LMFDB - Embedded newform 120.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [120,6,Mod(1,120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("120.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 120 = 2^{3} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	120.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:	$19.2460583776$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{2161}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 540 $ x^2 - x - 540
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-22.7433$ of defining polynomial
Character	$\chi$	$=$	120.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +25.0000 q^{5} +181.946 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +25.0000 q^{5} +181.946 q^{7} +81.0000 q^{9} -127.892 q^{11} -251.839 q^{13} +225.000 q^{15} +244.054 q^{17} +1997.41 q^{19} +1637.52 q^{21} -925.301 q^{23} +625.000 q^{25} +729.000 q^{27} +4836.92 q^{29} -1088.33 q^{31} -1151.03 q^{33} +4548.66 q^{35} -458.333 q^{37} -2266.55 q^{39} +11181.2 q^{41} +17327.1 q^{43} +2025.00 q^{45} -2971.56 q^{47} +16297.4 q^{49} +2196.48 q^{51} +494.409 q^{53} -3197.31 q^{55} +17976.7 q^{57} -40773.1 q^{59} +43276.9 q^{61} +14737.6 q^{63} -6295.97 q^{65} -64560.0 q^{67} -8327.71 q^{69} -59141.0 q^{71} +53599.0 q^{73} +5625.00 q^{75} -23269.6 q^{77} -33285.3 q^{79} +6561.00 q^{81} -66084.6 q^{83} +6101.34 q^{85} +43532.3 q^{87} -103742. q^{89} -45821.1 q^{91} -9795.00 q^{93} +49935.2 q^{95} -106653. q^{97} -10359.3 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 18 q^{3} + 50 q^{5} - 8 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q + 18 * q^3 + 50 * q^5 - 8 * q^7 + 162 * q^9	$ 2 q + 18 q^{3} + 50 q^{5} - 8 q^{7} + 162 q^{9} + 488 q^{11} + 612 q^{13} + 450 q^{15} + 860 q^{17} - 96 q^{19} - 72 q^{21} + 2984 q^{23} + 1250 q^{25} + 1458 q^{27} + 2236 q^{29} + 9352 q^{31} + 4392 q^{33} - 200 q^{35} + 10612 q^{37} + 5508 q^{39} + 17156 q^{41} + 440 q^{43} + 4050 q^{45} - 16728 q^{47} + 35570 q^{49} + 7740 q^{51} + 31484 q^{53} + 12200 q^{55} - 864 q^{57} - 61464 q^{59} + 51596 q^{61} - 648 q^{63} + 15300 q^{65} - 45816 q^{67} + 26856 q^{69} - 97456 q^{71} + 58852 q^{73} + 11250 q^{75} - 140256 q^{77} - 116776 q^{79} + 13122 q^{81} - 136632 q^{83} + 21500 q^{85} + 20124 q^{87} - 124924 q^{89} - 209904 q^{91} + 84168 q^{93} - 2400 q^{95} - 39260 q^{97} + 39528 q^{99}+O(q^{100}) $ 2 * q + 18 * q^3 + 50 * q^5 - 8 * q^7 + 162 * q^9 + 488 * q^11 + 612 * q^13 + 450 * q^15 + 860 * q^17 - 96 * q^19 - 72 * q^21 + 2984 * q^23 + 1250 * q^25 + 1458 * q^27 + 2236 * q^29 + 9352 * q^31 + 4392 * q^33 - 200 * q^35 + 10612 * q^37 + 5508 * q^39 + 17156 * q^41 + 440 * q^43 + 4050 * q^45 - 16728 * q^47 + 35570 * q^49 + 7740 * q^51 + 31484 * q^53 + 12200 * q^55 - 864 * q^57 - 61464 * q^59 + 51596 * q^61 - 648 * q^63 + 15300 * q^65 - 45816 * q^67 + 26856 * q^69 - 97456 * q^71 + 58852 * q^73 + 11250 * q^75 - 140256 * q^77 - 116776 * q^79 + 13122 * q^81 - 136632 * q^83 + 21500 * q^85 + 20124 * q^87 - 124924 * q^89 - 209904 * q^91 + 84168 * q^93 - 2400 * q^95 - 39260 * q^97 + 39528 * 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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	181.946	1.40345	0.701727	−	0.712446i	$-0.252413\pi$
$7$	181.946	1.40345	0.701727	+	0.712446i	$0.252413\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−127.892	−0.318686	−0.159343	−	0.987223i	$-0.550938\pi$
$11$	−127.892	−0.318686	−0.159343	+	0.987223i	$0.550938\pi$
$12$	0	0
$13$	−251.839	−0.413299	−0.206649	−	0.978415i	$-0.566256\pi$
$13$	−251.839	−0.413299	−0.206649	+	0.978415i	$0.566256\pi$
$14$	0	0
$15$	225.000	0.258199
$16$	0	0
$17$	244.054	0.204816	0.102408	−	0.994742i	$-0.467345\pi$
$17$	244.054	0.204816	0.102408	+	0.994742i	$0.467345\pi$
$18$	0	0
$19$	1997.41	1.26935	0.634677	−	0.772777i	$-0.281133\pi$
$19$	1997.41	1.26935	0.634677	+	0.772777i	$0.281133\pi$
$20$	0	0
$21$	1637.52	0.810284
$22$	0	0
$23$	−925.301	−0.364723	−0.182362	−	0.983232i	$-0.558374\pi$
$23$	−925.301	−0.364723	−0.182362	+	0.983232i	$0.558374\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	4836.92	1.06801	0.534004	−	0.845482i	$-0.320687\pi$
$29$	4836.92	1.06801	0.534004	+	0.845482i	$0.320687\pi$
$30$	0	0
$31$	−1088.33	−0.203403	−0.101702	−	0.994815i	$-0.532429\pi$
$31$	−1088.33	−0.203403	−0.101702	+	0.994815i	$0.532429\pi$
$32$	0	0
$33$	−1151.03	−0.183994
$34$	0	0
$35$	4548.66	0.627643
$36$	0	0
$37$	−458.333	−0.0550398	−0.0275199	−	0.999621i	$-0.508761\pi$
$37$	−458.333	−0.0550398	−0.0275199	+	0.999621i	$0.508761\pi$
$38$	0	0
$39$	−2266.55	−0.238618
$40$	0	0
$41$	11181.2	1.03880	0.519398	−	0.854532i	$-0.326156\pi$
$41$	11181.2	1.03880	0.519398	+	0.854532i	$0.326156\pi$
$42$	0	0
$43$	17327.1	1.42907	0.714535	−	0.699600i	$-0.246638\pi$
$43$	17327.1	1.42907	0.714535	+	0.699600i	$0.246638\pi$
$44$	0	0
$45$	2025.00	0.149071
$46$	0	0
$47$	−2971.56	−0.196218	−0.0981092	−	0.995176i	$-0.531279\pi$
$47$	−2971.56	−0.196218	−0.0981092	+	0.995176i	$0.531279\pi$
$48$	0	0
$49$	16297.4	0.969681
$50$	0	0
$51$	2196.48	0.118250
$52$	0	0
$53$	494.409	0.0241767	0.0120883	−	0.999927i	$-0.496152\pi$
$53$	494.409	0.0241767	0.0120883	+	0.999927i	$0.496152\pi$
$54$	0	0
$55$	−3197.31	−0.142521
$56$	0	0
$57$	17976.7	0.732862
$58$	0	0
$59$	−40773.1	−1.52491	−0.762454	−	0.647042i	$-0.776006\pi$
$59$	−40773.1	−1.52491	−0.762454	+	0.647042i	$0.776006\pi$
$60$	0	0
$61$	43276.9	1.48913	0.744564	−	0.667551i	$-0.232657\pi$
$61$	43276.9	1.48913	0.744564	+	0.667551i	$0.232657\pi$
$62$	0	0
$63$	14737.6	0.467818
$64$	0	0
$65$	−6295.97	−0.184833
$66$	0	0
$67$	−64560.0	−1.75702	−0.878509	−	0.477726i	$-0.841461\pi$
$67$	−64560.0	−1.75702	−0.878509	+	0.477726i	$0.841461\pi$
$68$	0	0
$69$	−8327.71	−0.210573
$70$	0	0
$71$	−59141.0	−1.39233	−0.696166	−	0.717881i	$-0.745112\pi$
$71$	−59141.0	−1.39233	−0.696166	+	0.717881i	$0.745112\pi$
$72$	0	0
$73$	53599.0	1.17720	0.588599	−	0.808425i	$-0.299680\pi$
$73$	53599.0	1.17720	0.588599	+	0.808425i	$0.299680\pi$
$74$	0	0
$75$	5625.00	0.115470
$76$	0	0
$77$	−23269.6	−0.447261
$78$	0	0
$79$	−33285.3	−0.600046	−0.300023	−	0.953932i	$-0.596994\pi$
$79$	−33285.3	−0.600046	−0.300023	+	0.953932i	$0.596994\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−66084.6	−1.05294	−0.526472	−	0.850192i	$-0.676486\pi$
$83$	−66084.6	−1.05294	−0.526472	+	0.850192i	$0.676486\pi$
$84$	0	0
$85$	6101.34	0.0915964
$86$	0	0
$87$	43532.3	0.616614
$88$	0	0
$89$	−103742.	−1.38829	−0.694144	−	0.719836i	$-0.744217\pi$
$89$	−103742.	−1.38829	−0.694144	+	0.719836i	$0.744217\pi$
$90$	0	0
$91$	−45821.1	−0.580045
$92$	0	0
$93$	−9795.00	−0.117435
$94$	0	0
$95$	49935.2	0.567673
$96$	0	0
$97$	−106653.	−1.15091	−0.575457	−	0.817832i	$-0.695176\pi$
$97$	−106653.	−1.15091	−0.575457	+	0.817832i	$0.695176\pi$
$98$	0	0
$99$	−10359.3	−0.106229
$100$	0	0
$101$	−129882.	−1.26691	−0.633455	−	0.773780i	$-0.718364\pi$
$101$	−129882.	−1.26691	−0.633455	+	0.773780i	$0.718364\pi$
$102$	0	0
$103$	−30389.0	−0.282243	−0.141122	−	0.989992i	$-0.545071\pi$
$103$	−30389.0	−0.282243	−0.141122	+	0.989992i	$0.545071\pi$
$104$	0	0
$105$	40937.9	0.362370
$106$	0	0
$107$	−84051.4	−0.709718	−0.354859	−	0.934920i	$-0.615471\pi$
$107$	−84051.4	−0.709718	−0.354859	+	0.934920i	$0.615471\pi$
$108$	0	0
$109$	221326.	1.78429	0.892146	−	0.451748i	$-0.149199\pi$
$109$	221326.	1.78429	0.892146	+	0.451748i	$0.149199\pi$
$110$	0	0
$111$	−4125.00	−0.0317772
$112$	0	0
$113$	87193.0	0.642370	0.321185	−	0.947016i	$-0.395919\pi$
$113$	87193.0	0.642370	0.321185	+	0.947016i	$0.395919\pi$
$114$	0	0
$115$	−23132.5	−0.163109
$116$	0	0
$117$	−20398.9	−0.137766
$118$	0	0
$119$	44404.7	0.287449
$120$	0	0
$121$	−144695.	−0.898439
$122$	0	0
$123$	100631.	0.599750
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	233065.	1.28224	0.641118	−	0.767442i	$-0.278471\pi$
$127$	233065.	1.28224	0.641118	+	0.767442i	$0.278471\pi$
$128$	0	0
$129$	155943.	0.825074
$130$	0	0
$131$	−219115.	−1.11556	−0.557780	−	0.829989i	$-0.688347\pi$
$131$	−219115.	−1.11556	−0.557780	+	0.829989i	$0.688347\pi$
$132$	0	0
$133$	363421.	1.78148
$134$	0	0
$135$	18225.0	0.0860663
$136$	0	0
$137$	−286397.	−1.30367	−0.651834	−	0.758362i	$-0.726000\pi$
$137$	−286397.	−1.30367	−0.651834	+	0.758362i	$0.726000\pi$
$138$	0	0
$139$	−44070.1	−0.193467	−0.0967336	−	0.995310i	$-0.530839\pi$
$139$	−44070.1	−0.193467	−0.0967336	+	0.995310i	$0.530839\pi$
$140$	0	0
$141$	−26744.0	−0.113287
$142$	0	0
$143$	32208.3	0.131713
$144$	0	0
$145$	120923.	0.477627
$146$	0	0
$147$	146677.	0.559846
$148$	0	0
$149$	−288846.	−1.06586	−0.532931	−	0.846159i	$-0.678909\pi$
$149$	−288846.	−1.06586	−0.532931	+	0.846159i	$0.678909\pi$
$150$	0	0
$151$	−152460.	−0.544144	−0.272072	−	0.962277i	$-0.587709\pi$
$151$	−152460.	−0.544144	−0.272072	+	0.962277i	$0.587709\pi$
$152$	0	0
$153$	19768.4	0.0682719
$154$	0	0
$155$	−27208.3	−0.0909646
$156$	0	0
$157$	179248.	0.580371	0.290185	−	0.956970i	$-0.406283\pi$
$157$	179248.	0.580371	0.290185	+	0.956970i	$0.406283\pi$
$158$	0	0
$159$	4449.68	0.0139584
$160$	0	0
$161$	−168355.	−0.511872
$162$	0	0
$163$	633499.	1.86757	0.933785	−	0.357833i	$-0.116484\pi$
$163$	633499.	1.86757	0.933785	+	0.357833i	$0.116484\pi$
$164$	0	0
$165$	−28775.8	−0.0822844
$166$	0	0
$167$	373688.	1.03685	0.518427	−	0.855122i	$-0.326518\pi$
$167$	373688.	1.03685	0.518427	+	0.855122i	$0.326518\pi$
$168$	0	0
$169$	−307870.	−0.829184
$170$	0	0
$171$	161790.	0.423118
$172$	0	0
$173$	592830.	1.50597	0.752983	−	0.658040i	$-0.228614\pi$
$173$	592830.	1.50597	0.752983	+	0.658040i	$0.228614\pi$
$174$	0	0
$175$	113716.	0.280691
$176$	0	0
$177$	−366958.	−0.880406
$178$	0	0
$179$	745714.	1.73956	0.869781	−	0.493438i	$-0.164260\pi$
$179$	745714.	1.73956	0.869781	+	0.493438i	$0.164260\pi$
$180$	0	0
$181$	201158.	0.456396	0.228198	−	0.973615i	$-0.426717\pi$
$181$	201158.	0.456396	0.228198	+	0.973615i	$0.426717\pi$
$182$	0	0
$183$	389493.	0.859748
$184$	0	0
$185$	−11458.3	−0.0246146
$186$	0	0
$187$	−31212.6	−0.0652719
$188$	0	0
$189$	132639.	0.270095
$190$	0	0
$191$	−430919.	−0.854696	−0.427348	−	0.904087i	$-0.640552\pi$
$191$	−430919.	−0.854696	−0.427348	+	0.904087i	$0.640552\pi$
$192$	0	0
$193$	100810.	0.194809	0.0974046	−	0.995245i	$-0.468946\pi$
$193$	100810.	0.194809	0.0974046	+	0.995245i	$0.468946\pi$
$194$	0	0
$195$	−56663.7	−0.106713
$196$	0	0
$197$	−413109.	−0.758402	−0.379201	−	0.925314i	$-0.623801\pi$
$197$	−413109.	−0.758402	−0.379201	+	0.925314i	$0.623801\pi$
$198$	0	0
$199$	−1.11025e6	−1.98742	−0.993710	−	0.111985i	$-0.964279\pi$
$199$	−1.11025e6	−1.98742	−0.993710	+	0.111985i	$0.964279\pi$
$200$	0	0
$201$	−581040.	−1.01441
$202$	0	0
$203$	880060.	1.49890
$204$	0	0
$205$	279531.	0.464564
$206$	0	0
$207$	−74949.4	−0.121574
$208$	0	0
$209$	−255453.	−0.404526
$210$	0	0
$211$	−1.18177e6	−1.82738	−0.913689	−	0.406414i	$-0.866779\pi$
$211$	−1.18177e6	−1.82738	−0.913689	+	0.406414i	$0.866779\pi$
$212$	0	0
$213$	−532269.	−0.803863
$214$	0	0
$215$	433176.	0.639100
$216$	0	0
$217$	−198018.	−0.285467
$218$	0	0
$219$	482391.	0.679656
$220$	0	0
$221$	−61462.2	−0.0846501
$222$	0	0
$223$	50731.0	0.0683143	0.0341571	−	0.999416i	$-0.489125\pi$
$223$	50731.0	0.0683143	0.0341571	+	0.999416i	$0.489125\pi$
$224$	0	0
$225$	50625.0	0.0666667
$226$	0	0
$227$	−390819.	−0.503397	−0.251699	−	0.967806i	$-0.580989\pi$
$227$	−390819.	−0.503397	−0.251699	+	0.967806i	$0.580989\pi$
$228$	0	0
$229$	−210736.	−0.265553	−0.132776	−	0.991146i	$-0.542389\pi$
$229$	−210736.	−0.265553	−0.132776	+	0.991146i	$0.542389\pi$
$230$	0	0
$231$	−209426.	−0.258226
$232$	0	0
$233$	−899004.	−1.08486	−0.542428	−	0.840103i	$-0.682495\pi$
$233$	−899004.	−1.08486	−0.542428	+	0.840103i	$0.682495\pi$
$234$	0	0
$235$	−74289.0	−0.0877515
$236$	0	0
$237$	−299567.	−0.346436
$238$	0	0
$239$	683426.	0.773921	0.386961	−	0.922096i	$-0.373525\pi$
$239$	683426.	0.773921	0.386961	+	0.922096i	$0.373525\pi$
$240$	0	0
$241$	171032.	0.189686	0.0948431	−	0.995492i	$-0.469765\pi$
$241$	171032.	0.189686	0.0948431	+	0.995492i	$0.469765\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	407436.	0.433655
$246$	0	0
$247$	−503025.	−0.524623
$248$	0	0
$249$	−594762.	−0.607918
$250$	0	0
$251$	−605609.	−0.606748	−0.303374	−	0.952872i	$-0.598113\pi$
$251$	−605609.	−0.606748	−0.303374	+	0.952872i	$0.598113\pi$
$252$	0	0
$253$	118339.	0.116232
$254$	0	0
$255$	54912.1	0.0528832
$256$	0	0
$257$	−1.44532e6	−1.36499	−0.682496	−	0.730890i	$-0.739105\pi$
$257$	−1.44532e6	−1.36499	−0.682496	+	0.730890i	$0.739105\pi$
$258$	0	0
$259$	−83392.0	−0.0772458
$260$	0	0
$261$	391791.	0.356003
$262$	0	0
$263$	−258342.	−0.230306	−0.115153	−	0.993348i	$-0.536736\pi$
$263$	−258342.	−0.230306	−0.115153	+	0.993348i	$0.536736\pi$
$264$	0	0
$265$	12360.2	0.0108121
$266$	0	0
$267$	−933679.	−0.801529
$268$	0	0
$269$	656692.	0.553326	0.276663	−	0.960967i	$-0.410771\pi$
$269$	656692.	0.553326	0.276663	+	0.960967i	$0.410771\pi$
$270$	0	0
$271$	−1.39116e6	−1.15068	−0.575338	−	0.817915i	$-0.695130\pi$
$271$	−1.39116e6	−1.15068	−0.575338	+	0.817915i	$0.695130\pi$
$272$	0	0
$273$	−412390.	−0.334889
$274$	0	0
$275$	−79932.8	−0.0637372
$276$	0	0
$277$	−183333.	−0.143562	−0.0717812	−	0.997420i	$-0.522868\pi$
$277$	−183333.	−0.143562	−0.0717812	+	0.997420i	$0.522868\pi$
$278$	0	0
$279$	−88155.0	−0.0678010
$280$	0	0
$281$	1.76964e6	1.33696	0.668482	−	0.743728i	$-0.266944\pi$
$281$	1.76964e6	1.33696	0.668482	+	0.743728i	$0.266944\pi$
$282$	0	0
$283$	558663.	0.414652	0.207326	−	0.978272i	$-0.433524\pi$
$283$	558663.	0.414652	0.207326	+	0.978272i	$0.433524\pi$
$284$	0	0
$285$	449417.	0.327746
$286$	0	0
$287$	2.03439e6	1.45790
$288$	0	0
$289$	−1.36029e6	−0.958051
$290$	0	0
$291$	−959876.	−0.664481
$292$	0	0
$293$	2.70821e6	1.84295	0.921475	−	0.388439i	$-0.126985\pi$
$293$	2.70821e6	1.84295	0.921475	+	0.388439i	$0.126985\pi$
$294$	0	0
$295$	−1.01933e6	−0.681960
$296$	0	0
$297$	−93233.6	−0.0613312
$298$	0	0
$299$	233027.	0.150740
$300$	0	0
$301$	3.15259e6	2.00563
$302$	0	0
$303$	−1.16894e6	−0.731451
$304$	0	0
$305$	1.08192e6	0.665958
$306$	0	0
$307$	−1.36276e6	−0.825229	−0.412614	−	0.910906i	$-0.635384\pi$
$307$	−1.36276e6	−0.825229	−0.412614	+	0.910906i	$0.635384\pi$
$308$	0	0
$309$	−273501.	−0.162953
$310$	0	0
$311$	551497.	0.323327	0.161664	−	0.986846i	$-0.448314\pi$
$311$	551497.	0.323327	0.161664	+	0.986846i	$0.448314\pi$
$312$	0	0
$313$	2.13970e6	1.23450	0.617250	−	0.786767i	$-0.288247\pi$
$313$	2.13970e6	1.23450	0.617250	+	0.786767i	$0.288247\pi$
$314$	0	0
$315$	368441.	0.209214
$316$	0	0
$317$	2.49183e6	1.39274	0.696371	−	0.717682i	$-0.254797\pi$
$317$	2.49183e6	1.39274	0.696371	+	0.717682i	$0.254797\pi$
$318$	0	0
$319$	−618606.	−0.340359
$320$	0	0
$321$	−756463.	−0.409756
$322$	0	0
$323$	487475.	0.259984
$324$	0	0
$325$	−157399.	−0.0826597
$326$	0	0
$327$	1.99193e6	1.03016
$328$	0	0
$329$	−540664.	−0.275383
$330$	0	0
$331$	510421.	0.256070	0.128035	−	0.991770i	$-0.459133\pi$
$331$	510421.	0.256070	0.128035	+	0.991770i	$0.459133\pi$
$332$	0	0
$333$	−37125.0	−0.0183466
$334$	0	0
$335$	−1.61400e6	−0.785762
$336$	0	0
$337$	−222751.	−0.106843	−0.0534214	−	0.998572i	$-0.517013\pi$
$337$	−222751.	−0.106843	−0.0534214	+	0.998572i	$0.517013\pi$
$338$	0	0
$339$	784737.	0.370873
$340$	0	0
$341$	139190.	0.0648218
$342$	0	0
$343$	−92714.3	−0.0425512
$344$	0	0
$345$	−208193.	−0.0941711
$346$	0	0
$347$	1.32623e6	0.591285	0.295642	−	0.955299i	$-0.404466\pi$
$347$	1.32623e6	0.591285	0.295642	+	0.955299i	$0.404466\pi$
$348$	0	0
$349$	−419785.	−0.184486	−0.0922429	−	0.995737i	$-0.529404\pi$
$349$	−419785.	−0.184486	−0.0922429	+	0.995737i	$0.529404\pi$
$350$	0	0
$351$	−183590.	−0.0795394
$352$	0	0
$353$	−3.43878e6	−1.46882	−0.734408	−	0.678709i	$-0.762540\pi$
$353$	−3.43878e6	−1.46882	−0.734408	+	0.678709i	$0.762540\pi$
$354$	0	0
$355$	−1.47852e6	−0.622670
$356$	0	0
$357$	399642.	0.165959
$358$	0	0
$359$	1.10582e6	0.452842	0.226421	−	0.974030i	$-0.427297\pi$
$359$	1.10582e6	0.452842	0.226421	+	0.974030i	$0.427297\pi$
$360$	0	0
$361$	1.51354e6	0.611261
$362$	0	0
$363$	−1.30225e6	−0.518714
$364$	0	0
$365$	1.33998e6	0.526459
$366$	0	0
$367$	1.25253e6	0.485428	0.242714	−	0.970098i	$-0.421962\pi$
$367$	1.25253e6	0.485428	0.242714	+	0.970098i	$0.421962\pi$
$368$	0	0
$369$	905681.	0.346266
$370$	0	0
$371$	89955.9	0.0339309
$372$	0	0
$373$	2.64185e6	0.983188	0.491594	−	0.870824i	$-0.336414\pi$
$373$	2.64185e6	0.983188	0.491594	+	0.870824i	$0.336414\pi$
$374$	0	0
$375$	140625.	0.0516398
$376$	0	0
$377$	−1.21812e6	−0.441406
$378$	0	0
$379$	1.56431e6	0.559403	0.279702	−	0.960087i	$-0.409764\pi$
$379$	1.56431e6	0.559403	0.279702	+	0.960087i	$0.409764\pi$
$380$	0	0
$381$	2.09759e6	0.740299
$382$	0	0
$383$	−2.61018e6	−0.909230	−0.454615	−	0.890688i	$-0.650223\pi$
$383$	−2.61018e6	−0.909230	−0.454615	+	0.890688i	$0.650223\pi$
$384$	0	0
$385$	−581739.	−0.200021
$386$	0	0
$387$	1.40349e6	0.476357
$388$	0	0
$389$	3.62371e6	1.21417	0.607085	−	0.794637i	$-0.292339\pi$
$389$	3.62371e6	1.21417	0.607085	+	0.794637i	$0.292339\pi$
$390$	0	0
$391$	−225823.	−0.0747010
$392$	0	0
$393$	−1.97203e6	−0.644069
$394$	0	0
$395$	−832131.	−0.268349
$396$	0	0
$397$	−2.74601e6	−0.874433	−0.437216	−	0.899356i	$-0.644036\pi$
$397$	−2.74601e6	−0.874433	−0.437216	+	0.899356i	$0.644036\pi$
$398$	0	0
$399$	3.27079e6	1.02854
$400$	0	0
$401$	−1.95684e6	−0.607706	−0.303853	−	0.952719i	$-0.598273\pi$
$401$	−1.95684e6	−0.607706	−0.303853	+	0.952719i	$0.598273\pi$
$402$	0	0
$403$	274084.	0.0840662
$404$	0	0
$405$	164025.	0.0496904
$406$	0	0
$407$	58617.3	0.0175404
$408$	0	0
$409$	−2.24202e6	−0.662720	−0.331360	−	0.943504i	$-0.607508\pi$
$409$	−2.24202e6	−0.662720	−0.331360	+	0.943504i	$0.607508\pi$
$410$	0	0
$411$	−2.57757e6	−0.752673
$412$	0	0
$413$	−7.41851e6	−2.14014
$414$	0	0
$415$	−1.65212e6	−0.470891
$416$	0	0
$417$	−396631.	−0.111698
$418$	0	0
$419$	−1.47312e6	−0.409925	−0.204962	−	0.978770i	$-0.565707\pi$
$419$	−1.47312e6	−0.409925	−0.204962	+	0.978770i	$0.565707\pi$
$420$	0	0
$421$	5.12480e6	1.40920	0.704598	−	0.709607i	$-0.251127\pi$
$421$	5.12480e6	1.40920	0.704598	+	0.709607i	$0.251127\pi$
$422$	0	0
$423$	−240696.	−0.0654061
$424$	0	0
$425$	152534.	0.0409631
$426$	0	0
$427$	7.87408e6	2.08992
$428$	0	0
$429$	289874.	0.0760443
$430$	0	0
$431$	−1.27555e6	−0.330754	−0.165377	−	0.986230i	$-0.552884\pi$
$431$	−1.27555e6	−0.330754	−0.165377	+	0.986230i	$0.552884\pi$
$432$	0	0
$433$	−5.88552e6	−1.50857	−0.754285	−	0.656547i	$-0.772016\pi$
$433$	−5.88552e6	−1.50857	−0.754285	+	0.656547i	$0.772016\pi$
$434$	0	0
$435$	1.08831e6	0.275758
$436$	0	0
$437$	−1.84820e6	−0.462963
$438$	0	0
$439$	216284.	0.0535629	0.0267815	−	0.999641i	$-0.491474\pi$
$439$	216284.	0.0535629	0.0267815	+	0.999641i	$0.491474\pi$
$440$	0	0
$441$	1.32009e6	0.323227
$442$	0	0
$443$	−5.26511e6	−1.27467	−0.637336	−	0.770586i	$-0.719964\pi$
$443$	−5.26511e6	−1.27467	−0.637336	+	0.770586i	$0.719964\pi$
$444$	0	0
$445$	−2.59355e6	−0.620862
$446$	0	0
$447$	−2.59961e6	−0.615375
$448$	0	0
$449$	238753.	0.0558898	0.0279449	−	0.999609i	$-0.491104\pi$
$449$	238753.	0.0558898	0.0279449	+	0.999609i	$0.491104\pi$
$450$	0	0
$451$	−1.43000e6	−0.331050
$452$	0	0
$453$	−1.37214e6	−0.314162
$454$	0	0
$455$	−1.14553e6	−0.259404
$456$	0	0
$457$	1.30963e6	0.293330	0.146665	−	0.989186i	$-0.453146\pi$
$457$	1.30963e6	0.293330	0.146665	+	0.989186i	$0.453146\pi$
$458$	0	0
$459$	177915.	0.0394168
$460$	0	0
$461$	1.94428e6	0.426095	0.213048	−	0.977042i	$-0.431661\pi$
$461$	1.94428e6	0.426095	0.213048	+	0.977042i	$0.431661\pi$
$462$	0	0
$463$	2.19401e6	0.475648	0.237824	−	0.971308i	$-0.423566\pi$
$463$	2.19401e6	0.475648	0.237824	+	0.971308i	$0.423566\pi$
$464$	0	0
$465$	−244875.	−0.0525185
$466$	0	0
$467$	1.68386e6	0.357284	0.178642	−	0.983914i	$-0.442830\pi$
$467$	1.68386e6	0.357284	0.178642	+	0.983914i	$0.442830\pi$
$468$	0	0
$469$	−1.17464e7	−2.46589
$470$	0	0
$471$	1.61323e6	0.335077
$472$	0	0
$473$	−2.21600e6	−0.455425
$474$	0	0
$475$	1.24838e6	0.253871
$476$	0	0
$477$	40047.1	0.00805890
$478$	0	0
$479$	2.64042e6	0.525818	0.262909	−	0.964821i	$-0.415318\pi$
$479$	2.64042e6	0.525818	0.262909	+	0.964821i	$0.415318\pi$
$480$	0	0
$481$	115426.	0.0227479
$482$	0	0
$483$	−1.51520e6	−0.295529
$484$	0	0
$485$	−2.66632e6	−0.514705
$486$	0	0
$487$	5.71035e6	1.09104	0.545520	−	0.838098i	$-0.316332\pi$
$487$	5.71035e6	1.09104	0.545520	+	0.838098i	$0.316332\pi$
$488$	0	0
$489$	5.70149e6	1.07824
$490$	0	0
$491$	−7.05533e6	−1.32073	−0.660364	−	0.750946i	$-0.729598\pi$
$491$	−7.05533e6	−1.32073	−0.660364	+	0.750946i	$0.729598\pi$
$492$	0	0
$493$	1.18047e6	0.218745
$494$	0	0
$495$	−258982.	−0.0475069
$496$	0	0
$497$	−1.07605e7	−1.95407
$498$	0	0
$499$	−6.81218e6	−1.22471	−0.612357	−	0.790581i	$-0.709779\pi$
$499$	−6.81218e6	−1.22471	−0.612357	+	0.790581i	$0.709779\pi$
$500$	0	0
$501$	3.36319e6	0.598628
$502$	0	0
$503$	−5.41746e6	−0.954720	−0.477360	−	0.878708i	$-0.658406\pi$
$503$	−5.41746e6	−0.954720	−0.477360	+	0.878708i	$0.658406\pi$
$504$	0	0
$505$	−3.24705e6	−0.566579
$506$	0	0
$507$	−2.77083e6	−0.478730
$508$	0	0
$509$	−836069.	−0.143037	−0.0715184	−	0.997439i	$-0.522784\pi$
$509$	−836069.	−0.143037	−0.0715184	+	0.997439i	$0.522784\pi$
$510$	0	0
$511$	9.75214e6	1.65214
$512$	0	0
$513$	1.45611e6	0.244287
$514$	0	0
$515$	−759725.	−0.126223
$516$	0	0
$517$	380040.	0.0625321
$518$	0	0
$519$	5.33547e6	0.869469
$520$	0	0
$521$	9.06522e6	1.46313	0.731567	−	0.681770i	$-0.238790\pi$
$521$	9.06522e6	1.46313	0.731567	+	0.681770i	$0.238790\pi$
$522$	0	0
$523$	1.14796e7	1.83515	0.917577	−	0.397558i	$-0.130142\pi$
$523$	1.14796e7	1.83515	0.917577	+	0.397558i	$0.130142\pi$
$524$	0	0
$525$	1.02345e6	0.162057
$526$	0	0
$527$	−265612.	−0.0416601
$528$	0	0
$529$	−5.58016e6	−0.866977
$530$	0	0
$531$	−3.30262e6	−0.508303
$532$	0	0
$533$	−2.81587e6	−0.429333
$534$	0	0
$535$	−2.10129e6	−0.317395
$536$	0	0
$537$	6.71143e6	1.00434
$538$	0	0
$539$	−2.08432e6	−0.309024
$540$	0	0
$541$	1.15712e7	1.69975	0.849876	−	0.526983i	$-0.176677\pi$
$541$	1.15712e7	1.69975	0.849876	+	0.526983i	$0.176677\pi$
$542$	0	0
$543$	1.81042e6	0.263500
$544$	0	0
$545$	5.53315e6	0.797959
$546$	0	0
$547$	9.81460e6	1.40250	0.701252	−	0.712913i	$-0.252625\pi$
$547$	9.81460e6	1.40250	0.701252	+	0.712913i	$0.252625\pi$
$548$	0	0
$549$	3.50543e6	0.496376
$550$	0	0
$551$	9.66131e6	1.35568
$552$	0	0
$553$	−6.05613e6	−0.842136
$554$	0	0
$555$	−103125.	−0.0142112
$556$	0	0
$557$	−486078.	−0.0663847	−0.0331923	−	0.999449i	$-0.510567\pi$
$557$	−486078.	−0.0663847	−0.0331923	+	0.999449i	$0.510567\pi$
$558$	0	0
$559$	−4.36362e6	−0.590633
$560$	0	0
$561$	−280914.	−0.0376848
$562$	0	0
$563$	−2.10066e6	−0.279309	−0.139654	−	0.990200i	$-0.544599\pi$
$563$	−2.10066e6	−0.279309	−0.139654	+	0.990200i	$0.544599\pi$
$564$	0	0
$565$	2.17982e6	0.287277
$566$	0	0
$567$	1.19375e6	0.155939
$568$	0	0
$569$	8.39323e6	1.08680	0.543398	−	0.839475i	$-0.317137\pi$
$569$	8.39323e6	1.08680	0.543398	+	0.839475i	$0.317137\pi$
$570$	0	0
$571$	9.34672e6	1.19969	0.599845	−	0.800116i	$-0.295229\pi$
$571$	9.34672e6	1.19969	0.599845	+	0.800116i	$0.295229\pi$
$572$	0	0
$573$	−3.87827e6	−0.493459
$574$	0	0
$575$	−578313.	−0.0729446
$576$	0	0
$577$	1.32427e6	0.165592	0.0827958	−	0.996567i	$-0.473615\pi$
$577$	1.32427e6	0.165592	0.0827958	+	0.996567i	$0.473615\pi$
$578$	0	0
$579$	907288.	0.112473
$580$	0	0
$581$	−1.20239e7	−1.47776
$582$	0	0
$583$	−63231.2	−0.00770478
$584$	0	0
$585$	−509973.	−0.0616109
$586$	0	0
$587$	7.81460e6	0.936077	0.468039	−	0.883708i	$-0.344961\pi$
$587$	7.81460e6	0.936077	0.468039	+	0.883708i	$0.344961\pi$
$588$	0	0
$589$	−2.17385e6	−0.258191
$590$	0	0
$591$	−3.71798e6	−0.437863
$592$	0	0
$593$	1.54880e7	1.80867	0.904336	−	0.426822i	$-0.140367\pi$
$593$	1.54880e7	1.80867	0.904336	+	0.426822i	$0.140367\pi$
$594$	0	0
$595$	1.11012e6	0.128551
$596$	0	0
$597$	−9.99229e6	−1.14744
$598$	0	0
$599$	1.39080e7	1.58378	0.791892	−	0.610661i	$-0.209096\pi$
$599$	1.39080e7	1.58378	0.791892	+	0.610661i	$0.209096\pi$
$600$	0	0
$601$	1.75908e6	0.198655	0.0993276	−	0.995055i	$-0.468331\pi$
$601$	1.75908e6	0.198655	0.0993276	+	0.995055i	$0.468331\pi$
$602$	0	0
$603$	−5.22936e6	−0.585673
$604$	0	0
$605$	−3.61736e6	−0.401794
$606$	0	0
$607$	−1.39483e7	−1.53656	−0.768282	−	0.640111i	$-0.778888\pi$
$607$	−1.39483e7	−1.53656	−0.768282	+	0.640111i	$0.778888\pi$
$608$	0	0
$609$	7.92054e6	0.865390
$610$	0	0
$611$	748354.	0.0810968
$612$	0	0
$613$	3.35511e6	0.360625	0.180313	−	0.983609i	$-0.442289\pi$
$613$	3.35511e6	0.360625	0.180313	+	0.983609i	$0.442289\pi$
$614$	0	0
$615$	2.51578e6	0.268216
$616$	0	0
$617$	−2.13423e6	−0.225699	−0.112849	−	0.993612i	$-0.535998\pi$
$617$	−2.13423e6	−0.225699	−0.112849	+	0.993612i	$0.535998\pi$
$618$	0	0
$619$	178637.	0.0187389	0.00936945	−	0.999956i	$-0.497018\pi$
$619$	178637.	0.0187389	0.00936945	+	0.999956i	$0.497018\pi$
$620$	0	0
$621$	−674544.	−0.0701910
$622$	0	0
$623$	−1.88755e7	−1.94840
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−2.29908e6	−0.233553
$628$	0	0
$629$	−111858.	−0.0112730
$630$	0	0
$631$	−1.17072e7	−1.17052	−0.585262	−	0.810844i	$-0.699008\pi$
$631$	−1.17072e7	−1.17052	−0.585262	+	0.810844i	$0.699008\pi$
$632$	0	0
$633$	−1.06360e7	−1.05504
$634$	0	0
$635$	5.82663e6	0.573433
$636$	0	0
$637$	−4.10432e6	−0.400768
$638$	0	0
$639$	−4.79042e6	−0.464111
$640$	0	0
$641$	4.68837e6	0.450688	0.225344	−	0.974279i	$-0.427649\pi$
$641$	4.68837e6	0.450688	0.225344	+	0.974279i	$0.427649\pi$
$642$	0	0
$643$	1.37226e7	1.30890	0.654452	−	0.756104i	$-0.272899\pi$
$643$	1.37226e7	1.30890	0.654452	+	0.756104i	$0.272899\pi$
$644$	0	0
$645$	3.89859e6	0.368984
$646$	0	0
$647$	684806.	0.0643142	0.0321571	−	0.999483i	$-0.489762\pi$
$647$	684806.	0.0643142	0.0321571	+	0.999483i	$0.489762\pi$
$648$	0	0
$649$	5.21457e6	0.485967
$650$	0	0
$651$	−1.78216e6	−0.164814
$652$	0	0
$653$	1.67593e7	1.53806	0.769028	−	0.639215i	$-0.220740\pi$
$653$	1.67593e7	1.53806	0.769028	+	0.639215i	$0.220740\pi$
$654$	0	0
$655$	−5.47787e6	−0.498894
$656$	0	0
$657$	4.34152e6	0.392399
$658$	0	0
$659$	2.36751e6	0.212363	0.106181	−	0.994347i	$-0.466138\pi$
$659$	2.36751e6	0.212363	0.106181	+	0.994347i	$0.466138\pi$
$660$	0	0
$661$	−1.90982e7	−1.70016	−0.850080	−	0.526654i	$-0.823446\pi$
$661$	−1.90982e7	−1.70016	−0.850080	+	0.526654i	$0.823446\pi$
$662$	0	0
$663$	−553160.	−0.0488727
$664$	0	0
$665$	9.08552e6	0.796702
$666$	0	0
$667$	−4.47561e6	−0.389527
$668$	0	0
$669$	456579.	0.0394413
$670$	0	0
$671$	−5.53479e6	−0.474564
$672$	0	0
$673$	2.23974e7	1.90616	0.953079	−	0.302721i	$-0.0978949\pi$
$673$	2.23974e7	1.90616	0.953079	+	0.302721i	$0.0978949\pi$
$674$	0	0
$675$	455625.	0.0384900
$676$	0	0
$677$	2.03473e7	1.70622	0.853108	−	0.521734i	$-0.174715\pi$
$677$	2.03473e7	1.70622	0.853108	+	0.521734i	$0.174715\pi$
$678$	0	0
$679$	−1.94051e7	−1.61525
$680$	0	0
$681$	−3.51737e6	−0.290636
$682$	0	0
$683$	−1.78196e7	−1.46166	−0.730830	−	0.682560i	$-0.760867\pi$
$683$	−1.78196e7	−1.46166	−0.730830	+	0.682560i	$0.760867\pi$
$684$	0	0
$685$	−7.15992e6	−0.583018
$686$	0	0
$687$	−1.89663e6	−0.153317
$688$	0	0
$689$	−124511.	−0.00999220
$690$	0	0
$691$	2.42200e6	0.192965	0.0964827	−	0.995335i	$-0.469241\pi$
$691$	2.42200e6	0.192965	0.0964827	+	0.995335i	$0.469241\pi$
$692$	0	0
$693$	−1.88483e6	−0.149087
$694$	0	0
$695$	−1.10175e6	−0.0865212
$696$	0	0
$697$	2.72883e6	0.212762
$698$	0	0
$699$	−8.09104e6	−0.626341
$700$	0	0
$701$	−1.21906e7	−0.936981	−0.468491	−	0.883469i	$-0.655202\pi$
$701$	−1.21906e7	−0.936981	−0.468491	+	0.883469i	$0.655202\pi$
$702$	0	0
$703$	−915478.	−0.0698650
$704$	0	0
$705$	−668601.	−0.0506634
$706$	0	0
$707$	−2.36315e7	−1.77805
$708$	0	0
$709$	−8.30736e6	−0.620651	−0.310325	−	0.950630i	$-0.600438\pi$
$709$	−8.30736e6	−0.620651	−0.310325	+	0.950630i	$0.600438\pi$
$710$	0	0
$711$	−2.69611e6	−0.200015
$712$	0	0
$713$	1.00704e6	0.0741858
$714$	0	0
$715$	805207.	0.0589037
$716$	0	0
$717$	6.15084e6	0.446824
$718$	0	0
$719$	−4.12547e6	−0.297613	−0.148806	−	0.988866i	$-0.547543\pi$
$719$	−4.12547e6	−0.297613	−0.148806	+	0.988866i	$0.547543\pi$
$720$	0	0
$721$	−5.52917e6	−0.396115
$722$	0	0
$723$	1.53929e6	0.109515
$724$	0	0
$725$	3.02308e6	0.213602
$726$	0	0
$727$	−4.58996e6	−0.322087	−0.161044	−	0.986947i	$-0.551486\pi$
$727$	−4.58996e6	−0.322087	−0.161044	+	0.986947i	$0.551486\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	4.22873e6	0.292696
$732$	0	0
$733$	−5.01972e6	−0.345080	−0.172540	−	0.985003i	$-0.555197\pi$
$733$	−5.01972e6	−0.345080	−0.172540	+	0.985003i	$0.555197\pi$
$734$	0	0
$735$	3.66692e6	0.250371
$736$	0	0
$737$	8.25673e6	0.559937
$738$	0	0
$739$	−1.35567e7	−0.913150	−0.456575	−	0.889685i	$-0.650924\pi$
$739$	−1.35567e7	−0.913150	−0.456575	+	0.889685i	$0.650924\pi$
$740$	0	0
$741$	−4.52722e6	−0.302891
$742$	0	0
$743$	2.70196e7	1.79559	0.897795	−	0.440414i	$-0.145168\pi$
$743$	2.70196e7	1.79559	0.897795	+	0.440414i	$0.145168\pi$
$744$	0	0
$745$	−7.22115e6	−0.476668
$746$	0	0
$747$	−5.35286e6	−0.350981
$748$	0	0
$749$	−1.52928e7	−0.996056
$750$	0	0
$751$	−1.53513e7	−0.993219	−0.496609	−	0.867974i	$-0.665422\pi$
$751$	−1.53513e7	−0.993219	−0.496609	+	0.867974i	$0.665422\pi$
$752$	0	0
$753$	−5.45048e6	−0.350306
$754$	0	0
$755$	−3.81150e6	−0.243349
$756$	0	0
$757$	−2.36598e7	−1.50062	−0.750311	−	0.661085i	$-0.770096\pi$
$757$	−2.36598e7	−1.50062	−0.750311	+	0.661085i	$0.770096\pi$
$758$	0	0
$759$	1.06505e6	0.0671067
$760$	0	0
$761$	−1.71252e7	−1.07195	−0.535976	−	0.844233i	$-0.680056\pi$
$761$	−1.71252e7	−1.07195	−0.535976	+	0.844233i	$0.680056\pi$
$762$	0	0
$763$	4.02694e7	2.50417
$764$	0	0
$765$	494209.	0.0305321
$766$	0	0
$767$	1.02682e7	0.630243
$768$	0	0
$769$	9.39886e6	0.573138	0.286569	−	0.958060i	$-0.407485\pi$
$769$	9.39886e6	0.573138	0.286569	+	0.958060i	$0.407485\pi$
$770$	0	0
$771$	−1.30078e7	−0.788078
$772$	0	0
$773$	2.90528e7	1.74880	0.874399	−	0.485207i	$-0.161256\pi$
$773$	2.90528e7	1.74880	0.874399	+	0.485207i	$0.161256\pi$
$774$	0	0
$775$	−680208.	−0.0406806
$776$	0	0
$777$	−750528.	−0.0445979
$778$	0	0
$779$	2.23335e7	1.31860
$780$	0	0
$781$	7.56369e6	0.443717
$782$	0	0
$783$	3.52612e6	0.205538
$784$	0	0
$785$	4.48120e6	0.259550
$786$	0	0
$787$	−2.24192e7	−1.29028	−0.645139	−	0.764066i	$-0.723200\pi$
$787$	−2.24192e7	−1.29028	−0.645139	+	0.764066i	$0.723200\pi$
$788$	0	0
$789$	−2.32508e6	−0.132967
$790$	0	0
$791$	1.58644e7	0.901537
$792$	0	0
$793$	−1.08988e7	−0.615455
$794$	0	0
$795$	111242.	0.00624240
$796$	0	0
$797$	2.78046e6	0.155050	0.0775248	−	0.996990i	$-0.475298\pi$
$797$	2.78046e6	0.155050	0.0775248	+	0.996990i	$0.475298\pi$
$798$	0	0
$799$	−725220.	−0.0401886
$800$	0	0
$801$	−8.40311e6	−0.462763
$802$	0	0
$803$	−6.85491e6	−0.375157
$804$	0	0
$805$	−4.20888e6	−0.228916
$806$	0	0
$807$	5.91023e6	0.319463
$808$	0	0
$809$	1.48340e7	0.796871	0.398435	−	0.917196i	$-0.369553\pi$
$809$	1.48340e7	0.796871	0.398435	+	0.917196i	$0.369553\pi$
$810$	0	0
$811$	−1.08786e7	−0.580792	−0.290396	−	0.956907i	$-0.593787\pi$
$811$	−1.08786e7	−0.580792	−0.290396	+	0.956907i	$0.593787\pi$
$812$	0	0
$813$	−1.25204e7	−0.664344
$814$	0	0
$815$	1.58375e7	0.835203
$816$	0	0
$817$	3.46092e7	1.81400
$818$	0	0
$819$	−3.71151e6	−0.193348
$820$	0	0
$821$	−3.48339e7	−1.80361	−0.901807	−	0.432139i	$-0.857759\pi$
$821$	−3.48339e7	−1.80361	−0.901807	+	0.432139i	$0.857759\pi$
$822$	0	0
$823$	1.17801e7	0.606246	0.303123	−	0.952951i	$-0.401971\pi$
$823$	1.17801e7	0.606246	0.303123	+	0.952951i	$0.401971\pi$
$824$	0	0
$825$	−719395.	−0.0367987
$826$	0	0
$827$	−5.24580e6	−0.266715	−0.133358	−	0.991068i	$-0.542576\pi$
$827$	−5.24580e6	−0.266715	−0.133358	+	0.991068i	$0.542576\pi$
$828$	0	0
$829$	−3.36697e7	−1.70158	−0.850791	−	0.525504i	$-0.823877\pi$
$829$	−3.36697e7	−1.70158	−0.850791	+	0.525504i	$0.823877\pi$
$830$	0	0
$831$	−1.65000e6	−0.0828858
$832$	0	0
$833$	3.97745e6	0.198606
$834$	0	0
$835$	9.34220e6	0.463696
$836$	0	0
$837$	−793395.	−0.0391449
$838$	0	0
$839$	1.13862e7	0.558439	0.279219	−	0.960227i	$-0.409924\pi$
$839$	1.13862e7	0.558439	0.279219	+	0.960227i	$0.409924\pi$
$840$	0	0
$841$	2.88469e6	0.140640
$842$	0	0
$843$	1.59268e7	0.771897
$844$	0	0
$845$	−7.69676e6	−0.370822
$846$	0	0
$847$	−2.63266e7	−1.26092
$848$	0	0
$849$	5.02797e6	0.239399
$850$	0	0
$851$	424096.	0.0200743
$852$	0	0
$853$	2.17941e7	1.02557	0.512786	−	0.858516i	$-0.328613\pi$
$853$	2.17941e7	1.02557	0.512786	+	0.858516i	$0.328613\pi$
$854$	0	0
$855$	4.04475e6	0.189224
$856$	0	0
$857$	−4.01574e6	−0.186773	−0.0933865	−	0.995630i	$-0.529769\pi$
$857$	−4.01574e6	−0.186773	−0.0933865	+	0.995630i	$0.529769\pi$
$858$	0	0
$859$	2.38283e7	1.10182	0.550909	−	0.834565i	$-0.314281\pi$
$859$	2.38283e7	1.10182	0.550909	+	0.834565i	$0.314281\pi$
$860$	0	0
$861$	1.83095e7	0.841721
$862$	0	0
$863$	1.22115e6	0.0558139	0.0279070	−	0.999611i	$-0.491116\pi$
$863$	1.22115e6	0.0558139	0.0279070	+	0.999611i	$0.491116\pi$
$864$	0	0
$865$	1.48208e7	0.673488
$866$	0	0
$867$	−1.22427e7	−0.553131
$868$	0	0
$869$	4.25693e6	0.191226
$870$	0	0
$871$	1.62587e7	0.726173
$872$	0	0
$873$	−8.63888e6	−0.383638
$874$	0	0
$875$	2.84291e6	0.125529
$876$	0	0
$877$	3.98937e7	1.75148	0.875740	−	0.482783i	$-0.160374\pi$
$877$	3.98937e7	1.75148	0.875740	+	0.482783i	$0.160374\pi$
$878$	0	0
$879$	2.43739e7	1.06403
$880$	0	0
$881$	−2.39399e7	−1.03916	−0.519581	−	0.854421i	$-0.673912\pi$
$881$	−2.39399e7	−1.03916	−0.519581	+	0.854421i	$0.673912\pi$
$882$	0	0
$883$	−1.28403e7	−0.554209	−0.277105	−	0.960840i	$-0.589375\pi$
$883$	−1.28403e7	−0.554209	−0.277105	+	0.960840i	$0.589375\pi$
$884$	0	0
$885$	−9.17395e6	−0.393730
$886$	0	0
$887$	2.06681e7	0.882044	0.441022	−	0.897496i	$-0.354616\pi$
$887$	2.06681e7	0.882044	0.441022	+	0.897496i	$0.354616\pi$
$888$	0	0
$889$	4.24053e7	1.79956
$890$	0	0
$891$	−839102.	−0.0354096
$892$	0	0
$893$	−5.93542e6	−0.249071
$894$	0	0
$895$	1.86429e7	0.777956
$896$	0	0
$897$	2.09724e6	0.0870296
$898$	0	0
$899$	−5.26419e6	−0.217236
$900$	0	0
$901$	120662.	0.00495177
$902$	0	0
$903$	2.83733e7	1.15795
$904$	0	0
$905$	5.02896e6	0.204106
$906$	0	0
$907$	6.83620e6	0.275929	0.137964	−	0.990437i	$-0.455944\pi$
$907$	6.83620e6	0.275929	0.137964	+	0.990437i	$0.455944\pi$
$908$	0	0
$909$	−1.05204e7	−0.422303
$910$	0	0
$911$	4.23905e7	1.69228	0.846140	−	0.532960i	$-0.178920\pi$
$911$	4.23905e7	1.69228	0.846140	+	0.532960i	$0.178920\pi$
$912$	0	0
$913$	8.45173e6	0.335559
$914$	0	0
$915$	9.73731e6	0.384491
$916$	0	0
$917$	−3.98671e7	−1.56564
$918$	0	0
$919$	−3.67892e7	−1.43692	−0.718458	−	0.695570i	$-0.755152\pi$
$919$	−3.67892e7	−1.43692	−0.718458	+	0.695570i	$0.755152\pi$
$920$	0	0
$921$	−1.22649e7	−0.476446
$922$	0	0
$923$	1.48940e7	0.575449
$924$	0	0
$925$	−286458.	−0.0110080
$926$	0	0
$927$	−2.46151e6	−0.0940811
$928$	0	0
$929$	9.13861e6	0.347409	0.173704	−	0.984798i	$-0.444426\pi$
$929$	9.13861e6	0.347409	0.173704	+	0.984798i	$0.444426\pi$
$930$	0	0
$931$	3.25526e7	1.23087
$932$	0	0
$933$	4.96347e6	0.186673
$934$	0	0
$935$	−780316.	−0.0291905
$936$	0	0
$937$	7.25527e6	0.269963	0.134982	−	0.990848i	$-0.456902\pi$
$937$	7.25527e6	0.269963	0.134982	+	0.990848i	$0.456902\pi$
$938$	0	0
$939$	1.92573e7	0.712739
$940$	0	0
$941$	−3.77242e7	−1.38882	−0.694410	−	0.719580i	$-0.744335\pi$
$941$	−3.77242e7	−1.38882	−0.694410	+	0.719580i	$0.744335\pi$
$942$	0	0
$943$	−1.03460e7	−0.378873
$944$	0	0
$945$	3.31597e6	0.120790
$946$	0	0
$947$	−3.24400e7	−1.17545	−0.587727	−	0.809060i	$-0.699977\pi$
$947$	−3.24400e7	−1.17545	−0.587727	+	0.809060i	$0.699977\pi$
$948$	0	0
$949$	−1.34983e7	−0.486535
$950$	0	0
$951$	2.24265e7	0.804100
$952$	0	0
$953$	−2.02261e7	−0.721407	−0.360703	−	0.932681i	$-0.617463\pi$
$953$	−2.02261e7	−0.721407	−0.360703	+	0.932681i	$0.617463\pi$
$954$	0	0
$955$	−1.07730e7	−0.382232
$956$	0	0
$957$	−5.56746e6	−0.196506
$958$	0	0
$959$	−5.21088e7	−1.82964
$960$	0	0
$961$	−2.74447e7	−0.958627
$962$	0	0
$963$	−6.80816e6	−0.236573
$964$	0	0
$965$	2.52024e6	0.0871213
$966$	0	0
$967$	−4.30858e6	−0.148173	−0.0740864	−	0.997252i	$-0.523604\pi$
$967$	−4.30858e6	−0.148173	−0.0740864	+	0.997252i	$0.523604\pi$
$968$	0	0
$969$	4.38728e6	0.150102
$970$	0	0
$971$	4.16423e7	1.41738	0.708690	−	0.705520i	$-0.249287\pi$
$971$	4.16423e7	1.41738	0.708690	+	0.705520i	$0.249287\pi$
$972$	0	0
$973$	−8.01840e6	−0.271522
$974$	0	0
$975$	−1.41659e6	−0.0477236
$976$	0	0
$977$	−3.76585e7	−1.26219	−0.631097	−	0.775704i	$-0.717395\pi$
$977$	−3.76585e7	−1.26219	−0.631097	+	0.775704i	$0.717395\pi$
$978$	0	0
$979$	1.32678e7	0.442428
$980$	0	0
$981$	1.79274e7	0.594764
$982$	0	0
$983$	−3.31025e7	−1.09264	−0.546321	−	0.837576i	$-0.683972\pi$
$983$	−3.31025e7	−1.09264	−0.546321	+	0.837576i	$0.683972\pi$
$984$	0	0
$985$	−1.03277e7	−0.339168
$986$	0	0
$987$	−4.86598e6	−0.158993
$988$	0	0
$989$	−1.60327e7	−0.521215
$990$	0	0
$991$	−5.77669e7	−1.86851	−0.934254	−	0.356609i	$-0.883933\pi$
$991$	−5.77669e7	−1.86851	−0.934254	+	0.356609i	$0.883933\pi$
$992$	0	0
$993$	4.59378e6	0.147842
$994$	0	0
$995$	−2.77563e7	−0.888801
$996$	0	0
$997$	3.67091e7	1.16960	0.584798	−	0.811179i	$-0.301174\pi$
$997$	3.67091e7	1.16960	0.584798	+	0.811179i	$0.301174\pi$
$998$	0	0
$999$	−334125.	−0.0105924

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.6.a.h.1.2	✓	2
3.2	odd	2		360.6.a.k.1.2		2
4.3	odd	2		240.6.a.p.1.1		2
5.2	odd	4		600.6.f.m.49.2		4
5.3	odd	4		600.6.f.m.49.3		4
5.4	even	2		600.6.a.l.1.1		2
8.3	odd	2		960.6.a.bk.1.1		2
8.5	even	2		960.6.a.be.1.2		2
12.11	even	2		720.6.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.6.a.h.1.2	✓	2	1.1	even	1	trivial
240.6.a.p.1.1		2	4.3	odd	2
360.6.a.k.1.2		2	3.2	odd	2
600.6.a.l.1.1		2	5.4	even	2
600.6.f.m.49.2		4	5.2	odd	4
600.6.f.m.49.3		4	5.3	odd	4
720.6.a.ba.1.1		2	12.11	even	2
960.6.a.be.1.2		2	8.5	even	2
960.6.a.bk.1.1		2	8.3	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:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	120.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +25.0000 q^{5} +181.946 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +25.0000 q^{5} +181.946 q^{7} +81.0000 q^{9} -127.892 q^{11} -251.839 q^{13} +225.000 q^{15} +244.054 q^{17} +1997.41 q^{19} +1637.52 q^{21} -925.301 q^{23} +625.000 q^{25} +729.000 q^{27} +4836.92 q^{29} -1088.33 q^{31} -1151.03 q^{33} +4548.66 q^{35} -458.333 q^{37} -2266.55 q^{39} +11181.2 q^{41} +17327.1 q^{43} +2025.00 q^{45} -2971.56 q^{47} +16297.4 q^{49} +2196.48 q^{51} +494.409 q^{53} -3197.31 q^{55} +17976.7 q^{57} -40773.1 q^{59} +43276.9 q^{61} +14737.6 q^{63} -6295.97 q^{65} -64560.0 q^{67} -8327.71 q^{69} -59141.0 q^{71} +53599.0 q^{73} +5625.00 q^{75} -23269.6 q^{77} -33285.3 q^{79} +6561.00 q^{81} -66084.6 q^{83} +6101.34 q^{85} +43532.3 q^{87} -103742. q^{89} -45821.1 q^{91} -9795.00 q^{93} +49935.2 q^{95} -106653. q^{97} -10359.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 18 q^{3} + 50 q^{5} - 8 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q + 18 * q^3 + 50 * q^5 - 8 * q^7 + 162 * q^9	\( 2 q + 18 q^{3} + 50 q^{5} - 8 q^{7} + 162 q^{9} + 488 q^{11} + 612 q^{13} + 450 q^{15} + 860 q^{17} - 96 q^{19} - 72 q^{21} + 2984 q^{23} + 1250 q^{25} + 1458 q^{27} + 2236 q^{29} + 9352 q^{31} + 4392 q^{33} - 200 q^{35} + 10612 q^{37} + 5508 q^{39} + 17156 q^{41} + 440 q^{43} + 4050 q^{45} - 16728 q^{47} + 35570 q^{49} + 7740 q^{51} + 31484 q^{53} + 12200 q^{55} - 864 q^{57} - 61464 q^{59} + 51596 q^{61} - 648 q^{63} + 15300 q^{65} - 45816 q^{67} + 26856 q^{69} - 97456 q^{71} + 58852 q^{73} + 11250 q^{75} - 140256 q^{77} - 116776 q^{79} + 13122 q^{81} - 136632 q^{83} + 21500 q^{85} + 20124 q^{87} - 124924 q^{89} - 209904 q^{91} + 84168 q^{93} - 2400 q^{95} - 39260 q^{97} + 39528 q^{99}+O(q^{100}) \) 2 * q + 18 * q^3 + 50 * q^5 - 8 * q^7 + 162 * q^9 + 488 * q^11 + 612 * q^13 + 450 * q^15 + 860 * q^17 - 96 * q^19 - 72 * q^21 + 2984 * q^23 + 1250 * q^25 + 1458 * q^27 + 2236 * q^29 + 9352 * q^31 + 4392 * q^33 - 200 * q^35 + 10612 * q^37 + 5508 * q^39 + 17156 * q^41 + 440 * q^43 + 4050 * q^45 - 16728 * q^47 + 35570 * q^49 + 7740 * q^51 + 31484 * q^53 + 12200 * q^55 - 864 * q^57 - 61464 * q^59 + 51596 * q^61 - 648 * q^63 + 15300 * q^65 - 45816 * q^67 + 26856 * q^69 - 97456 * q^71 + 58852 * q^73 + 11250 * q^75 - 140256 * q^77 - 116776 * q^79 + 13122 * q^81 - 136632 * q^83 + 21500 * q^85 + 20124 * q^87 - 124924 * q^89 - 209904 * q^91 + 84168 * q^93 - 2400 * q^95 - 39260 * q^97 + 39528 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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