e11	2.14996
$236$	0	0
$237$	−4.27847e10	−0.880888
$238$	0	0
$239$	2.76719e10	0.548591	0.274295	−	0.961645i	$-0.411555\pi$
$239$	2.76719e10	0.548591	0.274295	+	0.961645i	$0.411555\pi$
$240$	0	0
$241$	5.23953e10	1.00050	0.500248	−	0.865882i	$-0.333242\pi$
$241$	5.23953e10	1.00050	0.500248	+	0.865882i	$0.333242\pi$
$242$	0	0
$243$	−6.87757e9	−0.126534
$244$	0	0
$245$	−5.92571e10	−1.05073
$246$	0	0
$247$	1.98935e10	0.340075
$248$	0	0
$249$	−6.67076e10	−1.09971
$250$	0	0
$251$	−5.45608e10	−0.867659	−0.433829	−	0.900995i	$-0.642838\pi$
$251$	−5.45608e10	−0.867659	−0.433829	+	0.900995i	$0.642838\pi$
$252$	0	0
$253$	1.61513e10	0.247836
$254$	0	0
$255$	−6.16363e10	−0.912863
$256$	0	0
$257$	1.72020e10	0.245969	0.122985	−	0.992409i	$-0.460753\pi$
$257$	1.72020e10	0.245969	0.122985	+	0.992409i	$0.460753\pi$
$258$	0	0
$259$	6.36506e10	0.878928
$260$	0	0
$261$	−6.52996e9	−0.0871020
$262$	0	0
$263$	2.32552e10	0.299722	0.149861	−	0.988707i	$-0.452117\pi$
$263$	2.32552e10	0.299722	0.149861	+	0.988707i	$0.452117\pi$
$264$	0	0
$265$	−1.48255e11	−1.84672
$266$	0	0
$267$	6.70735e10	0.807700
$268$	0	0
$269$	2.82157e10	0.328554	0.164277	−	0.986414i	$-0.447471\pi$
$269$	2.82157e10	0.328554	0.164277	+	0.986414i	$0.447471\pi$
$270$	0	0
$271$	−1.37207e11	−1.54530	−0.772651	−	0.634831i	$-0.781070\pi$
$271$	−1.37207e11	−1.54530	−0.772651	+	0.634831i	$0.781070\pi$
$272$	0	0
$273$	8.50551e10	0.926762
$274$	0	0
$275$	−1.40450e10	−0.148090
$276$	0	0
$277$	1.85402e11	1.89215	0.946074	−	0.323949i	$-0.105011\pi$
$277$	1.85402e11	1.89215	0.946074	+	0.323949i	$0.105011\pi$
$278$	0	0
$279$	1.04556e10	0.103307
$280$	0	0
$281$	−5.20765e10	−0.498268	−0.249134	−	0.968469i	$-0.580146\pi$
$281$	−5.20765e10	−0.498268	−0.249134	+	0.968469i	$0.580146\pi$
$282$	0	0
$283$	−1.03796e11	−0.961925	−0.480963	−	0.876741i	$-0.659713\pi$
$283$	−1.03796e11	−0.961925	−0.480963	+	0.876741i	$0.659713\pi$
$284$	0	0
$285$	−7.23870e10	−0.649918
$286$	0	0
$287$	8.77352e10	0.763318
$288$	0	0
$289$	−5.62650e10	−0.474458
$290$	0	0
$291$	1.54606e10	0.126388
$292$	0	0
$293$	−8.51536e10	−0.674992	−0.337496	−	0.941327i	$-0.609580\pi$
$293$	−8.51536e10	−0.674992	−0.337496	+	0.941327i	$0.609580\pi$
$294$	0	0
$295$	−3.11113e11	−2.39177
$296$	0	0
$297$	3.90499e10	0.291217
$298$	0	0
$299$	−7.48513e10	−0.541600
$300$	0	0
$301$	2.22116e11	1.55966
$302$	0	0
$303$	2.00460e11	1.36627
$304$	0	0
$305$	−2.08381e11	−1.37882
$306$	0	0
$307$	−2.79447e11	−1.79547	−0.897734	−	0.440539i	$-0.854787\pi$
$307$	−2.79447e11	−1.79547	−0.897734	+	0.440539i	$0.854787\pi$
$308$	0	0
$309$	2.10125e11	1.31118
$310$	0	0
$311$	−2.40111e11	−1.45543	−0.727714	−	0.685881i	$-0.759417\pi$
$311$	−2.40111e11	−1.45543	−0.727714	+	0.685881i	$0.759417\pi$
$312$	0	0
$313$	9.68635e10	0.570441	0.285220	−	0.958462i	$-0.407933\pi$
$313$	9.68635e10	0.570441	0.285220	+	0.958462i	$0.407933\pi$
$314$	0	0
$315$	−1.84411e10	−0.105533
$316$	0	0
$317$	−1.70124e11	−0.946237	−0.473118	−	0.880999i	$-0.656872\pi$
$317$	−1.70124e11	−0.946237	−0.473118	+	0.880999i	$0.656872\pi$
$318$	0	0
$319$	7.66606e10	0.414490
$320$	0	0
$321$	−2.08874e11	−1.09803
$322$	0	0
$323$	7.31934e10	0.374163
$324$	0	0
$325$	6.50900e10	0.323623
$326$	0	0
$327$	3.74232e11	1.80999
$328$	0	0
$329$	−5.10342e11	−2.40148
$330$	0	0
$331$	−2.45318e10	−0.112332	−0.0561661	−	0.998421i	$-0.517888\pi$
$331$	−2.45318e10	−0.112332	−0.0561661	+	0.998421i	$0.517888\pi$
$332$	0	0
$333$	9.16136e9	0.0408282
$334$	0	0
$335$	−1.16138e11	−0.503818
$336$	0	0
$337$	9.64453e10	0.407330	0.203665	−	0.979041i	$-0.434715\pi$
$337$	9.64453e10	0.407330	0.203665	+	0.979041i	$0.434715\pi$
$338$	0	0
$339$	1.64349e11	0.675879
$340$	0	0
$341$	−1.22747e11	−0.491605
$342$	0	0
$343$	−4.87901e10	−0.190330
$344$	0	0
$345$	2.72364e11	1.03505
$346$	0	0
$347$	1.02569e11	0.379783	0.189891	−	0.981805i	$-0.439186\pi$
$347$	1.02569e11	0.379783	0.189891	+	0.981805i	$0.439186\pi$
$348$	0	0
$349$	−3.59960e11	−1.29879	−0.649397	−	0.760450i	$-0.724979\pi$
$349$	−3.59960e11	−1.29879	−0.649397	+	0.760450i	$0.724979\pi$
$350$	0	0
$351$	−1.80973e11	−0.636401
$352$	0	0
$353$	−6.23410e10	−0.213692	−0.106846	−	0.994276i	$-0.534075\pi$
$353$	−6.23410e10	−0.213692	−0.106846	+	0.994276i	$0.534075\pi$
$354$	0	0
$355$	−2.85594e11	−0.954380
$356$	0	0
$357$	3.12940e11	1.01966
$358$	0	0
$359$	1.12478e11	0.357391	0.178696	−	0.983904i	$-0.442812\pi$
$359$	1.12478e11	0.357391	0.178696	+	0.983904i	$0.442812\pi$
$360$	0	0
$361$	−2.36728e11	−0.733613
$362$	0	0
$363$	3.10118e10	0.0937449
$364$	0	0
$365$	2.59056e11	0.763970
$366$	0	0
$367$	−1.51938e11	−0.437189	−0.218594	−	0.975816i	$-0.570147\pi$
$367$	−1.51938e11	−0.437189	−0.218594	+	0.975816i	$0.570147\pi$
$368$	0	0
$369$	1.26279e10	0.0354579
$370$	0	0
$371$	7.52719e11	2.06277
$372$	0	0
$373$	1.79048e11	0.478938	0.239469	−	0.970904i	$-0.423027\pi$
$373$	1.79048e11	0.478938	0.239469	+	0.970904i	$0.423027\pi$
$374$	0	0
$375$	2.45372e11	0.640744
$376$	0	0
$377$	−3.55275e11	−0.905792
$378$	0	0
$379$	−2.31277e11	−0.575780	−0.287890	−	0.957663i	$-0.592954\pi$
$379$	−2.31277e11	−0.575780	−0.287890	+	0.957663i	$0.592954\pi$
$380$	0	0
$381$	4.74306e11	1.15318
$382$	0	0
$383$	−3.53920e11	−0.840448	−0.420224	−	0.907420i	$-0.638049\pi$
$383$	−3.53920e11	−0.840448	−0.420224	+	0.907420i	$0.638049\pi$
$384$	0	0
$385$	2.16496e11	0.502199
$386$	0	0
$387$	3.19696e10	0.0724500
$388$	0	0
$389$	4.20470e11	0.931027	0.465513	−	0.885041i	$-0.345870\pi$
$389$	4.20470e11	0.931027	0.465513	+	0.885041i	$0.345870\pi$
$390$	0	0
$391$	−2.75398e11	−0.595888
$392$	0	0
$393$	−2.61938e11	−0.553901
$394$	0	0
$395$	5.04695e11	1.04314
$396$	0	0
$397$	−3.43330e11	−0.693672	−0.346836	−	0.937926i	$-0.612744\pi$
$397$	−3.43330e11	−0.693672	−0.346836	+	0.937926i	$0.612744\pi$
$398$	0	0
$399$	3.67524e11	0.725951
$400$	0	0
$401$	5.15897e11	0.996354	0.498177	−	0.867075i	$-0.334003\pi$
$401$	5.15897e11	0.996354	0.498177	+	0.867075i	$0.334003\pi$
$402$	0	0
$403$	5.68857e11	1.07431
$404$	0	0
$405$	7.00402e11	1.29360
$406$	0	0
$407$	−1.07553e11	−0.194288
$408$	0	0
$409$	−4.68357e11	−0.827604	−0.413802	−	0.910367i	$-0.635799\pi$
$409$	−4.68357e11	−0.827604	−0.413802	+	0.910367i	$0.635799\pi$
$410$	0	0
$411$	−4.61964e11	−0.798582
$412$	0	0
$413$	1.57958e12	2.67158
$414$	0	0
$415$	7.86894e11	1.30227
$416$	0	0
$417$	1.20161e11	0.194603
$418$	0	0
$419$	1.15054e12	1.82364	0.911820	−	0.410590i	$-0.134677\pi$
$419$	1.15054e12	1.82364	0.911820	+	0.410590i	$0.134677\pi$
$420$	0	0
$421$	−2.49749e11	−0.387467	−0.193734	−	0.981054i	$-0.562060\pi$
$421$	−2.49749e11	−0.387467	−0.193734	+	0.981054i	$0.562060\pi$
$422$	0	0
$423$	−7.34545e10	−0.111554
$424$	0	0
$425$	2.39483e11	0.356062
$426$	0	0
$427$	1.05799e12	1.54013
$428$	0	0
$429$	−1.43721e11	−0.204862
$430$	0	0
$431$	−2.80621e11	−0.391717	−0.195858	−	0.980632i	$-0.562749\pi$
$431$	−2.80621e11	−0.391717	−0.195858	+	0.980632i	$0.562749\pi$
$432$	0	0
$433$	1.07421e12	1.46857	0.734285	−	0.678841i	$-0.237518\pi$
$433$	1.07421e12	1.46857	0.734285	+	0.678841i	$0.237518\pi$
$434$	0	0
$435$	1.29275e12	1.73106
$436$	0	0
$437$	−3.23433e11	−0.424246
$438$	0	0
$439$	−9.45531e11	−1.21503	−0.607513	−	0.794310i	$-0.707833\pi$
$439$	−9.45531e11	−1.21503	−0.607513	+	0.794310i	$0.707833\pi$
$440$	0	0
$441$	4.33034e10	0.0545191
$442$	0	0
$443$	−1.41348e12	−1.74370	−0.871852	−	0.489769i	$-0.837081\pi$
$443$	−1.41348e12	−1.74370	−0.871852	+	0.489769i	$0.837081\pi$
$444$	0	0
$445$	−7.91210e11	−0.956471
$446$	0	0
$447$	1.27873e12	1.51493
$448$	0	0
$449$	3.24374e11	0.376650	0.188325	−	0.982107i	$-0.439694\pi$
$449$	3.24374e11	0.376650	0.188325	+	0.982107i	$0.439694\pi$
$450$	0	0
$451$	−1.48250e11	−0.168733
$452$	0	0
$453$	8.75936e11	0.977305
$454$	0	0
$455$	−1.00332e12	−1.09746
$456$	0	0
$457$	−1.44780e11	−0.155269	−0.0776344	−	0.996982i	$-0.524737\pi$
$457$	−1.44780e11	−0.155269	−0.0776344	+	0.996982i	$0.524737\pi$
$458$	0	0
$459$	−6.65846e11	−0.700191
$460$	0	0
$461$	2.94094e11	0.303271	0.151636	−	0.988436i	$-0.451546\pi$
$461$	2.94094e11	0.303271	0.151636	+	0.988436i	$0.451546\pi$
$462$	0	0
$463$	−8.78478e11	−0.888416	−0.444208	−	0.895924i	$-0.646515\pi$
$463$	−8.78478e11	−0.888416	−0.444208	+	0.895924i	$0.646515\pi$
$464$	0	0
$465$	−2.06992e12	−2.05312
$466$	0	0
$467$	−6.31625e11	−0.614516	−0.307258	−	0.951626i	$-0.599411\pi$
$467$	−6.31625e11	−0.614516	−0.307258	+	0.951626i	$0.599411\pi$
$468$	0	0
$469$	5.89658e11	0.562759
$470$	0	0
$471$	−4.65008e11	−0.435378
$472$	0	0
$473$	−3.75318e11	−0.344766
$474$	0	0
$475$	2.81255e11	0.253500
$476$	0	0
$477$	1.08340e11	0.0958203
$478$	0	0
$479$	1.70132e12	1.47664	0.738321	−	0.674449i	$-0.235619\pi$
$479$	1.70132e12	1.47664	0.738321	+	0.674449i	$0.235619\pi$
$480$	0	0
$481$	4.98441e11	0.424581
$482$	0	0
$483$	−1.38284e12	−1.15614
$484$	0	0
$485$	−1.82375e11	−0.149668
$486$	0	0
$487$	−2.17815e11	−0.175472	−0.0877358	−	0.996144i	$-0.527963\pi$
$487$	−2.17815e11	−0.175472	−0.0877358	+	0.996144i	$0.527963\pi$
$488$	0	0
$489$	−4.85399e11	−0.383892
$490$	0	0
$491$	−2.45219e11	−0.190409	−0.0952045	−	0.995458i	$-0.530351\pi$
$491$	−2.45219e11	−0.190409	−0.0952045	+	0.995458i	$0.530351\pi$
$492$	0	0
$493$	−1.30715e12	−0.996585
$494$	0	0
$495$	3.11606e10	0.0233283
$496$	0	0
$497$	1.45002e12	1.06603
$498$	0	0
$499$	2.14365e12	1.54775	0.773876	−	0.633337i	$-0.218315\pi$
$499$	2.14365e12	1.54775	0.773876	+	0.633337i	$0.218315\pi$
$500$	0	0
$501$	−1.63777e12	−1.16140
$502$	0	0
$503$	−7.47725e11	−0.520818	−0.260409	−	0.965498i	$-0.583857\pi$
$503$	−7.47725e11	−0.520818	−0.260409	+	0.965498i	$0.583857\pi$
$504$	0	0
$505$	−2.36466e12	−1.61792
$506$	0	0
$507$	−8.68121e11	−0.583505
$508$	0	0
$509$	3.60539e11	0.238079	0.119040	−	0.992889i	$-0.462018\pi$
$509$	3.60539e11	0.238079	0.119040	+	0.992889i	$0.462018\pi$
$510$	0	0
$511$	−1.31528e12	−0.853345
$512$	0	0
$513$	−7.81984e11	−0.498505
$514$	0	0
$515$	−2.47867e12	−1.55269
$516$	0	0
$517$	8.62344e11	0.530852
$518$	0	0
$519$	1.79511e12	1.08602
$520$	0	0
$521$	−1.98989e12	−1.18320	−0.591602	−	0.806230i	$-0.701504\pi$
$521$	−1.98989e12	−1.18320	−0.591602	+	0.806230i	$0.701504\pi$
$522$	0	0
$523$	−1.26127e12	−0.737140	−0.368570	−	0.929600i	$-0.620153\pi$
$523$	−1.26127e12	−0.737140	−0.368570	+	0.929600i	$0.620153\pi$
$524$	0	0
$525$	1.20251e12	0.690831
$526$	0	0
$527$	2.09297e12	1.18200
$528$	0	0
$529$	−5.84204e11	−0.324350
$530$	0	0
$531$	2.27353e11	0.124101
$532$	0	0
$533$	6.87046e11	0.368734
$534$	0	0
$535$	2.46392e12	1.30027
$536$	0	0
$537$	2.58310e12	1.34047
$538$	0	0
$539$	−5.08375e11	−0.259439
$540$	0	0
$541$	3.48950e12	1.75136	0.875679	−	0.482893i	$-0.160414\pi$
$541$	3.48950e12	1.75136	0.875679	+	0.482893i	$0.160414\pi$
$542$	0	0
$543$	3.90318e12	1.92672
$544$	0	0
$545$	−4.41450e12	−2.14337
$546$	0	0
$547$	2.03037e12	0.969687	0.484844	−	0.874601i	$-0.338877\pi$
$547$	2.03037e12	0.969687	0.484844	+	0.874601i	$0.338877\pi$
$548$	0	0
$549$	1.52279e11	0.0715426
$550$	0	0
$551$	−1.53515e12	−0.709525
$552$	0	0
$553$	−2.56244e12	−1.16517
$554$	0	0
$555$	−1.81369e12	−0.811419
$556$	0	0
$557$	2.94692e12	1.29724	0.648620	−	0.761112i	$-0.275346\pi$
$557$	2.94692e12	1.29724	0.648620	+	0.761112i	$0.275346\pi$
$558$	0	0
$559$	1.73937e12	0.753423
$560$	0	0
$561$	−5.28787e11	−0.225397
$562$	0	0
$563$	7.94805e11	0.333405	0.166703	−	0.986007i	$-0.446688\pi$
$563$	7.94805e11	0.333405	0.166703	+	0.986007i	$0.446688\pi$
$564$	0	0
$565$	−1.93869e12	−0.800369
$566$	0	0
$567$	−3.55608e12	−1.44493
$568$	0	0
$569$	1.51857e12	0.607338	0.303669	−	0.952778i	$-0.401788\pi$
$569$	1.51857e12	0.607338	0.303669	+	0.952778i	$0.401788\pi$
$570$	0	0
$571$	2.97779e11	0.117228	0.0586140	−	0.998281i	$-0.481332\pi$
$571$	2.97779e11	0.117228	0.0586140	+	0.998281i	$0.481332\pi$
$572$	0	0
$573$	2.73002e12	1.05796
$574$	0	0
$575$	−1.05825e12	−0.403722
$576$	0	0
$577$	1.81913e12	0.683241	0.341620	−	0.939838i	$-0.389024\pi$
$577$	1.81913e12	0.683241	0.341620	+	0.939838i	$0.389024\pi$
$578$	0	0
$579$	−2.36929e12	−0.876121
$580$	0	0
$581$	−3.99522e12	−1.45462
$582$	0	0
$583$	−1.27190e12	−0.455978
$584$	0	0
$585$	−1.44410e11	−0.0509797
$586$	0	0
$587$	−2.78354e12	−0.967666	−0.483833	−	0.875160i	$-0.660756\pi$
$587$	−2.78354e12	−0.967666	−0.483833	+	0.875160i	$0.660756\pi$
$588$	0	0
$589$	2.45803e12	0.841529
$590$	0	0
$591$	−3.49205e12	−1.17744
$592$	0	0
$593$	−5.56438e11	−0.184787	−0.0923933	−	0.995723i	$-0.529452\pi$
$593$	−5.56438e11	−0.184787	−0.0923933	+	0.995723i	$0.529452\pi$
$594$	0	0
$595$	−3.69149e12	−1.20747
$596$	0	0
$597$	1.01050e10	0.00325576
$598$	0	0
$599$	5.73449e12	1.82001	0.910006	−	0.414595i	$-0.136077\pi$
$599$	5.73449e12	1.82001	0.910006	+	0.414595i	$0.136077\pi$
$600$	0	0
$601$	5.50049e11	0.171975	0.0859877	−	0.996296i	$-0.472595\pi$
$601$	5.50049e11	0.171975	0.0859877	+	0.996296i	$0.472595\pi$
$602$	0	0
$603$	8.48707e10	0.0261415
$604$	0	0
$605$	−3.65821e11	−0.111012
$606$	0	0
$607$	−3.73380e12	−1.11635	−0.558177	−	0.829722i	$-0.688499\pi$
$607$	−3.73380e12	−1.11635	−0.558177	+	0.829722i	$0.688499\pi$
$608$	0	0
$609$	−6.56355e12	−1.93358
$610$	0	0
$611$	−3.99644e12	−1.16008
$612$	0	0
$613$	3.83812e12	1.09786	0.548929	−	0.835869i	$-0.315036\pi$
$613$	3.83812e12	1.09786	0.548929	+	0.835869i	$0.315036\pi$
$614$	0	0
$615$	−2.49997e12	−0.704689
$616$	0	0
$617$	−2.53329e12	−0.703723	−0.351861	−	0.936052i	$-0.614451\pi$
$617$	−2.53329e12	−0.703723	−0.351861	+	0.936052i	$0.614451\pi$
$618$	0	0
$619$	−3.23255e12	−0.884988	−0.442494	−	0.896772i	$-0.645906\pi$
$619$	−3.23255e12	−0.884988	−0.442494	+	0.896772i	$0.645906\pi$
$620$	0	0
$621$	2.94229e12	0.793914
$622$	0	0
$623$	4.01713e12	1.06837
$624$	0	0
$625$	−4.76807e12	−1.24992
$626$	0	0
$627$	−6.21019e11	−0.160473
$628$	0	0
$629$	1.83390e12	0.467140
$630$	0	0
$631$	−3.95011e11	−0.0991921	−0.0495961	−	0.998769i	$-0.515793\pi$
$631$	−3.95011e11	−0.0991921	−0.0495961	+	0.998769i	$0.515793\pi$
$632$	0	0
$633$	−4.83836e12	−1.19779
$634$	0	0
$635$	−5.59500e12	−1.36558
$636$	0	0
$637$	2.35600e12	0.566955
$638$	0	0
$639$	2.08704e11	0.0495196
$640$	0	0
$641$	−3.96414e12	−0.927444	−0.463722	−	0.885981i	$-0.653486\pi$
$641$	−3.96414e12	−0.927444	−0.463722	+	0.885981i	$0.653486\pi$
$642$	0	0
$643$	7.33426e12	1.69203	0.846013	−	0.533163i	$-0.178997\pi$
$643$	7.33426e12	1.69203	0.846013	+	0.533163i	$0.178997\pi$
$644$	0	0
$645$	−6.32910e12	−1.43987
$646$	0	0
$647$	1.01763e12	0.228308	0.114154	−	0.993463i	$-0.463584\pi$
$647$	1.01763e12	0.228308	0.114154	+	0.993463i	$0.463584\pi$
$648$	0	0
$649$	−2.66908e12	−0.590556
$650$	0	0
$651$	1.05094e13	2.29331
$652$	0	0
$653$	6.78338e12	1.45995	0.729973	−	0.683476i	$-0.239533\pi$
$653$	6.78338e12	1.45995	0.729973	+	0.683476i	$0.239533\pi$
$654$	0	0
$655$	3.08987e12	0.655925
$656$	0	0
$657$	−1.89311e11	−0.0396399
$658$	0	0
$659$	−1.83983e12	−0.380008	−0.190004	−	0.981783i	$-0.560850\pi$
$659$	−1.83983e12	−0.380008	−0.190004	+	0.981783i	$0.560850\pi$
$660$	0	0
$661$	−3.08220e12	−0.627992	−0.313996	−	0.949424i	$-0.601668\pi$
$661$	−3.08220e12	−0.627992	−0.313996	+	0.949424i	$0.601668\pi$
$662$	0	0
$663$	2.45060e12	0.492563
$664$	0	0
$665$	−4.33537e12	−0.859664
$666$	0	0
$667$	5.77614e12	1.12998
$668$	0	0
$669$	2.52868e12	0.488063
$670$	0	0
$671$	−1.78773e12	−0.340448
$672$	0	0
$673$	−7.07648e12	−1.32969	−0.664844	−	0.746983i	$-0.731502\pi$
$673$	−7.07648e12	−1.32969	−0.664844	+	0.746983i	$0.731502\pi$
$674$	0	0
$675$	−2.55859e12	−0.474389
$676$	0	0
$677$	1.18522e11	0.0216846	0.0108423	−	0.999941i	$-0.496549\pi$
$677$	1.18522e11	0.0216846	0.0108423	+	0.999941i	$0.496549\pi$
$678$	0	0
$679$	9.25956e11	0.167177
$680$	0	0
$681$	2.88697e12	0.514376
$682$	0	0
$683$	4.08882e12	0.718961	0.359480	−	0.933153i	$-0.382954\pi$
$683$	4.08882e12	0.718961	0.359480	+	0.933153i	$0.382954\pi$
$684$	0	0
$685$	5.44940e12	0.945674
$686$	0	0
$687$	−1.07888e13	−1.84786
$688$	0	0
$689$	5.89446e12	0.996455
$690$	0	0
$691$	−2.48976e12	−0.415438	−0.207719	−	0.978189i	$-0.566604\pi$
$691$	−2.48976e12	−0.415438	−0.207719	+	0.978189i	$0.566604\pi$
$692$	0	0
$693$	−1.58209e11	−0.0260574
$694$	0	0
$695$	−1.41744e12	−0.230447
$696$	0	0
$697$	2.52782e12	0.405694
$698$	0	0
$699$	−7.00694e12	−1.11015
$700$	0	0
$701$	6.94398e12	1.08612	0.543059	−	0.839694i	$-0.317266\pi$
$701$	6.94398e12	1.08612	0.543059	+	0.839694i	$0.317266\pi$
$702$	0	0
$703$	2.15377e12	0.332583
$704$	0	0
$705$	1.45419e13	2.21703
$706$	0	0
$707$	1.20059e13	1.80720
$708$	0	0
$709$	−1.00274e13	−1.49032	−0.745159	−	0.666887i	$-0.767626\pi$
$709$	−1.00274e13	−1.49032	−0.745159	+	0.666887i	$0.767626\pi$
$710$	0	0
$711$	−3.68817e11	−0.0541250
$712$	0	0
$713$	−9.24861e12	−1.34021
$714$	0	0
$715$	1.69535e12	0.242596
$716$	0	0
$717$	4.00336e12	0.565703
$718$	0	0
$719$	−8.41668e12	−1.17452	−0.587261	−	0.809398i	$-0.699794\pi$
$719$	−8.41668e12	−1.17452	−0.587261	+	0.809398i	$0.699794\pi$
$720$	0	0
$721$	1.25847e13	1.73434
$722$	0	0
$723$	7.58015e12	1.03171
$724$	0	0
$725$	−5.02289e12	−0.675200
$726$	0	0
$727$	4.69707e10	0.00623622	0.00311811	−	0.999995i	$-0.499007\pi$
$727$	4.69707e10	0.00623622	0.00311811	+	0.999995i	$0.499007\pi$
$728$	0	0
$729$	7.08315e12	0.928865
$730$	0	0
$731$	6.39960e12	0.828943
$732$	0	0
$733$	6.70015e12	0.857269	0.428634	−	0.903478i	$-0.358995\pi$
$733$	6.70015e12	0.857269	0.428634	+	0.903478i	$0.358995\pi$
$734$	0	0
$735$	−8.57286e12	−1.08351
$736$	0	0
$737$	−9.96367e11	−0.124399
$738$	0	0
$739$	1.71079e11	0.0211007	0.0105503	−	0.999944i	$-0.496642\pi$
$739$	1.71079e11	0.0211007	0.0105503	+	0.999944i	$0.496642\pi$
$740$	0	0
$741$	2.87804e12	0.350683
$742$	0	0
$743$	−6.82102e12	−0.821106	−0.410553	−	0.911837i	$-0.634664\pi$
$743$	−6.82102e12	−0.821106	−0.410553	+	0.911837i	$0.634664\pi$
$744$	0	0
$745$	−1.50841e13	−1.79397
$746$	0	0
$747$	−5.75040e11	−0.0675702
$748$	0	0
$749$	−1.25098e13	−1.45239
$750$	0	0
$751$	−5.72654e12	−0.656920	−0.328460	−	0.944518i	$-0.606530\pi$
$751$	−5.72654e12	−0.656920	−0.328460	+	0.944518i	$0.606530\pi$
$752$	0	0
$753$	−7.89345e12	−0.894724
$754$	0	0
$755$	−1.03327e13	−1.15732
$756$	0	0
$757$	2.98205e12	0.330052	0.165026	−	0.986289i	$-0.447229\pi$
$757$	2.98205e12	0.330052	0.165026	+	0.986289i	$0.447229\pi$
$758$	0	0
$759$	2.33665e12	0.255567
$760$	0	0
$761$	−8.46525e12	−0.914974	−0.457487	−	0.889216i	$-0.651250\pi$
$761$	−8.46525e12	−0.914974	−0.457487	+	0.889216i	$0.651250\pi$
$762$	0	0
$763$	2.24133e13	2.39412
$764$	0	0
$765$	−5.31324e11	−0.0560897
$766$	0	0
$767$	1.23696e13	1.29055
$768$	0	0
$769$	1.77112e13	1.82633	0.913164	−	0.407592i	$-0.133631\pi$
$769$	1.77112e13	1.82633	0.913164	+	0.407592i	$0.133631\pi$
$770$	0	0
$771$	2.48866e12	0.253642
$772$	0	0
$773$	9.83204e12	0.990458	0.495229	−	0.868763i	$-0.335084\pi$
$773$	9.83204e12	0.990458	0.495229	+	0.868763i	$0.335084\pi$
$774$	0	0
$775$	8.04251e12	0.800818
$776$	0	0
$777$	9.20848e12	0.906345
$778$	0	0
$779$	2.96873e12	0.288837
$780$	0	0
$781$	−2.45015e12	−0.235648
$782$	0	0
$783$	1.39653e13	1.32777
$784$	0	0
$785$	5.48531e12	0.515570
$786$	0	0
$787$	1.78869e12	0.166207	0.0831036	−	0.996541i	$-0.473517\pi$
$787$	1.78869e12	0.166207	0.0831036	+	0.996541i	$0.473517\pi$
$788$	0	0
$789$	3.36439e12	0.309072
$790$	0	0
$791$	9.84312e12	0.894002
$792$	0	0
$793$	8.28504e12	0.743987
$794$	0	0
$795$	−2.14484e13	−1.90433
$796$	0	0
$797$	1.04170e13	0.914496	0.457248	−	0.889339i	$-0.348835\pi$
$797$	1.04170e13	0.914496	0.457248	+	0.889339i	$0.348835\pi$
$798$	0	0
$799$	−1.47039e13	−1.27636
$800$	0	0
$801$	5.78194e11	0.0496281
$802$	0	0
$803$	2.22248e12	0.188633
$804$	0	0
$805$	1.63123e13	1.36909
$806$	0	0
$807$	4.08204e12	0.338802
$808$	0	0
$809$	−7.91008e10	−0.00649251	−0.00324625	−	0.999995i	$-0.501033\pi$
$809$	−7.91008e10	−0.00649251	−0.00324625	+	0.999995i	$0.501033\pi$
$810$	0	0
$811$	−7.52015e12	−0.610425	−0.305213	−	0.952284i	$-0.598728\pi$
$811$	−7.52015e12	−0.610425	−0.305213	+	0.952284i	$0.598728\pi$
$812$	0	0
$813$	−1.98500e13	−1.59351
$814$	0	0
$815$	5.72585e12	0.454601
$816$	0	0
$817$	7.51583e12	0.590171
$818$	0	0
$819$	7.33201e11	0.0569437
$820$	0	0
$821$	4.67347e12	0.359000	0.179500	−	0.983758i	$-0.442552\pi$
$821$	4.67347e12	0.359000	0.179500	+	0.983758i	$0.442552\pi$
$822$	0	0
$823$	4.06045e12	0.308514	0.154257	−	0.988031i	$-0.450702\pi$
$823$	4.06045e12	0.308514	0.154257	+	0.988031i	$0.450702\pi$
$824$	0	0
$825$	−2.03193e12	−0.152709
$826$	0	0
$827$	2.53074e12	0.188136	0.0940682	−	0.995566i	$-0.470013\pi$
$827$	2.53074e12	0.188136	0.0940682	+	0.995566i	$0.470013\pi$
$828$	0	0
$829$	−2.26450e12	−0.166524	−0.0832620	−	0.996528i	$-0.526534\pi$
$829$	−2.26450e12	−0.166524	−0.0832620	+	0.996528i	$0.526534\pi$
$830$	0	0
$831$	2.68226e13	1.95117
$832$	0	0
$833$	8.66836e12	0.623784
$834$	0	0
$835$	1.93194e13	1.37532
$836$	0	0
$837$	−2.23609e13	−1.57480
$838$	0	0
$839$	−7.42242e12	−0.517150	−0.258575	−	0.965991i	$-0.583253\pi$
$839$	−7.42242e12	−0.517150	−0.258575	+	0.965991i	$0.583253\pi$
$840$	0	0
$841$	1.29088e13	0.889825
$842$	0	0
$843$	−7.53404e12	−0.513811
$844$	0	0
$845$	1.02405e13	0.690981
$846$	0	0
$847$	1.85735e12	0.123999
$848$	0	0
$849$	−1.50164e13	−0.991931
$850$	0	0
$851$	−8.10377e12	−0.529668
$852$	0	0
$853$	1.60812e13	1.04004	0.520019	−	0.854155i	$-0.325925\pi$
$853$	1.60812e13	1.04004	0.520019	+	0.854155i	$0.325925\pi$
$854$	0	0
$855$	−6.23999e11	−0.0399334
$856$	0	0
$857$	8.68029e11	0.0549693	0.0274847	−	0.999622i	$-0.491250\pi$
$857$	8.68029e11	0.0549693	0.0274847	+	0.999622i	$0.491250\pi$
$858$	0	0
$859$	1.16610e13	0.730748	0.365374	−	0.930861i	$-0.380941\pi$
$859$	1.16610e13	0.730748	0.365374	+	0.930861i	$0.380941\pi$
$860$	0	0
$861$	1.26929e13	0.787129
$862$	0	0
$863$	−2.04030e13	−1.25212	−0.626060	−	0.779775i	$-0.715334\pi$
$863$	−2.04030e13	−1.25212	−0.626060	+	0.779775i	$0.715334\pi$
$864$	0	0
$865$	−2.11755e13	−1.28606
$866$	0	0
$867$	−8.13999e12	−0.489258
$868$	0	0
$869$	4.32985e12	0.257563
$870$	0	0
$871$	4.61755e12	0.271850
$872$	0	0
$873$	1.33275e11	0.00776576
$874$	0	0
$875$	1.46957e13	0.847529
$876$	0	0
$877$	−3.26576e13	−1.86417	−0.932085	−	0.362239i	$-0.882012\pi$
$877$	−3.26576e13	−1.86417	−0.932085	+	0.362239i	$0.882012\pi$
$878$	0	0
$879$	−1.23194e13	−0.696048
$880$	0	0
$881$	1.89169e13	1.05793	0.528966	−	0.848643i	$-0.322580\pi$
$881$	1.89169e13	1.05793	0.528966	+	0.848643i	$0.322580\pi$
$882$	0	0
$883$	4.52036e12	0.250236	0.125118	−	0.992142i	$-0.460069\pi$
$883$	4.52036e12	0.250236	0.125118	+	0.992142i	$0.460069\pi$
$884$	0	0
$885$	−4.50095e13	−2.46638
$886$	0	0
$887$	−1.67209e13	−0.906994	−0.453497	−	0.891258i	$-0.649824\pi$
$887$	−1.67209e13	−0.906994	−0.453497	+	0.891258i	$0.649824\pi$
$888$	0	0
$889$	2.84069e13	1.52534
$890$	0	0
$891$	6.00885e12	0.319405
$892$	0	0
$893$	−1.72686e13	−0.908712
$894$	0	0
$895$	−3.04707e13	−1.58737
$896$	0	0
$897$	−1.08289e13	−0.558495
$898$	0	0
$899$	−4.38977e13	−2.24142
$900$	0	0
$901$	2.16873e13	1.09634
$902$	0	0
$903$	3.21341e13	1.60832
$904$	0	0
$905$	−4.60426e13	−2.28161
$906$	0	0
$907$	2.68525e13	1.31751	0.658753	−	0.752359i	$-0.271084\pi$
$907$	2.68525e13	1.31751	0.658753	+	0.752359i	$0.271084\pi$
$908$	0	0
$909$	1.72803e12	0.0839487
$910$	0	0
$911$	−8.77113e12	−0.421913	−0.210957	−	0.977495i	$-0.567658\pi$
$911$	−8.77113e12	−0.421913	−0.210957	+	0.977495i	$0.567658\pi$
$912$	0	0
$913$	6.75088e12	0.321545
$914$	0	0
$915$	−3.01470e13	−1.42184
$916$	0	0
$917$	−1.56879e13	−0.732660
$918$	0	0
$919$	8.62159e11	0.0398720	0.0199360	−	0.999801i	$-0.493654\pi$
$919$	8.62159e11	0.0398720	0.0199360	+	0.999801i	$0.493654\pi$
$920$	0	0
$921$	−4.04284e13	−1.85147
$922$	0	0
$923$	1.13549e13	0.514965
$924$	0	0
$925$	7.04697e12	0.316493
$926$	0	0
$927$	1.81134e12	0.0805640
$928$	0	0
$929$	1.65304e12	0.0728138	0.0364069	−	0.999337i	$-0.488409\pi$
$929$	1.65304e12	0.0728138	0.0364069	+	0.999337i	$0.488409\pi$
$930$	0	0
$931$	1.01803e13	0.444107
$932$	0	0
$933$	−3.47375e13	−1.50083
$934$	0	0
$935$	6.23766e12	0.266913
$936$	0	0
$937$	−2.41842e13	−1.02495	−0.512476	−	0.858702i	$-0.671272\pi$
$937$	−2.41842e13	−1.02495	−0.512476	+	0.858702i	$0.671272\pi$
$938$	0	0
$939$	1.40135e13	0.588235
$940$	0	0
$941$	3.40036e13	1.41375	0.706874	−	0.707340i	$-0.250105\pi$
$941$	3.40036e13	1.41375	0.706874	+	0.707340i	$0.250105\pi$
$942$	0	0
$943$	−1.11701e13	−0.459998
$944$	0	0
$945$	3.94392e13	1.60874
$946$	0	0
$947$	−1.65859e13	−0.670138	−0.335069	−	0.942194i	$-0.608760\pi$
$947$	−1.65859e13	−0.670138	−0.335069	+	0.942194i	$0.608760\pi$
$948$	0	0
$949$	−1.02998e13	−0.412223
$950$	0	0
$951$	−2.46123e13	−0.975753
$952$	0	0
$953$	−2.57430e13	−1.01098	−0.505489	−	0.862833i	$-0.668688\pi$
$953$	−2.57430e13	−1.01098	−0.505489	+	0.862833i	$0.668688\pi$
$954$	0	0
$955$	−3.22038e13	−1.25283
$956$	0	0
$957$	1.10907e13	0.427420
$958$	0	0
$959$	−2.76677e13	−1.05631
$960$	0	0
$961$	4.38482e13	1.65843
$962$	0	0
$963$	−1.80056e12	−0.0674668
$964$	0	0
$965$	2.79485e13	1.03749
$966$	0	0
$967$	−4.94862e12	−0.181997	−0.0909986	−	0.995851i	$-0.529006\pi$
$967$	−4.94862e12	−0.181997	−0.0909986	+	0.995851i	$0.529006\pi$
$968$	0	0
$969$	1.05891e13	0.385834
$970$	0	0
$971$	−1.45094e13	−0.523799	−0.261899	−	0.965095i	$-0.584349\pi$
$971$	−1.45094e13	−0.523799	−0.261899	+	0.965095i	$0.584349\pi$
$972$	0	0
$973$	7.19661e12	0.257407
$974$	0	0
$975$	9.41674e12	0.333718
$976$	0	0
$977$	−8.00736e12	−0.281167	−0.140583	−	0.990069i	$-0.544898\pi$
$977$	−8.00736e12	−0.281167	−0.140583	+	0.990069i	$0.544898\pi$
$978$	0	0
$979$	−6.78790e12	−0.236164
$980$	0	0
$981$	3.22599e12	0.111212
$982$	0	0
$983$	3.29697e13	1.12622	0.563111	−	0.826381i	$-0.309604\pi$
$983$	3.29697e13	1.12622	0.563111	+	0.826381i	$0.309604\pi$
$984$	0	0
$985$	4.11929e13	1.39431
$986$	0	0
$987$	−7.38324e13	−2.47639
$988$	0	0
$989$	−2.82791e13	−0.939900
$990$	0	0
$991$	−1.22142e12	−0.0402286	−0.0201143	−	0.999798i	$-0.506403\pi$
$991$	−1.22142e12	−0.0402286	−0.0201143	+	0.999798i	$0.506403\pi$
$992$	0	0
$993$	−3.54908e12	−0.115836
$994$	0	0
$995$	−1.19200e11	−0.00385544
$996$	0	0
$997$	−3.27455e13	−1.04960	−0.524800	−	0.851226i	$-0.675860\pi$
$997$	−3.27455e13	−1.04960	−0.524800	+	0.851226i	$0.675860\pi$
$998$	0	0
$999$	−1.95930e13	−0.622380

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	176.10.a.e.1.2		2
4.3	odd	2		22.10.a.d.1.1	✓	2
12.11	even	2		198.10.a.n.1.2		2
44.43	even	2		242.10.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
22.10.a.d.1.1	✓	2	4.3	odd	2
176.10.a.e.1.2		2	1.1	even	1	trivial
198.10.a.n.1.2		2	12.11	even	2
242.10.a.e.1.1		2	44.43	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:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	176.a (trivial)

\(f(q)\)	\(=\)	\(q+144.672 q^{3} -1706.58 q^{5} +8664.66 q^{7} +1247.12 q^{9} +O(q^{10})\)	\(q+144.672 q^{3} -1706.58 q^{5} +8664.66 q^{7} +1247.12 q^{9} -14641.0 q^{11} +67852.1 q^{13} -246895. q^{15} +249646. q^{17} +293189. q^{19} +1.25354e6 q^{21} -1.10315e6 q^{23} +959294. q^{25} -2.66716e6 q^{27} -5.23603e6 q^{29} +8.38378e6 q^{31} -2.11815e6 q^{33} -1.47869e7 q^{35} +7.34600e6 q^{37} +9.81632e6 q^{39} +1.01256e7 q^{41} +2.56347e7 q^{43} -2.12831e6 q^{45} -5.88993e7 q^{47} +3.47227e7 q^{49} +3.61168e7 q^{51} +8.68723e7 q^{53} +2.49861e7 q^{55} +4.24164e7 q^{57} +1.82302e8 q^{59} +1.22104e8 q^{61} +1.08059e7 q^{63} -1.15795e8 q^{65} +6.80532e7 q^{67} -1.59596e8 q^{69} +1.67349e8 q^{71} -1.51799e8 q^{73} +1.38783e8 q^{75} -1.26859e8 q^{77} -2.95735e8 q^{79} -4.10412e8 q^{81} -4.61094e8 q^{83} -4.26040e8 q^{85} -7.57509e8 q^{87} +4.63623e8 q^{89} +5.87915e8 q^{91} +1.21290e9 q^{93} -5.00351e8 q^{95} +1.06866e8 q^{97} -1.82591e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 21 q^{3} - 521 q^{5} + 7490 q^{7} - 3141 q^{9}+O(q^{10}) \) 2 * q + 21 * q^3 - 521 * q^5 + 7490 * q^7 - 3141 * q^9	\( 2 q + 21 q^{3} - 521 q^{5} + 7490 q^{7} - 3141 q^{9} - 29282 q^{11} + 150314 q^{13} - 393519 q^{15} + 690472 q^{17} - 511212 q^{19} + 1398810 q^{21} - 874751 q^{23} + 411771 q^{25} + 309771 q^{27} - 2951058 q^{29} + 5818705 q^{31} - 307461 q^{33} - 16179590 q^{35} + 2658905 q^{37} - 381948 q^{39} + 13427994 q^{41} + 17820762 q^{43} - 7330788 q^{45} - 56044104 q^{47} - 4251114 q^{49} - 18401250 q^{51} + 96842752 q^{53} + 7627961 q^{55} + 141898680 q^{57} + 119136183 q^{59} - 90424326 q^{61} + 15960420 q^{63} - 18029712 q^{65} + 295944891 q^{67} - 187843215 q^{69} + 322953267 q^{71} - 255975514 q^{73} + 206496864 q^{75} - 109661090 q^{77} + 889658 q^{79} - 692205750 q^{81} + 277699042 q^{83} + 96595042 q^{85} - 1040096232 q^{87} + 1363672217 q^{89} + 491050280 q^{91} + 1530132009 q^{93} - 1454033872 q^{95} + 1398434043 q^{97} + 45987381 q^{99}+O(q^{100}) \) 2 * q + 21 * q^3 - 521 * q^5 + 7490 * q^7 - 3141 * q^9 - 29282 * q^11 + 150314 * q^13 - 393519 * q^15 + 690472 * q^17 - 511212 * q^19 + 1398810 * q^21 - 874751 * q^23 + 411771 * q^25 + 309771 * q^27 - 2951058 * q^29 + 5818705 * q^31 - 307461 * q^33 - 16179590 * q^35 + 2658905 * q^37 - 381948 * q^39 + 13427994 * q^41 + 17820762 * q^43 - 7330788 * q^45 - 56044104 * q^47 - 4251114 * q^49 - 18401250 * q^51 + 96842752 * q^53 + 7627961 * q^55 + 141898680 * q^57 + 119136183 * q^59 - 90424326 * q^61 + 15960420 * q^63 - 18029712 * q^65 + 295944891 * q^67 - 187843215 * q^69 + 322953267 * q^71 - 255975514 * q^73 + 206496864 * q^75 - 109661090 * q^77 + 889658 * q^79 - 692205750 * q^81 + 277699042 * q^83 + 96595042 * q^85 - 1040096232 * q^87 + 1363672217 * q^89 + 491050280 * q^91 + 1530132009 * q^93 - 1454033872 * q^95 + 1398434043 * q^97 + 45987381 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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