LMFDB - Embedded newform 304.8.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,8,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, 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("304.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	304.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:	$94.9650477472$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{633}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 158 $ x^2 - x - 158
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$13.0797$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23924 q^{3} -312.472 q^{5} +766.767 q^{7} -2176.51 q^{9} +O(q^{10})$	$q-3.23924 q^{3} -312.472 q^{5} +766.767 q^{7} -2176.51 q^{9} -252.042 q^{11} -1065.19 q^{13} +1012.17 q^{15} -18769.7 q^{17} +6859.00 q^{19} -2483.74 q^{21} -11559.9 q^{23} +19513.8 q^{25} +14134.4 q^{27} -46290.0 q^{29} -46848.2 q^{31} +816.422 q^{33} -239593. q^{35} -182916. q^{37} +3450.42 q^{39} +819661. q^{41} -477471. q^{43} +680098. q^{45} -992580. q^{47} -235611. q^{49} +60799.7 q^{51} -852516. q^{53} +78756.0 q^{55} -22217.9 q^{57} +1.92404e6 q^{59} +209564. q^{61} -1.66887e6 q^{63} +332843. q^{65} +2.32447e6 q^{67} +37445.1 q^{69} +5.37237e6 q^{71} -3.71614e6 q^{73} -63209.9 q^{75} -193257. q^{77} +1.30851e6 q^{79} +4.71424e6 q^{81} +6.51683e6 q^{83} +5.86502e6 q^{85} +149944. q^{87} +3.99586e6 q^{89} -816756. q^{91} +151753. q^{93} -2.14325e6 q^{95} -6.90391e6 q^{97} +548570. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 69 q^{3} + 155 q^{5} + 2238 q^{7} + 855 q^{9}+O(q^{10}) $ 2 * q + 69 * q^3 + 155 * q^5 + 2238 * q^7 + 855 * q^9	$ 2 q + 69 q^{3} + 155 q^{5} + 2238 q^{7} + 855 q^{9} + 3295 q^{11} - 13427 q^{13} + 34782 q^{15} - 32256 q^{17} + 13718 q^{19} + 103797 q^{21} + 82525 q^{23} + 159919 q^{25} + 75141 q^{27} - 12749 q^{29} - 258944 q^{31} + 257052 q^{33} + 448167 q^{35} - 149260 q^{37} - 889557 q^{39} + 339130 q^{41} + 83869 q^{43} + 2097243 q^{45} - 1471025 q^{47} + 1105372 q^{49} - 913437 q^{51} - 945643 q^{53} + 1736899 q^{55} + 473271 q^{57} + 969009 q^{59} - 1506755 q^{61} + 2791179 q^{63} - 5445956 q^{65} + 1848219 q^{67} + 6834063 q^{69} + 3417184 q^{71} - 2499822 q^{73} + 10079553 q^{75} + 5025267 q^{77} - 2636926 q^{79} + 2491398 q^{81} + 10059354 q^{83} - 439425 q^{85} + 2572923 q^{87} - 3506160 q^{89} - 19003851 q^{91} - 15169884 q^{93} + 1063145 q^{95} + 5893526 q^{97} + 11301453 q^{99}+O(q^{100}) $ 2 * q + 69 * q^3 + 155 * q^5 + 2238 * q^7 + 855 * q^9 + 3295 * q^11 - 13427 * q^13 + 34782 * q^15 - 32256 * q^17 + 13718 * q^19 + 103797 * q^21 + 82525 * q^23 + 159919 * q^25 + 75141 * q^27 - 12749 * q^29 - 258944 * q^31 + 257052 * q^33 + 448167 * q^35 - 149260 * q^37 - 889557 * q^39 + 339130 * q^41 + 83869 * q^43 + 2097243 * q^45 - 1471025 * q^47 + 1105372 * q^49 - 913437 * q^51 - 945643 * q^53 + 1736899 * q^55 + 473271 * q^57 + 969009 * q^59 - 1506755 * q^61 + 2791179 * q^63 - 5445956 * q^65 + 1848219 * q^67 + 6834063 * q^69 + 3417184 * q^71 - 2499822 * q^73 + 10079553 * q^75 + 5025267 * q^77 - 2636926 * q^79 + 2491398 * q^81 + 10059354 * q^83 - 439425 * q^85 + 2572923 * q^87 - 3506160 * q^89 - 19003851 * q^91 - 15169884 * q^93 + 1063145 * q^95 + 5893526 * q^97 + 11301453 * 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
$3$	−3.23924	−0.0692657	−0.0346329	−	0.999400i	$-0.511026\pi$
$3$	−3.23924	−0.0692657	−0.0346329	+	0.999400i	$0.511026\pi$
$4$	0	0
$5$	−312.472	−1.11793	−0.558967	−	0.829190i	$-0.688802\pi$
$5$	−312.472	−1.11793	−0.558967	+	0.829190i	$0.688802\pi$
$6$	0	0
$7$	766.767	0.844929	0.422465	−	0.906379i	$-0.361165\pi$
$7$	766.767	0.844929	0.422465	+	0.906379i	$0.361165\pi$
$8$	0	0
$9$	−2176.51	−0.995202
$10$	0	0
$11$	−252.042	−0.0570950	−0.0285475	−	0.999592i	$-0.509088\pi$
$11$	−252.042	−0.0570950	−0.0285475	+	0.999592i	$0.509088\pi$
$12$	0	0
$13$	−1065.19	−0.134471	−0.0672353	−	0.997737i	$-0.521418\pi$
$13$	−1065.19	−0.134471	−0.0672353	+	0.997737i	$0.521418\pi$
$14$	0	0
$15$	1012.17	0.0774345
$16$	0	0
$17$	−18769.7	−0.926589	−0.463295	−	0.886204i	$-0.653333\pi$
$17$	−18769.7	−0.926589	−0.463295	+	0.886204i	$0.653333\pi$
$18$	0	0
$19$	6859.00	0.229416
$19$	6859.00	0.229416
$20$	0	0
$21$	−2483.74	−0.0585246
$22$	0	0
$23$	−11559.9	−0.198109	−0.0990546	−	0.995082i	$-0.531582\pi$
$23$	−11559.9	−0.198109	−0.0990546	+	0.995082i	$0.531582\pi$
$24$	0	0
$25$	19513.8	0.249777
$26$	0	0
$27$	14134.4	0.138199
$28$	0	0
$29$	−46290.0	−0.352448	−0.176224	−	0.984350i	$-0.556388\pi$
$29$	−46290.0	−0.352448	−0.176224	+	0.984350i	$0.556388\pi$
$30$	0	0
$31$	−46848.2	−0.282441	−0.141220	−	0.989978i	$-0.545103\pi$
$31$	−46848.2	−0.282441	−0.141220	+	0.989978i	$0.545103\pi$
$32$	0	0
$33$	816.422	0.00395472
$34$	0	0
$35$	−239593.	−0.944575
$36$	0	0
$37$	−182916.	−0.593672	−0.296836	−	0.954928i	$-0.595932\pi$
$37$	−182916.	−0.593672	−0.296836	+	0.954928i	$0.595932\pi$
$38$	0	0
$39$	3450.42	0.00931420
$40$	0	0
$41$	819661.	1.85734	0.928669	−	0.370910i	$-0.120954\pi$
$41$	819661.	1.85734	0.928669	+	0.370910i	$0.120954\pi$
$42$	0	0
$43$	−477471.	−0.915814	−0.457907	−	0.889000i	$-0.651401\pi$
$43$	−477471.	−0.915814	−0.457907	+	0.889000i	$0.651401\pi$
$44$	0	0
$45$	680098.	1.11257
$46$	0	0
$47$	−992580.	−1.39451	−0.697257	−	0.716821i	$-0.745596\pi$
$47$	−992580.	−1.39451	−0.697257	+	0.716821i	$0.745596\pi$
$48$	0	0
$49$	−235611.	−0.286095
$50$	0	0
$51$	60799.7	0.0641809
$52$	0	0
$53$	−852516.	−0.786569	−0.393285	−	0.919417i	$-0.628661\pi$
$53$	−852516.	−0.786569	−0.393285	+	0.919417i	$0.628661\pi$
$54$	0	0
$55$	78756.0	0.0638284
$56$	0	0
$57$	−22217.9	−0.0158906
$58$	0	0
$59$	1.92404e6	1.21964	0.609821	−	0.792539i	$-0.291241\pi$
$59$	1.92404e6	1.21964	0.609821	+	0.792539i	$0.291241\pi$
$60$	0	0
$61$	209564.	0.118212	0.0591062	−	0.998252i	$-0.481175\pi$
$61$	209564.	0.118212	0.0591062	+	0.998252i	$0.481175\pi$
$62$	0	0
$63$	−1.66887e6	−0.840875
$64$	0	0
$65$	332843.	0.150329
$66$	0	0
$67$	2.32447e6	0.944198	0.472099	−	0.881546i	$-0.343497\pi$
$67$	2.32447e6	0.944198	0.472099	+	0.881546i	$0.343497\pi$
$68$	0	0
$69$	37445.1	0.0137222
$70$	0	0
$71$	5.37237e6	1.78140	0.890700	−	0.454591i	$-0.150215\pi$
$71$	5.37237e6	1.78140	0.890700	+	0.454591i	$0.150215\pi$
$72$	0	0
$73$	−3.71614e6	−1.11805	−0.559027	−	0.829150i	$-0.688825\pi$
$73$	−3.71614e6	−1.11805	−0.559027	+	0.829150i	$0.688825\pi$
$74$	0	0
$75$	−63209.9	−0.0173010
$76$	0	0
$77$	−193257.	−0.0482412
$78$	0	0
$79$	1.30851e6	0.298596	0.149298	−	0.988792i	$-0.452299\pi$
$79$	1.30851e6	0.298596	0.149298	+	0.988792i	$0.452299\pi$
$80$	0	0
$81$	4.71424e6	0.985630
$82$	0	0
$83$	6.51683e6	1.25102	0.625508	−	0.780218i	$-0.284892\pi$
$83$	6.51683e6	1.25102	0.625508	+	0.780218i	$0.284892\pi$
$84$	0	0
$85$	5.86502e6	1.03587
$86$	0	0
$87$	149944.	0.0244125
$88$	0	0
$89$	3.99586e6	0.600822	0.300411	−	0.953810i	$-0.402876\pi$
$89$	3.99586e6	0.600822	0.300411	+	0.953810i	$0.402876\pi$
$90$	0	0
$91$	−816756.	−0.113618
$92$	0	0
$93$	151753.	0.0195635
$94$	0	0
$95$	−2.14325e6	−0.256472
$96$	0	0
$97$	−6.90391e6	−0.768058	−0.384029	−	0.923321i	$-0.625464\pi$
$97$	−6.90391e6	−0.768058	−0.384029	+	0.923321i	$0.625464\pi$
$98$	0	0
$99$	548570.	0.0568210
$100$	0	0
$101$	2.81610e6	0.271971	0.135986	−	0.990711i	$-0.456580\pi$
$101$	2.81610e6	0.271971	0.135986	+	0.990711i	$0.456580\pi$
$102$	0	0
$103$	−7.86273e6	−0.708995	−0.354497	−	0.935057i	$-0.615348\pi$
$103$	−7.86273e6	−0.708995	−0.354497	+	0.935057i	$0.615348\pi$
$104$	0	0
$105$	776100.	0.0654267
$106$	0	0
$107$	−611626.	−0.0482662	−0.0241331	−	0.999709i	$-0.507683\pi$
$107$	−611626.	−0.0482662	−0.0241331	+	0.999709i	$0.507683\pi$
$108$	0	0
$109$	5.23210e6	0.386976	0.193488	−	0.981103i	$-0.438020\pi$
$109$	5.23210e6	0.386976	0.193488	+	0.981103i	$0.438020\pi$
$110$	0	0
$111$	592510.	0.0411211
$112$	0	0
$113$	−3.58527e6	−0.233747	−0.116874	−	0.993147i	$-0.537287\pi$
$113$	−3.58527e6	−0.233747	−0.116874	+	0.993147i	$0.537287\pi$
$114$	0	0
$115$	3.61213e6	0.221473
$116$	0	0
$117$	2.31840e6	0.133825
$118$	0	0
$119$	−1.43920e7	−0.782902
$120$	0	0
$121$	−1.94236e7	−0.996740
$122$	0	0
$123$	−2.65508e6	−0.128650
$124$	0	0
$125$	1.83144e7	0.838700
$126$	0	0
$127$	2.99465e7	1.29728	0.648639	−	0.761096i	$-0.275339\pi$
$127$	2.99465e7	1.29728	0.648639	+	0.761096i	$0.275339\pi$
$128$	0	0
$129$	1.54664e6	0.0634345
$130$	0	0
$131$	8.11292e6	0.315303	0.157651	−	0.987495i	$-0.449608\pi$
$131$	8.11292e6	0.315303	0.157651	+	0.987495i	$0.449608\pi$
$132$	0	0
$133$	5.25926e6	0.193840
$134$	0	0
$135$	−4.41662e6	−0.154498
$136$	0	0
$137$	1.44157e7	0.478977	0.239488	−	0.970899i	$-0.423020\pi$
$137$	1.44157e7	0.478977	0.239488	+	0.970899i	$0.423020\pi$
$138$	0	0
$139$	3.77446e7	1.19207	0.596036	−	0.802957i	$-0.296741\pi$
$139$	3.77446e7	1.19207	0.596036	+	0.802957i	$0.296741\pi$
$140$	0	0
$141$	3.21520e6	0.0965920
$142$	0	0
$143$	268473.	0.00767759
$144$	0	0
$145$	1.44643e7	0.394013
$146$	0	0
$147$	763200.	0.0198165
$148$	0	0
$149$	7.30679e7	1.80957	0.904784	−	0.425872i	$-0.140033\pi$
$149$	7.30679e7	1.80957	0.904784	+	0.425872i	$0.140033\pi$
$150$	0	0
$151$	4.14056e7	0.978678	0.489339	−	0.872094i	$-0.337238\pi$
$151$	4.14056e7	0.978678	0.489339	+	0.872094i	$0.337238\pi$
$152$	0	0
$153$	4.08525e7	0.922144
$154$	0	0
$155$	1.46388e7	0.315750
$156$	0	0
$157$	2.93945e7	0.606203	0.303101	−	0.952958i	$-0.401978\pi$
$157$	2.93945e7	0.606203	0.303101	+	0.952958i	$0.401978\pi$
$158$	0	0
$159$	2.76150e6	0.0544823
$160$	0	0
$161$	−8.86371e6	−0.167388
$162$	0	0
$163$	−9.03924e6	−0.163484	−0.0817420	−	0.996654i	$-0.526048\pi$
$163$	−9.03924e6	−0.163484	−0.0817420	+	0.996654i	$0.526048\pi$
$164$	0	0
$165$	−255109.	−0.00442112
$166$	0	0
$167$	8.16728e7	1.35697	0.678484	−	0.734615i	$-0.262637\pi$
$167$	8.16728e7	1.35697	0.678484	+	0.734615i	$0.262637\pi$
$168$	0	0
$169$	−6.16139e7	−0.981918
$170$	0	0
$171$	−1.49287e7	−0.228315
$172$	0	0
$173$	1.33191e8	1.95575	0.977874	−	0.209197i	$-0.0670849\pi$
$173$	1.33191e8	1.95575	0.977874	+	0.209197i	$0.0670849\pi$
$174$	0	0
$175$	1.49626e7	0.211044
$176$	0	0
$177$	−6.23243e6	−0.0844794
$178$	0	0
$179$	4.90910e7	0.639759	0.319879	−	0.947458i	$-0.396358\pi$
$179$	4.90910e7	0.639759	0.319879	+	0.947458i	$0.396358\pi$
$180$	0	0
$181$	8.11979e7	1.01782	0.508908	−	0.860821i	$-0.330049\pi$
$181$	8.11979e7	1.01782	0.508908	+	0.860821i	$0.330049\pi$
$182$	0	0
$183$	−678828.	−0.00818806
$184$	0	0
$185$	5.71563e7	0.663686
$186$	0	0
$187$	4.73076e6	0.0529036
$188$	0	0
$189$	1.08378e7	0.116768
$190$	0	0
$191$	−7.40713e7	−0.769190	−0.384595	−	0.923086i	$-0.625659\pi$
$191$	−7.40713e7	−0.769190	−0.384595	+	0.923086i	$0.625659\pi$
$192$	0	0
$193$	−9.70125e7	−0.971353	−0.485676	−	0.874139i	$-0.661427\pi$
$193$	−9.70125e7	−0.971353	−0.485676	+	0.874139i	$0.661427\pi$
$194$	0	0
$195$	−1.07816e6	−0.0104127
$196$	0	0
$197$	−4.34770e7	−0.405161	−0.202581	−	0.979266i	$-0.564933\pi$
$197$	−4.34770e7	−0.405161	−0.202581	+	0.979266i	$0.564933\pi$
$198$	0	0
$199$	−3.89651e7	−0.350502	−0.175251	−	0.984524i	$-0.556074\pi$
$199$	−3.89651e7	−0.350502	−0.175251	+	0.984524i	$0.556074\pi$
$200$	0	0
$201$	−7.52952e6	−0.0654005
$202$	0	0
$203$	−3.54937e7	−0.297793
$204$	0	0
$205$	−2.56121e8	−2.07638
$206$	0	0
$207$	2.51601e7	0.197159
$208$	0	0
$209$	−1.72875e6	−0.0130985
$210$	0	0
$211$	3.85264e7	0.282339	0.141169	−	0.989985i	$-0.454914\pi$
$211$	3.85264e7	0.282339	0.141169	+	0.989985i	$0.454914\pi$
$212$	0	0
$213$	−1.74024e7	−0.123390
$214$	0	0
$215$	1.49196e8	1.02382
$216$	0	0
$217$	−3.59217e7	−0.238642
$218$	0	0
$219$	1.20375e7	0.0774428
$220$	0	0
$221$	1.99934e7	0.124599
$222$	0	0
$223$	2.03401e8	1.22825	0.614124	−	0.789209i	$-0.289509\pi$
$223$	2.03401e8	1.22825	0.614124	+	0.789209i	$0.289509\pi$
$224$	0	0
$225$	−4.24720e7	−0.248579
$226$	0	0
$227$	−2.24944e8	−1.27639	−0.638195	−	0.769875i	$-0.720319\pi$
$227$	−2.24944e8	−1.27639	−0.638195	+	0.769875i	$0.720319\pi$
$228$	0	0
$229$	−2.38882e8	−1.31450	−0.657248	−	0.753674i	$-0.728280\pi$
$229$	−2.38882e8	−1.31450	−0.657248	+	0.753674i	$0.728280\pi$
$230$	0	0
$231$	626006.	0.00334146
$232$	0	0
$233$	3.23019e8	1.67295	0.836475	−	0.548005i	$-0.184613\pi$
$233$	3.23019e8	1.67295	0.836475	+	0.548005i	$0.184613\pi$
$234$	0	0
$235$	3.10153e8	1.55897
$236$	0	0
$237$	−4.23859e6	−0.0206825
$238$	0	0
$239$	5.18160e7	0.245511	0.122756	−	0.992437i	$-0.460827\pi$
$239$	5.18160e7	0.245511	0.122756	+	0.992437i	$0.460827\pi$
$240$	0	0
$241$	1.15926e8	0.533484	0.266742	−	0.963768i	$-0.414053\pi$
$241$	1.15926e8	0.533484	0.266742	+	0.963768i	$0.414053\pi$
$242$	0	0
$243$	−4.61825e7	−0.206469
$244$	0	0
$245$	7.36219e7	0.319835
$246$	0	0
$247$	−7.30617e6	−0.0308497
$248$	0	0
$249$	−2.11096e7	−0.0866525
$250$	0	0
$251$	4.03761e7	0.161163	0.0805817	−	0.996748i	$-0.474322\pi$
$251$	4.03761e7	0.161163	0.0805817	+	0.996748i	$0.474322\pi$
$252$	0	0
$253$	2.91356e6	0.0113110
$254$	0	0
$255$	−1.89982e7	−0.0717500
$256$	0	0
$257$	2.67661e8	0.983602	0.491801	−	0.870708i	$-0.336339\pi$
$257$	2.67661e8	0.983602	0.491801	+	0.870708i	$0.336339\pi$
$258$	0	0
$259$	−1.40254e8	−0.501611
$260$	0	0
$261$	1.00751e8	0.350757
$262$	0	0
$263$	−5.12778e8	−1.73814	−0.869068	−	0.494692i	$-0.835281\pi$
$263$	−5.12778e8	−1.73814	−0.869068	+	0.494692i	$0.835281\pi$
$264$	0	0
$265$	2.66387e8	0.879333
$266$	0	0
$267$	−1.29435e7	−0.0416163
$268$	0	0
$269$	5.75792e8	1.80357	0.901784	−	0.432187i	$-0.142258\pi$
$269$	5.75792e8	1.80357	0.901784	+	0.432187i	$0.142258\pi$
$270$	0	0
$271$	−5.24839e8	−1.60189	−0.800946	−	0.598737i	$-0.795670\pi$
$271$	−5.24839e8	−1.60189	−0.800946	+	0.598737i	$0.795670\pi$
$272$	0	0
$273$	2.64567e6	0.00786984
$274$	0	0
$275$	−4.91829e6	−0.0142610
$276$	0	0
$277$	−4.64620e8	−1.31347	−0.656733	−	0.754123i	$-0.728062\pi$
$277$	−4.64620e8	−1.31347	−0.656733	+	0.754123i	$0.728062\pi$
$278$	0	0
$279$	1.01966e8	0.281086
$280$	0	0
$281$	−1.75429e8	−0.471660	−0.235830	−	0.971794i	$-0.575781\pi$
$281$	−1.75429e8	−0.471660	−0.235830	+	0.971794i	$0.575781\pi$
$282$	0	0
$283$	−1.44948e8	−0.380154	−0.190077	−	0.981769i	$-0.560874\pi$
$283$	−1.44948e8	−0.380154	−0.190077	+	0.981769i	$0.560874\pi$
$284$	0	0
$285$	6.94248e6	0.0177647
$286$	0	0
$287$	6.28489e8	1.56932
$288$	0	0
$289$	−5.80353e7	−0.141433
$290$	0	0
$291$	2.23634e7	0.0532001
$292$	0	0
$293$	−4.24522e8	−0.985970	−0.492985	−	0.870038i	$-0.664094\pi$
$293$	−4.24522e8	−0.985970	−0.492985	+	0.870038i	$0.664094\pi$
$294$	0	0
$295$	−6.01210e8	−1.36348
$296$	0	0
$297$	−3.56247e6	−0.00789047
$298$	0	0
$299$	1.23135e7	0.0266399
$300$	0	0
$301$	−3.66109e8	−0.773798
$302$	0	0
$303$	−9.12200e6	−0.0188383
$304$	0	0
$305$	−6.54830e7	−0.132154
$306$	0	0
$307$	−4.27528e8	−0.843297	−0.421649	−	0.906759i	$-0.638548\pi$
$307$	−4.27528e8	−0.843297	−0.421649	+	0.906759i	$0.638548\pi$
$308$	0	0
$309$	2.54692e7	0.0491090
$310$	0	0
$311$	1.70275e8	0.320989	0.160495	−	0.987037i	$-0.448691\pi$
$311$	1.70275e8	0.320989	0.160495	+	0.987037i	$0.448691\pi$
$312$	0	0
$313$	−4.02143e8	−0.741269	−0.370634	−	0.928779i	$-0.620860\pi$
$313$	−4.02143e8	−0.741269	−0.370634	+	0.928779i	$0.620860\pi$
$314$	0	0
$315$	5.21477e8	0.940043
$316$	0	0
$317$	−1.90630e8	−0.336112	−0.168056	−	0.985777i	$-0.553749\pi$
$317$	−1.90630e8	−0.336112	−0.168056	+	0.985777i	$0.553749\pi$
$318$	0	0
$319$	1.16670e7	0.0201230
$320$	0	0
$321$	1.98120e6	0.00334319
$322$	0	0
$323$	−1.28742e8	−0.212574
$324$	0	0
$325$	−2.07860e7	−0.0335876
$326$	0	0
$327$	−1.69480e7	−0.0268041
$328$	0	0
$329$	−7.61077e8	−1.17827
$330$	0	0
$331$	4.12008e8	0.624464	0.312232	−	0.950006i	$-0.398923\pi$
$331$	4.12008e8	0.624464	0.312232	+	0.950006i	$0.398923\pi$
$332$	0	0
$333$	3.98119e8	0.590824
$334$	0	0
$335$	−7.26333e8	−1.05555
$336$	0	0
$337$	−6.99272e8	−0.995271	−0.497636	−	0.867386i	$-0.665798\pi$
$337$	−6.99272e8	−0.995271	−0.497636	+	0.867386i	$0.665798\pi$
$338$	0	0
$339$	1.16135e7	0.0161907
$340$	0	0
$341$	1.18077e7	0.0161259
$342$	0	0
$343$	−8.12125e8	−1.08666
$344$	0	0
$345$	−1.17005e7	−0.0153405
$346$	0	0
$347$	9.29885e8	1.19475	0.597374	−	0.801963i	$-0.296211\pi$
$347$	9.29885e8	1.19475	0.597374	+	0.801963i	$0.296211\pi$
$348$	0	0
$349$	−4.43776e8	−0.558823	−0.279412	−	0.960171i	$-0.590139\pi$
$349$	−4.43776e8	−0.558823	−0.279412	+	0.960171i	$0.590139\pi$
$350$	0	0
$351$	−1.50559e7	−0.0185837
$352$	0	0
$353$	−1.49388e9	−1.80761	−0.903804	−	0.427946i	$-0.859237\pi$
$353$	−1.49388e9	−1.80761	−0.903804	+	0.427946i	$0.859237\pi$
$354$	0	0
$355$	−1.67872e9	−1.99149
$356$	0	0
$357$	4.66192e7	0.0542283
$358$	0	0
$359$	1.77937e8	0.202972	0.101486	−	0.994837i	$-0.467640\pi$
$359$	1.77937e8	0.202972	0.101486	+	0.994837i	$0.467640\pi$
$360$	0	0
$361$	4.70459e7	0.0526316
$362$	0	0
$363$	6.29178e7	0.0690399
$364$	0	0
$365$	1.16119e9	1.24991
$366$	0	0
$367$	1.37515e9	1.45218	0.726088	−	0.687601i	$-0.241336\pi$
$367$	1.37515e9	1.45218	0.726088	+	0.687601i	$0.241336\pi$
$368$	0	0
$369$	−1.78400e9	−1.84843
$370$	0	0
$371$	−6.53681e8	−0.664595
$372$	0	0
$373$	1.55845e9	1.55493	0.777466	−	0.628925i	$-0.216505\pi$
$373$	1.55845e9	1.55493	0.777466	+	0.628925i	$0.216505\pi$
$374$	0	0
$375$	−5.93245e7	−0.0580932
$376$	0	0
$377$	4.93079e7	0.0473938
$378$	0	0
$379$	−8.84549e8	−0.834613	−0.417306	−	0.908766i	$-0.637026\pi$
$379$	−8.84549e8	−0.834613	−0.417306	+	0.908766i	$0.637026\pi$
$380$	0	0
$381$	−9.70038e7	−0.0898569
$382$	0	0
$383$	−1.44573e9	−1.31490	−0.657449	−	0.753499i	$-0.728365\pi$
$383$	−1.44573e9	−1.31490	−0.657449	+	0.753499i	$0.728365\pi$
$384$	0	0
$385$	6.03875e7	0.0539305
$386$	0	0
$387$	1.03922e9	0.911420
$388$	0	0
$389$	−1.02164e9	−0.879986	−0.439993	−	0.898001i	$-0.645019\pi$
$389$	−1.02164e9	−0.879986	−0.439993	+	0.898001i	$0.645019\pi$
$390$	0	0
$391$	2.16975e8	0.183566
$392$	0	0
$393$	−2.62797e7	−0.0218397
$394$	0	0
$395$	−4.08874e8	−0.333811
$396$	0	0
$397$	1.80050e8	0.144419	0.0722097	−	0.997389i	$-0.476995\pi$
$397$	1.80050e8	0.144419	0.0722097	+	0.997389i	$0.476995\pi$
$398$	0	0
$399$	−1.70360e7	−0.0134265
$400$	0	0
$401$	5.06670e8	0.392392	0.196196	−	0.980565i	$-0.437141\pi$
$401$	5.06670e8	0.392392	0.196196	+	0.980565i	$0.437141\pi$
$402$	0	0
$403$	4.99025e7	0.0379799
$404$	0	0
$405$	−1.47307e9	−1.10187
$406$	0	0
$407$	4.61026e7	0.0338957
$408$	0	0
$409$	1.14898e9	0.830388	0.415194	−	0.909733i	$-0.363714\pi$
$409$	1.14898e9	0.830388	0.415194	+	0.909733i	$0.363714\pi$
$410$	0	0
$411$	−4.66959e7	−0.0331767
$412$	0	0
$413$	1.47529e9	1.03051
$414$	0	0
$415$	−2.03633e9	−1.39855
$416$	0	0
$417$	−1.22264e8	−0.0825698
$418$	0	0
$419$	−2.02289e9	−1.34346	−0.671729	−	0.740797i	$-0.734448\pi$
$419$	−2.02289e9	−1.34346	−0.671729	+	0.740797i	$0.734448\pi$
$420$	0	0
$421$	2.55503e9	1.66882	0.834408	−	0.551147i	$-0.185810\pi$
$421$	2.55503e9	1.66882	0.834408	+	0.551147i	$0.185810\pi$
$422$	0	0
$423$	2.16036e9	1.38782
$424$	0	0
$425$	−3.66269e8	−0.231441
$426$	0	0
$427$	1.60687e8	0.0998811
$428$	0	0
$429$	−869648.	−0.000531794 0
$430$	0	0
$431$	−1.47316e9	−0.886295	−0.443147	−	0.896449i	$-0.646138\pi$
$431$	−1.47316e9	−0.886295	−0.443147	+	0.896449i	$0.646138\pi$
$432$	0	0
$433$	−9.33243e8	−0.552443	−0.276221	−	0.961094i	$-0.589082\pi$
$433$	−9.33243e8	−0.552443	−0.276221	+	0.961094i	$0.589082\pi$
$434$	0	0
$435$	−4.68534e7	−0.0272916
$436$	0	0
$437$	−7.92890e7	−0.0454494
$438$	0	0
$439$	1.94411e9	1.09672	0.548358	−	0.836244i	$-0.315253\pi$
$439$	1.94411e9	1.09672	0.548358	+	0.836244i	$0.315253\pi$
$440$	0	0
$441$	5.12809e8	0.284722
$442$	0	0
$443$	2.35592e9	1.28750	0.643751	−	0.765235i	$-0.277377\pi$
$443$	2.35592e9	1.28750	0.643751	+	0.765235i	$0.277377\pi$
$444$	0	0
$445$	−1.24860e9	−0.671679
$446$	0	0
$447$	−2.36684e8	−0.125341
$448$	0	0
$449$	−1.83374e9	−0.956042	−0.478021	−	0.878349i	$-0.658646\pi$
$449$	−1.83374e9	−0.956042	−0.478021	+	0.878349i	$0.658646\pi$
$450$	0	0
$451$	−2.06589e8	−0.106045
$452$	0	0
$453$	−1.34123e8	−0.0677889
$454$	0	0
$455$	2.55213e8	0.127018
$456$	0	0
$457$	−3.76683e9	−1.84616	−0.923079	−	0.384611i	$-0.874336\pi$
$457$	−3.76683e9	−1.84616	−0.923079	+	0.384611i	$0.874336\pi$
$458$	0	0
$459$	−2.65300e8	−0.128054
$460$	0	0
$461$	3.30560e9	1.57144	0.785720	−	0.618583i	$-0.212293\pi$
$461$	3.30560e9	1.57144	0.785720	+	0.618583i	$0.212293\pi$
$462$	0	0
$463$	3.66326e9	1.71528	0.857638	−	0.514253i	$-0.171931\pi$
$463$	3.66326e9	1.71528	0.857638	+	0.514253i	$0.171931\pi$
$464$	0	0
$465$	−4.74184e7	−0.0218707
$466$	0	0
$467$	2.22674e9	1.01172	0.505860	−	0.862616i	$-0.331175\pi$
$467$	2.22674e9	1.01172	0.505860	+	0.862616i	$0.331175\pi$
$468$	0	0
$469$	1.78233e9	0.797780
$470$	0	0
$471$	−9.52159e7	−0.0419891
$472$	0	0
$473$	1.20342e8	0.0522883
$474$	0	0
$475$	1.33845e8	0.0573028
$476$	0	0
$477$	1.85551e9	0.782796
$478$	0	0
$479$	−1.43508e9	−0.596625	−0.298313	−	0.954468i	$-0.596424\pi$
$479$	−1.43508e9	−0.596625	−0.298313	+	0.954468i	$0.596424\pi$
$480$	0	0
$481$	1.94842e8	0.0798314
$482$	0	0
$483$	2.87117e7	0.0115943
$484$	0	0
$485$	2.15728e9	0.858638
$486$	0	0
$487$	2.11642e9	0.830330	0.415165	−	0.909746i	$-0.363724\pi$
$487$	2.11642e9	0.830330	0.415165	+	0.909746i	$0.363724\pi$
$488$	0	0
$489$	2.92802e7	0.0113238
$490$	0	0
$491$	−4.07105e9	−1.55211	−0.776053	−	0.630668i	$-0.782781\pi$
$491$	−4.07105e9	−1.55211	−0.776053	+	0.630668i	$0.782781\pi$
$492$	0	0
$493$	8.68852e8	0.326574
$494$	0	0
$495$	−1.71413e8	−0.0635222
$496$	0	0
$497$	4.11935e9	1.50516
$498$	0	0
$499$	−3.93039e9	−1.41607	−0.708034	−	0.706178i	$-0.750418\pi$
$499$	−3.93039e9	−1.41607	−0.708034	+	0.706178i	$0.750418\pi$
$500$	0	0
$501$	−2.64558e8	−0.0939914
$502$	0	0
$503$	1.70030e9	0.595714	0.297857	−	0.954610i	$-0.403728\pi$
$503$	1.70030e9	0.595714	0.297857	+	0.954610i	$0.403728\pi$
$504$	0	0
$505$	−8.79952e8	−0.304046
$506$	0	0
$507$	1.99582e8	0.0680132
$508$	0	0
$509$	−2.62955e8	−0.0883830	−0.0441915	−	0.999023i	$-0.514071\pi$
$509$	−2.62955e8	−0.0883830	−0.0441915	+	0.999023i	$0.514071\pi$
$510$	0	0
$511$	−2.84942e9	−0.944676
$512$	0	0
$513$	9.69481e7	0.0317051
$514$	0	0
$515$	2.45688e9	0.792609
$516$	0	0
$517$	2.50171e8	0.0796197
$518$	0	0
$519$	−4.31436e8	−0.135466
$520$	0	0
$521$	−1.86990e9	−0.579277	−0.289639	−	0.957136i	$-0.593535\pi$
$521$	−1.86990e9	−0.579277	−0.289639	+	0.957136i	$0.593535\pi$
$522$	0	0
$523$	5.49460e9	1.67950	0.839750	−	0.542972i	$-0.182701\pi$
$523$	5.49460e9	1.67950	0.839750	+	0.542972i	$0.182701\pi$
$524$	0	0
$525$	−4.84673e7	−0.0146181
$526$	0	0
$527$	8.79329e8	0.261706
$528$	0	0
$529$	−3.27120e9	−0.960753
$530$	0	0
$531$	−4.18769e9	−1.21379
$532$	0	0
$533$	−8.73098e8	−0.249757
$534$	0	0
$535$	1.91116e8	0.0539584
$536$	0	0
$537$	−1.59017e8	−0.0443133
$538$	0	0
$539$	5.93838e7	0.0163346
$540$	0	0
$541$	6.97892e9	1.89495	0.947475	−	0.319830i	$-0.103626\pi$
$541$	6.97892e9	1.89495	0.947475	+	0.319830i	$0.103626\pi$
$542$	0	0
$543$	−2.63019e8	−0.0704998
$544$	0	0
$545$	−1.63489e9	−0.432613
$546$	0	0
$547$	5.87859e9	1.53574	0.767870	−	0.640606i	$-0.221317\pi$
$547$	5.87859e9	1.53574	0.767870	+	0.640606i	$0.221317\pi$
$548$	0	0
$549$	−4.56118e8	−0.117645
$550$	0	0
$551$	−3.17503e8	−0.0808570
$552$	0	0
$553$	1.00333e9	0.252292
$554$	0	0
$555$	−1.85143e8	−0.0459707
$556$	0	0
$557$	−2.03155e9	−0.498119	−0.249060	−	0.968488i	$-0.580122\pi$
$557$	−2.03155e9	−0.498119	−0.249060	+	0.968488i	$0.580122\pi$
$558$	0	0
$559$	5.08599e8	0.123150
$560$	0	0
$561$	−1.53240e7	−0.00366440
$562$	0	0
$563$	−1.68161e9	−0.397142	−0.198571	−	0.980087i	$-0.563630\pi$
$563$	−1.68161e9	−0.397142	−0.198571	+	0.980087i	$0.563630\pi$
$564$	0	0
$565$	1.12030e9	0.261314
$566$	0	0
$567$	3.61472e9	0.832787
$568$	0	0
$569$	−4.57746e9	−1.04167	−0.520837	−	0.853656i	$-0.674380\pi$
$569$	−4.57746e9	−1.04167	−0.520837	+	0.853656i	$0.674380\pi$
$570$	0	0
$571$	5.79221e9	1.30202	0.651010	−	0.759069i	$-0.274345\pi$
$571$	5.79221e9	1.30202	0.651010	+	0.759069i	$0.274345\pi$
$572$	0	0
$573$	2.39935e8	0.0532785
$574$	0	0
$575$	−2.25577e8	−0.0494831
$576$	0	0
$577$	−3.74692e9	−0.812005	−0.406003	−	0.913872i	$-0.633078\pi$
$577$	−3.74692e9	−0.812005	−0.406003	+	0.913872i	$0.633078\pi$
$578$	0	0
$579$	3.14246e8	0.0672814
$580$	0	0
$581$	4.99689e9	1.05702
$582$	0	0
$583$	2.14869e8	0.0449091
$584$	0	0
$585$	−7.24436e8	−0.149608
$586$	0	0
$587$	5.69193e9	1.16152	0.580760	−	0.814075i	$-0.302756\pi$
$587$	5.69193e9	1.16152	0.580760	+	0.814075i	$0.302756\pi$
$588$	0	0
$589$	−3.21332e8	−0.0647963
$590$	0	0
$591$	1.40832e8	0.0280638
$592$	0	0
$593$	1.45596e9	0.286720	0.143360	−	0.989671i	$-0.454209\pi$
$593$	1.45596e9	0.286720	0.143360	+	0.989671i	$0.454209\pi$
$594$	0	0
$595$	4.49711e9	0.875233
$596$	0	0
$597$	1.26217e8	0.0242778
$598$	0	0
$599$	6.21735e9	1.18198	0.590992	−	0.806677i	$-0.298736\pi$
$599$	6.21735e9	1.18198	0.590992	+	0.806677i	$0.298736\pi$
$600$	0	0
$601$	9.98776e9	1.87675	0.938377	−	0.345613i	$-0.112329\pi$
$601$	9.98776e9	1.87675	0.938377	+	0.345613i	$0.112329\pi$
$602$	0	0
$603$	−5.05924e9	−0.939668
$604$	0	0
$605$	6.06935e9	1.11429
$606$	0	0
$607$	−3.17827e9	−0.576807	−0.288404	−	0.957509i	$-0.593124\pi$
$607$	−3.17827e9	−0.576807	−0.288404	+	0.957509i	$0.593124\pi$
$608$	0	0
$609$	1.14972e8	0.0206269
$610$	0	0
$611$	1.05729e9	0.187521
$612$	0	0
$613$	−6.34969e8	−0.111337	−0.0556687	−	0.998449i	$-0.517729\pi$
$613$	−6.34969e8	−0.111337	−0.0556687	+	0.998449i	$0.517729\pi$
$614$	0	0
$615$	8.29637e8	0.143822
$616$	0	0
$617$	−4.48835e7	−0.00769287	−0.00384643	−	0.999993i	$-0.501224\pi$
$617$	−4.48835e7	−0.00769287	−0.00384643	+	0.999993i	$0.501224\pi$
$618$	0	0
$619$	9.79132e8	0.165930	0.0829648	−	0.996552i	$-0.473561\pi$
$619$	9.79132e8	0.165930	0.0829648	+	0.996552i	$0.473561\pi$
$620$	0	0
$621$	−1.63392e8	−0.0273785
$622$	0	0
$623$	3.06390e9	0.507652
$624$	0	0
$625$	−7.24724e9	−1.18739
$626$	0	0
$627$	5.59984e6	0.000907276 0
$628$	0	0
$629$	3.43330e9	0.550090
$630$	0	0
$631$	2.42563e9	0.384345	0.192172	−	0.981361i	$-0.438447\pi$
$631$	2.42563e9	0.384345	0.192172	+	0.981361i	$0.438447\pi$
$632$	0	0
$633$	−1.24796e8	−0.0195564
$634$	0	0
$635$	−9.35744e9	−1.45027
$636$	0	0
$637$	2.50972e8	0.0384713
$638$	0	0
$639$	−1.16930e10	−1.77285
$640$	0	0
$641$	9.23132e8	0.138440	0.0692199	−	0.997601i	$-0.477949\pi$
$641$	9.23132e8	0.138440	0.0692199	+	0.997601i	$0.477949\pi$
$642$	0	0
$643$	−6.86334e9	−1.01812	−0.509058	−	0.860732i	$-0.670006\pi$
$643$	−6.86334e9	−1.01812	−0.509058	+	0.860732i	$0.670006\pi$
$644$	0	0
$645$	−4.83282e8	−0.0709156
$646$	0	0
$647$	−5.02399e8	−0.0729262	−0.0364631	−	0.999335i	$-0.511609\pi$
$647$	−5.02399e8	−0.0729262	−0.0364631	+	0.999335i	$0.511609\pi$
$648$	0	0
$649$	−4.84939e8	−0.0696355
$650$	0	0
$651$	1.16359e8	0.0165297
$652$	0	0
$653$	4.23523e9	0.595225	0.297612	−	0.954687i	$-0.403810\pi$
$653$	4.23523e9	0.595225	0.297612	+	0.954687i	$0.403810\pi$
$654$	0	0
$655$	−2.53506e9	−0.352488
$656$	0	0
$657$	8.08822e9	1.11269
$658$	0	0
$659$	1.34554e10	1.83146	0.915731	−	0.401791i	$-0.131612\pi$
$659$	1.34554e10	1.83146	0.915731	+	0.401791i	$0.131612\pi$
$660$	0	0
$661$	1.23518e10	1.66351	0.831753	−	0.555146i	$-0.187338\pi$
$661$	1.23518e10	1.66351	0.831753	+	0.555146i	$0.187338\pi$
$662$	0	0
$663$	−6.47634e7	−0.00863043
$664$	0	0
$665$	−1.64337e9	−0.216700
$666$	0	0
$667$	5.35106e8	0.0698231
$668$	0	0
$669$	−6.58864e8	−0.0850755
$670$	0	0
$671$	−5.28189e7	−0.00674933
$672$	0	0
$673$	−1.52638e9	−0.193024	−0.0965119	−	0.995332i	$-0.530769\pi$
$673$	−1.52638e9	−0.193024	−0.0965119	+	0.995332i	$0.530769\pi$
$674$	0	0
$675$	2.75817e8	0.0345189
$676$	0	0
$677$	−1.55415e9	−0.192500	−0.0962502	−	0.995357i	$-0.530685\pi$
$677$	−1.55415e9	−0.192500	−0.0962502	+	0.995357i	$0.530685\pi$
$678$	0	0
$679$	−5.29369e9	−0.648954
$680$	0	0
$681$	7.28646e8	0.0884100
$682$	0	0
$683$	3.21824e9	0.386497	0.193249	−	0.981150i	$-0.438098\pi$
$683$	3.21824e9	0.386497	0.193249	+	0.981150i	$0.438098\pi$
$684$	0	0
$685$	−4.50451e9	−0.535465
$686$	0	0
$687$	7.73796e8	0.0910495
$688$	0	0
$689$	9.08095e8	0.105770
$690$	0	0
$691$	−9.15999e9	−1.05614	−0.528071	−	0.849200i	$-0.677084\pi$
$691$	−9.15999e9	−1.05614	−0.528071	+	0.849200i	$0.677084\pi$
$692$	0	0
$693$	4.20626e8	0.0480098
$694$	0	0
$695$	−1.17941e10	−1.33266
$696$	0	0
$697$	−1.53848e10	−1.72099
$698$	0	0
$699$	−1.04634e9	−0.115878
$700$	0	0
$701$	−7.97717e9	−0.874653	−0.437327	−	0.899303i	$-0.644075\pi$
$701$	−7.97717e9	−0.874653	−0.437327	+	0.899303i	$0.644075\pi$
$702$	0	0
$703$	−1.25462e9	−0.136198
$704$	0	0
$705$	−1.00466e9	−0.107984
$706$	0	0
$707$	2.15929e9	0.229796
$708$	0	0
$709$	7.97010e9	0.839851	0.419925	−	0.907559i	$-0.362056\pi$
$709$	7.97010e9	0.839851	0.419925	+	0.907559i	$0.362056\pi$
$710$	0	0
$711$	−2.84799e9	−0.297163
$712$	0	0
$713$	5.41559e8	0.0559541
$714$	0	0
$715$	−8.38904e7	−0.00858304
$716$	0	0
$717$	−1.67844e8	−0.0170055
$718$	0	0
$719$	−1.00263e10	−1.00598	−0.502991	−	0.864292i	$-0.667767\pi$
$719$	−1.00263e10	−1.00598	−0.502991	+	0.864292i	$0.667767\pi$
$720$	0	0
$721$	−6.02888e9	−0.599050
$722$	0	0
$723$	−3.75512e8	−0.0369522
$724$	0	0
$725$	−9.03295e8	−0.0880333
$726$	0	0
$727$	1.65401e10	1.59650	0.798248	−	0.602329i	$-0.205760\pi$
$727$	1.65401e10	1.59650	0.798248	+	0.602329i	$0.205760\pi$
$728$	0	0
$729$	−1.01604e10	−0.971329
$730$	0	0
$731$	8.96200e9	0.848583
$732$	0	0
$733$	9.81994e9	0.920969	0.460484	−	0.887668i	$-0.347676\pi$
$733$	9.81994e9	0.920969	0.460484	+	0.887668i	$0.347676\pi$
$734$	0	0
$735$	−2.38479e8	−0.0221536
$736$	0	0
$737$	−5.85864e8	−0.0539089
$738$	0	0
$739$	−5.41304e9	−0.493385	−0.246692	−	0.969094i	$-0.579344\pi$
$739$	−5.41304e9	−0.493385	−0.246692	+	0.969094i	$0.579344\pi$
$740$	0	0
$741$	2.36664e7	0.00213682
$742$	0	0
$743$	8.26105e9	0.738881	0.369440	−	0.929254i	$-0.379549\pi$
$743$	8.26105e9	0.738881	0.369440	+	0.929254i	$0.379549\pi$
$744$	0	0
$745$	−2.28317e10	−2.02298
$746$	0	0
$747$	−1.41839e10	−1.24501
$748$	0	0
$749$	−4.68975e8	−0.0407815
$750$	0	0
$751$	1.44332e10	1.24343	0.621715	−	0.783243i	$-0.286436\pi$
$751$	1.44332e10	1.24343	0.621715	+	0.783243i	$0.286436\pi$
$752$	0	0
$753$	−1.30788e8	−0.0111631
$754$	0	0
$755$	−1.29381e10	−1.09410
$756$	0	0
$757$	−8.31294e9	−0.696496	−0.348248	−	0.937402i	$-0.613223\pi$
$757$	−8.31294e9	−0.696496	−0.348248	+	0.937402i	$0.613223\pi$
$758$	0	0
$759$	−9.43772e6	−0.000783467 0
$760$	0	0
$761$	−1.96636e10	−1.61739	−0.808697	−	0.588226i	$-0.799827\pi$
$761$	−1.96636e10	−1.61739	−0.808697	+	0.588226i	$0.799827\pi$
$762$	0	0
$763$	4.01180e9	0.326967
$764$	0	0
$765$	−1.27653e10	−1.03090
$766$	0	0
$767$	−2.04948e9	−0.164006
$768$	0	0
$769$	1.27102e10	1.00788	0.503942	−	0.863738i	$-0.331883\pi$
$769$	1.27102e10	1.00788	0.503942	+	0.863738i	$0.331883\pi$
$770$	0	0
$771$	−8.67018e8	−0.0681299
$772$	0	0
$773$	−2.87614e9	−0.223966	−0.111983	−	0.993710i	$-0.535720\pi$
$773$	−2.87614e9	−0.223966	−0.111983	+	0.993710i	$0.535720\pi$
$774$	0	0
$775$	−9.14188e8	−0.0705472
$776$	0	0
$777$	4.54317e8	0.0347444
$778$	0	0
$779$	5.62206e9	0.426102
$780$	0	0
$781$	−1.35406e9	−0.101709
$782$	0	0
$783$	−6.54283e8	−0.0487079
$784$	0	0
$785$	−9.18498e9	−0.677695
$786$	0	0
$787$	−5.61813e9	−0.410847	−0.205424	−	0.978673i	$-0.565857\pi$
$787$	−5.61813e9	−0.410847	−0.205424	+	0.978673i	$0.565857\pi$
$788$	0	0
$789$	1.66101e9	0.120393
$790$	0	0
$791$	−2.74906e9	−0.197500
$792$	0	0
$793$	−2.23227e8	−0.0158961
$794$	0	0
$795$	−8.62892e8	−0.0609076
$796$	0	0
$797$	−4.98926e9	−0.349085	−0.174543	−	0.984650i	$-0.555845\pi$
$797$	−4.98926e9	−0.349085	−0.174543	+	0.984650i	$0.555845\pi$
$798$	0	0
$799$	1.86305e10	1.29214
$800$	0	0
$801$	−8.69703e9	−0.597939
$802$	0	0
$803$	9.36623e8	0.0638352
$804$	0	0
$805$	2.76966e9	0.187129
$806$	0	0
$807$	−1.86513e9	−0.124925
$808$	0	0
$809$	2.60176e10	1.72762	0.863809	−	0.503820i	$-0.168073\pi$
$809$	2.60176e10	1.72762	0.863809	+	0.503820i	$0.168073\pi$
$810$	0	0
$811$	−1.36033e10	−0.895513	−0.447756	−	0.894156i	$-0.647777\pi$
$811$	−1.36033e10	−0.895513	−0.447756	+	0.894156i	$0.647777\pi$
$812$	0	0
$813$	1.70008e9	0.110956
$814$	0	0
$815$	2.82451e9	0.182764
$816$	0	0
$817$	−3.27497e9	−0.210102
$818$	0	0
$819$	1.77768e9	0.113073
$820$	0	0
$821$	5.59803e9	0.353048	0.176524	−	0.984296i	$-0.443515\pi$
$821$	5.59803e9	0.353048	0.176524	+	0.984296i	$0.443515\pi$
$822$	0	0
$823$	−2.83609e10	−1.77346	−0.886728	−	0.462292i	$-0.847027\pi$
$823$	−2.83609e10	−1.77346	−0.886728	+	0.462292i	$0.847027\pi$
$824$	0	0
$825$	1.59315e7	0.000987799 0
$826$	0	0
$827$	1.94087e10	1.19324	0.596619	−	0.802524i	$-0.296510\pi$
$827$	1.94087e10	1.19324	0.596619	+	0.802524i	$0.296510\pi$
$828$	0	0
$829$	1.25701e10	0.766301	0.383150	−	0.923686i	$-0.374839\pi$
$829$	1.25701e10	0.766301	0.383150	+	0.923686i	$0.374839\pi$
$830$	0	0
$831$	1.50501e9	0.0909781
$832$	0	0
$833$	4.42236e9	0.265092
$834$	0	0
$835$	−2.55205e10	−1.51700
$836$	0	0
$837$	−6.62173e8	−0.0390331
$838$	0	0
$839$	−1.95535e10	−1.14303	−0.571515	−	0.820592i	$-0.693644\pi$
$839$	−1.95535e10	−1.14303	−0.571515	+	0.820592i	$0.693644\pi$
$840$	0	0
$841$	−1.51071e10	−0.875781
$842$	0	0
$843$	5.68256e8	0.0326699
$844$	0	0
$845$	1.92526e10	1.09772
$846$	0	0
$847$	−1.48934e10	−0.842175
$848$	0	0
$849$	4.69521e8	0.0263316
$850$	0	0
$851$	2.11449e9	0.117612
$852$	0	0
$853$	1.14196e10	0.629986	0.314993	−	0.949094i	$-0.397998\pi$
$853$	1.14196e10	0.629986	0.314993	+	0.949094i	$0.397998\pi$
$854$	0	0
$855$	4.66479e9	0.255241
$856$	0	0
$857$	−1.94401e10	−1.05503	−0.527516	−	0.849545i	$-0.676877\pi$
$857$	−1.94401e10	−1.05503	−0.527516	+	0.849545i	$0.676877\pi$
$858$	0	0
$859$	−1.07841e10	−0.580509	−0.290254	−	0.956950i	$-0.593740\pi$
$859$	−1.07841e10	−0.580509	−0.290254	+	0.956950i	$0.593740\pi$
$860$	0	0
$861$	−2.03583e9	−0.108700
$862$	0	0
$863$	8.45236e8	0.0447651	0.0223826	−	0.999749i	$-0.492875\pi$
$863$	8.45236e8	0.0447651	0.0223826	+	0.999749i	$0.492875\pi$
$864$	0	0
$865$	−4.16184e10	−2.18640
$866$	0	0
$867$	1.87990e8	0.00979643
$868$	0	0
$869$	−3.29800e8	−0.0170483
$870$	0	0
$871$	−2.47602e9	−0.126967
$872$	0	0
$873$	1.50264e10	0.764373
$874$	0	0
$875$	1.40428e10	0.708642
$876$	0	0
$877$	2.17563e10	1.08915	0.544574	−	0.838713i	$-0.316691\pi$
$877$	2.17563e10	1.08915	0.544574	+	0.838713i	$0.316691\pi$
$878$	0	0
$879$	1.37513e9	0.0682939
$880$	0	0
$881$	2.18056e10	1.07436	0.537182	−	0.843466i	$-0.319489\pi$
$881$	2.18056e10	1.07436	0.537182	+	0.843466i	$0.319489\pi$
$882$	0	0
$883$	−2.25662e9	−0.110305	−0.0551526	−	0.998478i	$-0.517565\pi$
$883$	−2.25662e9	−0.110305	−0.0551526	+	0.998478i	$0.517565\pi$
$884$	0	0
$885$	1.94746e9	0.0944424
$886$	0	0
$887$	1.72271e10	0.828857	0.414428	−	0.910082i	$-0.363982\pi$
$887$	1.72271e10	0.828857	0.414428	+	0.910082i	$0.363982\pi$
$888$	0	0
$889$	2.29620e10	1.09611
$890$	0	0
$891$	−1.18818e9	−0.0562745
$892$	0	0
$893$	−6.80810e9	−0.319923
$894$	0	0
$895$	−1.53396e10	−0.715208
$896$	0	0
$897$	−3.98863e7	−0.00184523
$898$	0	0
$899$	2.16861e9	0.0995455
$900$	0	0
$901$	1.60015e10	0.728827
$902$	0	0
$903$	1.18591e9	0.0535976
$904$	0	0
$905$	−2.53721e10	−1.13785
$906$	0	0
$907$	−2.32557e10	−1.03491	−0.517456	−	0.855710i	$-0.673121\pi$
$907$	−2.32557e10	−1.03491	−0.517456	+	0.855710i	$0.673121\pi$
$908$	0	0
$909$	−6.12926e9	−0.270666
$910$	0	0
$911$	4.22775e10	1.85266	0.926329	−	0.376715i	$-0.122946\pi$
$911$	4.22775e10	1.85266	0.926329	+	0.376715i	$0.122946\pi$
$912$	0	0
$913$	−1.64251e9	−0.0714267
$914$	0	0
$915$	2.12115e8	0.00915372
$916$	0	0
$917$	6.22072e9	0.266408
$918$	0	0
$919$	−2.74897e10	−1.16833	−0.584164	−	0.811635i	$-0.698578\pi$
$919$	−2.74897e10	−1.16833	−0.584164	+	0.811635i	$0.698578\pi$
$920$	0	0
$921$	1.38487e9	0.0584116
$922$	0	0
$923$	−5.72261e9	−0.239546
$924$	0	0
$925$	−3.56940e9	−0.148286
$926$	0	0
$927$	1.71133e10	0.705593
$928$	0	0
$929$	−2.42814e10	−0.993614	−0.496807	−	0.867861i	$-0.665494\pi$
$929$	−2.42814e10	−0.993614	−0.496807	+	0.867861i	$0.665494\pi$
$930$	0	0
$931$	−1.61606e9	−0.0656346
$932$	0	0
$933$	−5.51563e8	−0.0222336
$934$	0	0
$935$	−1.47823e9	−0.0591427
$936$	0	0
$937$	−3.53708e10	−1.40461	−0.702305	−	0.711876i	$-0.747846\pi$
$937$	−3.53708e10	−1.40461	−0.702305	+	0.711876i	$0.747846\pi$
$938$	0	0
$939$	1.30264e9	0.0513445
$940$	0	0
$941$	−3.60255e10	−1.40944	−0.704719	−	0.709486i	$-0.748927\pi$
$941$	−3.60255e10	−1.40944	−0.704719	+	0.709486i	$0.748927\pi$
$942$	0	0
$943$	−9.47516e9	−0.367956
$944$	0	0
$945$	−3.38652e9	−0.130539
$946$	0	0
$947$	2.20930e10	0.845336	0.422668	−	0.906285i	$-0.361094\pi$
$947$	2.20930e10	0.845336	0.422668	+	0.906285i	$0.361094\pi$
$948$	0	0
$949$	3.95842e9	0.150345
$950$	0	0
$951$	6.17496e8	0.0232810
$952$	0	0
$953$	−2.36409e9	−0.0884787	−0.0442393	−	0.999021i	$-0.514086\pi$
$953$	−2.36409e9	−0.0884787	−0.0442393	+	0.999021i	$0.514086\pi$
$954$	0	0
$955$	2.31452e10	0.859903
$956$	0	0
$957$	−3.77922e7	−0.00139383
$958$	0	0
$959$	1.10535e10	0.404702
$960$	0	0
$961$	−2.53179e10	−0.920227
$962$	0	0
$963$	1.33121e9	0.0480346
$964$	0	0
$965$	3.03137e10	1.08591
$966$	0	0
$967$	−2.69098e10	−0.957014	−0.478507	−	0.878084i	$-0.658822\pi$
$967$	−2.69098e10	−0.957014	−0.478507	+	0.878084i	$0.658822\pi$
$968$	0	0
$969$	4.17025e8	0.0147241
$970$	0	0
$971$	4.71594e9	0.165311	0.0826553	−	0.996578i	$-0.473660\pi$
$971$	4.71594e9	0.165311	0.0826553	+	0.996578i	$0.473660\pi$
$972$	0	0
$973$	2.89413e10	1.00722
$974$	0	0
$975$	6.73308e7	0.00232647
$976$	0	0
$977$	1.79514e10	0.615839	0.307919	−	0.951412i	$-0.400367\pi$
$977$	1.79514e10	0.615839	0.307919	+	0.951412i	$0.400367\pi$
$978$	0	0
$979$	−1.00712e9	−0.0343039
$980$	0	0
$981$	−1.13877e10	−0.385119
$982$	0	0
$983$	−2.39450e10	−0.804039	−0.402019	−	0.915631i	$-0.631692\pi$
$983$	−2.39450e10	−0.804039	−0.402019	+	0.915631i	$0.631692\pi$
$984$	0	0
$985$	1.35854e10	0.452944
$986$	0	0
$987$	2.46531e9	0.0816134
$988$	0	0
$989$	5.51949e9	0.181431
$990$	0	0
$991$	−1.07956e10	−0.352362	−0.176181	−	0.984358i	$-0.556374\pi$
$991$	−1.07956e10	−0.352362	−0.176181	+	0.984358i	$0.556374\pi$
$992$	0	0
$993$	−1.33459e9	−0.0432540
$994$	0	0
$995$	1.21755e10	0.391838
$996$	0	0
$997$	−3.54933e10	−1.13426	−0.567131	−	0.823627i	$-0.691947\pi$
$997$	−3.54933e10	−1.13426	−0.567131	+	0.823627i	$0.691947\pi$
$998$	0	0
$999$	−2.58542e9	−0.0820450

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.8.a.d.1.1		2
4.3	odd	2		38.8.a.d.1.2	✓	2
12.11	even	2		342.8.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.a.d.1.2	✓	2	4.3	odd	2
304.8.a.d.1.1		2	1.1	even	1	trivial
342.8.a.g.1.2		2	12.11	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q-3.23924 q^{3} -312.472 q^{5} +766.767 q^{7} -2176.51 q^{9} +O(q^{10})\)	\(q-3.23924 q^{3} -312.472 q^{5} +766.767 q^{7} -2176.51 q^{9} -252.042 q^{11} -1065.19 q^{13} +1012.17 q^{15} -18769.7 q^{17} +6859.00 q^{19} -2483.74 q^{21} -11559.9 q^{23} +19513.8 q^{25} +14134.4 q^{27} -46290.0 q^{29} -46848.2 q^{31} +816.422 q^{33} -239593. q^{35} -182916. q^{37} +3450.42 q^{39} +819661. q^{41} -477471. q^{43} +680098. q^{45} -992580. q^{47} -235611. q^{49} +60799.7 q^{51} -852516. q^{53} +78756.0 q^{55} -22217.9 q^{57} +1.92404e6 q^{59} +209564. q^{61} -1.66887e6 q^{63} +332843. q^{65} +2.32447e6 q^{67} +37445.1 q^{69} +5.37237e6 q^{71} -3.71614e6 q^{73} -63209.9 q^{75} -193257. q^{77} +1.30851e6 q^{79} +4.71424e6 q^{81} +6.51683e6 q^{83} +5.86502e6 q^{85} +149944. q^{87} +3.99586e6 q^{89} -816756. q^{91} +151753. q^{93} -2.14325e6 q^{95} -6.90391e6 q^{97} +548570. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 69 q^{3} + 155 q^{5} + 2238 q^{7} + 855 q^{9}+O(q^{10}) \) 2 * q + 69 * q^3 + 155 * q^5 + 2238 * q^7 + 855 * q^9	\( 2 q + 69 q^{3} + 155 q^{5} + 2238 q^{7} + 855 q^{9} + 3295 q^{11} - 13427 q^{13} + 34782 q^{15} - 32256 q^{17} + 13718 q^{19} + 103797 q^{21} + 82525 q^{23} + 159919 q^{25} + 75141 q^{27} - 12749 q^{29} - 258944 q^{31} + 257052 q^{33} + 448167 q^{35} - 149260 q^{37} - 889557 q^{39} + 339130 q^{41} + 83869 q^{43} + 2097243 q^{45} - 1471025 q^{47} + 1105372 q^{49} - 913437 q^{51} - 945643 q^{53} + 1736899 q^{55} + 473271 q^{57} + 969009 q^{59} - 1506755 q^{61} + 2791179 q^{63} - 5445956 q^{65} + 1848219 q^{67} + 6834063 q^{69} + 3417184 q^{71} - 2499822 q^{73} + 10079553 q^{75} + 5025267 q^{77} - 2636926 q^{79} + 2491398 q^{81} + 10059354 q^{83} - 439425 q^{85} + 2572923 q^{87} - 3506160 q^{89} - 19003851 q^{91} - 15169884 q^{93} + 1063145 q^{95} + 5893526 q^{97} + 11301453 q^{99}+O(q^{100}) \) 2 * q + 69 * q^3 + 155 * q^5 + 2238 * q^7 + 855 * q^9 + 3295 * q^11 - 13427 * q^13 + 34782 * q^15 - 32256 * q^17 + 13718 * q^19 + 103797 * q^21 + 82525 * q^23 + 159919 * q^25 + 75141 * q^27 - 12749 * q^29 - 258944 * q^31 + 257052 * q^33 + 448167 * q^35 - 149260 * q^37 - 889557 * q^39 + 339130 * q^41 + 83869 * q^43 + 2097243 * q^45 - 1471025 * q^47 + 1105372 * q^49 - 913437 * q^51 - 945643 * q^53 + 1736899 * q^55 + 473271 * q^57 + 969009 * q^59 - 1506755 * q^61 + 2791179 * q^63 - 5445956 * q^65 + 1848219 * q^67 + 6834063 * q^69 + 3417184 * q^71 - 2499822 * q^73 + 10079553 * q^75 + 5025267 * q^77 - 2636926 * q^79 + 2491398 * q^81 + 10059354 * q^83 - 439425 * q^85 + 2572923 * q^87 - 3506160 * q^89 - 19003851 * q^91 - 15169884 * q^93 + 1063145 * q^95 + 5893526 * q^97 + 11301453 * q^99

Introduction

L-functions

Modular forms

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Database

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