LMFDB - Embedded newform 656.6.a.h.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [656,6,Mod(1,656)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(656, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("656.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 656 = 2^{4} \cdot 41 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	656.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:	$105.211785797$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 1807 x^{8} - 1186 x^{7} + 1075622 x^{6} + 1575146 x^{5} - 242812142 x^{4} - 535064182 x^{3} + \cdots - 425549129499 $ x^10 - 1807x^8 - 1186x^7 + 1075622x^6 + 1575146x^5 - 242812142x^4 - 535064182x^3 + 19880651457x^2 + 41753967582x - 425549129499
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{11} $
Twist minimal:	no (minimal twist has level 164)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-10.6097$ of defining polynomial
Character	$\chi$	$=$	656.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.60967 q^{3} -7.92999 q^{5} -170.769 q^{7} -150.654 q^{9} +O(q^{10})$	$q+9.60967 q^{3} -7.92999 q^{5} -170.769 q^{7} -150.654 q^{9} -106.609 q^{11} -524.606 q^{13} -76.2046 q^{15} +429.180 q^{17} +1559.42 q^{19} -1641.03 q^{21} -3355.22 q^{23} -3062.12 q^{25} -3782.89 q^{27} +7794.51 q^{29} -4511.03 q^{31} -1024.48 q^{33} +1354.20 q^{35} +7974.29 q^{37} -5041.29 q^{39} -1681.00 q^{41} -21987.8 q^{43} +1194.69 q^{45} +20747.3 q^{47} +12355.0 q^{49} +4124.27 q^{51} +35770.9 q^{53} +845.413 q^{55} +14985.5 q^{57} +29171.8 q^{59} +36134.0 q^{61} +25727.0 q^{63} +4160.12 q^{65} -57966.3 q^{67} -32242.6 q^{69} -55416.5 q^{71} +16010.2 q^{73} -29425.9 q^{75} +18205.6 q^{77} +75725.1 q^{79} +256.686 q^{81} +55388.5 q^{83} -3403.39 q^{85} +74902.7 q^{87} -80396.2 q^{89} +89586.3 q^{91} -43349.5 q^{93} -12366.2 q^{95} +148384. q^{97} +16061.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 32 q^{5} - 88 q^{7} + 1194 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 32 * q^5 - 88 * q^7 + 1194 * q^9	$ 10 q - 10 q^{3} + 32 q^{5} - 88 q^{7} + 1194 q^{9} + 476 q^{11} - 456 q^{13} + 2 q^{15} + 1456 q^{17} - 2094 q^{19} + 7048 q^{21} - 7380 q^{23} + 15158 q^{25} - 9550 q^{27} + 9948 q^{29} + 840 q^{31} + 34828 q^{33} - 31214 q^{35} + 21780 q^{37} - 17832 q^{39} - 16810 q^{41} - 56636 q^{43} + 95584 q^{45} - 72666 q^{47} + 76574 q^{49} - 115660 q^{51} + 47528 q^{53} - 14182 q^{55} + 60356 q^{57} - 87380 q^{59} + 97364 q^{61} - 66998 q^{63} + 65716 q^{65} - 5724 q^{67} + 80692 q^{69} + 2834 q^{71} + 11228 q^{73} + 50282 q^{75} + 22400 q^{77} - 90094 q^{79} + 212530 q^{81} + 16132 q^{83} + 88840 q^{85} + 318756 q^{87} + 79872 q^{89} - 62004 q^{91} + 33652 q^{93} + 574026 q^{95} - 167548 q^{97} + 441774 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 32 * q^5 - 88 * q^7 + 1194 * q^9 + 476 * q^11 - 456 * q^13 + 2 * q^15 + 1456 * q^17 - 2094 * q^19 + 7048 * q^21 - 7380 * q^23 + 15158 * q^25 - 9550 * q^27 + 9948 * q^29 + 840 * q^31 + 34828 * q^33 - 31214 * q^35 + 21780 * q^37 - 17832 * q^39 - 16810 * q^41 - 56636 * q^43 + 95584 * q^45 - 72666 * q^47 + 76574 * q^49 - 115660 * q^51 + 47528 * q^53 - 14182 * q^55 + 60356 * q^57 - 87380 * q^59 + 97364 * q^61 - 66998 * q^63 + 65716 * q^65 - 5724 * q^67 + 80692 * q^69 + 2834 * q^71 + 11228 * q^73 + 50282 * q^75 + 22400 * q^77 - 90094 * q^79 + 212530 * q^81 + 16132 * q^83 + 88840 * q^85 + 318756 * q^87 + 79872 * q^89 - 62004 * q^91 + 33652 * q^93 + 574026 * q^95 - 167548 * q^97 + 441774 * 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
$3$	9.60967	0.616461	0.308230	−	0.951312i	$-0.400263\pi$
$3$	9.60967	0.616461	0.308230	+	0.951312i	$0.400263\pi$
$4$	0	0
$5$	−7.92999	−0.141856	−0.0709280	−	0.997481i	$-0.522596\pi$
$5$	−7.92999	−0.141856	−0.0709280	+	0.997481i	$0.522596\pi$
$6$	0	0
$7$	−170.769	−1.31724	−0.658618	−	0.752478i	$-0.728859\pi$
$7$	−170.769	−1.31724	−0.658618	+	0.752478i	$0.728859\pi$
$8$	0	0
$9$	−150.654	−0.619976
$10$	0	0
$11$	−106.609	−0.265653	−0.132826	−	0.991139i	$-0.542405\pi$
$11$	−106.609	−0.265653	−0.132826	+	0.991139i	$0.542405\pi$
$12$	0	0
$13$	−524.606	−0.860944	−0.430472	−	0.902604i	$-0.641653\pi$
$13$	−524.606	−0.860944	−0.430472	+	0.902604i	$0.641653\pi$
$14$	0	0
$15$	−76.2046	−0.0874487
$16$	0	0
$17$	429.180	0.360178	0.180089	−	0.983650i	$-0.442361\pi$
$17$	429.180	0.360178	0.180089	+	0.983650i	$0.442361\pi$
$18$	0	0
$19$	1559.42	0.991011	0.495506	−	0.868605i	$-0.334983\pi$
$19$	1559.42	0.991011	0.495506	+	0.868605i	$0.334983\pi$
$20$	0	0
$21$	−1641.03	−0.812024
$22$	0	0
$23$	−3355.22	−1.32252	−0.661259	−	0.750158i	$-0.729978\pi$
$23$	−3355.22	−1.32252	−0.661259	+	0.750158i	$0.729978\pi$
$24$	0	0
$25$	−3062.12	−0.979877
$26$	0	0
$27$	−3782.89	−0.998652
$28$	0	0
$29$	7794.51	1.72105	0.860526	−	0.509407i	$-0.170135\pi$
$29$	7794.51	1.72105	0.860526	+	0.509407i	$0.170135\pi$
$30$	0	0
$31$	−4511.03	−0.843084	−0.421542	−	0.906809i	$-0.638511\pi$
$31$	−4511.03	−0.843084	−0.421542	+	0.906809i	$0.638511\pi$
$32$	0	0
$33$	−1024.48	−0.163764
$34$	0	0
$35$	1354.20	0.186858
$36$	0	0
$37$	7974.29	0.957607	0.478804	−	0.877922i	$-0.341071\pi$
$37$	7974.29	0.957607	0.478804	+	0.877922i	$0.341071\pi$
$38$	0	0
$39$	−5041.29	−0.530738
$40$	0	0
$41$	−1681.00	−0.156174
$41$	−1681.00	−0.156174
$42$	0	0
$43$	−21987.8	−1.81347	−0.906737	−	0.421698i	$-0.861434\pi$
$43$	−21987.8	−1.81347	−0.906737	+	0.421698i	$0.861434\pi$
$44$	0	0
$45$	1194.69	0.0879474
$46$	0	0
$47$	20747.3	1.36999	0.684996	−	0.728547i	$-0.259804\pi$
$47$	20747.3	1.36999	0.684996	+	0.728547i	$0.259804\pi$
$48$	0	0
$49$	12355.0	0.735109
$50$	0	0
$51$	4124.27	0.222035
$52$	0	0
$53$	35770.9	1.74920	0.874600	−	0.484845i	$-0.161124\pi$
$53$	35770.9	1.74920	0.874600	+	0.484845i	$0.161124\pi$
$54$	0	0
$55$	845.413	0.0376844
$56$	0	0
$57$	14985.5	0.610919
$58$	0	0
$59$	29171.8	1.09102	0.545510	−	0.838104i	$-0.316336\pi$
$59$	29171.8	1.09102	0.545510	+	0.838104i	$0.316336\pi$
$60$	0	0
$61$	36134.0	1.24335	0.621673	−	0.783277i	$-0.286453\pi$
$61$	36134.0	1.24335	0.621673	+	0.783277i	$0.286453\pi$
$62$	0	0
$63$	25727.0	0.816655
$64$	0	0
$65$	4160.12	0.122130
$66$	0	0
$67$	−57966.3	−1.57757	−0.788785	−	0.614670i	$-0.789289\pi$
$67$	−57966.3	−1.57757	−0.788785	+	0.614670i	$0.789289\pi$
$68$	0	0
$69$	−32242.6	−0.815280
$70$	0	0
$71$	−55416.5	−1.30465	−0.652324	−	0.757940i	$-0.726206\pi$
$71$	−55416.5	−1.30465	−0.652324	+	0.757940i	$0.726206\pi$
$72$	0	0
$73$	16010.2	0.351634	0.175817	−	0.984423i	$-0.443743\pi$
$73$	16010.2	0.351634	0.175817	+	0.984423i	$0.443743\pi$
$74$	0	0
$75$	−29425.9	−0.604055
$76$	0	0
$77$	18205.6	0.349927
$78$	0	0
$79$	75725.1	1.36512	0.682562	−	0.730828i	$-0.260866\pi$
$79$	75725.1	1.36512	0.682562	+	0.730828i	$0.260866\pi$
$80$	0	0
$81$	256.686	0.00434701
$82$	0	0
$83$	55388.5	0.882519	0.441260	−	0.897379i	$-0.354532\pi$
$83$	55388.5	0.882519	0.441260	+	0.897379i	$0.354532\pi$
$84$	0	0
$85$	−3403.39	−0.0510934
$86$	0	0
$87$	74902.7	1.06096
$88$	0	0
$89$	−80396.2	−1.07587	−0.537936	−	0.842986i	$-0.680796\pi$
$89$	−80396.2	−1.07587	−0.537936	+	0.842986i	$0.680796\pi$
$90$	0	0
$91$	89586.3	1.13407
$92$	0	0
$93$	−43349.5	−0.519728
$94$	0	0
$95$	−12366.2	−0.140581
$96$	0	0
$97$	148384.	1.60125	0.800624	−	0.599168i	$-0.204502\pi$
$97$	148384.	1.60125	0.800624	+	0.599168i	$0.204502\pi$
$98$	0	0
$99$	16061.2	0.164698
$100$	0	0
$101$	99569.8	0.971235	0.485618	−	0.874171i	$-0.338595\pi$
$101$	99569.8	0.971235	0.485618	+	0.874171i	$0.338595\pi$
$102$	0	0
$103$	−143577.	−1.33349	−0.666746	−	0.745285i	$-0.732314\pi$
$103$	−143577.	−1.33349	−0.666746	+	0.745285i	$0.732314\pi$
$104$	0	0
$105$	13013.4	0.115190
$106$	0	0
$107$	147820.	1.24817	0.624085	−	0.781356i	$-0.285472\pi$
$107$	147820.	1.24817	0.624085	+	0.781356i	$0.285472\pi$
$108$	0	0
$109$	56191.4	0.453006	0.226503	−	0.974011i	$-0.427271\pi$
$109$	56191.4	0.453006	0.226503	+	0.974011i	$0.427271\pi$
$110$	0	0
$111$	76630.2	0.590327
$112$	0	0
$113$	−115224.	−0.848878	−0.424439	−	0.905457i	$-0.639529\pi$
$113$	−115224.	−0.848878	−0.424439	+	0.905457i	$0.639529\pi$
$114$	0	0
$115$	26606.9	0.187607
$116$	0	0
$117$	79034.1	0.533765
$118$	0	0
$119$	−73290.5	−0.474439
$120$	0	0
$121$	−149685.	−0.929429
$122$	0	0
$123$	−16153.9	−0.0962750
$124$	0	0
$125$	49063.8	0.280858
$126$	0	0
$127$	2194.66	0.0120742	0.00603710	−	0.999982i	$-0.498078\pi$
$127$	2194.66	0.0120742	0.00603710	+	0.999982i	$0.498078\pi$
$128$	0	0
$129$	−211296.	−1.11793
$130$	0	0
$131$	−12330.9	−0.0627794	−0.0313897	−	0.999507i	$-0.509993\pi$
$131$	−12330.9	−0.0627794	−0.0313897	+	0.999507i	$0.509993\pi$
$132$	0	0
$133$	−266300.	−1.30540
$134$	0	0
$135$	29998.3	0.141665
$136$	0	0
$137$	−262570.	−1.19521	−0.597604	−	0.801791i	$-0.703881\pi$
$137$	−262570.	−1.19521	−0.597604	+	0.801791i	$0.703881\pi$
$138$	0	0
$139$	411933.	1.80838	0.904190	−	0.427130i	$-0.140475\pi$
$139$	411933.	1.80838	0.904190	+	0.427130i	$0.140475\pi$
$140$	0	0
$141$	199375.	0.844546
$142$	0	0
$143$	55928.0	0.228712
$144$	0	0
$145$	−61810.4	−0.244142
$146$	0	0
$147$	118727.	0.453166
$148$	0	0
$149$	107503.	0.396694	0.198347	−	0.980132i	$-0.436443\pi$
$149$	107503.	0.396694	0.198347	+	0.980132i	$0.436443\pi$
$150$	0	0
$151$	−121217.	−0.432636	−0.216318	−	0.976323i	$-0.569405\pi$
$151$	−121217.	−0.432636	−0.216318	+	0.976323i	$0.569405\pi$
$152$	0	0
$153$	−64657.7	−0.223302
$154$	0	0
$155$	35772.4	0.119597
$156$	0	0
$157$	−98686.4	−0.319528	−0.159764	−	0.987155i	$-0.551073\pi$
$157$	−98686.4	−0.319528	−0.159764	+	0.987155i	$0.551073\pi$
$158$	0	0
$159$	343746.	1.07831
$160$	0	0
$161$	572967.	1.74207
$162$	0	0
$163$	−529682.	−1.56152	−0.780758	−	0.624833i	$-0.785167\pi$
$163$	−529682.	−1.56152	−0.780758	+	0.624833i	$0.785167\pi$
$164$	0	0
$165$	8124.14	0.0232310
$166$	0	0
$167$	181065.	0.502394	0.251197	−	0.967936i	$-0.419176\pi$
$167$	181065.	0.502394	0.251197	+	0.967936i	$0.419176\pi$
$168$	0	0
$169$	−96081.6	−0.258776
$170$	0	0
$171$	−234933.	−0.614404
$172$	0	0
$173$	245674.	0.624086	0.312043	−	0.950068i	$-0.398987\pi$
$173$	245674.	0.624086	0.312043	+	0.950068i	$0.398987\pi$
$174$	0	0
$175$	522914.	1.29073
$176$	0	0
$177$	280331.	0.672571
$178$	0	0
$179$	551218.	1.28585	0.642926	−	0.765928i	$-0.277720\pi$
$179$	551218.	1.28585	0.642926	+	0.765928i	$0.277720\pi$
$180$	0	0
$181$	590385.	1.33949	0.669744	−	0.742592i	$-0.266404\pi$
$181$	590385.	1.33949	0.669744	+	0.742592i	$0.266404\pi$
$182$	0	0
$183$	347236.	0.766473
$184$	0	0
$185$	−63236.0	−0.135842
$186$	0	0
$187$	−45754.6	−0.0956822
$188$	0	0
$189$	645999.	1.31546
$190$	0	0
$191$	−301285.	−0.597577	−0.298788	−	0.954319i	$-0.596582\pi$
$191$	−301285.	−0.597577	−0.298788	+	0.954319i	$0.596582\pi$
$192$	0	0
$193$	−34600.6	−0.0668637	−0.0334318	−	0.999441i	$-0.510644\pi$
$193$	−34600.6	−0.0668637	−0.0334318	+	0.999441i	$0.510644\pi$
$194$	0	0
$195$	39977.4	0.0752884
$196$	0	0
$197$	773755.	1.42049	0.710245	−	0.703955i	$-0.248584\pi$
$197$	773755.	1.42049	0.710245	+	0.703955i	$0.248584\pi$
$198$	0	0
$199$	−323519.	−0.579118	−0.289559	−	0.957160i	$-0.593509\pi$
$199$	−323519.	−0.579118	−0.289559	+	0.957160i	$0.593509\pi$
$200$	0	0
$201$	−557037.	−0.972509
$202$	0	0
$203$	−1.33106e6	−2.26703
$204$	0	0
$205$	13330.3	0.0221542
$206$	0	0
$207$	505478.	0.819930
$208$	0	0
$209$	−166249.	−0.263265
$210$	0	0
$211$	−216262.	−0.334405	−0.167203	−	0.985923i	$-0.553473\pi$
$211$	−216262.	−0.334405	−0.167203	+	0.985923i	$0.553473\pi$
$212$	0	0
$213$	−532534.	−0.804264
$214$	0	0
$215$	174363.	0.257252
$216$	0	0
$217$	770342.	1.11054
$218$	0	0
$219$	153853.	0.216768
$220$	0	0
$221$	−225150.	−0.310093
$222$	0	0
$223$	−1.15492e6	−1.55521	−0.777605	−	0.628753i	$-0.783566\pi$
$223$	−1.15492e6	−1.55521	−0.777605	+	0.628753i	$0.783566\pi$
$224$	0	0
$225$	461321.	0.607500
$226$	0	0
$227$	434014.	0.559035	0.279518	−	0.960141i	$-0.409825\pi$
$227$	434014.	0.559035	0.279518	+	0.960141i	$0.409825\pi$
$228$	0	0
$229$	1.20918e6	1.52371	0.761855	−	0.647747i	$-0.224289\pi$
$229$	1.20918e6	1.52371	0.761855	+	0.647747i	$0.224289\pi$
$230$	0	0
$231$	174950.	0.215716
$232$	0	0
$233$	804660.	0.971007	0.485504	−	0.874235i	$-0.338636\pi$
$233$	804660.	0.971007	0.485504	+	0.874235i	$0.338636\pi$
$234$	0	0
$235$	−164526.	−0.194342
$236$	0	0
$237$	727693.	0.841545
$238$	0	0
$239$	1.20872e6	1.36878	0.684388	−	0.729118i	$-0.260069\pi$
$239$	1.20872e6	1.36878	0.684388	+	0.729118i	$0.260069\pi$
$240$	0	0
$241$	−345423.	−0.383097	−0.191549	−	0.981483i	$-0.561351\pi$
$241$	−345423.	−0.383097	−0.191549	+	0.981483i	$0.561351\pi$
$242$	0	0
$243$	921708.	1.00133
$244$	0	0
$245$	−97974.9	−0.104280
$246$	0	0
$247$	−818080.	−0.853205
$248$	0	0
$249$	532265.	0.544038
$250$	0	0
$251$	980369.	0.982212	0.491106	−	0.871100i	$-0.336593\pi$
$251$	980369.	0.982212	0.491106	+	0.871100i	$0.336593\pi$
$252$	0	0
$253$	357698.	0.351330
$254$	0	0
$255$	−32705.5	−0.0314971
$256$	0	0
$257$	92402.6	0.0872673	0.0436337	−	0.999048i	$-0.486107\pi$
$257$	92402.6	0.0872673	0.0436337	+	0.999048i	$0.486107\pi$
$258$	0	0
$259$	−1.36176e6	−1.26139
$260$	0	0
$261$	−1.17428e6	−1.06701
$262$	0	0
$263$	495467.	0.441698	0.220849	−	0.975308i	$-0.429117\pi$
$263$	495467.	0.441698	0.220849	+	0.975308i	$0.429117\pi$
$264$	0	0
$265$	−283663.	−0.248135
$266$	0	0
$267$	−772581.	−0.663233
$268$	0	0
$269$	−866828.	−0.730386	−0.365193	−	0.930932i	$-0.618997\pi$
$269$	−866828.	−0.730386	−0.365193	+	0.930932i	$0.618997\pi$
$270$	0	0
$271$	1.91793e6	1.58639	0.793196	−	0.608967i	$-0.208416\pi$
$271$	1.91793e6	1.58639	0.793196	+	0.608967i	$0.208416\pi$
$272$	0	0
$273$	860895.	0.699107
$274$	0	0
$275$	326451.	0.260307
$276$	0	0
$277$	−1.90076e6	−1.48843	−0.744214	−	0.667941i	$-0.767176\pi$
$277$	−1.90076e6	−1.48843	−0.744214	+	0.667941i	$0.767176\pi$
$278$	0	0
$279$	679605.	0.522692
$280$	0	0
$281$	2.31390e6	1.74815	0.874074	−	0.485792i	$-0.161469\pi$
$281$	2.31390e6	1.74815	0.874074	+	0.485792i	$0.161469\pi$
$282$	0	0
$283$	−964349.	−0.715761	−0.357881	−	0.933767i	$-0.616501\pi$
$283$	−964349.	−0.715761	−0.357881	+	0.933767i	$0.616501\pi$
$284$	0	0
$285$	−118835.	−0.0866626
$286$	0	0
$287$	287062.	0.205718
$288$	0	0
$289$	−1.23566e6	−0.870272
$290$	0	0
$291$	1.42592e6	0.987106
$292$	0	0
$293$	−2.69644e6	−1.83494	−0.917468	−	0.397809i	$-0.869771\pi$
$293$	−2.69644e6	−1.83494	−0.917468	+	0.397809i	$0.869771\pi$
$294$	0	0
$295$	−231332.	−0.154768
$296$	0	0
$297$	403292.	0.265294
$298$	0	0
$299$	1.76017e6	1.13861
$300$	0	0
$301$	3.75483e6	2.38877
$302$	0	0
$303$	956833.	0.598728
$304$	0	0
$305$	−286543.	−0.176376
$306$	0	0
$307$	356043.	0.215604	0.107802	−	0.994172i	$-0.465619\pi$
$307$	356043.	0.215604	0.107802	+	0.994172i	$0.465619\pi$
$308$	0	0
$309$	−1.37972e6	−0.822046
$310$	0	0
$311$	−384955.	−0.225689	−0.112844	−	0.993613i	$-0.535996\pi$
$311$	−384955.	−0.225689	−0.112844	+	0.993613i	$0.535996\pi$
$312$	0	0
$313$	1.78491e6	1.02981	0.514903	−	0.857248i	$-0.327828\pi$
$313$	1.78491e6	1.02981	0.514903	+	0.857248i	$0.327828\pi$
$314$	0	0
$315$	−204015.	−0.115847
$316$	0	0
$317$	1.68674e6	0.942759	0.471380	−	0.881930i	$-0.343756\pi$
$317$	1.68674e6	0.942759	0.471380	+	0.881930i	$0.343756\pi$
$318$	0	0
$319$	−830969.	−0.457202
$320$	0	0
$321$	1.42050e6	0.769448
$322$	0	0
$323$	669271.	0.356940
$324$	0	0
$325$	1.60640e6	0.843619
$326$	0	0
$327$	539981.	0.279260
$328$	0	0
$329$	−3.54300e6	−1.80460
$330$	0	0
$331$	−306476.	−0.153754	−0.0768769	−	0.997041i	$-0.524495\pi$
$331$	−306476.	−0.153754	−0.0768769	+	0.997041i	$0.524495\pi$
$332$	0	0
$333$	−1.20136e6	−0.593694
$334$	0	0
$335$	459672.	0.223788
$336$	0	0
$337$	−3.78377e6	−1.81489	−0.907445	−	0.420171i	$-0.861970\pi$
$337$	−3.78377e6	−1.81489	−0.907445	+	0.420171i	$0.861970\pi$
$338$	0	0
$339$	−1.10726e6	−0.523300
$340$	0	0
$341$	480918.	0.223968
$342$	0	0
$343$	760266.	0.348924
$344$	0	0
$345$	255683.	0.115652
$346$	0	0
$347$	3.07937e6	1.37290	0.686448	−	0.727179i	$-0.259169\pi$
$347$	3.07937e6	1.37290	0.686448	+	0.727179i	$0.259169\pi$
$348$	0	0
$349$	1.26495e6	0.555916	0.277958	−	0.960593i	$-0.410342\pi$
$349$	1.26495e6	0.555916	0.277958	+	0.960593i	$0.410342\pi$
$350$	0	0
$351$	1.98453e6	0.859783
$352$	0	0
$353$	−1.12695e6	−0.481358	−0.240679	−	0.970605i	$-0.577370\pi$
$353$	−1.12695e6	−0.481358	−0.240679	+	0.970605i	$0.577370\pi$
$354$	0	0
$355$	439453.	0.185072
$356$	0	0
$357$	−704297.	−0.292473
$358$	0	0
$359$	−933632.	−0.382331	−0.191166	−	0.981558i	$-0.561227\pi$
$359$	−933632.	−0.382331	−0.191166	+	0.981558i	$0.561227\pi$
$360$	0	0
$361$	−44313.2	−0.0178964
$362$	0	0
$363$	−1.43843e6	−0.572956
$364$	0	0
$365$	−126961.	−0.0498814
$366$	0	0
$367$	−321634.	−0.124651	−0.0623257	−	0.998056i	$-0.519852\pi$
$367$	−321634.	−0.124651	−0.0623257	+	0.998056i	$0.519852\pi$
$368$	0	0
$369$	253250.	0.0968240
$370$	0	0
$371$	−6.10855e6	−2.30411
$372$	0	0
$373$	−1.04407e6	−0.388560	−0.194280	−	0.980946i	$-0.562237\pi$
$373$	−1.04407e6	−0.388560	−0.194280	+	0.980946i	$0.562237\pi$
$374$	0	0
$375$	471487.	0.173138
$376$	0	0
$377$	−4.08905e6	−1.48173
$378$	0	0
$379$	130463.	0.0466540	0.0233270	−	0.999728i	$-0.492574\pi$
$379$	130463.	0.0466540	0.0233270	+	0.999728i	$0.492574\pi$
$380$	0	0
$381$	21090.0	0.00744327
$382$	0	0
$383$	4.47557e6	1.55902	0.779510	−	0.626390i	$-0.215468\pi$
$383$	4.47557e6	1.55902	0.779510	+	0.626390i	$0.215468\pi$
$384$	0	0
$385$	−144370.	−0.0496393
$386$	0	0
$387$	3.31256e6	1.12431
$388$	0	0
$389$	5.51543e6	1.84801	0.924007	−	0.382377i	$-0.124894\pi$
$389$	5.51543e6	1.84801	0.924007	+	0.382377i	$0.124894\pi$
$390$	0	0
$391$	−1.43999e6	−0.476342
$392$	0	0
$393$	−118496.	−0.0387011
$394$	0	0
$395$	−600499.	−0.193651
$396$	0	0
$397$	−3.03791e6	−0.967382	−0.483691	−	0.875239i	$-0.660704\pi$
$397$	−3.03791e6	−0.967382	−0.483691	+	0.875239i	$0.660704\pi$
$398$	0	0
$399$	−2.55905e6	−0.804725
$400$	0	0
$401$	−2.94819e6	−0.915576	−0.457788	−	0.889061i	$-0.651358\pi$
$401$	−2.94819e6	−0.915576	−0.457788	+	0.889061i	$0.651358\pi$
$402$	0	0
$403$	2.36651e6	0.725848
$404$	0	0
$405$	−2035.52	−0.000616649 0
$406$	0	0
$407$	−850134.	−0.254391
$408$	0	0
$409$	2.88521e6	0.852844	0.426422	−	0.904524i	$-0.359774\pi$
$409$	2.88521e6	0.852844	0.426422	+	0.904524i	$0.359774\pi$
$410$	0	0
$411$	−2.52321e6	−0.736799
$412$	0	0
$413$	−4.98163e6	−1.43713
$414$	0	0
$415$	−439230.	−0.125191
$416$	0	0
$417$	3.95854e6	1.11480
$418$	0	0
$419$	−2.20472e6	−0.613506	−0.306753	−	0.951789i	$-0.599242\pi$
$419$	−2.20472e6	−0.613506	−0.306753	+	0.951789i	$0.599242\pi$
$420$	0	0
$421$	963200.	0.264857	0.132428	−	0.991193i	$-0.457723\pi$
$421$	963200.	0.264857	0.132428	+	0.991193i	$0.457723\pi$
$422$	0	0
$423$	−3.12568e6	−0.849363
$424$	0	0
$425$	−1.31420e6	−0.352930
$426$	0	0
$427$	−6.17056e6	−1.63778
$428$	0	0
$429$	537449.	0.140992
$430$	0	0
$431$	−6.64602e6	−1.72333	−0.861665	−	0.507477i	$-0.830578\pi$
$431$	−6.64602e6	−1.72333	−0.861665	+	0.507477i	$0.830578\pi$
$432$	0	0
$433$	3.94282e6	1.01062	0.505309	−	0.862939i	$-0.331379\pi$
$433$	3.94282e6	1.01062	0.505309	+	0.862939i	$0.331379\pi$
$434$	0	0
$435$	−593978.	−0.150504
$436$	0	0
$437$	−5.23219e6	−1.31063
$438$	0	0
$439$	−1.23392e6	−0.305580	−0.152790	−	0.988259i	$-0.548826\pi$
$439$	−1.23392e6	−0.305580	−0.152790	+	0.988259i	$0.548826\pi$
$440$	0	0
$441$	−1.86133e6	−0.455750
$442$	0	0
$443$	184351.	0.0446309	0.0223154	−	0.999751i	$-0.492896\pi$
$443$	184351.	0.0446309	0.0223154	+	0.999751i	$0.492896\pi$
$444$	0	0
$445$	637542.	0.152619
$446$	0	0
$447$	1.03307e6	0.244546
$448$	0	0
$449$	−4.16970e6	−0.976088	−0.488044	−	0.872819i	$-0.662289\pi$
$449$	−4.16970e6	−0.976088	−0.488044	+	0.872819i	$0.662289\pi$
$450$	0	0
$451$	179211.	0.0414880
$452$	0	0
$453$	−1.16486e6	−0.266703
$454$	0	0
$455$	−710419.	−0.160874
$456$	0	0
$457$	−1.47636e6	−0.330675	−0.165337	−	0.986237i	$-0.552871\pi$
$457$	−1.47636e6	−0.330675	−0.165337	+	0.986237i	$0.552871\pi$
$458$	0	0
$459$	−1.62354e6	−0.359692
$460$	0	0
$461$	2.03195e6	0.445308	0.222654	−	0.974898i	$-0.428528\pi$
$461$	2.03195e6	0.445308	0.222654	+	0.974898i	$0.428528\pi$
$462$	0	0
$463$	−2.44120e6	−0.529238	−0.264619	−	0.964353i	$-0.585246\pi$
$463$	−2.44120e6	−0.529238	−0.264619	+	0.964353i	$0.585246\pi$
$464$	0	0
$465$	343761.	0.0737266
$466$	0	0
$467$	−1.72471e6	−0.365951	−0.182976	−	0.983117i	$-0.558573\pi$
$467$	−1.72471e6	−0.365951	−0.182976	+	0.983117i	$0.558573\pi$
$468$	0	0
$469$	9.89883e6	2.07803
$470$	0	0
$471$	−948344.	−0.196976
$472$	0	0
$473$	2.34411e6	0.481754
$474$	0	0
$475$	−4.77512e6	−0.971069
$476$	0	0
$477$	−5.38903e6	−1.08446
$478$	0	0
$479$	700104.	0.139420	0.0697098	−	0.997567i	$-0.477793\pi$
$479$	700104.	0.139420	0.0697098	+	0.997567i	$0.477793\pi$
$480$	0	0
$481$	−4.18336e6	−0.824446
$482$	0	0
$483$	5.50603e6	1.07392
$484$	0	0
$485$	−1.17669e6	−0.227147
$486$	0	0
$487$	−2.40153e6	−0.458845	−0.229423	−	0.973327i	$-0.573684\pi$
$487$	−2.40153e6	−0.458845	−0.229423	+	0.973327i	$0.573684\pi$
$488$	0	0
$489$	−5.09007e6	−0.962613
$490$	0	0
$491$	−7.57305e6	−1.41764	−0.708822	−	0.705387i	$-0.750773\pi$
$491$	−7.57305e6	−1.41764	−0.708822	+	0.705387i	$0.750773\pi$
$492$	0	0
$493$	3.34525e6	0.619884
$494$	0	0
$495$	−127365.	−0.0233635
$496$	0	0
$497$	9.46341e6	1.71853
$498$	0	0
$499$	−1.90388e6	−0.342286	−0.171143	−	0.985246i	$-0.554746\pi$
$499$	−1.90388e6	−0.342286	−0.171143	+	0.985246i	$0.554746\pi$
$500$	0	0
$501$	1.73998e6	0.309706
$502$	0	0
$503$	2.20321e6	0.388272	0.194136	−	0.980975i	$-0.437810\pi$
$503$	2.20321e6	0.388272	0.194136	+	0.980975i	$0.437810\pi$
$504$	0	0
$505$	−789588.	−0.137776
$506$	0	0
$507$	−923312.	−0.159525
$508$	0	0
$509$	−986279.	−0.168735	−0.0843676	−	0.996435i	$-0.526887\pi$
$509$	−986279.	−0.168735	−0.0843676	+	0.996435i	$0.526887\pi$
$510$	0	0
$511$	−2.73405e6	−0.463185
$512$	0	0
$513$	−5.89910e6	−0.989675
$514$	0	0
$515$	1.13856e6	0.189164
$516$	0	0
$517$	−2.21186e6	−0.363942
$518$	0	0
$519$	2.36085e6	0.384724
$520$	0	0
$521$	−9.91304e6	−1.59997	−0.799986	−	0.600019i	$-0.795160\pi$
$521$	−9.91304e6	−1.59997	−0.799986	+	0.600019i	$0.795160\pi$
$522$	0	0
$523$	−3.87006e6	−0.618676	−0.309338	−	0.950952i	$-0.600107\pi$
$523$	−3.87006e6	−0.618676	−0.309338	+	0.950952i	$0.600107\pi$
$524$	0	0
$525$	5.02503e6	0.795683
$526$	0	0
$527$	−1.93604e6	−0.303660
$528$	0	0
$529$	4.82117e6	0.749054
$530$	0	0
$531$	−4.39485e6	−0.676407
$532$	0	0
$533$	881863.	0.134457
$534$	0	0
$535$	−1.17221e6	−0.177060
$536$	0	0
$537$	5.29703e6	0.792677
$538$	0	0
$539$	−1.31716e6	−0.195284
$540$	0	0
$541$	1.55378e6	0.228242	0.114121	−	0.993467i	$-0.463595\pi$
$541$	1.55378e6	0.228242	0.114121	+	0.993467i	$0.463595\pi$
$542$	0	0
$543$	5.67341e6	0.825742
$544$	0	0
$545$	−445598.	−0.0642616
$546$	0	0
$547$	−1.28830e7	−1.84098	−0.920488	−	0.390770i	$-0.872209\pi$
$547$	−1.28830e7	−1.84098	−0.920488	+	0.390770i	$0.872209\pi$
$548$	0	0
$549$	−5.44374e6	−0.770845
$550$	0	0
$551$	1.21549e7	1.70558
$552$	0	0
$553$	−1.29315e7	−1.79819
$554$	0	0
$555$	−607677.	−0.0837415
$556$	0	0
$557$	−3.76940e6	−0.514795	−0.257397	−	0.966306i	$-0.582865\pi$
$557$	−3.76940e6	−0.514795	−0.257397	+	0.966306i	$0.582865\pi$
$558$	0	0
$559$	1.15349e7	1.56130
$560$	0	0
$561$	−439687.	−0.0589843
$562$	0	0
$563$	5.91654e6	0.786678	0.393339	−	0.919394i	$-0.371320\pi$
$563$	5.91654e6	0.786678	0.393339	+	0.919394i	$0.371320\pi$
$564$	0	0
$565$	913722.	0.120418
$566$	0	0
$567$	−43834.0	−0.00572603
$568$	0	0
$569$	−2.03078e6	−0.262955	−0.131478	−	0.991319i	$-0.541972\pi$
$569$	−2.03078e6	−0.262955	−0.131478	+	0.991319i	$0.541972\pi$
$570$	0	0
$571$	−1.80624e6	−0.231838	−0.115919	−	0.993259i	$-0.536981\pi$
$571$	−1.80624e6	−0.231838	−0.115919	+	0.993259i	$0.536981\pi$
$572$	0	0
$573$	−2.89525e6	−0.368382
$574$	0	0
$575$	1.02741e7	1.29590
$576$	0	0
$577$	1.40265e6	0.175392	0.0876961	−	0.996147i	$-0.472050\pi$
$577$	1.40265e6	0.175392	0.0876961	+	0.996147i	$0.472050\pi$
$578$	0	0
$579$	−332500.	−0.0412188
$580$	0	0
$581$	−9.45862e6	−1.16249
$582$	0	0
$583$	−3.81351e6	−0.464680
$584$	0	0
$585$	−626740.	−0.0757178
$586$	0	0
$587$	5.11365e6	0.612541	0.306271	−	0.951944i	$-0.400919\pi$
$587$	5.11365e6	0.612541	0.306271	+	0.951944i	$0.400919\pi$
$588$	0	0
$589$	−7.03458e6	−0.835506
$590$	0	0
$591$	7.43553e6	0.875676
$592$	0	0
$593$	6.79370e6	0.793359	0.396680	−	0.917957i	$-0.370162\pi$
$593$	6.79370e6	0.793359	0.396680	+	0.917957i	$0.370162\pi$
$594$	0	0
$595$	581193.	0.0673020
$596$	0	0
$597$	−3.10891e6	−0.357003
$598$	0	0
$599$	−1.67570e6	−0.190822	−0.0954111	−	0.995438i	$-0.530417\pi$
$599$	−1.67570e6	−0.190822	−0.0954111	+	0.995438i	$0.530417\pi$
$600$	0	0
$601$	5.22134e6	0.589652	0.294826	−	0.955551i	$-0.404738\pi$
$601$	5.22134e6	0.589652	0.294826	+	0.955551i	$0.404738\pi$
$602$	0	0
$603$	8.73287e6	0.978056
$604$	0	0
$605$	1.18700e6	0.131845
$606$	0	0
$607$	1.50746e7	1.66063	0.830317	−	0.557291i	$-0.188159\pi$
$607$	1.50746e7	1.66063	0.830317	+	0.557291i	$0.188159\pi$
$608$	0	0
$609$	−1.27910e7	−1.39753
$610$	0	0
$611$	−1.08842e7	−1.17949
$612$	0	0
$613$	9.48684e6	1.01970	0.509848	−	0.860265i	$-0.329702\pi$
$613$	9.48684e6	1.01970	0.509848	+	0.860265i	$0.329702\pi$
$614$	0	0
$615$	128100.	0.0136572
$616$	0	0
$617$	−288438.	−0.0305028	−0.0152514	−	0.999884i	$-0.504855\pi$
$617$	−288438.	−0.0305028	−0.0152514	+	0.999884i	$0.504855\pi$
$618$	0	0
$619$	−1.46926e7	−1.54124	−0.770622	−	0.637293i	$-0.780054\pi$
$619$	−1.46926e7	−1.54124	−0.770622	+	0.637293i	$0.780054\pi$
$620$	0	0
$621$	1.26924e7	1.32073
$622$	0	0
$623$	1.37292e7	1.41718
$624$	0	0
$625$	9.18003e6	0.940036
$626$	0	0
$627$	−1.59760e6	−0.162292
$628$	0	0
$629$	3.42240e6	0.344909
$630$	0	0
$631$	3.90441e6	0.390375	0.195188	−	0.980766i	$-0.437468\pi$
$631$	3.90441e6	0.390375	0.195188	+	0.980766i	$0.437468\pi$
$632$	0	0
$633$	−2.07820e6	−0.206148
$634$	0	0
$635$	−17403.7	−0.00171280
$636$	0	0
$637$	−6.48150e6	−0.632888
$638$	0	0
$639$	8.34873e6	0.808851
$640$	0	0
$641$	1.48802e7	1.43042	0.715211	−	0.698908i	$-0.246330\pi$
$641$	1.48802e7	1.43042	0.715211	+	0.698908i	$0.246330\pi$
$642$	0	0
$643$	1.10564e6	0.105460	0.0527298	−	0.998609i	$-0.483208\pi$
$643$	1.10564e6	0.105460	0.0527298	+	0.998609i	$0.483208\pi$
$644$	0	0
$645$	1.67557e6	0.158586
$646$	0	0
$647$	−1.16287e7	−1.09212	−0.546062	−	0.837745i	$-0.683874\pi$
$647$	−1.16287e7	−1.09212	−0.546062	+	0.837745i	$0.683874\pi$
$648$	0	0
$649$	−3.10999e6	−0.289832
$650$	0	0
$651$	7.40274e6	0.684605
$652$	0	0
$653$	−1.72836e7	−1.58618	−0.793089	−	0.609106i	$-0.791528\pi$
$653$	−1.72836e7	−1.58618	−0.793089	+	0.609106i	$0.791528\pi$
$654$	0	0
$655$	97784.2	0.00890565
$656$	0	0
$657$	−2.41201e6	−0.218005
$658$	0	0
$659$	−9.44369e6	−0.847087	−0.423544	−	0.905876i	$-0.639214\pi$
$659$	−9.44369e6	−0.847087	−0.423544	+	0.905876i	$0.639214\pi$
$660$	0	0
$661$	2.01668e6	0.179528	0.0897641	−	0.995963i	$-0.471389\pi$
$661$	2.01668e6	0.179528	0.0897641	+	0.995963i	$0.471389\pi$
$662$	0	0
$663$	−2.16362e6	−0.191160
$664$	0	0
$665$	2.11176e6	0.185178
$666$	0	0
$667$	−2.61523e7	−2.27612
$668$	0	0
$669$	−1.10984e7	−0.958726
$670$	0	0
$671$	−3.85223e6	−0.330298
$672$	0	0
$673$	1.26067e6	0.107291	0.0536454	−	0.998560i	$-0.482916\pi$
$673$	1.26067e6	0.107291	0.0536454	+	0.998560i	$0.482916\pi$
$674$	0	0
$675$	1.15836e7	0.978556
$676$	0	0
$677$	−1.62099e7	−1.35928	−0.679639	−	0.733546i	$-0.737864\pi$
$677$	−1.62099e7	−1.35928	−0.679639	+	0.733546i	$0.737864\pi$
$678$	0	0
$679$	−2.53394e7	−2.10922
$680$	0	0
$681$	4.17073e6	0.344623
$682$	0	0
$683$	3.97103e6	0.325726	0.162863	−	0.986649i	$-0.447927\pi$
$683$	3.97103e6	0.325726	0.162863	+	0.986649i	$0.447927\pi$
$684$	0	0
$685$	2.08218e6	0.169548
$686$	0	0
$687$	1.16198e7	0.939308
$688$	0	0
$689$	−1.87656e7	−1.50596
$690$	0	0
$691$	1.37573e7	1.09607	0.548033	−	0.836457i	$-0.315377\pi$
$691$	1.37573e7	1.09607	0.548033	+	0.836457i	$0.315377\pi$
$692$	0	0
$693$	−2.74275e6	−0.216946
$694$	0	0
$695$	−3.26663e6	−0.256530
$696$	0	0
$697$	−721451.	−0.0562503
$698$	0	0
$699$	7.73252e6	0.598588
$700$	0	0
$701$	2.13291e6	0.163937	0.0819685	−	0.996635i	$-0.473879\pi$
$701$	2.13291e6	0.163937	0.0819685	+	0.996635i	$0.473879\pi$
$702$	0	0
$703$	1.24352e7	0.949000
$704$	0	0
$705$	−1.58104e6	−0.119804
$706$	0	0
$707$	−1.70034e7	−1.27935
$708$	0	0
$709$	2.11167e6	0.157765	0.0788824	−	0.996884i	$-0.474865\pi$
$709$	2.11167e6	0.157765	0.0788824	+	0.996884i	$0.474865\pi$
$710$	0	0
$711$	−1.14083e7	−0.846344
$712$	0	0
$713$	1.51355e7	1.11499
$714$	0	0
$715$	−443508.	−0.0324442
$716$	0	0
$717$	1.16154e7	0.843797
$718$	0	0
$719$	3.77456e6	0.272298	0.136149	−	0.990688i	$-0.456527\pi$
$719$	3.77456e6	0.272298	0.136149	+	0.990688i	$0.456527\pi$
$720$	0	0
$721$	2.45184e7	1.75652
$722$	0	0
$723$	−3.31941e6	−0.236164
$724$	0	0
$725$	−2.38677e7	−1.68642
$726$	0	0
$727$	2.59009e7	1.81752	0.908760	−	0.417318i	$-0.137030\pi$
$727$	2.59009e7	1.81752	0.908760	+	0.417318i	$0.137030\pi$
$728$	0	0
$729$	8.79494e6	0.612934
$730$	0	0
$731$	−9.43673e6	−0.653173
$732$	0	0
$733$	−3.40291e6	−0.233932	−0.116966	−	0.993136i	$-0.537317\pi$
$733$	−3.40291e6	−0.233932	−0.116966	+	0.993136i	$0.537317\pi$
$734$	0	0
$735$	−941507.	−0.0642843
$736$	0	0
$737$	6.17975e6	0.419085
$738$	0	0
$739$	2.34859e7	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$739$	2.34859e7	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$740$	0	0
$741$	−7.86148e6	−0.525967
$742$	0	0
$743$	−1.26089e7	−0.837922	−0.418961	−	0.908004i	$-0.637606\pi$
$743$	−1.26089e7	−0.837922	−0.418961	+	0.908004i	$0.637606\pi$
$744$	0	0
$745$	−852500.	−0.0562735
$746$	0	0
$747$	−8.34451e6	−0.547141
$748$	0	0
$749$	−2.52430e7	−1.64413
$750$	0	0
$751$	1.27658e7	0.825939	0.412970	−	0.910745i	$-0.364492\pi$
$751$	1.27658e7	0.825939	0.412970	+	0.910745i	$0.364492\pi$
$752$	0	0
$753$	9.42102e6	0.605495
$754$	0	0
$755$	961253.	0.0613720
$756$	0	0
$757$	2.57982e7	1.63625	0.818125	−	0.575041i	$-0.195014\pi$
$757$	2.57982e7	1.63625	0.818125	+	0.575041i	$0.195014\pi$
$758$	0	0
$759$	3.43736e6	0.216581
$760$	0	0
$761$	3.16315e7	1.97997	0.989983	−	0.141189i	$-0.0450926\pi$
$761$	3.16315e7	1.97997	0.989983	+	0.141189i	$0.0450926\pi$
$762$	0	0
$763$	−9.59574e6	−0.596715
$764$	0	0
$765$	512736.	0.0316767
$766$	0	0
$767$	−1.53037e7	−0.939307
$768$	0	0
$769$	2.25502e7	1.37510	0.687552	−	0.726135i	$-0.258685\pi$
$769$	2.25502e7	1.37510	0.687552	+	0.726135i	$0.258685\pi$
$770$	0	0
$771$	887959.	0.0537969
$772$	0	0
$773$	2.79590e7	1.68295	0.841477	−	0.540293i	$-0.181686\pi$
$773$	2.79590e7	1.68295	0.841477	+	0.540293i	$0.181686\pi$
$774$	0	0
$775$	1.38133e7	0.826119
$776$	0	0
$777$	−1.30861e7	−0.777600
$778$	0	0
$779$	−2.62138e6	−0.154770
$780$	0	0
$781$	5.90792e6	0.346583
$782$	0	0
$783$	−2.94858e7	−1.71873
$784$	0	0
$785$	782583.	0.0453269
$786$	0	0
$787$	−2.95631e7	−1.70142	−0.850712	−	0.525633i	$-0.823829\pi$
$787$	−2.95631e7	−1.70142	−0.850712	+	0.525633i	$0.823829\pi$
$788$	0	0
$789$	4.76127e6	0.272289
$790$	0	0
$791$	1.96766e7	1.11817
$792$	0	0
$793$	−1.89561e7	−1.07045
$794$	0	0
$795$	−2.72590e6	−0.152965
$796$	0	0
$797$	−1.41028e7	−0.786428	−0.393214	−	0.919447i	$-0.628637\pi$
$797$	−1.41028e7	−0.786428	−0.393214	+	0.919447i	$0.628637\pi$
$798$	0	0
$799$	8.90434e6	0.493441
$800$	0	0
$801$	1.21120e7	0.667015
$802$	0	0
$803$	−1.70684e6	−0.0934125
$804$	0	0
$805$	−4.54363e6	−0.247123
$806$	0	0
$807$	−8.32993e6	−0.450254
$808$	0	0
$809$	9.27703e6	0.498353	0.249177	−	0.968458i	$-0.419840\pi$
$809$	9.27703e6	0.498353	0.249177	+	0.968458i	$0.419840\pi$
$810$	0	0
$811$	2.36175e7	1.26090	0.630451	−	0.776229i	$-0.282870\pi$
$811$	2.36175e7	1.26090	0.630451	+	0.776229i	$0.282870\pi$
$812$	0	0
$813$	1.84307e7	0.977948
$814$	0	0
$815$	4.20038e6	0.221511
$816$	0	0
$817$	−3.42882e7	−1.79717
$818$	0	0
$819$	−1.34966e7	−0.703094
$820$	0	0
$821$	−4.72731e6	−0.244769	−0.122385	−	0.992483i	$-0.539054\pi$
$821$	−4.72731e6	−0.244769	−0.122385	+	0.992483i	$0.539054\pi$
$822$	0	0
$823$	3.07918e7	1.58466	0.792329	−	0.610094i	$-0.208868\pi$
$823$	3.07918e7	1.58466	0.792329	+	0.610094i	$0.208868\pi$
$824$	0	0
$825$	3.13708e6	0.160469
$826$	0	0
$827$	3.26432e6	0.165970	0.0829848	−	0.996551i	$-0.473555\pi$
$827$	3.26432e6	0.165970	0.0829848	+	0.996551i	$0.473555\pi$
$828$	0	0
$829$	2.55332e7	1.29038	0.645192	−	0.764020i	$-0.276777\pi$
$829$	2.55332e7	1.29038	0.645192	+	0.764020i	$0.276777\pi$
$830$	0	0
$831$	−1.82657e7	−0.917558
$832$	0	0
$833$	5.30251e6	0.264770
$834$	0	0
$835$	−1.43585e6	−0.0712676
$836$	0	0
$837$	1.70647e7	0.841948
$838$	0	0
$839$	1.44891e7	0.710621	0.355310	−	0.934748i	$-0.384375\pi$
$839$	1.44891e7	0.710621	0.355310	+	0.934748i	$0.384375\pi$
$840$	0	0
$841$	4.02432e7	1.96202
$842$	0	0
$843$	2.22358e7	1.07766
$844$	0	0
$845$	761927.	0.0367089
$846$	0	0
$847$	2.55616e7	1.22428
$848$	0	0
$849$	−9.26708e6	−0.441239
$850$	0	0
$851$	−2.67555e7	−1.26645
$852$	0	0
$853$	−1.22710e7	−0.577439	−0.288719	−	0.957414i	$-0.593229\pi$
$853$	−1.22710e7	−0.577439	−0.288719	+	0.957414i	$0.593229\pi$
$854$	0	0
$855$	1.86302e6	0.0871569
$856$	0	0
$857$	2.00160e7	0.930948	0.465474	−	0.885061i	$-0.345884\pi$
$857$	2.00160e7	0.930948	0.465474	+	0.885061i	$0.345884\pi$
$858$	0	0
$859$	2.47500e6	0.114444	0.0572219	−	0.998361i	$-0.481776\pi$
$859$	2.47500e6	0.114444	0.0572219	+	0.998361i	$0.481776\pi$
$860$	0	0
$861$	2.75857e6	0.126817
$862$	0	0
$863$	−6.10239e6	−0.278916	−0.139458	−	0.990228i	$-0.544536\pi$
$863$	−6.10239e6	−0.278916	−0.139458	+	0.990228i	$0.544536\pi$
$864$	0	0
$865$	−1.94819e6	−0.0885303
$866$	0	0
$867$	−1.18743e7	−0.536488
$868$	0	0
$869$	−8.07301e6	−0.362649
$870$	0	0
$871$	3.04095e7	1.35820
$872$	0	0
$873$	−2.23547e7	−0.992735
$874$	0	0
$875$	−8.37856e6	−0.369955
$876$	0	0
$877$	4.03494e6	0.177149	0.0885744	−	0.996070i	$-0.471769\pi$
$877$	4.03494e6	0.177149	0.0885744	+	0.996070i	$0.471769\pi$
$878$	0	0
$879$	−2.59119e7	−1.13117
$880$	0	0
$881$	−387854.	−0.0168356	−0.00841781	−	0.999965i	$-0.502680\pi$
$881$	−387854.	−0.0168356	−0.00841781	+	0.999965i	$0.502680\pi$
$882$	0	0
$883$	1.41282e7	0.609797	0.304898	−	0.952385i	$-0.401378\pi$
$883$	1.41282e7	0.609797	0.304898	+	0.952385i	$0.401378\pi$
$884$	0	0
$885$	−2.22302e6	−0.0954083
$886$	0	0
$887$	1.37356e7	0.586191	0.293095	−	0.956083i	$-0.405315\pi$
$887$	1.37356e7	0.586191	0.293095	+	0.956083i	$0.405315\pi$
$888$	0	0
$889$	−374780.	−0.0159046
$890$	0	0
$891$	−27365.2	−0.00115479
$892$	0	0
$893$	3.23538e7	1.35768
$894$	0	0
$895$	−4.37116e6	−0.182406
$896$	0	0
$897$	1.69146e7	0.701911
$898$	0	0
$899$	−3.51612e7	−1.45099
$900$	0	0
$901$	1.53521e7	0.630023
$902$	0	0
$903$	3.60827e7	1.47258
$904$	0	0
$905$	−4.68175e6	−0.190015
$906$	0	0
$907$	−1.34522e7	−0.542970	−0.271485	−	0.962443i	$-0.587515\pi$
$907$	−1.34522e7	−0.542970	−0.271485	+	0.962443i	$0.587515\pi$
$908$	0	0
$909$	−1.50006e7	−0.602143
$910$	0	0
$911$	−2.10371e6	−0.0839826	−0.0419913	−	0.999118i	$-0.513370\pi$
$911$	−2.10371e6	−0.0839826	−0.0419913	+	0.999118i	$0.513370\pi$
$912$	0	0
$913$	−5.90494e6	−0.234444
$914$	0	0
$915$	−2.75358e6	−0.108729
$916$	0	0
$917$	2.10574e6	0.0826953
$918$	0	0
$919$	−3.60753e7	−1.40903	−0.704517	−	0.709687i	$-0.748836\pi$
$919$	−3.60753e7	−1.40903	−0.704517	+	0.709687i	$0.748836\pi$
$920$	0	0
$921$	3.42146e6	0.132911
$922$	0	0
$923$	2.90718e7	1.12323
$924$	0	0
$925$	−2.44182e7	−0.938337
$926$	0	0
$927$	2.16304e7	0.826734
$928$	0	0
$929$	−3.00401e7	−1.14199	−0.570994	−	0.820954i	$-0.693442\pi$
$929$	−3.00401e7	−1.14199	−0.570994	+	0.820954i	$0.693442\pi$
$930$	0	0
$931$	1.92666e7	0.728502
$932$	0	0
$933$	−3.69929e6	−0.139128
$934$	0	0
$935$	362834.	0.0135731
$936$	0	0
$937$	−2.46334e6	−0.0916591	−0.0458296	−	0.998949i	$-0.514593\pi$
$937$	−2.46334e6	−0.0916591	−0.0458296	+	0.998949i	$0.514593\pi$
$938$	0	0
$939$	1.71524e7	0.634835
$940$	0	0
$941$	1.96878e7	0.724810	0.362405	−	0.932021i	$-0.381956\pi$
$941$	1.96878e7	0.724810	0.362405	+	0.932021i	$0.381956\pi$
$942$	0	0
$943$	5.64013e6	0.206543
$944$	0	0
$945$	−5.12277e6	−0.186606
$946$	0	0
$947$	−9.10534e6	−0.329930	−0.164965	−	0.986299i	$-0.552751\pi$
$947$	−9.10534e6	−0.329930	−0.164965	+	0.986299i	$0.552751\pi$
$948$	0	0
$949$	−8.39907e6	−0.302737
$950$	0	0
$951$	1.62090e7	0.581174
$952$	0	0
$953$	−2.52210e7	−0.899561	−0.449781	−	0.893139i	$-0.648498\pi$
$953$	−2.52210e7	−0.899561	−0.449781	+	0.893139i	$0.648498\pi$
$954$	0	0
$955$	2.38919e6	0.0847699
$956$	0	0
$957$	−7.98533e6	−0.281847
$958$	0	0
$959$	4.48388e7	1.57437
$960$	0	0
$961$	−8.27980e6	−0.289209
$962$	0	0
$963$	−2.22697e7	−0.773836
$964$	0	0
$965$	274383.	0.00948502
$966$	0	0
$967$	1.41170e7	0.485486	0.242743	−	0.970091i	$-0.421953\pi$
$967$	1.41170e7	0.485486	0.242743	+	0.970091i	$0.421953\pi$
$968$	0	0
$969$	6.43147e6	0.220040
$970$	0	0
$971$	4.63231e7	1.57670	0.788350	−	0.615227i	$-0.210936\pi$
$971$	4.63231e7	1.57670	0.788350	+	0.615227i	$0.210936\pi$
$972$	0	0
$973$	−7.03453e7	−2.38206
$974$	0	0
$975$	1.54370e7	0.520058
$976$	0	0
$977$	−4.63046e7	−1.55199	−0.775993	−	0.630742i	$-0.782751\pi$
$977$	−4.63046e7	−1.55199	−0.775993	+	0.630742i	$0.782751\pi$
$978$	0	0
$979$	8.57100e6	0.285808
$980$	0	0
$981$	−8.46547e6	−0.280853
$982$	0	0
$983$	3.09720e7	1.02232	0.511158	−	0.859487i	$-0.329217\pi$
$983$	3.09720e7	1.02232	0.511158	+	0.859487i	$0.329217\pi$
$984$	0	0
$985$	−6.13587e6	−0.201505
$986$	0	0
$987$	−3.40471e7	−1.11247
$988$	0	0
$989$	7.37740e7	2.39835
$990$	0	0
$991$	−3.48737e7	−1.12801	−0.564007	−	0.825770i	$-0.690741\pi$
$991$	−3.48737e7	−1.12801	−0.564007	+	0.825770i	$0.690741\pi$
$992$	0	0
$993$	−2.94513e6	−0.0947832
$994$	0	0
$995$	2.56550e6	0.0821514
$996$	0	0
$997$	5.35545e7	1.70631	0.853155	−	0.521657i	$-0.174686\pi$
$997$	5.35545e7	1.70631	0.853155	+	0.521657i	$0.174686\pi$
$998$	0	0
$999$	−3.01658e7	−0.956316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	656.6.a.h.1.7		10
4.3	odd	2		164.6.a.b.1.4	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.6.a.b.1.4	✓	10	4.3	odd	2
656.6.a.h.1.7		10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 656 = 2^{4} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	656.a (trivial)

\(f(q)\)	\(=\)	\(q+9.60967 q^{3} -7.92999 q^{5} -170.769 q^{7} -150.654 q^{9} +O(q^{10})\)	\(q+9.60967 q^{3} -7.92999 q^{5} -170.769 q^{7} -150.654 q^{9} -106.609 q^{11} -524.606 q^{13} -76.2046 q^{15} +429.180 q^{17} +1559.42 q^{19} -1641.03 q^{21} -3355.22 q^{23} -3062.12 q^{25} -3782.89 q^{27} +7794.51 q^{29} -4511.03 q^{31} -1024.48 q^{33} +1354.20 q^{35} +7974.29 q^{37} -5041.29 q^{39} -1681.00 q^{41} -21987.8 q^{43} +1194.69 q^{45} +20747.3 q^{47} +12355.0 q^{49} +4124.27 q^{51} +35770.9 q^{53} +845.413 q^{55} +14985.5 q^{57} +29171.8 q^{59} +36134.0 q^{61} +25727.0 q^{63} +4160.12 q^{65} -57966.3 q^{67} -32242.6 q^{69} -55416.5 q^{71} +16010.2 q^{73} -29425.9 q^{75} +18205.6 q^{77} +75725.1 q^{79} +256.686 q^{81} +55388.5 q^{83} -3403.39 q^{85} +74902.7 q^{87} -80396.2 q^{89} +89586.3 q^{91} -43349.5 q^{93} -12366.2 q^{95} +148384. q^{97} +16061.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 32 q^{5} - 88 q^{7} + 1194 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 32 * q^5 - 88 * q^7 + 1194 * q^9	\( 10 q - 10 q^{3} + 32 q^{5} - 88 q^{7} + 1194 q^{9} + 476 q^{11} - 456 q^{13} + 2 q^{15} + 1456 q^{17} - 2094 q^{19} + 7048 q^{21} - 7380 q^{23} + 15158 q^{25} - 9550 q^{27} + 9948 q^{29} + 840 q^{31} + 34828 q^{33} - 31214 q^{35} + 21780 q^{37} - 17832 q^{39} - 16810 q^{41} - 56636 q^{43} + 95584 q^{45} - 72666 q^{47} + 76574 q^{49} - 115660 q^{51} + 47528 q^{53} - 14182 q^{55} + 60356 q^{57} - 87380 q^{59} + 97364 q^{61} - 66998 q^{63} + 65716 q^{65} - 5724 q^{67} + 80692 q^{69} + 2834 q^{71} + 11228 q^{73} + 50282 q^{75} + 22400 q^{77} - 90094 q^{79} + 212530 q^{81} + 16132 q^{83} + 88840 q^{85} + 318756 q^{87} + 79872 q^{89} - 62004 q^{91} + 33652 q^{93} + 574026 q^{95} - 167548 q^{97} + 441774 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 32 * q^5 - 88 * q^7 + 1194 * q^9 + 476 * q^11 - 456 * q^13 + 2 * q^15 + 1456 * q^17 - 2094 * q^19 + 7048 * q^21 - 7380 * q^23 + 15158 * q^25 - 9550 * q^27 + 9948 * q^29 + 840 * q^31 + 34828 * q^33 - 31214 * q^35 + 21780 * q^37 - 17832 * q^39 - 16810 * q^41 - 56636 * q^43 + 95584 * q^45 - 72666 * q^47 + 76574 * q^49 - 115660 * q^51 + 47528 * q^53 - 14182 * q^55 + 60356 * q^57 - 87380 * q^59 + 97364 * q^61 - 66998 * q^63 + 65716 * q^65 - 5724 * q^67 + 80692 * q^69 + 2834 * q^71 + 11228 * q^73 + 50282 * q^75 + 22400 * q^77 - 90094 * q^79 + 212530 * q^81 + 16132 * q^83 + 88840 * q^85 + 318756 * q^87 + 79872 * q^89 - 62004 * q^91 + 33652 * q^93 + 574026 * q^95 - 167548 * q^97 + 441774 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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