LMFDB - Embedded newform 304.6.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,6,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, 6, names="a")

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

chi := DirichletCharacter("304.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
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:	$48.7566812231$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.15207$ 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)

$f(q)$	$=$	$q-16.4562 q^{3} -99.7603 q^{5} +57.1289 q^{7} +27.8066 q^{9} +O(q^{10})$	$q-16.4562 q^{3} -99.7603 q^{5} +57.1289 q^{7} +27.8066 q^{9} +438.977 q^{11} -477.590 q^{13} +1641.68 q^{15} +1671.39 q^{17} +361.000 q^{19} -940.125 q^{21} -459.514 q^{23} +6827.12 q^{25} +3541.27 q^{27} -2696.78 q^{29} +2314.43 q^{31} -7223.89 q^{33} -5699.20 q^{35} +3022.51 q^{37} +7859.32 q^{39} +5483.05 q^{41} -5101.80 q^{43} -2773.99 q^{45} -5803.63 q^{47} -13543.3 q^{49} -27504.7 q^{51} -23207.5 q^{53} -43792.5 q^{55} -5940.69 q^{57} +32849.5 q^{59} -46911.5 q^{61} +1588.56 q^{63} +47644.5 q^{65} +3565.12 q^{67} +7561.86 q^{69} -2346.44 q^{71} +57370.0 q^{73} -112349. q^{75} +25078.3 q^{77} +29715.8 q^{79} -65032.8 q^{81} -60526.0 q^{83} -166738. q^{85} +44378.8 q^{87} +19134.9 q^{89} -27284.2 q^{91} -38086.7 q^{93} -36013.5 q^{95} -43189.4 q^{97} +12206.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 7 q^{3} - 133 q^{5} - 72 q^{7} + 335 q^{9}+O(q^{10}) $ 2 * q + 7 * q^3 - 133 * q^5 - 72 * q^7 + 335 * q^9	$ 2 q + 7 q^{3} - 133 q^{5} - 72 q^{7} + 335 q^{9} + 705 q^{11} - 1341 q^{13} + 862 q^{15} + 2784 q^{17} + 722 q^{19} - 3969 q^{21} + 2713 q^{23} + 4807 q^{25} + 5047 q^{27} - 7775 q^{29} - 7132 q^{31} - 984 q^{33} - 1407 q^{35} - 6248 q^{37} - 12393 q^{39} - 4174 q^{41} - 25357 q^{43} - 12985 q^{45} - 11727 q^{47} - 13676 q^{49} - 1407 q^{51} - 29133 q^{53} - 52635 q^{55} + 2527 q^{57} + 64515 q^{59} - 40939 q^{61} - 38079 q^{63} + 76344 q^{65} - 19039 q^{67} + 81977 q^{69} + 70236 q^{71} + 67058 q^{73} - 159733 q^{75} - 9273 q^{77} + 32850 q^{79} - 104362 q^{81} - 71534 q^{83} - 203721 q^{85} - 74737 q^{87} - 87268 q^{89} + 84207 q^{91} - 259664 q^{93} - 48013 q^{95} - 62458 q^{97} + 93927 q^{99}+O(q^{100}) $ 2 * q + 7 * q^3 - 133 * q^5 - 72 * q^7 + 335 * q^9 + 705 * q^11 - 1341 * q^13 + 862 * q^15 + 2784 * q^17 + 722 * q^19 - 3969 * q^21 + 2713 * q^23 + 4807 * q^25 + 5047 * q^27 - 7775 * q^29 - 7132 * q^31 - 984 * q^33 - 1407 * q^35 - 6248 * q^37 - 12393 * q^39 - 4174 * q^41 - 25357 * q^43 - 12985 * q^45 - 11727 * q^47 - 13676 * q^49 - 1407 * q^51 - 29133 * q^53 - 52635 * q^55 + 2527 * q^57 + 64515 * q^59 - 40939 * q^61 - 38079 * q^63 + 76344 * q^65 - 19039 * q^67 + 81977 * q^69 + 70236 * q^71 + 67058 * q^73 - 159733 * q^75 - 9273 * q^77 + 32850 * q^79 - 104362 * q^81 - 71534 * q^83 - 203721 * q^85 - 74737 * q^87 - 87268 * q^89 + 84207 * q^91 - 259664 * q^93 - 48013 * q^95 - 62458 * q^97 + 93927 * 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
$3$	−16.4562	−1.05567	−0.527833	−	0.849348i	$-0.676995\pi$
$3$	−16.4562	−1.05567	−0.527833	+	0.849348i	$0.676995\pi$
$4$	0	0
$5$	−99.7603	−1.78457	−0.892284	−	0.451475i	$-0.850898\pi$
$5$	−99.7603	−1.78457	−0.892284	+	0.451475i	$0.850898\pi$
$6$	0	0
$7$	57.1289	0.440668	0.220334	−	0.975425i	$-0.429285\pi$
$7$	57.1289	0.440668	0.220334	+	0.975425i	$0.429285\pi$
$8$	0	0
$9$	27.8066	0.114430
$10$	0	0
$11$	438.977	1.09386	0.546928	−	0.837180i	$-0.315797\pi$
$11$	438.977	1.09386	0.546928	+	0.837180i	$0.315797\pi$
$12$	0	0
$13$	−477.590	−0.783785	−0.391892	−	0.920011i	$-0.628179\pi$
$13$	−477.590	−0.783785	−0.391892	+	0.920011i	$0.628179\pi$
$14$	0	0
$15$	1641.68	1.88391
$16$	0	0
$17$	1671.39	1.40267	0.701334	−	0.712833i	$-0.252588\pi$
$17$	1671.39	1.40267	0.701334	+	0.712833i	$0.252588\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	−940.125	−0.465198
$22$	0	0
$23$	−459.514	−0.181125	−0.0905627	−	0.995891i	$-0.528867\pi$
$23$	−459.514	−0.181125	−0.0905627	+	0.995891i	$0.528867\pi$
$24$	0	0
$25$	6827.12	2.18468
$26$	0	0
$27$	3541.27	0.934866
$28$	0	0
$29$	−2696.78	−0.595457	−0.297729	−	0.954651i	$-0.596229\pi$
$29$	−2696.78	−0.595457	−0.297729	+	0.954651i	$0.596229\pi$
$30$	0	0
$31$	2314.43	0.432553	0.216277	−	0.976332i	$-0.430609\pi$
$31$	2314.43	0.432553	0.216277	+	0.976332i	$0.430609\pi$
$32$	0	0
$33$	−7223.89	−1.15475
$34$	0	0
$35$	−5699.20	−0.786401
$36$	0	0
$37$	3022.51	0.362964	0.181482	−	0.983394i	$-0.441911\pi$
$37$	3022.51	0.362964	0.181482	+	0.983394i	$0.441911\pi$
$38$	0	0
$39$	7859.32	0.827415
$40$	0	0
$41$	5483.05	0.509404	0.254702	−	0.967020i	$-0.418023\pi$
$41$	5483.05	0.509404	0.254702	+	0.967020i	$0.418023\pi$
$42$	0	0
$43$	−5101.80	−0.420777	−0.210388	−	0.977618i	$-0.567473\pi$
$43$	−5101.80	−0.420777	−0.210388	+	0.977618i	$0.567473\pi$
$44$	0	0
$45$	−2773.99	−0.204209
$46$	0	0
$47$	−5803.63	−0.383226	−0.191613	−	0.981471i	$-0.561372\pi$
$47$	−5803.63	−0.383226	−0.191613	+	0.981471i	$0.561372\pi$
$48$	0	0
$49$	−13543.3	−0.805812
$50$	0	0
$51$	−27504.7	−1.48075
$52$	0	0
$53$	−23207.5	−1.13485	−0.567426	−	0.823424i	$-0.692061\pi$
$53$	−23207.5	−1.13485	−0.567426	+	0.823424i	$0.692061\pi$
$54$	0	0
$55$	−43792.5	−1.95206
$56$	0	0
$57$	−5940.69	−0.242186
$58$	0	0
$59$	32849.5	1.22857	0.614284	−	0.789085i	$-0.289445\pi$
$59$	32849.5	1.22857	0.614284	+	0.789085i	$0.289445\pi$
$60$	0	0
$61$	−46911.5	−1.61419	−0.807095	−	0.590422i	$-0.798961\pi$
$61$	−46911.5	−1.61419	−0.807095	+	0.590422i	$0.798961\pi$
$62$	0	0
$63$	1588.56	0.0504258
$64$	0	0
$65$	47644.5	1.39872
$66$	0	0
$67$	3565.12	0.0970257	0.0485128	−	0.998823i	$-0.484552\pi$
$67$	3565.12	0.0970257	0.0485128	+	0.998823i	$0.484552\pi$
$68$	0	0
$69$	7561.86	0.191208
$70$	0	0
$71$	−2346.44	−0.0552413	−0.0276207	−	0.999618i	$-0.508793\pi$
$71$	−2346.44	−0.0552413	−0.0276207	+	0.999618i	$0.508793\pi$
$72$	0	0
$73$	57370.0	1.26002	0.630010	−	0.776587i	$-0.283051\pi$
$73$	57370.0	1.26002	0.630010	+	0.776587i	$0.283051\pi$
$74$	0	0
$75$	−112349.	−2.30629
$76$	0	0
$77$	25078.3	0.482027
$78$	0	0
$79$	29715.8	0.535698	0.267849	−	0.963461i	$-0.413687\pi$
$79$	29715.8	0.535698	0.267849	+	0.963461i	$0.413687\pi$
$80$	0	0
$81$	−65032.8	−1.10134
$82$	0	0
$83$	−60526.0	−0.964377	−0.482188	−	0.876068i	$-0.660158\pi$
$83$	−60526.0	−0.964377	−0.482188	+	0.876068i	$0.660158\pi$
$84$	0	0
$85$	−166738.	−2.50315
$86$	0	0
$87$	44378.8	0.628604
$88$	0	0
$89$	19134.9	0.256066	0.128033	−	0.991770i	$-0.459134\pi$
$89$	19134.9	0.256066	0.128033	+	0.991770i	$0.459134\pi$
$90$	0	0
$91$	−27284.2	−0.345389
$92$	0	0
$93$	−38086.7	−0.456631
$94$	0	0
$95$	−36013.5	−0.409408
$96$	0	0
$97$	−43189.4	−0.466067	−0.233033	−	0.972469i	$-0.574865\pi$
$97$	−43189.4	−0.466067	−0.233033	+	0.972469i	$0.574865\pi$
$98$	0	0
$99$	12206.4	0.125170
$100$	0	0
$101$	−55330.5	−0.539711	−0.269855	−	0.962901i	$-0.586976\pi$
$101$	−55330.5	−0.539711	−0.269855	+	0.962901i	$0.586976\pi$
$102$	0	0
$103$	205641.	1.90993	0.954964	−	0.296722i	$-0.0958933\pi$
$103$	205641.	1.90993	0.954964	+	0.296722i	$0.0958933\pi$
$104$	0	0
$105$	93787.2	0.830177
$106$	0	0
$107$	−181889.	−1.53585	−0.767923	−	0.640543i	$-0.778709\pi$
$107$	−181889.	−1.53585	−0.767923	+	0.640543i	$0.778709\pi$
$108$	0	0
$109$	24288.8	0.195813	0.0979063	−	0.995196i	$-0.468785\pi$
$109$	24288.8	0.195813	0.0979063	+	0.995196i	$0.468785\pi$
$110$	0	0
$111$	−49739.0	−0.383169
$112$	0	0
$113$	−229667.	−1.69201	−0.846006	−	0.533174i	$-0.820999\pi$
$113$	−229667.	−1.69201	−0.846006	+	0.533174i	$0.820999\pi$
$114$	0	0
$115$	45841.3	0.323231
$116$	0	0
$117$	−13280.1	−0.0896888
$118$	0	0
$119$	95484.6	0.618110
$120$	0	0
$121$	31649.7	0.196520
$122$	0	0
$123$	−90230.2	−0.537761
$124$	0	0
$125$	−369325.	−2.11414
$126$	0	0
$127$	−8086.69	−0.0444899	−0.0222450	−	0.999753i	$-0.507081\pi$
$127$	−8086.69	−0.0444899	−0.0222450	+	0.999753i	$0.507081\pi$
$128$	0	0
$129$	83956.2	0.444200
$130$	0	0
$131$	−118712.	−0.604390	−0.302195	−	0.953246i	$-0.597719\pi$
$131$	−118712.	−0.604390	−0.302195	+	0.953246i	$0.597719\pi$
$132$	0	0
$133$	20623.5	0.101096
$134$	0	0
$135$	−353278.	−1.66833
$136$	0	0
$137$	65348.7	0.297465	0.148732	−	0.988877i	$-0.452481\pi$
$137$	65348.7	0.297465	0.148732	+	0.988877i	$0.452481\pi$
$138$	0	0
$139$	28437.6	0.124841	0.0624203	−	0.998050i	$-0.480118\pi$
$139$	28437.6	0.124841	0.0624203	+	0.998050i	$0.480118\pi$
$140$	0	0
$141$	95505.7	0.404559
$142$	0	0
$143$	−209651.	−0.857347
$144$	0	0
$145$	269032.	1.06263
$146$	0	0
$147$	222871.	0.850668
$148$	0	0
$149$	115852.	0.427502	0.213751	−	0.976888i	$-0.431432\pi$
$149$	115852.	0.427502	0.213751	+	0.976888i	$0.431432\pi$
$150$	0	0
$151$	−35980.9	−0.128419	−0.0642096	−	0.997936i	$-0.520453\pi$
$151$	−35980.9	−0.128419	−0.0642096	+	0.997936i	$0.520453\pi$
$152$	0	0
$153$	46475.6	0.160508
$154$	0	0
$155$	−230888.	−0.771920
$156$	0	0
$157$	566234.	1.83336	0.916678	−	0.399627i	$-0.130860\pi$
$157$	566234.	1.83336	0.916678	+	0.399627i	$0.130860\pi$
$158$	0	0
$159$	381908.	1.19802
$160$	0	0
$161$	−26251.6	−0.0798161
$162$	0	0
$163$	307841.	0.907523	0.453761	−	0.891123i	$-0.350082\pi$
$163$	307841.	0.907523	0.453761	+	0.891123i	$0.350082\pi$
$164$	0	0
$165$	720658.	2.06072
$166$	0	0
$167$	−126991.	−0.352357	−0.176179	−	0.984358i	$-0.556374\pi$
$167$	−126991.	−0.352357	−0.176179	+	0.984358i	$0.556374\pi$
$168$	0	0
$169$	−143201.	−0.385681
$170$	0	0
$171$	10038.2	0.0262521
$172$	0	0
$173$	−582515.	−1.47976	−0.739881	−	0.672738i	$-0.765118\pi$
$173$	−582515.	−1.47976	−0.739881	+	0.672738i	$0.765118\pi$
$174$	0	0
$175$	390026.	0.962718
$176$	0	0
$177$	−540579.	−1.29696
$178$	0	0
$179$	233819.	0.545440	0.272720	−	0.962093i	$-0.412077\pi$
$179$	233819.	0.545440	0.272720	+	0.962093i	$0.412077\pi$
$180$	0	0
$181$	318180.	0.721899	0.360950	−	0.932585i	$-0.382453\pi$
$181$	318180.	0.721899	0.360950	+	0.932585i	$0.382453\pi$
$182$	0	0
$183$	771985.	1.70404
$184$	0	0
$185$	−301527.	−0.647734
$186$	0	0
$187$	733700.	1.53432
$188$	0	0
$189$	202309.	0.411965
$190$	0	0
$191$	182032.	0.361047	0.180523	−	0.983571i	$-0.442221\pi$
$191$	182032.	0.361047	0.180523	+	0.983571i	$0.442221\pi$
$192$	0	0
$193$	286026.	0.552729	0.276365	−	0.961053i	$-0.410870\pi$
$193$	286026.	0.552729	0.276365	+	0.961053i	$0.410870\pi$
$194$	0	0
$195$	−784048.	−1.47658
$196$	0	0
$197$	−465880.	−0.855281	−0.427640	−	0.903949i	$-0.640655\pi$
$197$	−465880.	−0.855281	−0.427640	+	0.903949i	$0.640655\pi$
$198$	0	0
$199$	−1.06447e6	−1.90547	−0.952736	−	0.303798i	$-0.901745\pi$
$199$	−1.06447e6	−1.90547	−0.952736	+	0.303798i	$0.901745\pi$
$200$	0	0
$201$	−58668.3	−0.102427
$202$	0	0
$203$	−154064.	−0.262399
$204$	0	0
$205$	−546991.	−0.909067
$206$	0	0
$207$	−12777.5	−0.0207263
$208$	0	0
$209$	158471.	0.250948
$210$	0	0
$211$	−1.00586e6	−1.55536	−0.777679	−	0.628662i	$-0.783603\pi$
$211$	−1.00586e6	−1.55536	−0.777679	+	0.628662i	$0.783603\pi$
$212$	0	0
$213$	38613.5	0.0583164
$214$	0	0
$215$	508957.	0.750905
$216$	0	0
$217$	132221.	0.190612
$218$	0	0
$219$	−944092.	−1.33016
$220$	0	0
$221$	−798238.	−1.09939
$222$	0	0
$223$	377403.	0.508210	0.254105	−	0.967177i	$-0.418219\pi$
$223$	377403.	0.508210	0.254105	+	0.967177i	$0.418219\pi$
$224$	0	0
$225$	189839.	0.249994
$226$	0	0
$227$	1.28433e6	1.65429	0.827145	−	0.561989i	$-0.189964\pi$
$227$	1.28433e6	1.65429	0.827145	+	0.561989i	$0.189964\pi$
$228$	0	0
$229$	299103.	0.376905	0.188452	−	0.982082i	$-0.439653\pi$
$229$	299103.	0.376905	0.188452	+	0.982082i	$0.439653\pi$
$230$	0	0
$231$	−412693.	−0.508859
$232$	0	0
$233$	564942.	0.681732	0.340866	−	0.940112i	$-0.389280\pi$
$233$	564942.	0.681732	0.340866	+	0.940112i	$0.389280\pi$
$234$	0	0
$235$	578972.	0.683893
$236$	0	0
$237$	−489010.	−0.565518
$238$	0	0
$239$	1.03635e6	1.17358	0.586789	−	0.809740i	$-0.300392\pi$
$239$	1.03635e6	1.17358	0.586789	+	0.809740i	$0.300392\pi$
$240$	0	0
$241$	821274.	0.910847	0.455423	−	0.890275i	$-0.349488\pi$
$241$	821274.	0.910847	0.455423	+	0.890275i	$0.349488\pi$
$242$	0	0
$243$	209665.	0.227777
$244$	0	0
$245$	1.35108e6	1.43803
$246$	0	0
$247$	−172410.	−0.179813
$248$	0	0
$249$	996028.	1.01806
$250$	0	0
$251$	−1.23062e6	−1.23293	−0.616467	−	0.787380i	$-0.711437\pi$
$251$	−1.23062e6	−1.23293	−0.616467	+	0.787380i	$0.711437\pi$
$252$	0	0
$253$	−201716.	−0.198125
$254$	0	0
$255$	2.74388e6	2.64249
$256$	0	0
$257$	−1.05206e6	−0.993592	−0.496796	−	0.867867i	$-0.665490\pi$
$257$	−1.05206e6	−0.993592	−0.496796	+	0.867867i	$0.665490\pi$
$258$	0	0
$259$	172673.	0.159946
$260$	0	0
$261$	−74988.2	−0.0681384
$262$	0	0
$263$	−1.88157e6	−1.67738	−0.838688	−	0.544612i	$-0.816677\pi$
$263$	−1.88157e6	−1.67738	−0.838688	+	0.544612i	$0.816677\pi$
$264$	0	0
$265$	2.31519e6	2.02522
$266$	0	0
$267$	−314888.	−0.270320
$268$	0	0
$269$	−1.05389e6	−0.888002	−0.444001	−	0.896026i	$-0.646441\pi$
$269$	−1.05389e6	−0.888002	−0.444001	+	0.896026i	$0.646441\pi$
$270$	0	0
$271$	−688686.	−0.569637	−0.284818	−	0.958581i	$-0.591933\pi$
$271$	−688686.	−0.569637	−0.284818	+	0.958581i	$0.591933\pi$
$272$	0	0
$273$	448995.	0.364615
$274$	0	0
$275$	2.99695e6	2.38972
$276$	0	0
$277$	−522720.	−0.409326	−0.204663	−	0.978832i	$-0.565610\pi$
$277$	−522720.	−0.409326	−0.204663	+	0.978832i	$0.565610\pi$
$278$	0	0
$279$	64356.3	0.0494972
$280$	0	0
$281$	−2.00251e6	−1.51290	−0.756448	−	0.654053i	$-0.773067\pi$
$281$	−2.00251e6	−1.51290	−0.756448	+	0.654053i	$0.773067\pi$
$282$	0	0
$283$	2.25219e6	1.67163	0.835813	−	0.549014i	$-0.184997\pi$
$283$	2.25219e6	1.67163	0.835813	+	0.549014i	$0.184997\pi$
$284$	0	0
$285$	592645.	0.432198
$286$	0	0
$287$	313241.	0.224478
$288$	0	0
$289$	1.37368e6	0.967476
$290$	0	0
$291$	710734.	0.492011
$292$	0	0
$293$	1.76627e6	1.20195	0.600976	−	0.799267i	$-0.294779\pi$
$293$	1.76627e6	1.20195	0.600976	+	0.799267i	$0.294779\pi$
$294$	0	0
$295$	−3.27708e6	−2.19246
$296$	0	0
$297$	1.55453e6	1.02261
$298$	0	0
$299$	219459.	0.141963
$300$	0	0
$301$	−291460.	−0.185423
$302$	0	0
$303$	910530.	0.569754
$304$	0	0
$305$	4.67990e6	2.88063
$306$	0	0
$307$	1.71867e6	1.04075	0.520376	−	0.853937i	$-0.325792\pi$
$307$	1.71867e6	1.04075	0.520376	+	0.853937i	$0.325792\pi$
$308$	0	0
$309$	−3.38407e6	−2.01625
$310$	0	0
$311$	−1.58149e6	−0.927182	−0.463591	−	0.886049i	$-0.653439\pi$
$311$	−1.58149e6	−0.927182	−0.463591	+	0.886049i	$0.653439\pi$
$312$	0	0
$313$	54617.0	0.0315114	0.0157557	−	0.999876i	$-0.494985\pi$
$313$	54617.0	0.0315114	0.0157557	+	0.999876i	$0.494985\pi$
$314$	0	0
$315$	−158475.	−0.0899882
$316$	0	0
$317$	−1.03467e6	−0.578303	−0.289151	−	0.957283i	$-0.593373\pi$
$317$	−1.03467e6	−0.578303	−0.289151	+	0.957283i	$0.593373\pi$
$318$	0	0
$319$	−1.18382e6	−0.651344
$320$	0	0
$321$	2.99321e6	1.62134
$322$	0	0
$323$	603371.	0.321794
$324$	0	0
$325$	−3.26057e6	−1.71232
$326$	0	0
$327$	−399702.	−0.206713
$328$	0	0
$329$	−331555.	−0.168875
$330$	0	0
$331$	−2.03987e6	−1.02337	−0.511685	−	0.859173i	$-0.670979\pi$
$331$	−2.03987e6	−1.02337	−0.511685	+	0.859173i	$0.670979\pi$
$332$	0	0
$333$	84045.7	0.0415341
$334$	0	0
$335$	−355657.	−0.173149
$336$	0	0
$337$	1.22774e6	0.588887	0.294444	−	0.955669i	$-0.404866\pi$
$337$	1.22774e6	0.588887	0.294444	+	0.955669i	$0.404866\pi$
$338$	0	0
$339$	3.77945e6	1.78620
$340$	0	0
$341$	1.01598e6	0.473150
$342$	0	0
$343$	−1.73388e6	−0.795763
$344$	0	0
$345$	−754374.	−0.341223
$346$	0	0
$347$	−3.93267e6	−1.75333	−0.876665	−	0.481101i	$-0.840237\pi$
$347$	−3.93267e6	−1.75333	−0.876665	+	0.481101i	$0.840237\pi$
$348$	0	0
$349$	−3.46176e6	−1.52136	−0.760682	−	0.649125i	$-0.775135\pi$
$349$	−3.46176e6	−1.52136	−0.760682	+	0.649125i	$0.775135\pi$
$350$	0	0
$351$	−1.69127e6	−0.732734
$352$	0	0
$353$	−3.91227e6	−1.67106	−0.835531	−	0.549443i	$-0.814840\pi$
$353$	−3.91227e6	−1.67106	−0.835531	+	0.549443i	$0.814840\pi$
$354$	0	0
$355$	234082.	0.0985819
$356$	0	0
$357$	−1.57131e6	−0.652518
$358$	0	0
$359$	949856.	0.388975	0.194487	−	0.980905i	$-0.437696\pi$
$359$	949856.	0.388975	0.194487	+	0.980905i	$0.437696\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−520834.	−0.207459
$364$	0	0
$365$	−5.72325e6	−2.24859
$366$	0	0
$367$	−3.72584e6	−1.44397	−0.721986	−	0.691908i	$-0.756770\pi$
$367$	−3.72584e6	−1.44397	−0.721986	+	0.691908i	$0.756770\pi$
$368$	0	0
$369$	152465.	0.0582914
$370$	0	0
$371$	−1.32582e6	−0.500093
$372$	0	0
$373$	−4.15376e6	−1.54586	−0.772929	−	0.634493i	$-0.781209\pi$
$373$	−4.15376e6	−1.54586	−0.772929	+	0.634493i	$0.781209\pi$
$374$	0	0
$375$	6.07769e6	2.23183
$376$	0	0
$377$	1.28796e6	0.466710
$378$	0	0
$379$	−2.29264e6	−0.819857	−0.409928	−	0.912118i	$-0.634446\pi$
$379$	−2.29264e6	−0.819857	−0.409928	+	0.912118i	$0.634446\pi$
$380$	0	0
$381$	133076.	0.0469665
$382$	0	0
$383$	−3.09365e6	−1.07764	−0.538821	−	0.842420i	$-0.681130\pi$
$383$	−3.09365e6	−1.07764	−0.538821	+	0.842420i	$0.681130\pi$
$384$	0	0
$385$	−2.50182e6	−0.860209
$386$	0	0
$387$	−141864.	−0.0481497
$388$	0	0
$389$	−4.53732e6	−1.52029	−0.760144	−	0.649755i	$-0.774871\pi$
$389$	−4.53732e6	−1.52029	−0.760144	+	0.649755i	$0.774871\pi$
$390$	0	0
$391$	−768026.	−0.254059
$392$	0	0
$393$	1.95355e6	0.638034
$394$	0	0
$395$	−2.96446e6	−0.955989
$396$	0	0
$397$	−404604.	−0.128841	−0.0644205	−	0.997923i	$-0.520520\pi$
$397$	−404604.	−0.128841	−0.0644205	+	0.997923i	$0.520520\pi$
$398$	0	0
$399$	−339385.	−0.106724
$400$	0	0
$401$	−5.08752e6	−1.57996	−0.789978	−	0.613135i	$-0.789908\pi$
$401$	−5.08752e6	−1.57996	−0.789978	+	0.613135i	$0.789908\pi$
$402$	0	0
$403$	−1.10535e6	−0.339029
$404$	0	0
$405$	6.48769e6	1.96541
$406$	0	0
$407$	1.32681e6	0.397030
$408$	0	0
$409$	367772.	0.108710	0.0543551	−	0.998522i	$-0.482690\pi$
$409$	367772.	0.108710	0.0543551	+	0.998522i	$0.482690\pi$
$410$	0	0
$411$	−1.07539e6	−0.314024
$412$	0	0
$413$	1.87666e6	0.541390
$414$	0	0
$415$	6.03809e6	1.72099
$416$	0	0
$417$	−467975.	−0.131790
$418$	0	0
$419$	4.12612e6	1.14817	0.574086	−	0.818795i	$-0.305357\pi$
$419$	4.12612e6	1.14817	0.574086	+	0.818795i	$0.305357\pi$
$420$	0	0
$421$	−2.05918e6	−0.566224	−0.283112	−	0.959087i	$-0.591367\pi$
$421$	−2.05918e6	−0.566224	−0.283112	+	0.959087i	$0.591367\pi$
$422$	0	0
$423$	−161379.	−0.0438527
$424$	0	0
$425$	1.14108e7	3.06438
$426$	0	0
$427$	−2.68000e6	−0.711321
$428$	0	0
$429$	3.45006e6	0.905072
$430$	0	0
$431$	3.23089e6	0.837777	0.418889	−	0.908038i	$-0.362420\pi$
$431$	3.23089e6	0.837777	0.418889	+	0.908038i	$0.362420\pi$
$432$	0	0
$433$	−4.46477e6	−1.14440	−0.572202	−	0.820113i	$-0.693911\pi$
$433$	−4.46477e6	−1.14440	−0.572202	+	0.820113i	$0.693911\pi$
$434$	0	0
$435$	−4.42724e6	−1.12179
$436$	0	0
$437$	−165885.	−0.0415530
$438$	0	0
$439$	−3.96204e6	−0.981201	−0.490601	−	0.871385i	$-0.663223\pi$
$439$	−3.96204e6	−0.981201	−0.490601	+	0.871385i	$0.663223\pi$
$440$	0	0
$441$	−376592.	−0.0922094
$442$	0	0
$443$	−5.87689e6	−1.42278	−0.711391	−	0.702797i	$-0.751934\pi$
$443$	−5.87689e6	−1.42278	−0.711391	+	0.702797i	$0.751934\pi$
$444$	0	0
$445$	−1.90890e6	−0.456966
$446$	0	0
$447$	−1.90649e6	−0.451299
$448$	0	0
$449$	−2.62215e6	−0.613820	−0.306910	−	0.951739i	$-0.599295\pi$
$449$	−2.62215e6	−0.613820	−0.306910	+	0.951739i	$0.599295\pi$
$450$	0	0
$451$	2.40693e6	0.557215
$452$	0	0
$453$	592109.	0.135568
$454$	0	0
$455$	2.72188e6	0.616369
$456$	0	0
$457$	5.12578e6	1.14807	0.574037	−	0.818830i	$-0.305377\pi$
$457$	5.12578e6	1.14807	0.574037	+	0.818830i	$0.305377\pi$
$458$	0	0
$459$	5.91883e6	1.31131
$460$	0	0
$461$	2.65571e6	0.582008	0.291004	−	0.956722i	$-0.406011\pi$
$461$	2.65571e6	0.582008	0.291004	+	0.956722i	$0.406011\pi$
$462$	0	0
$463$	2.94632e6	0.638744	0.319372	−	0.947629i	$-0.396528\pi$
$463$	2.94632e6	0.638744	0.319372	+	0.947629i	$0.396528\pi$
$464$	0	0
$465$	3.79954e6	0.814890
$466$	0	0
$467$	−1.82563e6	−0.387364	−0.193682	−	0.981064i	$-0.562043\pi$
$467$	−1.82563e6	−0.387364	−0.193682	+	0.981064i	$0.562043\pi$
$468$	0	0
$469$	203671.	0.0427561
$470$	0	0
$471$	−9.31806e6	−1.93541
$472$	0	0
$473$	−2.23957e6	−0.460269
$474$	0	0
$475$	2.46459e6	0.501200
$476$	0	0
$477$	−645322.	−0.129862
$478$	0	0
$479$	−5.92850e6	−1.18061	−0.590304	−	0.807181i	$-0.700992\pi$
$479$	−5.92850e6	−1.18061	−0.590304	+	0.807181i	$0.700992\pi$
$480$	0	0
$481$	−1.44352e6	−0.284486
$482$	0	0
$483$	432001.	0.0842592
$484$	0	0
$485$	4.30859e6	0.831727
$486$	0	0
$487$	3.61261e6	0.690237	0.345118	−	0.938559i	$-0.387839\pi$
$487$	3.61261e6	0.690237	0.345118	+	0.938559i	$0.387839\pi$
$488$	0	0
$489$	−5.06589e6	−0.958041
$490$	0	0
$491$	4.32852e6	0.810281	0.405140	−	0.914255i	$-0.367223\pi$
$491$	4.32852e6	0.810281	0.405140	+	0.914255i	$0.367223\pi$
$492$	0	0
$493$	−4.50736e6	−0.835228
$494$	0	0
$495$	−1.21772e6	−0.223375
$496$	0	0
$497$	−134050.	−0.0243431
$498$	0	0
$499$	4.93615e6	0.887435	0.443718	−	0.896167i	$-0.353659\pi$
$499$	4.93615e6	0.887435	0.443718	+	0.896167i	$0.353659\pi$
$500$	0	0
$501$	2.08980e6	0.371972
$502$	0	0
$503$	−1.03023e6	−0.181557	−0.0907786	−	0.995871i	$-0.528936\pi$
$503$	−1.03023e6	−0.181557	−0.0907786	+	0.995871i	$0.528936\pi$
$504$	0	0
$505$	5.51979e6	0.963150
$506$	0	0
$507$	2.35654e6	0.407151
$508$	0	0
$509$	4.99120e6	0.853907	0.426954	−	0.904274i	$-0.359587\pi$
$509$	4.99120e6	0.853907	0.426954	+	0.904274i	$0.359587\pi$
$510$	0	0
$511$	3.27749e6	0.555250
$512$	0	0
$513$	1.27840e6	0.214473
$514$	0	0
$515$	−2.05148e7	−3.40839
$516$	0	0
$517$	−2.54766e6	−0.419194
$518$	0	0
$519$	9.58598e6	1.56213
$520$	0	0
$521$	9.78050e6	1.57858	0.789291	−	0.614020i	$-0.210449\pi$
$521$	9.78050e6	1.57858	0.789291	+	0.614020i	$0.210449\pi$
$522$	0	0
$523$	6.73157e6	1.07612	0.538062	−	0.842905i	$-0.319157\pi$
$523$	6.73157e6	1.07612	0.538062	+	0.842905i	$0.319157\pi$
$524$	0	0
$525$	−6.41835e6	−1.01631
$526$	0	0
$527$	3.86830e6	0.606728
$528$	0	0
$529$	−6.22519e6	−0.967194
$530$	0	0
$531$	913433.	0.140586
$532$	0	0
$533$	−2.61865e6	−0.399264
$534$	0	0
$535$	1.81453e7	2.74082
$536$	0	0
$537$	−3.84777e6	−0.575803
$538$	0	0
$539$	−5.94519e6	−0.881442
$540$	0	0
$541$	−6.68215e6	−0.981574	−0.490787	−	0.871280i	$-0.663291\pi$
$541$	−6.68215e6	−0.981574	−0.490787	+	0.871280i	$0.663291\pi$
$542$	0	0
$543$	−5.23603e6	−0.762084
$544$	0	0
$545$	−2.42306e6	−0.349441
$546$	0	0
$547$	235187.	0.0336081	0.0168041	−	0.999859i	$-0.494651\pi$
$547$	235187.	0.0336081	0.0168041	+	0.999859i	$0.494651\pi$
$548$	0	0
$549$	−1.30445e6	−0.184712
$550$	0	0
$551$	−973538.	−0.136607
$552$	0	0
$553$	1.69763e6	0.236065
$554$	0	0
$555$	4.96198e6	0.683790
$556$	0	0
$557$	−4.80203e6	−0.655824	−0.327912	−	0.944708i	$-0.606345\pi$
$557$	−4.80203e6	−0.655824	−0.327912	+	0.944708i	$0.606345\pi$
$558$	0	0
$559$	2.43657e6	0.329799
$560$	0	0
$561$	−1.20739e7	−1.61972
$562$	0	0
$563$	8.37831e6	1.11400	0.557000	−	0.830512i	$-0.311952\pi$
$563$	8.37831e6	1.11400	0.557000	+	0.830512i	$0.311952\pi$
$564$	0	0
$565$	2.29117e7	3.01951
$566$	0	0
$567$	−3.71525e6	−0.485323
$568$	0	0
$569$	−8.38130e6	−1.08525	−0.542626	−	0.839974i	$-0.682570\pi$
$569$	−8.38130e6	−1.08525	−0.542626	+	0.839974i	$0.682570\pi$
$570$	0	0
$571$	−3.97400e6	−0.510079	−0.255040	−	0.966931i	$-0.582088\pi$
$571$	−3.97400e6	−0.510079	−0.255040	+	0.966931i	$0.582088\pi$
$572$	0	0
$573$	−2.99555e6	−0.381145
$574$	0	0
$575$	−3.13716e6	−0.395701
$576$	0	0
$577$	1.40884e7	1.76166	0.880830	−	0.473432i	$-0.156985\pi$
$577$	1.40884e7	1.76166	0.880830	+	0.473432i	$0.156985\pi$
$578$	0	0
$579$	−4.70690e6	−0.583497
$580$	0	0
$581$	−3.45779e6	−0.424970
$582$	0	0
$583$	−1.01876e7	−1.24136
$584$	0	0
$585$	1.32483e6	0.160056
$586$	0	0
$587$	−3.14647e6	−0.376901	−0.188451	−	0.982083i	$-0.560347\pi$
$587$	−3.14647e6	−0.376901	−0.188451	+	0.982083i	$0.560347\pi$
$588$	0	0
$589$	835508.	0.0992345
$590$	0	0
$591$	7.66662e6	0.902891
$592$	0	0
$593$	−4.29402e6	−0.501450	−0.250725	−	0.968058i	$-0.580669\pi$
$593$	−4.29402e6	−0.501450	−0.250725	+	0.968058i	$0.580669\pi$
$594$	0	0
$595$	−9.52557e6	−1.10306
$596$	0	0
$597$	1.75172e7	2.01154
$598$	0	0
$599$	6.82650e6	0.777376	0.388688	−	0.921370i	$-0.372928\pi$
$599$	6.82650e6	0.777376	0.388688	+	0.921370i	$0.372928\pi$
$600$	0	0
$601$	1.82575e6	0.206184	0.103092	−	0.994672i	$-0.467126\pi$
$601$	1.82575e6	0.206184	0.103092	+	0.994672i	$0.467126\pi$
$602$	0	0
$603$	99133.7	0.0111027
$604$	0	0
$605$	−3.15738e6	−0.350703
$606$	0	0
$607$	2.44884e6	0.269767	0.134883	−	0.990861i	$-0.456934\pi$
$607$	2.44884e6	0.269767	0.134883	+	0.990861i	$0.456934\pi$
$608$	0	0
$609$	2.53531e6	0.277005
$610$	0	0
$611$	2.77176e6	0.300367
$612$	0	0
$613$	−6.05994e6	−0.651354	−0.325677	−	0.945481i	$-0.605592\pi$
$613$	−6.05994e6	−0.651354	−0.325677	+	0.945481i	$0.605592\pi$
$614$	0	0
$615$	9.00140e6	0.959670
$616$	0	0
$617$	9.33155e6	0.986827	0.493414	−	0.869795i	$-0.335749\pi$
$617$	9.33155e6	0.986827	0.493414	+	0.869795i	$0.335749\pi$
$618$	0	0
$619$	9.73252e6	1.02094	0.510468	−	0.859897i	$-0.329472\pi$
$619$	9.73252e6	1.02094	0.510468	+	0.859897i	$0.329472\pi$
$620$	0	0
$621$	−1.62726e6	−0.169328
$622$	0	0
$623$	1.09316e6	0.112840
$624$	0	0
$625$	1.55092e7	1.58815
$626$	0	0
$627$	−2.60783e6	−0.264917
$628$	0	0
$629$	5.05178e6	0.509118
$630$	0	0
$631$	−8.09784e6	−0.809647	−0.404824	−	0.914395i	$-0.632667\pi$
$631$	−8.09784e6	−0.809647	−0.404824	+	0.914395i	$0.632667\pi$
$632$	0	0
$633$	1.65526e7	1.64194
$634$	0	0
$635$	806731.	0.0793953
$636$	0	0
$637$	6.46814e6	0.631583
$638$	0	0
$639$	−65246.6	−0.00632129
$640$	0	0
$641$	−1.54883e7	−1.48887	−0.744436	−	0.667694i	$-0.767282\pi$
$641$	−1.54883e7	−1.48887	−0.744436	+	0.667694i	$0.767282\pi$
$642$	0	0
$643$	−1.43966e7	−1.37320	−0.686600	−	0.727036i	$-0.740897\pi$
$643$	−1.43966e7	−1.37320	−0.686600	+	0.727036i	$0.740897\pi$
$644$	0	0
$645$	−8.37550e6	−0.792704
$646$	0	0
$647$	6.47992e6	0.608568	0.304284	−	0.952581i	$-0.401583\pi$
$647$	6.47992e6	0.608568	0.304284	+	0.952581i	$0.401583\pi$
$648$	0	0
$649$	1.44202e7	1.34388
$650$	0	0
$651$	−2.17585e6	−0.201223
$652$	0	0
$653$	−7.89619e6	−0.724660	−0.362330	−	0.932050i	$-0.618019\pi$
$653$	−7.89619e6	−0.724660	−0.362330	+	0.932050i	$0.618019\pi$
$654$	0	0
$655$	1.18428e7	1.07857
$656$	0	0
$657$	1.59526e6	0.144185
$658$	0	0
$659$	1.08756e7	0.975530	0.487765	−	0.872975i	$-0.337812\pi$
$659$	1.08756e7	0.975530	0.487765	+	0.872975i	$0.337812\pi$
$660$	0	0
$661$	−8.09950e6	−0.721032	−0.360516	−	0.932753i	$-0.617399\pi$
$661$	−8.09950e6	−0.721032	−0.360516	+	0.932753i	$0.617399\pi$
$662$	0	0
$663$	1.31360e7	1.16059
$664$	0	0
$665$	−2.05741e6	−0.180413
$666$	0	0
$667$	1.23921e6	0.107852
$668$	0	0
$669$	−6.21062e6	−0.536500
$670$	0	0
$671$	−2.05930e7	−1.76569
$672$	0	0
$673$	6.23497e6	0.530636	0.265318	−	0.964161i	$-0.414523\pi$
$673$	6.23497e6	0.530636	0.265318	+	0.964161i	$0.414523\pi$
$674$	0	0
$675$	2.41767e7	2.04238
$676$	0	0
$677$	1.16588e7	0.977649	0.488824	−	0.872382i	$-0.337426\pi$
$677$	1.16588e7	0.977649	0.488824	+	0.872382i	$0.337426\pi$
$678$	0	0
$679$	−2.46737e6	−0.205380
$680$	0	0
$681$	−2.11352e7	−1.74638
$682$	0	0
$683$	1.22340e6	0.100350	0.0501748	−	0.998740i	$-0.484022\pi$
$683$	1.22340e6	0.100350	0.0501748	+	0.998740i	$0.484022\pi$
$684$	0	0
$685$	−6.51921e6	−0.530846
$686$	0	0
$687$	−4.92209e6	−0.397885
$688$	0	0
$689$	1.10837e7	0.889480
$690$	0	0
$691$	5.82015e6	0.463702	0.231851	−	0.972751i	$-0.425522\pi$
$691$	5.82015e6	0.463702	0.231851	+	0.972751i	$0.425522\pi$
$692$	0	0
$693$	697341.	0.0551585
$694$	0	0
$695$	−2.83695e6	−0.222787
$696$	0	0
$697$	9.16430e6	0.714525
$698$	0	0
$699$	−9.29680e6	−0.719682
$700$	0	0
$701$	9.44191e6	0.725713	0.362856	−	0.931845i	$-0.381802\pi$
$701$	9.44191e6	0.725713	0.362856	+	0.931845i	$0.381802\pi$
$702$	0	0
$703$	1.09113e6	0.0832696
$704$	0	0
$705$	−9.52768e6	−0.721962
$706$	0	0
$707$	−3.16097e6	−0.237833
$708$	0	0
$709$	7.88964e6	0.589443	0.294721	−	0.955583i	$-0.404773\pi$
$709$	7.88964e6	0.589443	0.294721	+	0.955583i	$0.404773\pi$
$710$	0	0
$711$	826296.	0.0613002
$712$	0	0
$713$	−1.06351e6	−0.0783464
$714$	0	0
$715$	2.09149e7	1.52999
$716$	0	0
$717$	−1.70544e7	−1.23891
$718$	0	0
$719$	−1.89500e7	−1.36706	−0.683530	−	0.729922i	$-0.739556\pi$
$719$	−1.89500e7	−1.36706	−0.683530	+	0.729922i	$0.739556\pi$
$720$	0	0
$721$	1.17481e7	0.841643
$722$	0	0
$723$	−1.35150e7	−0.961550
$724$	0	0
$725$	−1.84113e7	−1.30088
$726$	0	0
$727$	−6.46947e6	−0.453976	−0.226988	−	0.973898i	$-0.572888\pi$
$727$	−6.46947e6	−0.453976	−0.226988	+	0.973898i	$0.572888\pi$
$728$	0	0
$729$	1.23527e7	0.860879
$730$	0	0
$731$	−8.52707e6	−0.590210
$732$	0	0
$733$	6.39936e6	0.439923	0.219962	−	0.975509i	$-0.429407\pi$
$733$	6.39936e6	0.439923	0.219962	+	0.975509i	$0.429407\pi$
$734$	0	0
$735$	−2.22337e7	−1.51807
$736$	0	0
$737$	1.56500e6	0.106132
$738$	0	0
$739$	1.28565e7	0.865987	0.432993	−	0.901397i	$-0.357457\pi$
$739$	1.28565e7	0.865987	0.432993	+	0.901397i	$0.357457\pi$
$740$	0	0
$741$	2.83721e6	0.189822
$742$	0	0
$743$	2.19966e7	1.46179	0.730893	−	0.682492i	$-0.239104\pi$
$743$	2.19966e7	1.46179	0.730893	+	0.682492i	$0.239104\pi$
$744$	0	0
$745$	−1.15574e7	−0.762906
$746$	0	0
$747$	−1.68302e6	−0.110354
$748$	0	0
$749$	−1.03911e7	−0.676797
$750$	0	0
$751$	−4.73982e6	−0.306663	−0.153332	−	0.988175i	$-0.549000\pi$
$751$	−4.73982e6	−0.306663	−0.153332	+	0.988175i	$0.549000\pi$
$752$	0	0
$753$	2.02514e7	1.30157
$754$	0	0
$755$	3.58947e6	0.229173
$756$	0	0
$757$	−8.15849e6	−0.517452	−0.258726	−	0.965951i	$-0.583303\pi$
$757$	−8.15849e6	−0.517452	−0.258726	+	0.965951i	$0.583303\pi$
$758$	0	0
$759$	3.31948e6	0.209154
$760$	0	0
$761$	−2.40805e7	−1.50731	−0.753657	−	0.657268i	$-0.771712\pi$
$761$	−2.40805e7	−1.50731	−0.753657	+	0.657268i	$0.771712\pi$
$762$	0	0
$763$	1.38760e6	0.0862883
$764$	0	0
$765$	−4.63642e6	−0.286437
$766$	0	0
$767$	−1.56886e7	−0.962933
$768$	0	0
$769$	−3.57395e6	−0.217938	−0.108969	−	0.994045i	$-0.534755\pi$
$769$	−3.57395e6	−0.217938	−0.108969	+	0.994045i	$0.534755\pi$
$770$	0	0
$771$	1.73129e7	1.04890
$772$	0	0
$773$	1.38262e7	0.832251	0.416126	−	0.909307i	$-0.363388\pi$
$773$	1.38262e7	0.832251	0.416126	+	0.909307i	$0.363388\pi$
$774$	0	0
$775$	1.58009e7	0.944990
$776$	0	0
$777$	−2.84154e6	−0.168850
$778$	0	0
$779$	1.97938e6	0.116865
$780$	0	0
$781$	−1.03003e6	−0.0604260
$782$	0	0
$783$	−9.55002e6	−0.556672
$784$	0	0
$785$	−5.64877e7	−3.27175
$786$	0	0
$787$	−2.52043e7	−1.45057	−0.725283	−	0.688451i	$-0.758291\pi$
$787$	−2.52043e7	−1.45057	−0.725283	+	0.688451i	$0.758291\pi$
$788$	0	0
$789$	3.09635e7	1.77075
$790$	0	0
$791$	−1.31207e7	−0.745615
$792$	0	0
$793$	2.24045e7	1.26518
$794$	0	0
$795$	−3.80993e7	−2.13796
$796$	0	0
$797$	3.37562e7	1.88238	0.941190	−	0.337877i	$-0.109709\pi$
$797$	3.37562e7	1.88238	0.941190	+	0.337877i	$0.109709\pi$
$798$	0	0
$799$	−9.70011e6	−0.537539
$800$	0	0
$801$	532076.	0.0293017
$802$	0	0
$803$	2.51841e7	1.37828
$804$	0	0
$805$	2.61887e6	0.142437
$806$	0	0
$807$	1.73430e7	0.937433
$808$	0	0
$809$	−2.10318e7	−1.12981	−0.564906	−	0.825155i	$-0.691088\pi$
$809$	−2.10318e7	−1.12981	−0.564906	+	0.825155i	$0.691088\pi$
$810$	0	0
$811$	2.27692e7	1.21561	0.607807	−	0.794085i	$-0.292050\pi$
$811$	2.27692e7	1.21561	0.607807	+	0.794085i	$0.292050\pi$
$812$	0	0
$813$	1.13332e7	0.601346
$814$	0	0
$815$	−3.07103e7	−1.61954
$816$	0	0
$817$	−1.84175e6	−0.0965328
$818$	0	0
$819$	−758681.	−0.0395230
$820$	0	0
$821$	−2.30273e7	−1.19230	−0.596149	−	0.802874i	$-0.703303\pi$
$821$	−2.30273e7	−1.19230	−0.596149	+	0.802874i	$0.703303\pi$
$822$	0	0
$823$	1.70408e7	0.876979	0.438490	−	0.898736i	$-0.355514\pi$
$823$	1.70408e7	0.876979	0.438490	+	0.898736i	$0.355514\pi$
$824$	0	0
$825$	−4.93184e7	−2.52275
$826$	0	0
$827$	−3.62225e6	−0.184168	−0.0920840	−	0.995751i	$-0.529353\pi$
$827$	−3.62225e6	−0.184168	−0.0920840	+	0.995751i	$0.529353\pi$
$828$	0	0
$829$	1.36980e7	0.692261	0.346131	−	0.938186i	$-0.387495\pi$
$829$	1.36980e7	0.692261	0.346131	+	0.938186i	$0.387495\pi$
$830$	0	0
$831$	8.60198e6	0.432112
$832$	0	0
$833$	−2.26361e7	−1.13029
$834$	0	0
$835$	1.26687e7	0.628805
$836$	0	0
$837$	8.19600e6	0.404379
$838$	0	0
$839$	8.15272e6	0.399850	0.199925	−	0.979811i	$-0.435930\pi$
$839$	8.15272e6	0.399850	0.199925	+	0.979811i	$0.435930\pi$
$840$	0	0
$841$	−1.32385e7	−0.645431
$842$	0	0
$843$	3.29537e7	1.59711
$844$	0	0
$845$	1.42858e7	0.688274
$846$	0	0
$847$	1.80811e6	0.0865999
$848$	0	0
$849$	−3.70625e7	−1.76468
$850$	0	0
$851$	−1.38889e6	−0.0657420
$852$	0	0
$853$	−3.76514e7	−1.77178	−0.885888	−	0.463899i	$-0.846450\pi$
$853$	−3.76514e7	−1.77178	−0.885888	+	0.463899i	$0.846450\pi$
$854$	0	0
$855$	−1.00141e6	−0.0468487
$856$	0	0
$857$	1.18679e7	0.551980	0.275990	−	0.961160i	$-0.410994\pi$
$857$	1.18679e7	0.551980	0.275990	+	0.961160i	$0.410994\pi$
$858$	0	0
$859$	8.63985e6	0.399506	0.199753	−	0.979846i	$-0.435986\pi$
$859$	8.63985e6	0.399506	0.199753	+	0.979846i	$0.435986\pi$
$860$	0	0
$861$	−5.15476e6	−0.236974
$862$	0	0
$863$	1.39409e7	0.637181	0.318591	−	0.947892i	$-0.396790\pi$
$863$	1.39409e7	0.637181	0.318591	+	0.947892i	$0.396790\pi$
$864$	0	0
$865$	5.81119e7	2.64073
$866$	0	0
$867$	−2.26055e7	−1.02133
$868$	0	0
$869$	1.30446e7	0.585976
$870$	0	0
$871$	−1.70266e6	−0.0760473
$872$	0	0
$873$	−1.20095e6	−0.0533322
$874$	0	0
$875$	−2.10992e7	−0.931633
$876$	0	0
$877$	2.92561e7	1.28445	0.642224	−	0.766517i	$-0.278012\pi$
$877$	2.92561e7	1.28445	0.642224	+	0.766517i	$0.278012\pi$
$878$	0	0
$879$	−2.90661e7	−1.26886
$880$	0	0
$881$	−1.44235e7	−0.626081	−0.313041	−	0.949740i	$-0.601348\pi$
$881$	−1.44235e7	−0.626081	−0.313041	+	0.949740i	$0.601348\pi$
$882$	0	0
$883$	4.94360e6	0.213374	0.106687	−	0.994293i	$-0.465976\pi$
$883$	4.94360e6	0.213374	0.106687	+	0.994293i	$0.465976\pi$
$884$	0	0
$885$	5.39283e7	2.31451
$886$	0	0
$887$	1.32368e7	0.564903	0.282452	−	0.959282i	$-0.408852\pi$
$887$	1.32368e7	0.564903	0.282452	+	0.959282i	$0.408852\pi$
$888$	0	0
$889$	−461984.	−0.0196053
$890$	0	0
$891$	−2.85479e7	−1.20470
$892$	0	0
$893$	−2.09511e6	−0.0879181
$894$	0	0
$895$	−2.33259e7	−0.973375
$896$	0	0
$897$	−3.61147e6	−0.149866
$898$	0	0
$899$	−6.24150e6	−0.257567
$900$	0	0
$901$	−3.87888e7	−1.59182
$902$	0	0
$903$	4.79633e6	0.195744
$904$	0	0
$905$	−3.17417e7	−1.28828
$906$	0	0
$907$	−7.51989e6	−0.303524	−0.151762	−	0.988417i	$-0.548495\pi$
$907$	−7.51989e6	−0.303524	−0.151762	+	0.988417i	$0.548495\pi$
$908$	0	0
$909$	−1.53855e6	−0.0617593
$910$	0	0
$911$	−4.53248e7	−1.80942	−0.904710	−	0.426027i	$-0.859913\pi$
$911$	−4.53248e7	−1.80942	−0.904710	+	0.426027i	$0.859913\pi$
$912$	0	0
$913$	−2.65695e7	−1.05489
$914$	0	0
$915$	−7.70134e7	−3.04098
$916$	0	0
$917$	−6.78190e6	−0.266335
$918$	0	0
$919$	−1.47088e7	−0.574497	−0.287249	−	0.957856i	$-0.592741\pi$
$919$	−1.47088e7	−0.574497	−0.287249	+	0.957856i	$0.592741\pi$
$920$	0	0
$921$	−2.82828e7	−1.09869
$922$	0	0
$923$	1.12064e6	0.0432973
$924$	0	0
$925$	2.06351e7	0.792960
$926$	0	0
$927$	5.71818e6	0.218554
$928$	0	0
$929$	−1.35395e7	−0.514711	−0.257356	−	0.966317i	$-0.582851\pi$
$929$	−1.35395e7	−0.514711	−0.257356	+	0.966317i	$0.582851\pi$
$930$	0	0
$931$	−4.88913e6	−0.184866
$932$	0	0
$933$	2.60253e7	0.978795
$934$	0	0
$935$	−7.31942e7	−2.73809
$936$	0	0
$937$	−1.21203e7	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$937$	−1.21203e7	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$938$	0	0
$939$	−898788.	−0.0332655
$940$	0	0
$941$	2.61927e7	0.964288	0.482144	−	0.876092i	$-0.339858\pi$
$941$	2.61927e7	0.964288	0.482144	+	0.876092i	$0.339858\pi$
$942$	0	0
$943$	−2.51954e6	−0.0922661
$944$	0	0
$945$	−2.01824e7	−0.735179
$946$	0	0
$947$	3.47875e7	1.26051	0.630257	−	0.776386i	$-0.282949\pi$
$947$	3.47875e7	1.26051	0.630257	+	0.776386i	$0.282949\pi$
$948$	0	0
$949$	−2.73993e7	−0.987585
$950$	0	0
$951$	1.70268e7	0.610494
$952$	0	0
$953$	1.16268e7	0.414695	0.207347	−	0.978267i	$-0.433517\pi$
$953$	1.16268e7	0.414695	0.207347	+	0.978267i	$0.433517\pi$
$954$	0	0
$955$	−1.81595e7	−0.644312
$956$	0	0
$957$	1.94812e7	0.687602
$958$	0	0
$959$	3.73330e6	0.131083
$960$	0	0
$961$	−2.32726e7	−0.812898
$962$	0	0
$963$	−5.05772e6	−0.175747
$964$	0	0
$965$	−2.85341e7	−0.986383
$966$	0	0
$967$	2.70316e7	0.929622	0.464811	−	0.885410i	$-0.346122\pi$
$967$	2.70316e7	0.929622	0.464811	+	0.885410i	$0.346122\pi$
$968$	0	0
$969$	−9.92919e6	−0.339707
$970$	0	0
$971$	5.13472e6	0.174771	0.0873854	−	0.996175i	$-0.472149\pi$
$971$	5.13472e6	0.174771	0.0873854	+	0.996175i	$0.472149\pi$
$972$	0	0
$973$	1.62461e6	0.0550132
$974$	0	0
$975$	5.36565e7	1.80764
$976$	0	0
$977$	−5.11922e6	−0.171580	−0.0857902	−	0.996313i	$-0.527341\pi$
$977$	−5.11922e6	−0.171580	−0.0857902	+	0.996313i	$0.527341\pi$
$978$	0	0
$979$	8.39978e6	0.280099
$980$	0	0
$981$	675390.	0.0224069
$982$	0	0
$983$	−3.30087e7	−1.08954	−0.544771	−	0.838585i	$-0.683384\pi$
$983$	−3.30087e7	−1.08954	−0.544771	+	0.838585i	$0.683384\pi$
$984$	0	0
$985$	4.64764e7	1.52631
$986$	0	0
$987$	5.45614e6	0.178276
$988$	0	0
$989$	2.34435e6	0.0762134
$990$	0	0
$991$	3.54462e7	1.14653	0.573265	−	0.819370i	$-0.305676\pi$
$991$	3.54462e7	1.14653	0.573265	+	0.819370i	$0.305676\pi$
$992$	0	0
$993$	3.35685e7	1.08034
$994$	0	0
$995$	1.06192e8	3.40044
$996$	0	0
$997$	4.51197e7	1.43757	0.718784	−	0.695233i	$-0.244699\pi$
$997$	4.51197e7	1.43757	0.718784	+	0.695233i	$0.244699\pi$
$998$	0	0
$999$	1.07035e7	0.339322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.6.a.g.1.1		2
4.3	odd	2		19.6.a.c.1.1	✓	2
12.11	even	2		171.6.a.f.1.2		2
20.19	odd	2		475.6.a.d.1.2		2
28.27	even	2		931.6.a.c.1.1		2
76.75	even	2		361.6.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.6.a.c.1.1	✓	2	4.3	odd	2
171.6.a.f.1.2		2	12.11	even	2
304.6.a.g.1.1		2	1.1	even	1	trivial
361.6.a.d.1.2		2	76.75	even	2
475.6.a.d.1.2		2	20.19	odd	2
931.6.a.c.1.1		2	28.27	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q-16.4562 q^{3} -99.7603 q^{5} +57.1289 q^{7} +27.8066 q^{9} +O(q^{10})\)	\(q-16.4562 q^{3} -99.7603 q^{5} +57.1289 q^{7} +27.8066 q^{9} +438.977 q^{11} -477.590 q^{13} +1641.68 q^{15} +1671.39 q^{17} +361.000 q^{19} -940.125 q^{21} -459.514 q^{23} +6827.12 q^{25} +3541.27 q^{27} -2696.78 q^{29} +2314.43 q^{31} -7223.89 q^{33} -5699.20 q^{35} +3022.51 q^{37} +7859.32 q^{39} +5483.05 q^{41} -5101.80 q^{43} -2773.99 q^{45} -5803.63 q^{47} -13543.3 q^{49} -27504.7 q^{51} -23207.5 q^{53} -43792.5 q^{55} -5940.69 q^{57} +32849.5 q^{59} -46911.5 q^{61} +1588.56 q^{63} +47644.5 q^{65} +3565.12 q^{67} +7561.86 q^{69} -2346.44 q^{71} +57370.0 q^{73} -112349. q^{75} +25078.3 q^{77} +29715.8 q^{79} -65032.8 q^{81} -60526.0 q^{83} -166738. q^{85} +44378.8 q^{87} +19134.9 q^{89} -27284.2 q^{91} -38086.7 q^{93} -36013.5 q^{95} -43189.4 q^{97} +12206.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 7 q^{3} - 133 q^{5} - 72 q^{7} + 335 q^{9}+O(q^{10}) \) 2 * q + 7 * q^3 - 133 * q^5 - 72 * q^7 + 335 * q^9	\( 2 q + 7 q^{3} - 133 q^{5} - 72 q^{7} + 335 q^{9} + 705 q^{11} - 1341 q^{13} + 862 q^{15} + 2784 q^{17} + 722 q^{19} - 3969 q^{21} + 2713 q^{23} + 4807 q^{25} + 5047 q^{27} - 7775 q^{29} - 7132 q^{31} - 984 q^{33} - 1407 q^{35} - 6248 q^{37} - 12393 q^{39} - 4174 q^{41} - 25357 q^{43} - 12985 q^{45} - 11727 q^{47} - 13676 q^{49} - 1407 q^{51} - 29133 q^{53} - 52635 q^{55} + 2527 q^{57} + 64515 q^{59} - 40939 q^{61} - 38079 q^{63} + 76344 q^{65} - 19039 q^{67} + 81977 q^{69} + 70236 q^{71} + 67058 q^{73} - 159733 q^{75} - 9273 q^{77} + 32850 q^{79} - 104362 q^{81} - 71534 q^{83} - 203721 q^{85} - 74737 q^{87} - 87268 q^{89} + 84207 q^{91} - 259664 q^{93} - 48013 q^{95} - 62458 q^{97} + 93927 q^{99}+O(q^{100}) \) 2 * q + 7 * q^3 - 133 * q^5 - 72 * q^7 + 335 * q^9 + 705 * q^11 - 1341 * q^13 + 862 * q^15 + 2784 * q^17 + 722 * q^19 - 3969 * q^21 + 2713 * q^23 + 4807 * q^25 + 5047 * q^27 - 7775 * q^29 - 7132 * q^31 - 984 * q^33 - 1407 * q^35 - 6248 * q^37 - 12393 * q^39 - 4174 * q^41 - 25357 * q^43 - 12985 * q^45 - 11727 * q^47 - 13676 * q^49 - 1407 * q^51 - 29133 * q^53 - 52635 * q^55 + 2527 * q^57 + 64515 * q^59 - 40939 * q^61 - 38079 * q^63 + 76344 * q^65 - 19039 * q^67 + 81977 * q^69 + 70236 * q^71 + 67058 * q^73 - 159733 * q^75 - 9273 * q^77 + 32850 * q^79 - 104362 * q^81 - 71534 * q^83 - 203721 * q^85 - 74737 * q^87 - 87268 * q^89 + 84207 * q^91 - 259664 * q^93 - 48013 * q^95 - 62458 * q^97 + 93927 * q^99

Introduction

L-functions

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Database

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