LMFDB - Embedded newform 192.8.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,8,Mod(1,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("192.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.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:	$59.9779248930$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{46}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 46 $ x^2 - 46
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 96)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$6.78233$ of defining polynomial
Character	$\chi$	$=$	192.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-27.0000 q^{3} +227.552 q^{5} -724.656 q^{7} +729.000 q^{9} +O(q^{10})$	$q-27.0000 q^{3} +227.552 q^{5} -724.656 q^{7} +729.000 q^{9} +1125.31 q^{11} -2426.90 q^{13} -6143.90 q^{15} +28738.2 q^{17} -44674.2 q^{19} +19565.7 q^{21} +23822.7 q^{23} -26345.2 q^{25} -19683.0 q^{27} -11111.2 q^{29} +84678.5 q^{31} -30383.4 q^{33} -164897. q^{35} +199202. q^{37} +65526.2 q^{39} +272733. q^{41} -584037. q^{43} +165885. q^{45} +1.27056e6 q^{47} -298417. q^{49} -775932. q^{51} +371805. q^{53} +256067. q^{55} +1.20620e6 q^{57} -1.79416e6 q^{59} -2.22047e6 q^{61} -528274. q^{63} -552245. q^{65} -3.78620e6 q^{67} -643213. q^{69} -4.40108e6 q^{71} -4.62913e6 q^{73} +711319. q^{75} -815463. q^{77} -2.37397e6 q^{79} +531441. q^{81} -5.76865e6 q^{83} +6.53943e6 q^{85} +300002. q^{87} +1.61090e6 q^{89} +1.75866e6 q^{91} -2.28632e6 q^{93} -1.01657e7 q^{95} -81667.2 q^{97} +820352. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9}+O(q^{10}) $ 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9	$ 2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9} - 1656 q^{11} - 6156 q^{13} + 5292 q^{15} + 17108 q^{17} + 504 q^{19} - 13608 q^{21} + 51552 q^{23} + 74926 q^{25} - 39366 q^{27} + 199804 q^{29} + 257256 q^{31} + 44712 q^{33} - 685296 q^{35} + 468724 q^{37} + 166212 q^{39} - 106940 q^{41} - 1617336 q^{43} - 142884 q^{45} + 646416 q^{47} + 387634 q^{49} - 461916 q^{51} - 1469492 q^{53} + 1434096 q^{55} - 13608 q^{57} - 4541544 q^{59} + 481412 q^{61} + 367416 q^{63} + 1027224 q^{65} - 4775256 q^{67} - 1391904 q^{69} - 1094400 q^{71} - 5731884 q^{73} - 2023002 q^{75} - 4232736 q^{77} - 10402776 q^{79} + 1062882 q^{81} - 2212200 q^{83} + 11465432 q^{85} - 5394708 q^{87} - 3604364 q^{89} - 2823120 q^{91} - 6945912 q^{93} - 29300976 q^{95} - 7156188 q^{97} - 1207224 q^{99}+O(q^{100}) $ 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9 - 1656 * q^11 - 6156 * q^13 + 5292 * q^15 + 17108 * q^17 + 504 * q^19 - 13608 * q^21 + 51552 * q^23 + 74926 * q^25 - 39366 * q^27 + 199804 * q^29 + 257256 * q^31 + 44712 * q^33 - 685296 * q^35 + 468724 * q^37 + 166212 * q^39 - 106940 * q^41 - 1617336 * q^43 - 142884 * q^45 + 646416 * q^47 + 387634 * q^49 - 461916 * q^51 - 1469492 * q^53 + 1434096 * q^55 - 13608 * q^57 - 4541544 * q^59 + 481412 * q^61 + 367416 * q^63 + 1027224 * q^65 - 4775256 * q^67 - 1391904 * q^69 - 1094400 * q^71 - 5731884 * q^73 - 2023002 * q^75 - 4232736 * q^77 - 10402776 * q^79 + 1062882 * q^81 - 2212200 * q^83 + 11465432 * q^85 - 5394708 * q^87 - 3604364 * q^89 - 2823120 * q^91 - 6945912 * q^93 - 29300976 * q^95 - 7156188 * q^97 - 1207224 * 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
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	0	0
$5$	227.552	0.814114	0.407057	−	0.913403i	$-0.366555\pi$
$5$	227.552	0.814114	0.407057	+	0.913403i	$0.366555\pi$
$6$	0	0
$7$	−724.656	−0.798525	−0.399262	−	0.916837i	$-0.630734\pi$
$7$	−724.656	−0.798525	−0.399262	+	0.916837i	$0.630734\pi$
$8$	0	0
$9$	729.000	0.333333
$10$	0	0
$11$	1125.31	0.254917	0.127458	−	0.991844i	$-0.459318\pi$
$11$	1125.31	0.254917	0.127458	+	0.991844i	$0.459318\pi$
$12$	0	0
$13$	−2426.90	−0.306372	−0.153186	−	0.988197i	$-0.548953\pi$
$13$	−2426.90	−0.306372	−0.153186	+	0.988197i	$0.548953\pi$
$14$	0	0
$15$	−6143.90	−0.470029
$16$	0	0
$17$	28738.2	1.41869	0.709347	−	0.704860i	$-0.248990\pi$
$17$	28738.2	1.41869	0.709347	+	0.704860i	$0.248990\pi$
$18$	0	0
$19$	−44674.2	−1.49423	−0.747117	−	0.664692i	$-0.768563\pi$
$19$	−44674.2	−1.49423	−0.747117	+	0.664692i	$0.768563\pi$
$20$	0	0
$21$	19565.7	0.461029
$22$	0	0
$23$	23822.7	0.408266	0.204133	−	0.978943i	$-0.434562\pi$
$23$	23822.7	0.408266	0.204133	+	0.978943i	$0.434562\pi$
$24$	0	0
$25$	−26345.2	−0.337218
$26$	0	0
$27$	−19683.0	−0.192450
$28$	0	0
$29$	−11111.2	−0.0845994	−0.0422997	−	0.999105i	$-0.513468\pi$
$29$	−11111.2	−0.0845994	−0.0422997	+	0.999105i	$0.513468\pi$
$30$	0	0
$31$	84678.5	0.510514	0.255257	−	0.966873i	$-0.417840\pi$
$31$	84678.5	0.510514	0.255257	+	0.966873i	$0.417840\pi$
$32$	0	0
$33$	−30383.4	−0.147176
$34$	0	0
$35$	−164897.	−0.650090
$36$	0	0
$37$	199202.	0.646530	0.323265	−	0.946309i	$-0.395220\pi$
$37$	199202.	0.646530	0.323265	+	0.946309i	$0.395220\pi$
$38$	0	0
$39$	65526.2	0.176884
$40$	0	0
$41$	272733.	0.618008	0.309004	−	0.951061i	$-0.400004\pi$
$41$	272733.	0.618008	0.309004	+	0.951061i	$0.400004\pi$
$42$	0	0
$43$	−584037.	−1.12021	−0.560107	−	0.828420i	$-0.689240\pi$
$43$	−584037.	−1.12021	−0.560107	+	0.828420i	$0.689240\pi$
$44$	0	0
$45$	165885.	0.271371
$46$	0	0
$47$	1.27056e6	1.78506	0.892532	−	0.450983i	$-0.148927\pi$
$47$	1.27056e6	1.78506	0.892532	+	0.450983i	$0.148927\pi$
$48$	0	0
$49$	−298417.	−0.362358
$50$	0	0
$51$	−775932.	−0.819083
$52$	0	0
$53$	371805.	0.343044	0.171522	−	0.985180i	$-0.445132\pi$
$53$	371805.	0.343044	0.171522	+	0.985180i	$0.445132\pi$
$54$	0	0
$55$	256067.	0.207531
$56$	0	0
$57$	1.20620e6	0.862697
$58$	0	0
$59$	−1.79416e6	−1.13731	−0.568657	−	0.822575i	$-0.692537\pi$
$59$	−1.79416e6	−1.13731	−0.568657	+	0.822575i	$0.692537\pi$
$60$	0	0
$61$	−2.22047e6	−1.25253	−0.626267	−	0.779608i	$-0.715418\pi$
$61$	−2.22047e6	−1.25253	−0.626267	+	0.779608i	$0.715418\pi$
$62$	0	0
$63$	−528274.	−0.266175
$64$	0	0
$65$	−552245.	−0.249422
$66$	0	0
$67$	−3.78620e6	−1.53795	−0.768974	−	0.639280i	$-0.779232\pi$
$67$	−3.78620e6	−1.53795	−0.768974	+	0.639280i	$0.779232\pi$
$68$	0	0
$69$	−643213.	−0.235713
$70$	0	0
$71$	−4.40108e6	−1.45934	−0.729668	−	0.683802i	$-0.760325\pi$
$71$	−4.40108e6	−1.45934	−0.729668	+	0.683802i	$0.760325\pi$
$72$	0	0
$73$	−4.62913e6	−1.39274	−0.696369	−	0.717684i	$-0.745202\pi$
$73$	−4.62913e6	−1.39274	−0.696369	+	0.717684i	$0.745202\pi$
$74$	0	0
$75$	711319.	0.194693
$76$	0	0
$77$	−815463.	−0.203557
$78$	0	0
$79$	−2.37397e6	−0.541727	−0.270863	−	0.962618i	$-0.587309\pi$
$79$	−2.37397e6	−0.541727	−0.270863	+	0.962618i	$0.587309\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	−5.76865e6	−1.10739	−0.553696	−	0.832719i	$-0.686783\pi$
$83$	−5.76865e6	−1.10739	−0.553696	+	0.832719i	$0.686783\pi$
$84$	0	0
$85$	6.53943e6	1.15498
$86$	0	0
$87$	300002.	0.0488435
$88$	0	0
$89$	1.61090e6	0.242217	0.121108	−	0.992639i	$-0.461355\pi$
$89$	1.61090e6	0.242217	0.121108	+	0.992639i	$0.461355\pi$
$90$	0	0
$91$	1.75866e6	0.244646
$92$	0	0
$93$	−2.28632e6	−0.294745
$94$	0	0
$95$	−1.01657e7	−1.21648
$96$	0	0
$97$	−81667.2	−0.00908546	−0.00454273	−	0.999990i	$-0.501446\pi$
$97$	−81667.2	−0.00908546	−0.00454273	+	0.999990i	$0.501446\pi$
$98$	0	0
$99$	820352.	0.0849722
$100$	0	0
$101$	−9.09433e6	−0.878307	−0.439153	−	0.898412i	$-0.644722\pi$
$101$	−9.09433e6	−0.878307	−0.439153	+	0.898412i	$0.644722\pi$
$102$	0	0
$103$	−1.97220e7	−1.77836	−0.889181	−	0.457556i	$-0.848725\pi$
$103$	−1.97220e7	−1.77836	−0.889181	+	0.457556i	$0.848725\pi$
$104$	0	0
$105$	4.45221e6	0.375330
$106$	0	0
$107$	8.09689e6	0.638962	0.319481	−	0.947593i	$-0.396491\pi$
$107$	8.09689e6	0.638962	0.319481	+	0.947593i	$0.396491\pi$
$108$	0	0
$109$	1.90660e7	1.41016	0.705078	−	0.709130i	$-0.250912\pi$
$109$	1.90660e7	1.41016	0.705078	+	0.709130i	$0.250912\pi$
$110$	0	0
$111$	−5.37846e6	−0.373274
$112$	0	0
$113$	1.69299e7	1.10377	0.551886	−	0.833919i	$-0.313908\pi$
$113$	1.69299e7	1.10377	0.551886	+	0.833919i	$0.313908\pi$
$114$	0	0
$115$	5.42090e6	0.332375
$116$	0	0
$117$	−1.76921e6	−0.102124
$118$	0	0
$119$	−2.08253e7	−1.13286
$120$	0	0
$121$	−1.82208e7	−0.935018
$122$	0	0
$123$	−7.36379e6	−0.356807
$124$	0	0
$125$	−2.37724e7	−1.08865
$126$	0	0
$127$	−1.91925e7	−0.831418	−0.415709	−	0.909498i	$-0.636466\pi$
$127$	−1.91925e7	−0.831418	−0.415709	+	0.909498i	$0.636466\pi$
$128$	0	0
$129$	1.57690e7	0.646756
$130$	0	0
$131$	−2.48634e7	−0.966298	−0.483149	−	0.875538i	$-0.660507\pi$
$131$	−2.48634e7	−0.966298	−0.483149	+	0.875538i	$0.660507\pi$
$132$	0	0
$133$	3.23734e7	1.19318
$134$	0	0
$135$	−4.47890e6	−0.156676
$136$	0	0
$137$	−2.43232e7	−0.808164	−0.404082	−	0.914723i	$-0.632409\pi$
$137$	−2.43232e7	−0.808164	−0.404082	+	0.914723i	$0.632409\pi$
$138$	0	0
$139$	−3.83111e7	−1.20996	−0.604982	−	0.796239i	$-0.706820\pi$
$139$	−3.83111e7	−1.20996	−0.604982	+	0.796239i	$0.706820\pi$
$140$	0	0
$141$	−3.43052e7	−1.03061
$142$	0	0
$143$	−2.73101e6	−0.0780994
$144$	0	0
$145$	−2.52837e6	−0.0688735
$146$	0	0
$147$	8.05727e6	0.209207
$148$	0	0
$149$	2.80329e6	0.0694250	0.0347125	−	0.999397i	$-0.488948\pi$
$149$	2.80329e6	0.0694250	0.0347125	+	0.999397i	$0.488948\pi$
$150$	0	0
$151$	4.31756e7	1.02052	0.510258	−	0.860022i	$-0.329550\pi$
$151$	4.31756e7	1.02052	0.510258	+	0.860022i	$0.329550\pi$
$152$	0	0
$153$	2.09502e7	0.472898
$154$	0	0
$155$	1.92687e7	0.415616
$156$	0	0
$157$	7.86833e7	1.62268	0.811341	−	0.584573i	$-0.198738\pi$
$157$	7.86833e7	1.62268	0.811341	+	0.584573i	$0.198738\pi$
$158$	0	0
$159$	−1.00387e7	−0.198056
$160$	0	0
$161$	−1.72632e7	−0.326011
$162$	0	0
$163$	−9.04599e6	−0.163606	−0.0818031	−	0.996649i	$-0.526068\pi$
$163$	−9.04599e6	−0.163606	−0.0818031	+	0.996649i	$0.526068\pi$
$164$	0	0
$165$	−6.91380e6	−0.119818
$166$	0	0
$167$	1.02777e8	1.70761	0.853806	−	0.520592i	$-0.174289\pi$
$167$	1.02777e8	1.70761	0.853806	+	0.520592i	$0.174289\pi$
$168$	0	0
$169$	−5.68587e7	−0.906136
$170$	0	0
$171$	−3.25675e7	−0.498078
$172$	0	0
$173$	−1.19796e8	−1.75906	−0.879532	−	0.475839i	$-0.842145\pi$
$173$	−1.19796e8	−1.75906	−0.879532	+	0.475839i	$0.842145\pi$
$174$	0	0
$175$	1.90912e7	0.269277
$176$	0	0
$177$	4.84424e7	0.656628
$178$	0	0
$179$	−1.07336e7	−0.139882	−0.0699408	−	0.997551i	$-0.522281\pi$
$179$	−1.07336e7	−0.139882	−0.0699408	+	0.997551i	$0.522281\pi$
$180$	0	0
$181$	−1.57289e7	−0.197162	−0.0985812	−	0.995129i	$-0.531430\pi$
$181$	−1.57289e7	−0.197162	−0.0985812	+	0.995129i	$0.531430\pi$
$182$	0	0
$183$	5.99526e7	0.723151
$184$	0	0
$185$	4.53289e7	0.526349
$186$	0	0
$187$	3.23394e7	0.361649
$188$	0	0
$189$	1.42634e7	0.153676
$190$	0	0
$191$	1.01986e8	1.05906	0.529532	−	0.848290i	$-0.322368\pi$
$191$	1.01986e8	1.05906	0.529532	+	0.848290i	$0.322368\pi$
$192$	0	0
$193$	−8.34130e7	−0.835186	−0.417593	−	0.908634i	$-0.637126\pi$
$193$	−8.34130e7	−0.835186	−0.417593	+	0.908634i	$0.637126\pi$
$194$	0	0
$195$	1.49106e7	0.144004
$196$	0	0
$197$	−5.81982e7	−0.542348	−0.271174	−	0.962530i	$-0.587412\pi$
$197$	−5.81982e7	−0.542348	−0.271174	+	0.962530i	$0.587412\pi$
$198$	0	0
$199$	1.61506e8	1.45279	0.726393	−	0.687280i	$-0.241195\pi$
$199$	1.61506e8	1.45279	0.726393	+	0.687280i	$0.241195\pi$
$200$	0	0
$201$	1.02227e8	0.887935
$202$	0	0
$203$	8.05178e6	0.0675547
$204$	0	0
$205$	6.20609e7	0.503129
$206$	0	0
$207$	1.73667e7	0.136089
$208$	0	0
$209$	−5.02723e7	−0.380905
$210$	0	0
$211$	7.74548e7	0.567623	0.283811	−	0.958880i	$-0.408401\pi$
$211$	7.74548e7	0.567623	0.283811	+	0.958880i	$0.408401\pi$
$212$	0	0
$213$	1.18829e8	0.842548
$214$	0	0
$215$	−1.32899e8	−0.911982
$216$	0	0
$217$	−6.13627e7	−0.407658
$218$	0	0
$219$	1.24987e8	0.804098
$220$	0	0
$221$	−6.97447e7	−0.434648
$222$	0	0
$223$	−9.33465e7	−0.563678	−0.281839	−	0.959462i	$-0.590944\pi$
$223$	−9.33465e7	−0.563678	−0.281839	+	0.959462i	$0.590944\pi$
$224$	0	0
$225$	−1.92056e7	−0.112406
$226$	0	0
$227$	4.28306e7	0.243032	0.121516	−	0.992589i	$-0.461224\pi$
$227$	4.28306e7	0.243032	0.121516	+	0.992589i	$0.461224\pi$
$228$	0	0
$229$	1.81117e8	0.996630	0.498315	−	0.866996i	$-0.333952\pi$
$229$	1.81117e8	0.996630	0.498315	+	0.866996i	$0.333952\pi$
$230$	0	0
$231$	2.20175e7	0.117524
$232$	0	0
$233$	−2.73990e7	−0.141902	−0.0709512	−	0.997480i	$-0.522603\pi$
$233$	−2.73990e7	−0.141902	−0.0709512	+	0.997480i	$0.522603\pi$
$234$	0	0
$235$	2.89119e8	1.45325
$236$	0	0
$237$	6.40972e7	0.312766
$238$	0	0
$239$	−7.76036e7	−0.367696	−0.183848	−	0.982955i	$-0.558855\pi$
$239$	−7.76036e7	−0.367696	−0.183848	+	0.982955i	$0.558855\pi$
$240$	0	0
$241$	4.21193e8	1.93830	0.969150	−	0.246471i	$-0.0792710\pi$
$241$	4.21193e8	1.93830	0.969150	+	0.246471i	$0.0792710\pi$
$242$	0	0
$243$	−1.43489e7	−0.0641500
$244$	0	0
$245$	−6.79054e7	−0.295001
$246$	0	0
$247$	1.08420e8	0.457792
$248$	0	0
$249$	1.55754e8	0.639353
$250$	0	0
$251$	4.75375e8	1.89748	0.948741	−	0.316053i	$-0.102358\pi$
$251$	4.75375e8	1.89748	0.948741	+	0.316053i	$0.102358\pi$
$252$	0	0
$253$	2.68079e7	0.104074
$254$	0	0
$255$	−1.76565e8	−0.666827
$256$	0	0
$257$	−5.12886e8	−1.88476	−0.942378	−	0.334551i	$-0.891416\pi$
$257$	−5.12886e8	−1.88476	−0.942378	+	0.334551i	$0.891416\pi$
$258$	0	0
$259$	−1.44353e8	−0.516270
$260$	0	0
$261$	−8.10005e6	−0.0281998
$262$	0	0
$263$	2.07182e8	0.702275	0.351138	−	0.936324i	$-0.385795\pi$
$263$	2.07182e8	0.702275	0.351138	+	0.936324i	$0.385795\pi$
$264$	0	0
$265$	8.46048e7	0.279277
$266$	0	0
$267$	−4.34944e7	−0.139844
$268$	0	0
$269$	−3.95082e7	−0.123753	−0.0618763	−	0.998084i	$-0.519708\pi$
$269$	−3.95082e7	−0.123753	−0.0618763	+	0.998084i	$0.519708\pi$
$270$	0	0
$271$	−1.60355e8	−0.489430	−0.244715	−	0.969595i	$-0.578694\pi$
$271$	−1.60355e8	−0.489430	−0.244715	+	0.969595i	$0.578694\pi$
$272$	0	0
$273$	−4.74839e7	−0.141246
$274$	0	0
$275$	−2.96465e7	−0.0859625
$276$	0	0
$277$	−6.47081e8	−1.82928	−0.914638	−	0.404274i	$-0.867524\pi$
$277$	−6.47081e8	−1.82928	−0.914638	+	0.404274i	$0.867524\pi$
$278$	0	0
$279$	6.17306e7	0.170171
$280$	0	0
$281$	−3.78439e8	−1.01748	−0.508738	−	0.860922i	$-0.669888\pi$
$281$	−3.78439e8	−1.01748	−0.508738	+	0.860922i	$0.669888\pi$
$282$	0	0
$283$	4.46034e8	1.16981	0.584905	−	0.811102i	$-0.301132\pi$
$283$	4.46034e8	1.16981	0.584905	+	0.811102i	$0.301132\pi$
$284$	0	0
$285$	2.74474e8	0.702334
$286$	0	0
$287$	−1.97637e8	−0.493495
$288$	0	0
$289$	4.15546e8	1.01269
$290$	0	0
$291$	2.20502e6	0.00524549
$292$	0	0
$293$	−5.46621e8	−1.26955	−0.634775	−	0.772697i	$-0.718907\pi$
$293$	−5.46621e8	−1.26955	−0.634775	+	0.772697i	$0.718907\pi$
$294$	0	0
$295$	−4.08265e8	−0.925903
$296$	0	0
$297$	−2.21495e7	−0.0490587
$298$	0	0
$299$	−5.78152e7	−0.125081
$300$	0	0
$301$	4.23226e8	0.894519
$302$	0	0
$303$	2.45547e8	0.507091
$304$	0	0
$305$	−5.05271e8	−1.01971
$306$	0	0
$307$	−6.30513e8	−1.24368	−0.621842	−	0.783143i	$-0.713615\pi$
$307$	−6.30513e8	−1.24368	−0.621842	+	0.783143i	$0.713615\pi$
$308$	0	0
$309$	5.32493e8	1.02674
$310$	0	0
$311$	3.23661e8	0.610139	0.305069	−	0.952330i	$-0.401320\pi$
$311$	3.23661e8	0.610139	0.305069	+	0.952330i	$0.401320\pi$
$312$	0	0
$313$	8.60648e8	1.58643	0.793214	−	0.608943i	$-0.208406\pi$
$313$	8.60648e8	1.58643	0.793214	+	0.608943i	$0.208406\pi$
$314$	0	0
$315$	−1.20210e8	−0.216697
$316$	0	0
$317$	−2.23939e8	−0.394840	−0.197420	−	0.980319i	$-0.563256\pi$
$317$	−2.23939e8	−0.394840	−0.197420	+	0.980319i	$0.563256\pi$
$318$	0	0
$319$	−1.25035e7	−0.0215658
$320$	0	0
$321$	−2.18616e8	−0.368905
$322$	0	0
$323$	−1.28386e9	−2.11986
$324$	0	0
$325$	6.39370e7	0.103314
$326$	0	0
$327$	−5.14782e8	−0.814154
$328$	0	0
$329$	−9.20721e8	−1.42542
$330$	0	0
$331$	−1.12017e9	−1.69780	−0.848902	−	0.528550i	$-0.822736\pi$
$331$	−1.12017e9	−1.69780	−0.848902	+	0.528550i	$0.822736\pi$
$332$	0	0
$333$	1.45219e8	0.215510
$334$	0	0
$335$	−8.61556e8	−1.25207
$336$	0	0
$337$	3.53693e8	0.503410	0.251705	−	0.967804i	$-0.419009\pi$
$337$	3.53693e8	0.503410	0.251705	+	0.967804i	$0.419009\pi$
$338$	0	0
$339$	−4.57107e8	−0.637263
$340$	0	0
$341$	9.52897e7	0.130138
$342$	0	0
$343$	8.13035e8	1.08788
$344$	0	0
$345$	−1.46364e8	−0.191897
$346$	0	0
$347$	−1.88230e8	−0.241845	−0.120922	−	0.992662i	$-0.538585\pi$
$347$	−1.88230e8	−0.241845	−0.120922	+	0.992662i	$0.538585\pi$
$348$	0	0
$349$	8.58810e7	0.108145	0.0540727	−	0.998537i	$-0.482780\pi$
$349$	8.58810e7	0.108145	0.0540727	+	0.998537i	$0.482780\pi$
$350$	0	0
$351$	4.77686e7	0.0589614
$352$	0	0
$353$	4.55346e7	0.0550973	0.0275487	−	0.999620i	$-0.491230\pi$
$353$	4.55346e7	0.0550973	0.0275487	+	0.999620i	$0.491230\pi$
$354$	0	0
$355$	−1.00147e9	−1.18807
$356$	0	0
$357$	5.62283e8	0.654058
$358$	0	0
$359$	1.01706e9	1.16015	0.580077	−	0.814562i	$-0.303022\pi$
$359$	1.01706e9	1.16015	0.580077	+	0.814562i	$0.303022\pi$
$360$	0	0
$361$	1.10191e9	1.23274
$362$	0	0
$363$	4.91963e8	0.539833
$364$	0	0
$365$	−1.05337e9	−1.13385
$366$	0	0
$367$	−9.15304e8	−0.966571	−0.483286	−	0.875463i	$-0.660557\pi$
$367$	−9.15304e8	−0.966571	−0.483286	+	0.875463i	$0.660557\pi$
$368$	0	0
$369$	1.98822e8	0.206003
$370$	0	0
$371$	−2.69430e8	−0.273929
$372$	0	0
$373$	−6.99424e8	−0.697846	−0.348923	−	0.937151i	$-0.613452\pi$
$373$	−6.99424e8	−0.697846	−0.348923	+	0.937151i	$0.613452\pi$
$374$	0	0
$375$	6.41854e8	0.628531
$376$	0	0
$377$	2.69657e7	0.0259189
$378$	0	0
$379$	1.35218e9	1.27584	0.637921	−	0.770102i	$-0.279795\pi$
$379$	1.35218e9	1.27584	0.637921	+	0.770102i	$0.279795\pi$
$380$	0	0
$381$	5.18198e8	0.480019
$382$	0	0
$383$	−1.69797e9	−1.54431	−0.772155	−	0.635435i	$-0.780821\pi$
$383$	−1.69797e9	−1.54431	−0.772155	+	0.635435i	$0.780821\pi$
$384$	0	0
$385$	−1.85560e8	−0.165719
$386$	0	0
$387$	−4.25763e8	−0.373405
$388$	0	0
$389$	7.88879e8	0.679496	0.339748	−	0.940516i	$-0.389658\pi$
$389$	7.88879e8	0.679496	0.339748	+	0.940516i	$0.389658\pi$
$390$	0	0
$391$	6.84622e8	0.579204
$392$	0	0
$393$	6.71312e8	0.557892
$394$	0	0
$395$	−5.40201e8	−0.441028
$396$	0	0
$397$	−1.19931e9	−0.961978	−0.480989	−	0.876727i	$-0.659722\pi$
$397$	−1.19931e9	−0.961978	−0.480989	+	0.876727i	$0.659722\pi$
$398$	0	0
$399$	−8.74081e8	−0.688885
$400$	0	0
$401$	−1.04006e9	−0.805480	−0.402740	−	0.915314i	$-0.631942\pi$
$401$	−1.04006e9	−0.805480	−0.402740	+	0.915314i	$0.631942\pi$
$402$	0	0
$403$	−2.05506e8	−0.156407
$404$	0	0
$405$	1.20930e8	0.0904571
$406$	0	0
$407$	2.24165e8	0.164811
$408$	0	0
$409$	−4.04909e8	−0.292634	−0.146317	−	0.989238i	$-0.546742\pi$
$409$	−4.04909e8	−0.292634	−0.146317	+	0.989238i	$0.546742\pi$
$410$	0	0
$411$	6.56727e8	0.466594
$412$	0	0
$413$	1.30015e9	0.908173
$414$	0	0
$415$	−1.31267e9	−0.901543
$416$	0	0
$417$	1.03440e9	0.698573
$418$	0	0
$419$	2.46116e8	0.163452	0.0817261	−	0.996655i	$-0.473957\pi$
$419$	2.46116e8	0.163452	0.0817261	+	0.996655i	$0.473957\pi$
$420$	0	0
$421$	1.07285e9	0.700731	0.350365	−	0.936613i	$-0.386057\pi$
$421$	1.07285e9	0.700731	0.350365	+	0.936613i	$0.386057\pi$
$422$	0	0
$423$	9.26241e8	0.595022
$424$	0	0
$425$	−7.57113e8	−0.478409
$426$	0	0
$427$	1.60907e9	1.00018
$428$	0	0
$429$	7.37374e7	0.0450907
$430$	0	0
$431$	−8.24317e8	−0.495934	−0.247967	−	0.968768i	$-0.579762\pi$
$431$	−8.24317e8	−0.495934	−0.247967	+	0.968768i	$0.579762\pi$
$432$	0	0
$433$	−4.40778e8	−0.260923	−0.130462	−	0.991453i	$-0.541646\pi$
$433$	−4.40778e8	−0.260923	−0.130462	+	0.991453i	$0.541646\pi$
$434$	0	0
$435$	6.82660e7	0.0397642
$436$	0	0
$437$	−1.06426e9	−0.610045
$438$	0	0
$439$	−5.43490e8	−0.306596	−0.153298	−	0.988180i	$-0.548989\pi$
$439$	−5.43490e8	−0.306596	−0.153298	+	0.988180i	$0.548989\pi$
$440$	0	0
$441$	−2.17546e8	−0.120786
$442$	0	0
$443$	1.85228e9	1.01226	0.506130	−	0.862457i	$-0.331075\pi$
$443$	1.85228e9	1.01226	0.506130	+	0.862457i	$0.331075\pi$
$444$	0	0
$445$	3.66564e8	0.197192
$446$	0	0
$447$	−7.56888e7	−0.0400825
$448$	0	0
$449$	1.41843e9	0.739515	0.369758	−	0.929128i	$-0.379441\pi$
$449$	1.41843e9	0.739515	0.369758	+	0.929128i	$0.379441\pi$
$450$	0	0
$451$	3.06909e8	0.157541
$452$	0	0
$453$	−1.16574e9	−0.589195
$454$	0	0
$455$	4.00187e8	0.199170
$456$	0	0
$457$	−1.89446e9	−0.928493	−0.464247	−	0.885706i	$-0.653675\pi$
$457$	−1.89446e9	−0.928493	−0.464247	+	0.885706i	$0.653675\pi$
$458$	0	0
$459$	−5.65654e8	−0.273028
$460$	0	0
$461$	−3.41152e9	−1.62179	−0.810895	−	0.585192i	$-0.801019\pi$
$461$	−3.41152e9	−1.62179	−0.810895	+	0.585192i	$0.801019\pi$
$462$	0	0
$463$	−1.75923e9	−0.823736	−0.411868	−	0.911244i	$-0.635123\pi$
$463$	−1.75923e9	−0.823736	−0.411868	+	0.911244i	$0.635123\pi$
$464$	0	0
$465$	−5.20256e8	−0.239956
$466$	0	0
$467$	2.76328e9	1.25550	0.627749	−	0.778416i	$-0.283976\pi$
$467$	2.76328e9	1.25550	0.627749	+	0.778416i	$0.283976\pi$
$468$	0	0
$469$	2.74369e9	1.22809
$470$	0	0
$471$	−2.12445e9	−0.936856
$472$	0	0
$473$	−6.57224e8	−0.285561
$474$	0	0
$475$	1.17695e9	0.503883
$476$	0	0
$477$	2.71046e8	0.114348
$478$	0	0
$479$	9.32149e8	0.387535	0.193767	−	0.981047i	$-0.437929\pi$
$479$	9.32149e8	0.387535	0.193767	+	0.981047i	$0.437929\pi$
$480$	0	0
$481$	−4.83444e8	−0.198079
$482$	0	0
$483$	4.66108e8	0.188222
$484$	0	0
$485$	−1.85835e7	−0.00739660
$486$	0	0
$487$	3.53986e9	1.38878	0.694392	−	0.719597i	$-0.255674\pi$
$487$	3.53986e9	1.38878	0.694392	+	0.719597i	$0.255674\pi$
$488$	0	0
$489$	2.44242e8	0.0944581
$490$	0	0
$491$	4.04042e9	1.54043	0.770213	−	0.637786i	$-0.220150\pi$
$491$	4.04042e9	1.54043	0.770213	+	0.637786i	$0.220150\pi$
$492$	0	0
$493$	−3.19315e8	−0.120021
$494$	0	0
$495$	1.86673e8	0.0691771
$496$	0	0
$497$	3.18927e9	1.16532
$498$	0	0
$499$	−1.83149e9	−0.659860	−0.329930	−	0.944005i	$-0.607025\pi$
$499$	−1.83149e9	−0.659860	−0.329930	+	0.944005i	$0.607025\pi$
$500$	0	0
$501$	−2.77498e9	−0.985890
$502$	0	0
$503$	3.44153e9	1.20577	0.602884	−	0.797829i	$-0.294018\pi$
$503$	3.44153e9	1.20577	0.602884	+	0.797829i	$0.294018\pi$
$504$	0	0
$505$	−2.06943e9	−0.715042
$506$	0	0
$507$	1.53518e9	0.523158
$508$	0	0
$509$	5.73694e8	0.192827	0.0964136	−	0.995341i	$-0.469263\pi$
$509$	5.73694e8	0.192827	0.0964136	+	0.995341i	$0.469263\pi$
$510$	0	0
$511$	3.35453e9	1.11214
$512$	0	0
$513$	8.79321e8	0.287566
$514$	0	0
$515$	−4.48777e9	−1.44779
$516$	0	0
$517$	1.42978e9	0.455043
$518$	0	0
$519$	3.23450e9	1.01560
$520$	0	0
$521$	2.83121e9	0.877081	0.438540	−	0.898712i	$-0.355496\pi$
$521$	2.83121e9	0.877081	0.438540	+	0.898712i	$0.355496\pi$
$522$	0	0
$523$	3.02759e9	0.925426	0.462713	−	0.886508i	$-0.346876\pi$
$523$	3.02759e9	0.925426	0.462713	+	0.886508i	$0.346876\pi$
$524$	0	0
$525$	−5.15461e8	−0.155467
$526$	0	0
$527$	2.43351e9	0.724262
$528$	0	0
$529$	−2.83730e9	−0.833319
$530$	0	0
$531$	−1.30795e9	−0.379104
$532$	0	0
$533$	−6.61895e8	−0.189341
$534$	0	0
$535$	1.84246e9	0.520188
$536$	0	0
$537$	2.89808e8	0.0807606
$538$	0	0
$539$	−3.35812e8	−0.0923711
$540$	0	0
$541$	4.46669e9	1.21282	0.606408	−	0.795153i	$-0.292610\pi$
$541$	4.46669e9	1.21282	0.606408	+	0.795153i	$0.292610\pi$
$542$	0	0
$543$	4.24681e8	0.113832
$544$	0	0
$545$	4.33851e9	1.14803
$546$	0	0
$547$	1.49405e9	0.390311	0.195155	−	0.980772i	$-0.437479\pi$
$547$	1.49405e9	0.390311	0.195155	+	0.980772i	$0.437479\pi$
$548$	0	0
$549$	−1.61872e9	−0.417511
$550$	0	0
$551$	4.96382e8	0.126411
$552$	0	0
$553$	1.72031e9	0.432582
$554$	0	0
$555$	−1.22388e9	−0.303888
$556$	0	0
$557$	5.02645e9	1.23245	0.616224	−	0.787571i	$-0.288662\pi$
$557$	5.02645e9	1.23245	0.616224	+	0.787571i	$0.288662\pi$
$558$	0	0
$559$	1.41740e9	0.343202
$560$	0	0
$561$	−8.73165e8	−0.208798
$562$	0	0
$563$	−4.95585e9	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$563$	−4.95585e9	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$564$	0	0
$565$	3.85243e9	0.898597
$566$	0	0
$567$	−3.85112e8	−0.0887250
$568$	0	0
$569$	−2.06899e9	−0.470831	−0.235415	−	0.971895i	$-0.575645\pi$
$569$	−2.06899e9	−0.470831	−0.235415	+	0.971895i	$0.575645\pi$
$570$	0	0
$571$	−1.84156e9	−0.413961	−0.206980	−	0.978345i	$-0.566364\pi$
$571$	−1.84156e9	−0.413961	−0.206980	+	0.978345i	$0.566364\pi$
$572$	0	0
$573$	−2.75361e9	−0.611451
$574$	0	0
$575$	−6.27613e8	−0.137675
$576$	0	0
$577$	−5.22939e9	−1.13328	−0.566638	−	0.823967i	$-0.691756\pi$
$577$	−5.22939e9	−1.13328	−0.566638	+	0.823967i	$0.691756\pi$
$578$	0	0
$579$	2.25215e9	0.482195
$580$	0	0
$581$	4.18029e9	0.884280
$582$	0	0
$583$	4.18396e8	0.0874475
$584$	0	0
$585$	−4.02586e8	−0.0831407
$586$	0	0
$587$	3.29340e9	0.672065	0.336033	−	0.941850i	$-0.390915\pi$
$587$	3.29340e9	0.672065	0.336033	+	0.941850i	$0.390915\pi$
$588$	0	0
$589$	−3.78294e9	−0.762827
$590$	0	0
$591$	1.57135e9	0.313125
$592$	0	0
$593$	2.35508e8	0.0463783	0.0231892	−	0.999731i	$-0.492618\pi$
$593$	2.35508e8	0.0463783	0.0231892	+	0.999731i	$0.492618\pi$
$594$	0	0
$595$	−4.73884e9	−0.922279
$596$	0	0
$597$	−4.36065e9	−0.838766
$598$	0	0
$599$	−1.02534e8	−0.0194928	−0.00974639	−	0.999953i	$-0.503102\pi$
$599$	−1.02534e8	−0.0194928	−0.00974639	+	0.999953i	$0.503102\pi$
$600$	0	0
$601$	2.48675e9	0.467275	0.233637	−	0.972324i	$-0.424937\pi$
$601$	2.48675e9	0.467275	0.233637	+	0.972324i	$0.424937\pi$
$602$	0	0
$603$	−2.76014e9	−0.512649
$604$	0	0
$605$	−4.14619e9	−0.761211
$606$	0	0
$607$	−2.87128e9	−0.521092	−0.260546	−	0.965461i	$-0.583903\pi$
$607$	−2.87128e9	−0.521092	−0.260546	+	0.965461i	$0.583903\pi$
$608$	0	0
$609$	−2.17398e8	−0.0390027
$610$	0	0
$611$	−3.08353e9	−0.546894
$612$	0	0
$613$	−7.52444e9	−1.31936	−0.659679	−	0.751547i	$-0.729308\pi$
$613$	−7.52444e9	−1.31936	−0.659679	+	0.751547i	$0.729308\pi$
$614$	0	0
$615$	−1.67564e9	−0.290482
$616$	0	0
$617$	6.34114e8	0.108685	0.0543425	−	0.998522i	$-0.482694\pi$
$617$	6.34114e8	0.108685	0.0543425	+	0.998522i	$0.482694\pi$
$618$	0	0
$619$	−4.81321e9	−0.815675	−0.407838	−	0.913054i	$-0.633717\pi$
$619$	−4.81321e9	−0.815675	−0.407838	+	0.913054i	$0.633717\pi$
$620$	0	0
$621$	−4.68902e8	−0.0785709
$622$	0	0
$623$	−1.16735e9	−0.193416
$624$	0	0
$625$	−3.35123e9	−0.549066
$626$	0	0
$627$	1.35735e9	0.219916
$628$	0	0
$629$	5.72472e9	0.917228
$630$	0	0
$631$	−3.35568e9	−0.531714	−0.265857	−	0.964012i	$-0.585655\pi$
$631$	−3.35568e9	−0.531714	−0.265857	+	0.964012i	$0.585655\pi$
$632$	0	0
$633$	−2.09128e9	−0.327717
$634$	0	0
$635$	−4.36730e9	−0.676869
$636$	0	0
$637$	7.24228e8	0.111016
$638$	0	0
$639$	−3.20839e9	−0.486445
$640$	0	0
$641$	1.01582e10	1.52339	0.761696	−	0.647935i	$-0.224367\pi$
$641$	1.01582e10	1.52339	0.761696	+	0.647935i	$0.224367\pi$
$642$	0	0
$643$	5.05703e9	0.750165	0.375083	−	0.926991i	$-0.377614\pi$
$643$	5.05703e9	0.750165	0.375083	+	0.926991i	$0.377614\pi$
$644$	0	0
$645$	3.58827e9	0.526533
$646$	0	0
$647$	−1.05626e10	−1.53323	−0.766613	−	0.642109i	$-0.778060\pi$
$647$	−1.05626e10	−1.53323	−0.766613	+	0.642109i	$0.778060\pi$
$648$	0	0
$649$	−2.01899e9	−0.289920
$650$	0	0
$651$	1.65679e9	0.235361
$652$	0	0
$653$	−6.19593e8	−0.0870784	−0.0435392	−	0.999052i	$-0.513863\pi$
$653$	−6.19593e8	−0.0870784	−0.0435392	+	0.999052i	$0.513863\pi$
$654$	0	0
$655$	−5.65771e9	−0.786677
$656$	0	0
$657$	−3.37464e9	−0.464246
$658$	0	0
$659$	6.32359e9	0.860725	0.430363	−	0.902656i	$-0.358386\pi$
$659$	6.32359e9	0.860725	0.430363	+	0.902656i	$0.358386\pi$
$660$	0	0
$661$	−9.81233e9	−1.32150	−0.660750	−	0.750606i	$-0.729761\pi$
$661$	−9.81233e9	−1.32150	−0.660750	+	0.750606i	$0.729761\pi$
$662$	0	0
$663$	1.88311e9	0.250944
$664$	0	0
$665$	7.36662e9	0.971388
$666$	0	0
$667$	−2.64698e8	−0.0345391
$668$	0	0
$669$	2.52036e9	0.325440
$670$	0	0
$671$	−2.49871e9	−0.319292
$672$	0	0
$673$	−1.37509e10	−1.73892	−0.869458	−	0.494006i	$-0.835532\pi$
$673$	−1.37509e10	−1.73892	−0.869458	+	0.494006i	$0.835532\pi$
$674$	0	0
$675$	5.18552e8	0.0648976
$676$	0	0
$677$	8.30590e9	1.02879	0.514395	−	0.857553i	$-0.328017\pi$
$677$	8.30590e9	1.02879	0.514395	+	0.857553i	$0.328017\pi$
$678$	0	0
$679$	5.91806e7	0.00725496
$680$	0	0
$681$	−1.15643e9	−0.140315
$682$	0	0
$683$	2.10499e9	0.252800	0.126400	−	0.991979i	$-0.459658\pi$
$683$	2.10499e9	0.252800	0.126400	+	0.991979i	$0.459658\pi$
$684$	0	0
$685$	−5.53480e9	−0.657938
$686$	0	0
$687$	−4.89015e9	−0.575405
$688$	0	0
$689$	−9.02331e8	−0.105099
$690$	0	0
$691$	4.48072e9	0.516624	0.258312	−	0.966061i	$-0.416834\pi$
$691$	4.48072e9	0.516624	0.258312	+	0.966061i	$0.416834\pi$
$692$	0	0
$693$	−5.94472e8	−0.0678524
$694$	0	0
$695$	−8.71775e9	−0.985049
$696$	0	0
$697$	7.83786e9	0.876764
$698$	0	0
$699$	7.39774e8	0.0819274
$700$	0	0
$701$	1.62398e10	1.78060	0.890300	−	0.455374i	$-0.150494\pi$
$701$	1.62398e10	1.78060	0.890300	+	0.455374i	$0.150494\pi$
$702$	0	0
$703$	−8.89920e9	−0.966067
$704$	0	0
$705$	−7.80622e9	−0.839032
$706$	0	0
$707$	6.59026e9	0.701350
$708$	0	0
$709$	−8.62447e9	−0.908805	−0.454403	−	0.890796i	$-0.650147\pi$
$709$	−8.62447e9	−0.908805	−0.454403	+	0.890796i	$0.650147\pi$
$710$	0	0
$711$	−1.73062e9	−0.180576
$712$	0	0
$713$	2.01727e9	0.208425
$714$	0	0
$715$	−6.21447e8	−0.0635818
$716$	0	0
$717$	2.09530e9	0.212289
$718$	0	0
$719$	2.97175e9	0.298168	0.149084	−	0.988825i	$-0.452367\pi$
$719$	2.97175e9	0.298168	0.149084	+	0.988825i	$0.452367\pi$
$720$	0	0
$721$	1.42916e10	1.42007
$722$	0	0
$723$	−1.13722e10	−1.11908
$724$	0	0
$725$	2.92726e8	0.0285284
$726$	0	0
$727$	−1.52769e10	−1.47457	−0.737284	−	0.675583i	$-0.763892\pi$
$727$	−1.52769e10	−1.47457	−0.737284	+	0.675583i	$0.763892\pi$
$728$	0	0
$729$	3.87420e8	0.0370370
$730$	0	0
$731$	−1.67842e10	−1.58924
$732$	0	0
$733$	2.55786e9	0.239891	0.119945	−	0.992780i	$-0.461728\pi$
$733$	2.55786e9	0.239891	0.119945	+	0.992780i	$0.461728\pi$
$734$	0	0
$735$	1.83345e9	0.170319
$736$	0	0
$737$	−4.26065e9	−0.392048
$738$	0	0
$739$	−5.76099e9	−0.525099	−0.262550	−	0.964919i	$-0.584563\pi$
$739$	−5.76099e9	−0.525099	−0.262550	+	0.964919i	$0.584563\pi$
$740$	0	0
$741$	−2.92733e9	−0.264306
$742$	0	0
$743$	8.79171e9	0.786344	0.393172	−	0.919465i	$-0.371378\pi$
$743$	8.79171e9	0.786344	0.393172	+	0.919465i	$0.371378\pi$
$744$	0	0
$745$	6.37894e8	0.0565199
$746$	0	0
$747$	−4.20535e9	−0.369130
$748$	0	0
$749$	−5.86746e9	−0.510227
$750$	0	0
$751$	6.14505e9	0.529402	0.264701	−	0.964330i	$-0.414727\pi$
$751$	6.14505e9	0.529402	0.264701	+	0.964330i	$0.414727\pi$
$752$	0	0
$753$	−1.28351e10	−1.09551
$754$	0	0
$755$	9.82470e9	0.830816
$756$	0	0
$757$	−7.09591e9	−0.594528	−0.297264	−	0.954795i	$-0.596074\pi$
$757$	−7.09591e9	−0.594528	−0.297264	+	0.954795i	$0.596074\pi$
$758$	0	0
$759$	−7.23814e8	−0.0600871
$760$	0	0
$761$	−1.44986e10	−1.19256	−0.596280	−	0.802776i	$-0.703355\pi$
$761$	−1.44986e10	−1.19256	−0.596280	+	0.802776i	$0.703355\pi$
$762$	0	0
$763$	−1.38163e10	−1.12604
$764$	0	0
$765$	4.76725e9	0.384993
$766$	0	0
$767$	4.35425e9	0.348441
$768$	0	0
$769$	1.48341e10	1.17630	0.588151	−	0.808751i	$-0.299856\pi$
$769$	1.48341e10	1.17630	0.588151	+	0.808751i	$0.299856\pi$
$770$	0	0
$771$	1.38479e10	1.08816
$772$	0	0
$773$	1.64984e10	1.28473	0.642366	−	0.766398i	$-0.277953\pi$
$773$	1.64984e10	1.28473	0.642366	+	0.766398i	$0.277953\pi$
$774$	0	0
$775$	−2.23087e9	−0.172154
$776$	0	0
$777$	3.89753e9	0.298069
$778$	0	0
$779$	−1.21841e10	−0.923449
$780$	0	0
$781$	−4.95259e9	−0.372009
$782$	0	0
$783$	2.18701e8	0.0162812
$784$	0	0
$785$	1.79045e10	1.32105
$786$	0	0
$787$	4.05010e9	0.296179	0.148090	−	0.988974i	$-0.452688\pi$
$787$	4.05010e9	0.296179	0.148090	+	0.988974i	$0.452688\pi$
$788$	0	0
$789$	−5.59392e9	−0.405459
$790$	0	0
$791$	−1.22683e10	−0.881390
$792$	0	0
$793$	5.38884e9	0.383742
$794$	0	0
$795$	−2.28433e9	−0.161240
$796$	0	0
$797$	−1.61834e9	−0.113231	−0.0566156	−	0.998396i	$-0.518031\pi$
$797$	−1.61834e9	−0.113231	−0.0566156	+	0.998396i	$0.518031\pi$
$798$	0	0
$799$	3.65137e10	2.53246
$800$	0	0
$801$	1.17435e9	0.0807390
$802$	0	0
$803$	−5.20921e9	−0.355032
$804$	0	0
$805$	−3.92828e9	−0.265410
$806$	0	0
$807$	1.06672e9	0.0714486
$808$	0	0
$809$	−1.32510e10	−0.879889	−0.439945	−	0.898025i	$-0.645002\pi$
$809$	−1.32510e10	−0.879889	−0.439945	+	0.898025i	$0.645002\pi$
$810$	0	0
$811$	−5.29612e9	−0.348646	−0.174323	−	0.984689i	$-0.555774\pi$
$811$	−5.29612e9	−0.348646	−0.174323	+	0.984689i	$0.555774\pi$
$812$	0	0
$813$	4.32959e9	0.282573
$814$	0	0
$815$	−2.05843e9	−0.133194
$816$	0	0
$817$	2.60914e10	1.67386
$818$	0	0
$819$	1.28207e9	0.0815486
$820$	0	0
$821$	1.34687e10	0.849425	0.424712	−	0.905328i	$-0.360375\pi$
$821$	1.34687e10	0.849425	0.424712	+	0.905328i	$0.360375\pi$
$822$	0	0
$823$	5.91545e9	0.369903	0.184952	−	0.982748i	$-0.440787\pi$
$823$	5.91545e9	0.369903	0.184952	+	0.982748i	$0.440787\pi$
$824$	0	0
$825$	8.00455e8	0.0496305
$826$	0	0
$827$	−2.16343e9	−0.133007	−0.0665033	−	0.997786i	$-0.521184\pi$
$827$	−2.16343e9	−0.133007	−0.0665033	+	0.997786i	$0.521184\pi$
$828$	0	0
$829$	−1.03884e9	−0.0633301	−0.0316650	−	0.999499i	$-0.510081\pi$
$829$	−1.03884e9	−0.0633301	−0.0316650	+	0.999499i	$0.510081\pi$
$830$	0	0
$831$	1.74712e10	1.05613
$832$	0	0
$833$	−8.57598e9	−0.514075
$834$	0	0
$835$	2.33871e10	1.39019
$836$	0	0
$837$	−1.66673e9	−0.0982484
$838$	0	0
$839$	2.96059e10	1.73066	0.865329	−	0.501205i	$-0.167110\pi$
$839$	2.96059e10	1.73066	0.865329	+	0.501205i	$0.167110\pi$
$840$	0	0
$841$	−1.71264e10	−0.992843
$842$	0	0
$843$	1.02179e10	0.587440
$844$	0	0
$845$	−1.29383e10	−0.737698
$846$	0	0
$847$	1.32038e10	0.746635
$848$	0	0
$849$	−1.20429e10	−0.675391
$850$	0	0
$851$	4.74554e9	0.263956
$852$	0	0
$853$	−1.48670e10	−0.820163	−0.410082	−	0.912049i	$-0.634500\pi$
$853$	−1.48670e10	−0.820163	−0.410082	+	0.912049i	$0.634500\pi$
$854$	0	0
$855$	−7.41078e9	−0.405492
$856$	0	0
$857$	7.25984e8	0.0393998	0.0196999	−	0.999806i	$-0.493729\pi$
$857$	7.25984e8	0.0393998	0.0196999	+	0.999806i	$0.493729\pi$
$858$	0	0
$859$	−9.14667e9	−0.492365	−0.246182	−	0.969224i	$-0.579176\pi$
$859$	−9.14667e9	−0.492365	−0.246182	+	0.969224i	$0.579176\pi$
$860$	0	0
$861$	5.33621e9	0.284919
$862$	0	0
$863$	3.17870e10	1.68349	0.841746	−	0.539873i	$-0.181528\pi$
$863$	3.17870e10	1.68349	0.841746	+	0.539873i	$0.181528\pi$
$864$	0	0
$865$	−2.72599e10	−1.43208
$866$	0	0
$867$	−1.12197e10	−0.584677
$868$	0	0
$869$	−2.67145e9	−0.138095
$870$	0	0
$871$	9.18871e9	0.471185
$872$	0	0
$873$	−5.95354e7	−0.00302849
$874$	0	0
$875$	1.72268e10	0.869313
$876$	0	0
$877$	3.37560e10	1.68987	0.844933	−	0.534873i	$-0.179640\pi$
$877$	3.37560e10	1.68987	0.844933	+	0.534873i	$0.179640\pi$
$878$	0	0
$879$	1.47588e10	0.732975
$880$	0	0
$881$	−1.35501e10	−0.667617	−0.333809	−	0.942641i	$-0.608334\pi$
$881$	−1.35501e10	−0.667617	−0.333809	+	0.942641i	$0.608334\pi$
$882$	0	0
$883$	−3.32363e9	−0.162461	−0.0812307	−	0.996695i	$-0.525885\pi$
$883$	−3.32363e9	−0.162461	−0.0812307	+	0.996695i	$0.525885\pi$
$884$	0	0
$885$	1.10232e10	0.534570
$886$	0	0
$887$	−2.18728e10	−1.05238	−0.526190	−	0.850367i	$-0.676380\pi$
$887$	−2.18728e10	−1.05238	−0.526190	+	0.850367i	$0.676380\pi$
$888$	0	0
$889$	1.39080e10	0.663908
$890$	0	0
$891$	5.98036e8	0.0283241
$892$	0	0
$893$	−5.67614e10	−2.66731
$894$	0	0
$895$	−2.44245e9	−0.113880
$896$	0	0
$897$	1.56101e9	0.0722158
$898$	0	0
$899$	−9.40878e8	−0.0431891
$900$	0	0
$901$	1.06850e10	0.486674
$902$	0	0
$903$	−1.14271e10	−0.516451
$904$	0	0
$905$	−3.57915e9	−0.160513
$906$	0	0
$907$	2.92742e10	1.30275	0.651373	−	0.758757i	$-0.274193\pi$
$907$	2.92742e10	1.30275	0.651373	+	0.758757i	$0.274193\pi$
$908$	0	0
$909$	−6.62977e9	−0.292769
$910$	0	0
$911$	−7.06839e9	−0.309747	−0.154873	−	0.987934i	$-0.549497\pi$
$911$	−7.06839e9	−0.309747	−0.154873	+	0.987934i	$0.549497\pi$
$912$	0	0
$913$	−6.49153e9	−0.282292
$914$	0	0
$915$	1.36423e10	0.588728
$916$	0	0
$917$	1.80174e10	0.771613
$918$	0	0
$919$	−3.15970e10	−1.34289	−0.671446	−	0.741054i	$-0.734326\pi$
$919$	−3.15970e10	−1.34289	−0.671446	+	0.741054i	$0.734326\pi$
$920$	0	0
$921$	1.70239e10	0.718041
$922$	0	0
$923$	1.06810e10	0.447100
$924$	0	0
$925$	−5.24802e9	−0.218022
$926$	0	0
$927$	−1.43773e10	−0.592787
$928$	0	0
$929$	4.01061e10	1.64118	0.820589	−	0.571519i	$-0.193646\pi$
$929$	4.01061e10	1.64118	0.820589	+	0.571519i	$0.193646\pi$
$930$	0	0
$931$	1.33315e10	0.541448
$932$	0	0
$933$	−8.73884e9	−0.352264
$934$	0	0
$935$	7.35890e9	0.294423
$936$	0	0
$937$	1.56498e10	0.621470	0.310735	−	0.950497i	$-0.399425\pi$
$937$	1.56498e10	0.621470	0.310735	+	0.950497i	$0.399425\pi$
$938$	0	0
$939$	−2.32375e10	−0.915925
$940$	0	0
$941$	−7.90172e9	−0.309142	−0.154571	−	0.987982i	$-0.549400\pi$
$941$	−7.90172e9	−0.309142	−0.154571	+	0.987982i	$0.549400\pi$
$942$	0	0
$943$	6.49723e9	0.252312
$944$	0	0
$945$	3.24566e9	0.125110
$946$	0	0
$947$	5.04304e10	1.92960	0.964800	−	0.262985i	$-0.0847069\pi$
$947$	5.04304e10	1.92960	0.964800	+	0.262985i	$0.0847069\pi$
$948$	0	0
$949$	1.12344e10	0.426696
$950$	0	0
$951$	6.04634e9	0.227961
$952$	0	0
$953$	−5.01845e10	−1.87821	−0.939107	−	0.343626i	$-0.888345\pi$
$953$	−5.01845e10	−1.87821	−0.939107	+	0.343626i	$0.888345\pi$
$954$	0	0
$955$	2.32070e10	0.862199
$956$	0	0
$957$	3.37595e8	0.0124510
$958$	0	0
$959$	1.76260e10	0.645339
$960$	0	0
$961$	−2.03422e10	−0.739376
$962$	0	0
$963$	5.90264e9	0.212987
$964$	0	0
$965$	−1.89808e10	−0.679937
$966$	0	0
$967$	−2.67558e10	−0.951535	−0.475767	−	0.879571i	$-0.657830\pi$
$967$	−2.67558e10	−0.951535	−0.475767	+	0.879571i	$0.657830\pi$
$968$	0	0
$969$	3.46641e10	1.22390
$970$	0	0
$971$	−3.60085e10	−1.26223	−0.631115	−	0.775690i	$-0.717402\pi$
$971$	−3.60085e10	−1.26223	−0.631115	+	0.775690i	$0.717402\pi$
$972$	0	0
$973$	2.77623e10	0.966186
$974$	0	0
$975$	−1.72630e9	−0.0596485
$976$	0	0
$977$	2.09028e10	0.717089	0.358544	−	0.933513i	$-0.383273\pi$
$977$	2.09028e10	0.717089	0.358544	+	0.933513i	$0.383273\pi$
$978$	0	0
$979$	1.81277e9	0.0617451
$980$	0	0
$981$	1.38991e10	0.470052
$982$	0	0
$983$	−4.52039e10	−1.51788	−0.758942	−	0.651158i	$-0.774284\pi$
$983$	−4.52039e10	−1.51788	−0.758942	+	0.651158i	$0.774284\pi$
$984$	0	0
$985$	−1.32431e10	−0.441533
$986$	0	0
$987$	2.48595e10	0.822966
$988$	0	0
$989$	−1.39133e10	−0.457345
$990$	0	0
$991$	1.67128e10	0.545495	0.272747	−	0.962086i	$-0.412068\pi$
$991$	1.67128e10	0.545495	0.272747	+	0.962086i	$0.412068\pi$
$992$	0	0
$993$	3.02447e10	0.980227
$994$	0	0
$995$	3.67509e10	1.18273
$996$	0	0
$997$	8.89725e9	0.284330	0.142165	−	0.989843i	$-0.454594\pi$
$997$	8.89725e9	0.284330	0.142165	+	0.989843i	$0.454594\pi$
$998$	0	0
$999$	−3.92090e9	−0.124425

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.a.q.1.2		2
3.2	odd	2		576.8.a.bo.1.1		2
4.3	odd	2		192.8.a.t.1.2		2
8.3	odd	2		96.8.a.e.1.1	✓	2
8.5	even	2		96.8.a.h.1.1	yes	2
12.11	even	2		576.8.a.bn.1.1		2
24.5	odd	2		288.8.a.h.1.2		2
24.11	even	2		288.8.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
96.8.a.e.1.1	✓	2	8.3	odd	2
96.8.a.h.1.1	yes	2	8.5	even	2
192.8.a.q.1.2		2	1.1	even	1	trivial
192.8.a.t.1.2		2	4.3	odd	2
288.8.a.g.1.2		2	24.11	even	2
288.8.a.h.1.2		2	24.5	odd	2
576.8.a.bn.1.1		2	12.11	even	2
576.8.a.bo.1.1		2	3.2	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.a (trivial)

\(f(q)\)	\(=\)	\(q-27.0000 q^{3} +227.552 q^{5} -724.656 q^{7} +729.000 q^{9} +O(q^{10})\)	\(q-27.0000 q^{3} +227.552 q^{5} -724.656 q^{7} +729.000 q^{9} +1125.31 q^{11} -2426.90 q^{13} -6143.90 q^{15} +28738.2 q^{17} -44674.2 q^{19} +19565.7 q^{21} +23822.7 q^{23} -26345.2 q^{25} -19683.0 q^{27} -11111.2 q^{29} +84678.5 q^{31} -30383.4 q^{33} -164897. q^{35} +199202. q^{37} +65526.2 q^{39} +272733. q^{41} -584037. q^{43} +165885. q^{45} +1.27056e6 q^{47} -298417. q^{49} -775932. q^{51} +371805. q^{53} +256067. q^{55} +1.20620e6 q^{57} -1.79416e6 q^{59} -2.22047e6 q^{61} -528274. q^{63} -552245. q^{65} -3.78620e6 q^{67} -643213. q^{69} -4.40108e6 q^{71} -4.62913e6 q^{73} +711319. q^{75} -815463. q^{77} -2.37397e6 q^{79} +531441. q^{81} -5.76865e6 q^{83} +6.53943e6 q^{85} +300002. q^{87} +1.61090e6 q^{89} +1.75866e6 q^{91} -2.28632e6 q^{93} -1.01657e7 q^{95} -81667.2 q^{97} +820352. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9}+O(q^{10}) \) 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9	\( 2 q - 54 q^{3} - 196 q^{5} + 504 q^{7} + 1458 q^{9} - 1656 q^{11} - 6156 q^{13} + 5292 q^{15} + 17108 q^{17} + 504 q^{19} - 13608 q^{21} + 51552 q^{23} + 74926 q^{25} - 39366 q^{27} + 199804 q^{29} + 257256 q^{31} + 44712 q^{33} - 685296 q^{35} + 468724 q^{37} + 166212 q^{39} - 106940 q^{41} - 1617336 q^{43} - 142884 q^{45} + 646416 q^{47} + 387634 q^{49} - 461916 q^{51} - 1469492 q^{53} + 1434096 q^{55} - 13608 q^{57} - 4541544 q^{59} + 481412 q^{61} + 367416 q^{63} + 1027224 q^{65} - 4775256 q^{67} - 1391904 q^{69} - 1094400 q^{71} - 5731884 q^{73} - 2023002 q^{75} - 4232736 q^{77} - 10402776 q^{79} + 1062882 q^{81} - 2212200 q^{83} + 11465432 q^{85} - 5394708 q^{87} - 3604364 q^{89} - 2823120 q^{91} - 6945912 q^{93} - 29300976 q^{95} - 7156188 q^{97} - 1207224 q^{99}+O(q^{100}) \) 2 * q - 54 * q^3 - 196 * q^5 + 504 * q^7 + 1458 * q^9 - 1656 * q^11 - 6156 * q^13 + 5292 * q^15 + 17108 * q^17 + 504 * q^19 - 13608 * q^21 + 51552 * q^23 + 74926 * q^25 - 39366 * q^27 + 199804 * q^29 + 257256 * q^31 + 44712 * q^33 - 685296 * q^35 + 468724 * q^37 + 166212 * q^39 - 106940 * q^41 - 1617336 * q^43 - 142884 * q^45 + 646416 * q^47 + 387634 * q^49 - 461916 * q^51 - 1469492 * q^53 + 1434096 * q^55 - 13608 * q^57 - 4541544 * q^59 + 481412 * q^61 + 367416 * q^63 + 1027224 * q^65 - 4775256 * q^67 - 1391904 * q^69 - 1094400 * q^71 - 5731884 * q^73 - 2023002 * q^75 - 4232736 * q^77 - 10402776 * q^79 + 1062882 * q^81 - 2212200 * q^83 + 11465432 * q^85 - 5394708 * q^87 - 3604364 * q^89 - 2823120 * q^91 - 6945912 * q^93 - 29300976 * q^95 - 7156188 * q^97 - 1207224 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more