LMFDB - Embedded newform 325.6.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 163x^{4} - 8x^{3} + 6120x^{2} + 6624x - 19440 $ x^6 - 163x^4 - 8x^3 + 6120x^2 + 6624x - 19440
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-5.93318$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+5.93318 q^{2} +7.05430 q^{3} +3.20258 q^{4} +41.8544 q^{6} +185.746 q^{7} -170.860 q^{8} -193.237 q^{9} -353.912 q^{11} +22.5920 q^{12} -169.000 q^{13} +1102.06 q^{14} -1116.23 q^{16} -634.652 q^{17} -1146.51 q^{18} +1118.29 q^{19} +1310.31 q^{21} -2099.83 q^{22} -3509.85 q^{23} -1205.30 q^{24} -1002.71 q^{26} -3077.35 q^{27} +594.867 q^{28} -3765.67 q^{29} +2906.63 q^{31} -1155.24 q^{32} -2496.60 q^{33} -3765.50 q^{34} -618.857 q^{36} -283.305 q^{37} +6635.04 q^{38} -1192.18 q^{39} -13563.6 q^{41} +7774.29 q^{42} +5184.47 q^{43} -1133.43 q^{44} -20824.6 q^{46} +6781.50 q^{47} -7874.19 q^{48} +17694.6 q^{49} -4477.03 q^{51} -541.237 q^{52} -7664.43 q^{53} -18258.4 q^{54} -31736.6 q^{56} +7888.78 q^{57} -22342.4 q^{58} +2806.29 q^{59} -13764.7 q^{61} +17245.5 q^{62} -35893.0 q^{63} +28865.0 q^{64} -14812.8 q^{66} -67744.1 q^{67} -2032.53 q^{68} -24759.5 q^{69} -66519.0 q^{71} +33016.5 q^{72} -75902.7 q^{73} -1680.90 q^{74} +3581.43 q^{76} -65737.9 q^{77} -7073.39 q^{78} +101641. q^{79} +25248.0 q^{81} -80475.1 q^{82} +50882.7 q^{83} +4196.37 q^{84} +30760.4 q^{86} -26564.2 q^{87} +60469.5 q^{88} -52439.2 q^{89} -31391.1 q^{91} -11240.6 q^{92} +20504.2 q^{93} +40235.8 q^{94} -8149.42 q^{96} +142557. q^{97} +104985. q^{98} +68388.9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 38 q^{3} + 134 q^{4} + 318 q^{6} - 220 q^{7} - 24 q^{8} + 518 q^{9} - 170 q^{11} - 2238 q^{12} - 1014 q^{13} - 1440 q^{14} + 3506 q^{16} - 728 q^{17} - 7788 q^{18} + 1218 q^{19} - 396 q^{21} - 5154 q^{22}+ \cdots - 32270 q^{99}+O(q^{100}) $ 6 * q - 38 * q^3 + 134 * q^4 + 318 * q^6 - 220 * q^7 - 24 * q^8 + 518 * q^9 - 170 * q^11 - 2238 * q^12 - 1014 * q^13 - 1440 * q^14 + 3506 * q^16 - 728 * q^17 - 7788 * q^18 + 1218 * q^19 - 396 * q^21 - 5154 * q^22 - 8954 * q^23 + 12618 * q^24 - 13112 * q^27 - 13212 * q^28 + 8364 * q^29 + 2862 * q^31 + 29672 * q^32 + 3772 * q^33 - 21484 * q^34 - 2138 * q^36 - 13840 * q^37 + 27706 * q^38 + 6422 * q^39 + 2248 * q^41 + 47028 * q^42 - 3822 * q^43 - 73482 * q^44 - 39886 * q^46 + 10860 * q^47 - 13962 * q^48 + 85902 * q^49 - 55028 * q^51 - 22646 * q^52 - 6584 * q^53 + 45504 * q^54 - 188800 * q^56 - 11692 * q^57 + 88500 * q^58 - 28874 * q^59 + 972 * q^61 - 123466 * q^62 + 125236 * q^63 + 14022 * q^64 - 158040 * q^66 - 60040 * q^67 - 3504 * q^68 + 153088 * q^69 - 74174 * q^71 - 298668 * q^72 - 48960 * q^73 + 226112 * q^74 - 93422 * q^76 - 231588 * q^77 - 53742 * q^78 + 214508 * q^79 + 116006 * q^81 - 58732 * q^82 - 123260 * q^83 - 235268 * q^84 - 304666 * q^86 - 228144 * q^87 - 127622 * q^88 + 94028 * q^89 + 37180 * q^91 - 430298 * q^92 - 192892 * q^93 - 125920 * q^94 - 385858 * q^96 - 246284 * q^97 + 4736 * q^98 - 32270 * q^99

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 $

$2$	5.93318	1.04885	0.524424	−	0.851457i	$-0.324281\pi$
$2$	5.93318	1.04885	0.524424	+	0.851457i	$0.324281\pi$
$3$	7.05430	0.452534	0.226267	−	0.974065i	$-0.427348\pi$
$3$	7.05430	0.452534	0.226267	+	0.974065i	$0.427348\pi$
$4$	3.20258	0.100081
$5$	0	0
$5$	0	0
$6$	41.8544	0.474639
$7$	185.746	1.43276	0.716382	−	0.697708i	$-0.245797\pi$
$7$	185.746	1.43276	0.716382	+	0.697708i	$0.245797\pi$
$8$	−170.860	−0.943878
$9$	−193.237	−0.795213
$10$	0	0
$11$	−353.912	−0.881889	−0.440945	−	0.897534i	$-0.645357\pi$
$11$	−353.912	−0.881889	−0.440945	+	0.897534i	$0.645357\pi$
$12$	22.5920	0.0452899
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	1102.06	1.50275
$15$	0	0
$16$	−1116.23	−1.09006
$17$	−634.652	−0.532615	−0.266308	−	0.963888i	$-0.585804\pi$
$17$	−634.652	−0.532615	−0.266308	+	0.963888i	$0.585804\pi$
$18$	−1146.51	−0.834057
$19$	1118.29	0.710677	0.355338	−	0.934738i	$-0.384366\pi$
$19$	1118.29	0.710677	0.355338	+	0.934738i	$0.384366\pi$
$20$	0	0
$21$	1310.31	0.648374
$22$	−2099.83	−0.924967
$23$	−3509.85	−1.38347	−0.691734	−	0.722153i	$-0.743153\pi$
$23$	−3509.85	−1.38347	−0.691734	+	0.722153i	$0.743153\pi$
$24$	−1205.30	−0.427136
$25$	0	0
$26$	−1002.71	−0.290898
$27$	−3077.35	−0.812394
$28$	594.867	0.143392
$29$	−3765.67	−0.831471	−0.415735	−	0.909486i	$-0.636476\pi$
$29$	−3765.67	−0.831471	−0.415735	+	0.909486i	$0.636476\pi$
$30$	0	0
$31$	2906.63	0.543232	0.271616	−	0.962406i	$-0.412442\pi$
$31$	2906.63	0.543232	0.271616	+	0.962406i	$0.412442\pi$
$32$	−1155.24	−0.199433
$33$	−2496.60	−0.399085
$34$	−3765.50	−0.558632
$35$	0	0
$36$	−618.857	−0.0795855
$37$	−283.305	−0.0340213	−0.0170106	−	0.999855i	$-0.505415\pi$
$37$	−283.305	−0.0340213	−0.0170106	+	0.999855i	$0.505415\pi$
$38$	6635.04	0.745391
$39$	−1192.18	−0.125510
$40$	0	0
$41$	−13563.6	−1.26013	−0.630064	−	0.776544i	$-0.716971\pi$
$41$	−13563.6	−1.26013	−0.630064	+	0.776544i	$0.716971\pi$
$42$	7774.29	0.680045
$43$	5184.47	0.427596	0.213798	−	0.976878i	$-0.431417\pi$
$43$	5184.47	0.427596	0.213798	+	0.976878i	$0.431417\pi$
$44$	−1133.43	−0.0882601
$45$	0	0
$46$	−20824.6	−1.45105
$47$	6781.50	0.447797	0.223898	−	0.974612i	$-0.428122\pi$
$47$	6781.50	0.447797	0.223898	+	0.974612i	$0.428122\pi$
$48$	−7874.19	−0.493291
$49$	17694.6	1.05281
$50$	0	0
$51$	−4477.03	−0.241026
$52$	−541.237	−0.0277574
$53$	−7664.43	−0.374792	−0.187396	−	0.982284i	$-0.560005\pi$
$53$	−7664.43	−0.374792	−0.187396	+	0.982284i	$0.560005\pi$
$54$	−18258.4	−0.852078
$55$	0	0
$56$	−31736.6	−1.35235
$57$	7888.78	0.321605
$58$	−22342.4	−0.872086
$59$	2806.29	0.104955	0.0524773	−	0.998622i	$-0.483288\pi$
$59$	2806.29	0.104955	0.0524773	+	0.998622i	$0.483288\pi$
$60$	0	0
$61$	−13764.7	−0.473634	−0.236817	−	0.971554i	$-0.576104\pi$
$61$	−13764.7	−0.473634	−0.236817	+	0.971554i	$0.576104\pi$
$62$	17245.5	0.569768
$63$	−35893.0	−1.13935
$64$	28865.0	0.880889
$65$	0	0
$66$	−14812.8	−0.418579
$67$	−67744.1	−1.84368	−0.921838	−	0.387576i	$-0.873313\pi$
$67$	−67744.1	−1.84368	−0.921838	+	0.387576i	$0.873313\pi$
$68$	−2032.53	−0.0533045
$69$	−24759.5	−0.626065
$70$	0	0
$71$	−66519.0	−1.56603	−0.783014	−	0.622004i	$-0.786319\pi$
$71$	−66519.0	−1.56603	−0.783014	+	0.622004i	$0.786319\pi$
$72$	33016.5	0.750584
$73$	−75902.7	−1.66706	−0.833528	−	0.552478i	$-0.813682\pi$
$73$	−75902.7	−1.66706	−0.833528	+	0.552478i	$0.813682\pi$
$74$	−1680.90	−0.0356831
$75$	0	0
$76$	3581.43	0.0711250
$77$	−65737.9	−1.26354
$78$	−7073.39	−0.131641
$79$	101641.	1.83233	0.916163	−	0.400806i	$-0.131270\pi$
$79$	101641.	1.83233	0.916163	+	0.400806i	$0.131270\pi$
$80$	0	0
$81$	25248.0	0.427578
$82$	−80475.1	−1.32168
$83$	50882.7	0.810727	0.405363	−	0.914156i	$-0.367145\pi$
$83$	50882.7	0.810727	0.405363	+	0.914156i	$0.367145\pi$
$84$	4196.37	0.0648897
$85$	0	0
$86$	30760.4	0.448483
$87$	−26564.2	−0.376269
$88$	60469.5	0.832396
$89$	−52439.2	−0.701748	−0.350874	−	0.936423i	$-0.614115\pi$
$89$	−52439.2	−0.701748	−0.350874	+	0.936423i	$0.614115\pi$
$90$	0	0
$91$	−31391.1	−0.397377
$92$	−11240.6	−0.138458
$93$	20504.2	0.245831
$94$	40235.8	0.469671
$95$	0	0
$96$	−8149.42	−0.0902503
$97$	142557.	1.53837	0.769183	−	0.639028i	$-0.220663\pi$
$97$	142557.	1.53837	0.769183	+	0.639028i	$0.220663\pi$
$98$	104985.	1.10424
$99$	68388.9	0.701290
$100$	0	0
$101$	4751.74	0.0463499	0.0231750	−	0.999731i	$-0.492623\pi$
$101$	4751.74	0.0463499	0.0231750	+	0.999731i	$0.492623\pi$
$102$	−26563.0	−0.252800
$103$	59290.6	0.550672	0.275336	−	0.961348i	$-0.411211\pi$
$103$	59290.6	0.550672	0.275336	+	0.961348i	$0.411211\pi$
$104$	28875.4	0.261785
$105$	0	0
$106$	−45474.4	−0.393100
$107$	−157927.	−1.33351	−0.666756	−	0.745276i	$-0.732318\pi$
$107$	−157927.	−1.33351	−0.666756	+	0.745276i	$0.732318\pi$
$108$	−9855.46	−0.0813050
$109$	58878.4	0.474668	0.237334	−	0.971428i	$-0.423726\pi$
$109$	58878.4	0.474668	0.237334	+	0.971428i	$0.423726\pi$
$110$	0	0
$111$	−1998.52	−0.0153958
$112$	−207335.	−1.56180
$113$	−179734.	−1.32414	−0.662069	−	0.749443i	$-0.730322\pi$
$113$	−179734.	−1.32414	−0.662069	+	0.749443i	$0.730322\pi$
$114$	46805.5	0.337315
$115$	0	0
$116$	−12059.9	−0.0832142
$117$	32657.0	0.220553
$118$	16650.2	0.110081
$119$	−117884.	−0.763112
$120$	0	0
$121$	−35797.0	−0.222271
$122$	−81668.5	−0.496770
$123$	−95681.5	−0.570250
$124$	9308.72	0.0543671
$125$	0	0
$126$	−212959.	−1.19501
$127$	123741.	0.680774	0.340387	−	0.940286i	$-0.389442\pi$
$127$	123741.	0.680774	0.340387	+	0.940286i	$0.389442\pi$
$128$	208229.	1.12335
$129$	36572.8	0.193501
$130$	0	0
$131$	−43205.0	−0.219966	−0.109983	−	0.993933i	$-0.535080\pi$
$131$	−43205.0	−0.219966	−0.109983	+	0.993933i	$0.535080\pi$
$132$	−7995.58	−0.0399407
$133$	207719.	1.01823
$134$	−401938.	−1.93373
$135$	0	0
$136$	108437.	0.502724
$137$	188517.	0.858120	0.429060	−	0.903276i	$-0.358845\pi$
$137$	188517.	0.858120	0.429060	+	0.903276i	$0.358845\pi$
$138$	−146903.	−0.656647
$139$	344148.	1.51081	0.755403	−	0.655260i	$-0.227441\pi$
$139$	344148.	1.51081	0.755403	+	0.655260i	$0.227441\pi$
$140$	0	0
$141$	47838.7	0.202643
$142$	−394669.	−1.64253
$143$	59811.2	0.244592
$144$	215696.	0.866834
$145$	0	0
$146$	−450344.	−1.74849
$147$	124823.	0.476433
$148$	−907.309	−0.00340487
$149$	−177809.	−0.656126	−0.328063	−	0.944656i	$-0.606396\pi$
$149$	−177809.	−0.656126	−0.328063	+	0.944656i	$0.606396\pi$
$150$	0	0
$151$	554784.	1.98008	0.990038	−	0.140803i	$-0.0449685\pi$
$151$	554784.	1.98008	0.990038	+	0.140803i	$0.0449685\pi$
$152$	−191072.	−0.670792
$153$	122638.	0.423543
$154$	−390034.	−1.32526
$155$	0	0
$156$	−3818.05	−0.0125612
$157$	−255896.	−0.828542	−0.414271	−	0.910154i	$-0.635963\pi$
$157$	−255896.	−0.828542	−0.414271	+	0.910154i	$0.635963\pi$
$158$	603056.	1.92183
$159$	−54067.2	−0.169606
$160$	0	0
$161$	−651941.	−1.98218
$162$	149801.	0.448464
$163$	262686.	0.774404	0.387202	−	0.921995i	$-0.373442\pi$
$163$	262686.	0.774404	0.387202	+	0.921995i	$0.373442\pi$
$164$	−43438.5	−0.126114
$165$	0	0
$166$	301896.	0.850329
$167$	−287069.	−0.796517	−0.398259	−	0.917273i	$-0.630385\pi$
$167$	−287069.	−0.796517	−0.398259	+	0.917273i	$0.630385\pi$
$168$	−223880.	−0.611986
$169$	28561.0	0.0769231
$170$	0	0
$171$	−216096.	−0.565140
$172$	16603.7	0.0427941
$173$	−719663.	−1.82816	−0.914080	−	0.405534i	$-0.867085\pi$
$173$	−719663.	−1.82816	−0.914080	+	0.405534i	$0.867085\pi$
$174$	−157610.	−0.394648
$175$	0	0
$176$	395046.	0.961316
$177$	19796.4	0.0474955
$178$	−311131.	−0.736026
$179$	779772.	1.81901	0.909505	−	0.415693i	$-0.136461\pi$
$179$	779772.	1.81901	0.909505	+	0.415693i	$0.136461\pi$
$180$	0	0
$181$	336065.	0.762478	0.381239	−	0.924477i	$-0.375498\pi$
$181$	336065.	0.762478	0.381239	+	0.924477i	$0.375498\pi$
$182$	−186249.	−0.416788
$183$	−97100.5	−0.214335
$184$	599693.	1.30582
$185$	0	0
$186$	121655.	0.257839
$187$	224611.	0.469708
$188$	21718.3	0.0448158
$189$	−571605.	−1.16397
$190$	0	0
$191$	409479.	0.812173	0.406086	−	0.913835i	$-0.366893\pi$
$191$	409479.	0.812173	0.406086	+	0.913835i	$0.366893\pi$
$192$	203622.	0.398632
$193$	−339565.	−0.656189	−0.328095	−	0.944645i	$-0.606407\pi$
$193$	−339565.	−0.656189	−0.328095	+	0.944645i	$0.606407\pi$
$194$	845817.	1.61351
$195$	0	0
$196$	56668.4	0.105366
$197$	871469.	1.59988	0.799938	−	0.600082i	$-0.204865\pi$
$197$	871469.	1.59988	0.799938	+	0.600082i	$0.204865\pi$
$198$	405764.	0.735546
$199$	270952.	0.485019	0.242510	−	0.970149i	$-0.422029\pi$
$199$	270952.	0.485019	0.242510	+	0.970149i	$0.422029\pi$
$200$	0	0
$201$	−477887.	−0.834325
$202$	28192.9	0.0486140
$203$	−699458.	−1.19130
$204$	−14338.1	−0.0241221
$205$	0	0
$206$	351781.	0.577570
$207$	678232.	1.10015
$208$	188642.	0.302330
$209$	−395778.	−0.626738
$210$	0	0
$211$	181455.	0.280583	0.140292	−	0.990110i	$-0.455196\pi$
$211$	181455.	0.280583	0.140292	+	0.990110i	$0.455196\pi$
$212$	−24546.0	−0.0375095
$213$	−469245.	−0.708681
$214$	−937009.	−1.39865
$215$	0	0
$216$	525796.	0.766801
$217$	539895.	0.778323
$218$	349336.	0.497854
$219$	−535440.	−0.754398
$220$	0	0
$221$	107256.	0.147721
$222$	−11857.6	−0.0161478
$223$	−1.38761e6	−1.86855	−0.934276	−	0.356552i	$-0.883952\pi$
$223$	−1.38761e6	−1.86855	−0.934276	+	0.356552i	$0.883952\pi$
$224$	−214582.	−0.285741
$225$	0	0
$226$	−1.06639e6	−1.38882
$227$	690397.	0.889271	0.444636	−	0.895711i	$-0.353333\pi$
$227$	690397.	0.889271	0.444636	+	0.895711i	$0.353333\pi$
$228$	25264.5	0.0321865
$229$	−1.38257e6	−1.74221	−0.871104	−	0.491099i	$-0.836595\pi$
$229$	−1.38257e6	−1.74221	−0.871104	+	0.491099i	$0.836595\pi$
$230$	0	0
$231$	−463735.	−0.571794
$232$	643403.	0.784807
$233$	71911.6	0.0867780	0.0433890	−	0.999058i	$-0.486185\pi$
$233$	71911.6	0.0867780	0.0433890	+	0.999058i	$0.486185\pi$
$234$	193760.	0.231326
$235$	0	0
$236$	8987.36	0.0105039
$237$	717009.	0.829189
$238$	−699427.	−0.800387
$239$	−825442.	−0.934743	−0.467371	−	0.884061i	$-0.654799\pi$
$239$	−825442.	−0.934743	−0.467371	+	0.884061i	$0.654799\pi$
$240$	0	0
$241$	−615086.	−0.682171	−0.341086	−	0.940032i	$-0.610795\pi$
$241$	−615086.	−0.682171	−0.341086	+	0.940032i	$0.610795\pi$
$242$	−212390.	−0.233128
$243$	925902.	1.00589
$244$	−44082.7	−0.0474016
$245$	0	0
$246$	−567695.	−0.598105
$247$	−188992.	−0.197106
$248$	−496627.	−0.512745
$249$	358942.	0.366881
$250$	0	0
$251$	622589.	0.623759	0.311879	−	0.950122i	$-0.399041\pi$
$251$	622589.	0.623759	0.311879	+	0.950122i	$0.399041\pi$
$252$	−114950.	−0.114027
$253$	1.24218e6	1.22007
$254$	734174.	0.714028
$255$	0	0
$256$	311779.	0.297335
$257$	1.04334e6	0.985359	0.492680	−	0.870211i	$-0.336017\pi$
$257$	1.04334e6	0.985359	0.492680	+	0.870211i	$0.336017\pi$
$258$	216993.	0.202953
$259$	−52622.9	−0.0487444
$260$	0	0
$261$	727666.	0.661197
$262$	−256343.	−0.230711
$263$	1.31940e6	1.17621	0.588106	−	0.808784i	$-0.299874\pi$
$263$	1.31940e6	1.17621	0.588106	+	0.808784i	$0.299874\pi$
$264$	426570.	0.376687
$265$	0	0
$266$	1.23243e6	1.06797
$267$	−369922.	−0.317564
$268$	−216956.	−0.184516
$269$	369633.	0.311452	0.155726	−	0.987800i	$-0.450228\pi$
$269$	369633.	0.311452	0.155726	+	0.987800i	$0.450228\pi$
$270$	0	0
$271$	749291.	0.619765	0.309883	−	0.950775i	$-0.399710\pi$
$271$	749291.	0.619765	0.309883	+	0.950775i	$0.399710\pi$
$272$	708415.	0.580585
$273$	−221442.	−0.179826
$274$	1.11850e6	0.900037
$275$	0	0
$276$	−79294.5	−0.0626571
$277$	−1.64757e6	−1.29016	−0.645081	−	0.764114i	$-0.723176\pi$
$277$	−1.64757e6	−1.29016	−0.645081	+	0.764114i	$0.723176\pi$
$278$	2.04189e6	1.58461
$279$	−561668.	−0.431986
$280$	0	0
$281$	1.21917e6	0.921080	0.460540	−	0.887639i	$-0.347656\pi$
$281$	1.21917e6	0.921080	0.460540	+	0.887639i	$0.347656\pi$
$282$	283836.	0.212542
$283$	997517.	0.740379	0.370190	−	0.928956i	$-0.379293\pi$
$283$	997517.	0.740379	0.370190	+	0.928956i	$0.379293\pi$
$284$	−213033.	−0.156729
$285$	0	0
$286$	354870.	0.256540
$287$	−2.51938e6	−1.80546
$288$	223235.	0.158592
$289$	−1.01707e6	−0.716321
$290$	0	0
$291$	1.00564e6	0.696162
$292$	−243085.	−0.166840
$293$	−1.80793e6	−1.23031	−0.615153	−	0.788408i	$-0.710906\pi$
$293$	−1.80793e6	−1.23031	−0.615153	+	0.788408i	$0.710906\pi$
$294$	740597.	0.499705
$295$	0	0
$296$	48405.6	0.0321119
$297$	1.08911e6	0.716442
$298$	−1.05497e6	−0.688176
$299$	593165.	0.383705
$300$	0	0
$301$	962995.	0.612644
$302$	3.29163e6	2.07680
$303$	33520.2	0.0209749
$304$	−1.24827e6	−0.774683
$305$	0	0
$306$	727634.	0.444232
$307$	24494.5	0.0148328	0.00741638	−	0.999972i	$-0.497639\pi$
$307$	24494.5	0.0148328	0.00741638	+	0.999972i	$0.497639\pi$
$308$	−210531.	−0.126456
$309$	418254.	0.249197
$310$	0	0
$311$	1.48212e6	0.868924	0.434462	−	0.900690i	$-0.356939\pi$
$311$	1.48212e6	0.868924	0.434462	+	0.900690i	$0.356939\pi$
$312$	203695.	0.118466
$313$	348766.	0.201221	0.100611	−	0.994926i	$-0.467920\pi$
$313$	348766.	0.201221	0.100611	+	0.994926i	$0.467920\pi$
$314$	−1.51828e6	−0.869014
$315$	0	0
$316$	325515.	0.183380
$317$	406486.	0.227194	0.113597	−	0.993527i	$-0.463763\pi$
$317$	406486.	0.227194	0.113597	+	0.993527i	$0.463763\pi$
$318$	−320790.	−0.177891
$319$	1.33272e6	0.733265
$320$	0	0
$321$	−1.11406e6	−0.603459
$322$	−3.86808e6	−2.07901
$323$	−709728.	−0.378517
$324$	80858.9	0.0427923
$325$	0	0
$326$	1.55856e6	0.812231
$327$	415346.	0.214803
$328$	2.31747e6	1.18941
$329$	1.25964e6	0.641587
$330$	0	0
$331$	1.58614e6	0.795743	0.397871	−	0.917441i	$-0.369749\pi$
$331$	1.58614e6	0.795743	0.397871	+	0.917441i	$0.369749\pi$
$332$	162956.	0.0811381
$333$	54745.0	0.0270542
$334$	−1.70323e6	−0.835425
$335$	0	0
$336$	−1.46260e6	−0.706769
$337$	−2.43587e6	−1.16837	−0.584185	−	0.811621i	$-0.698586\pi$
$337$	−2.43587e6	−1.16837	−0.584185	+	0.811621i	$0.698586\pi$
$338$	169457.	0.0806806
$339$	−1.26790e6	−0.599217
$340$	0	0
$341$	−1.02869e6	−0.479071
$342$	−1.28213e6	−0.592745
$343$	164869.	0.0756664
$344$	−885820.	−0.403598
$345$	0	0
$346$	−4.26989e6	−1.91746
$347$	1.17786e6	0.525132	0.262566	−	0.964914i	$-0.415431\pi$
$347$	1.17786e6	0.525132	0.262566	+	0.964914i	$0.415431\pi$
$348$	−85073.9	−0.0376572
$349$	−338854.	−0.148919	−0.0744594	−	0.997224i	$-0.523723\pi$
$349$	−338854.	−0.148919	−0.0744594	+	0.997224i	$0.523723\pi$
$350$	0	0
$351$	520071.	0.225318
$352$	408854.	0.175878
$353$	−3.25607e6	−1.39077	−0.695387	−	0.718635i	$-0.744767\pi$
$353$	−3.25607e6	−1.39077	−0.695387	+	0.718635i	$0.744767\pi$
$354$	117455.	0.0498156
$355$	0	0
$356$	−167941.	−0.0702314
$357$	−831590.	−0.345334
$358$	4.62652e6	1.90786
$359$	2.81818e6	1.15407	0.577036	−	0.816719i	$-0.304209\pi$
$359$	2.81818e6	1.15407	0.577036	+	0.816719i	$0.304209\pi$
$360$	0	0
$361$	−1.22552e6	−0.494939
$362$	1.99393e6	0.799723
$363$	−252522.	−0.100585
$364$	−100533.	−0.0397698
$365$	0	0
$366$	−576114.	−0.224805
$367$	−3.09661e6	−1.20011	−0.600056	−	0.799958i	$-0.704855\pi$
$367$	−3.09661e6	−1.20011	−0.600056	+	0.799958i	$0.704855\pi$
$368$	3.91779e6	1.50807
$369$	2.62098e6	1.00207
$370$	0	0
$371$	−1.42364e6	−0.536988
$372$	65666.5	0.0246029
$373$	4.21455e6	1.56848	0.784240	−	0.620457i	$-0.213053\pi$
$373$	4.21455e6	1.56848	0.784240	+	0.620457i	$0.213053\pi$
$374$	1.33266e6	0.492652
$375$	0	0
$376$	−1.15869e6	−0.422666
$377$	636398.	0.230609
$378$	−3.39143e6	−1.22083
$379$	−1.26649e6	−0.452903	−0.226452	−	0.974022i	$-0.572712\pi$
$379$	−1.26649e6	−0.452903	−0.226452	+	0.974022i	$0.572712\pi$
$380$	0	0
$381$	872903.	0.308073
$382$	2.42951e6	0.851845
$383$	−5.66939e6	−1.97487	−0.987436	−	0.158017i	$-0.949490\pi$
$383$	−5.66939e6	−1.97487	−0.987436	+	0.158017i	$0.949490\pi$
$384$	1.46891e6	0.508354
$385$	0	0
$386$	−2.01470e6	−0.688242
$387$	−1.00183e6	−0.340030
$388$	456551.	0.153961
$389$	−5.49114e6	−1.83988	−0.919938	−	0.392063i	$-0.871762\pi$
$389$	−5.49114e6	−1.83988	−0.919938	+	0.392063i	$0.871762\pi$
$390$	0	0
$391$	2.22753e6	0.736856
$392$	−3.02330e6	−0.993726
$393$	−304781.	−0.0995420
$394$	5.17058e6	1.67803
$395$	0	0
$396$	219021.	0.0701856
$397$	1.53688e6	0.489398	0.244699	−	0.969599i	$-0.421311\pi$
$397$	1.53688e6	0.489398	0.244699	+	0.969599i	$0.421311\pi$
$398$	1.60760e6	0.508711
$399$	1.46531e6	0.460784
$400$	0	0
$401$	30028.4	0.00932547	0.00466273	−	0.999989i	$-0.498516\pi$
$401$	30028.4	0.00932547	0.00466273	+	0.999989i	$0.498516\pi$
$402$	−2.83539e6	−0.875080
$403$	−491220.	−0.150666
$404$	15217.8	0.00463873
$405$	0	0
$406$	−4.15001e6	−1.24949
$407$	100265.	0.0300030
$408$	764946.	0.227499
$409$	−5.54413e6	−1.63880	−0.819398	−	0.573225i	$-0.805692\pi$
$409$	−5.54413e6	−1.63880	−0.819398	+	0.573225i	$0.805692\pi$
$410$	0	0
$411$	1.32985e6	0.388328
$412$	189883.	0.0551116
$413$	521257.	0.150375
$414$	4.02407e6	1.15389
$415$	0	0
$416$	195236.	0.0553129
$417$	2.42773e6	0.683691
$418$	−2.34822e6	−0.657353
$419$	−1.29360e6	−0.359968	−0.179984	−	0.983670i	$-0.557605\pi$
$419$	−1.29360e6	−0.359968	−0.179984	+	0.983670i	$0.557605\pi$
$420$	0	0
$421$	−3.68620e6	−1.01362	−0.506809	−	0.862058i	$-0.669175\pi$
$421$	−3.68620e6	−1.01362	−0.506809	+	0.862058i	$0.669175\pi$
$422$	1.07660e6	0.294289
$423$	−1.31044e6	−0.356094
$424$	1.30955e6	0.353758
$425$	0	0
$426$	−2.78411e6	−0.743298
$427$	−2.55674e6	−0.678606
$428$	−505774.	−0.133459
$429$	421926.	0.110686
$430$	0	0
$431$	1.22645e6	0.318021	0.159010	−	0.987277i	$-0.449170\pi$
$431$	1.22645e6	0.318021	0.159010	+	0.987277i	$0.449170\pi$
$432$	3.43501e6	0.885562
$433$	−1.02459e6	−0.262621	−0.131311	−	0.991341i	$-0.541919\pi$
$433$	−1.02459e6	−0.262621	−0.131311	+	0.991341i	$0.541919\pi$
$434$	3.20329e6	0.816342
$435$	0	0
$436$	188563.	0.0475051
$437$	−3.92504e6	−0.983198
$438$	−3.17686e6	−0.791249
$439$	−5.04951e6	−1.25051	−0.625256	−	0.780420i	$-0.715005\pi$
$439$	−5.04951e6	−1.25051	−0.625256	+	0.780420i	$0.715005\pi$
$440$	0	0
$441$	−3.41925e6	−0.837210
$442$	636370.	0.154937
$443$	−6.30848e6	−1.52727	−0.763634	−	0.645649i	$-0.776587\pi$
$443$	−6.30848e6	−1.52727	−0.763634	+	0.645649i	$0.776587\pi$
$444$	−6400.43	−0.00154082
$445$	0	0
$446$	−8.23293e6	−1.95983
$447$	−1.25432e6	−0.296919
$448$	5.36156e6	1.26211
$449$	1.16391e6	0.272461	0.136231	−	0.990677i	$-0.456501\pi$
$449$	1.16391e6	0.272461	0.136231	+	0.990677i	$0.456501\pi$
$450$	0	0
$451$	4.80032e6	1.11129
$452$	−575612.	−0.132521
$453$	3.91361e6	0.896050
$454$	4.09625e6	0.932710
$455$	0	0
$456$	−1.34788e6	−0.303556
$457$	−1.48156e6	−0.331840	−0.165920	−	0.986139i	$-0.553059\pi$
$457$	−1.48156e6	−0.331840	−0.165920	+	0.986139i	$0.553059\pi$
$458$	−8.20306e6	−1.82731
$459$	1.95304e6	0.432693
$460$	0	0
$461$	−5.65392e6	−1.23908	−0.619538	−	0.784967i	$-0.712680\pi$
$461$	−5.65392e6	−1.23908	−0.619538	+	0.784967i	$0.712680\pi$
$462$	−2.75142e6	−0.599724
$463$	−2.09215e6	−0.453566	−0.226783	−	0.973945i	$-0.572821\pi$
$463$	−2.09215e6	−0.453566	−0.226783	+	0.973945i	$0.572821\pi$
$464$	4.20334e6	0.906357
$465$	0	0
$466$	426664.	0.0910168
$467$	7.48481e6	1.58814	0.794069	−	0.607827i	$-0.207959\pi$
$467$	7.48481e6	1.58814	0.794069	+	0.607827i	$0.207959\pi$
$468$	104587.	0.0220731
$469$	−1.25832e7	−2.64155
$470$	0	0
$471$	−1.80517e6	−0.374943
$472$	−479482.	−0.0990644
$473$	−1.83485e6	−0.377092
$474$	4.25414e6	0.869693
$475$	0	0
$476$	−377534.	−0.0763728
$477$	1.48105e6	0.298040
$478$	−4.89750e6	−0.980402
$479$	2.54779e6	0.507371	0.253685	−	0.967287i	$-0.418357\pi$
$479$	2.54779e6	0.507371	0.253685	+	0.967287i	$0.418357\pi$
$480$	0	0
$481$	47878.6	0.00943580
$482$	−3.64941e6	−0.715493
$483$	−4.59899e6	−0.897004
$484$	−114643.	−0.0222450
$485$	0	0
$486$	5.49354e6	1.05502
$487$	−3.67779e6	−0.702692	−0.351346	−	0.936246i	$-0.614276\pi$
$487$	−3.67779e6	−0.702692	−0.351346	+	0.936246i	$0.614276\pi$
$488$	2.35184e6	0.447053
$489$	1.85306e6	0.350444
$490$	0	0
$491$	−7.23294e6	−1.35398	−0.676988	−	0.735994i	$-0.736715\pi$
$491$	−7.23294e6	−1.35398	−0.676988	+	0.735994i	$0.736715\pi$
$492$	−306428.	−0.0570710
$493$	2.38989e6	0.442854
$494$	−1.12132e6	−0.206734
$495$	0	0
$496$	−3.24446e6	−0.592158
$497$	−1.23556e7	−2.24375
$498$	2.12966e6	0.384802
$499$	875124.	0.157332	0.0786662	−	0.996901i	$-0.474934\pi$
$499$	875124.	0.157332	0.0786662	+	0.996901i	$0.474934\pi$
$500$	0	0
$501$	−2.02507e6	−0.360451
$502$	3.69393e6	0.654228
$503$	2.95982e6	0.521609	0.260805	−	0.965392i	$-0.416012\pi$
$503$	2.95982e6	0.521609	0.260805	+	0.965392i	$0.416012\pi$
$504$	6.13268e6	1.07541
$505$	0	0
$506$	7.37007e6	1.27966
$507$	201478.	0.0348103
$508$	396289.	0.0681323
$509$	−1.12208e7	−1.91968	−0.959842	−	0.280540i	$-0.909487\pi$
$509$	−1.12208e7	−1.91968	−0.959842	+	0.280540i	$0.909487\pi$
$510$	0	0
$511$	−1.40986e7	−2.38850
$512$	−4.81348e6	−0.811493
$513$	−3.44138e6	−0.577350
$514$	6.19034e6	1.03349
$515$	0	0
$516$	117128.	0.0193658
$517$	−2.40006e6	−0.394907
$518$	−312221.	−0.0511255
$519$	−5.07672e6	−0.827304
$520$	0	0
$521$	8.92586e6	1.44064	0.720321	−	0.693641i	$-0.243995\pi$
$521$	8.92586e6	1.44064	0.720321	+	0.693641i	$0.243995\pi$
$522$	4.31737e6	0.693495
$523$	−6.44897e6	−1.03095	−0.515473	−	0.856906i	$-0.672384\pi$
$523$	−6.44897e6	−1.03095	−0.515473	+	0.856906i	$0.672384\pi$
$524$	−138368.	−0.0220144
$525$	0	0
$526$	7.82821e6	1.23367
$527$	−1.84470e6	−0.289334
$528$	2.78678e6	0.435028
$529$	5.88270e6	0.913982
$530$	0	0
$531$	−542278.	−0.0834614
$532$	665237.	0.101905
$533$	2.29224e6	0.349496
$534$	−2.19481e6	−0.333077
$535$	0	0
$536$	1.15748e7	1.74020
$537$	5.50075e6	0.823163
$538$	2.19310e6	0.326665
$539$	−6.26234e6	−0.928463
$540$	0	0
$541$	6.01652e6	0.883796	0.441898	−	0.897065i	$-0.354305\pi$
$541$	6.01652e6	0.883796	0.441898	+	0.897065i	$0.354305\pi$
$542$	4.44567e6	0.650039
$543$	2.37071e6	0.345047
$544$	733177.	0.106221
$545$	0	0
$546$	−1.31386e6	−0.188611
$547$	−928354.	−0.132662	−0.0663308	−	0.997798i	$-0.521129\pi$
$547$	−928354.	−0.132662	−0.0663308	+	0.997798i	$0.521129\pi$
$548$	603740.	0.0858813
$549$	2.65985e6	0.376640
$550$	0	0
$551$	−4.21113e6	−0.590907
$552$	4.23042e6	0.590929
$553$	1.88795e7	2.62529
$554$	−9.77532e6	−1.35318
$555$	0	0
$556$	1.10216e6	0.151203
$557$	−652347.	−0.0890924	−0.0445462	−	0.999007i	$-0.514184\pi$
$557$	−652347.	−0.0890924	−0.0445462	+	0.999007i	$0.514184\pi$
$558$	−3.33248e6	−0.453087
$559$	−876176.	−0.118594
$560$	0	0
$561$	1.58448e6	0.212558
$562$	7.23353e6	0.966072
$563$	992674.	0.131988	0.0659942	−	0.997820i	$-0.478978\pi$
$563$	992674.	0.131988	0.0659942	+	0.997820i	$0.478978\pi$
$564$	153208.	0.0202807
$565$	0	0
$566$	5.91844e6	0.776545
$567$	4.68972e6	0.612618
$568$	1.13654e7	1.47814
$569$	8.79703e6	1.13908	0.569541	−	0.821963i	$-0.307121\pi$
$569$	8.79703e6	1.13908	0.569541	+	0.821963i	$0.307121\pi$
$570$	0	0
$571$	−6.18261e6	−0.793563	−0.396782	−	0.917913i	$-0.629873\pi$
$571$	−6.18261e6	−0.793563	−0.396782	+	0.917913i	$0.629873\pi$
$572$	191550.	0.0244790
$573$	2.88859e6	0.367535
$574$	−1.49479e7	−1.89366
$575$	0	0
$576$	−5.57778e6	−0.700495
$577$	−1.42671e7	−1.78401	−0.892003	−	0.452030i	$-0.850700\pi$
$577$	−1.42671e7	−1.78401	−0.892003	+	0.452030i	$0.850700\pi$
$578$	−6.03448e6	−0.751312
$579$	−2.39539e6	−0.296948
$580$	0	0
$581$	9.45125e6	1.16158
$582$	5.96665e6	0.730168
$583$	2.71254e6	0.330525
$584$	1.29687e7	1.57350
$585$	0	0
$586$	−1.07268e7	−1.29040
$587$	7.84422e6	0.939625	0.469812	−	0.882766i	$-0.344322\pi$
$587$	7.84422e6	0.939625	0.469812	+	0.882766i	$0.344322\pi$
$588$	399756.	0.0476817
$589$	3.25047e6	0.386062
$590$	0	0
$591$	6.14761e6	0.723998
$592$	316233.	0.0370854
$593$	1.85555e6	0.216689	0.108344	−	0.994113i	$-0.465445\pi$
$593$	1.85555e6	0.216689	0.108344	+	0.994113i	$0.465445\pi$
$594$	6.46189e6	0.751438
$595$	0	0
$596$	−569447.	−0.0656656
$597$	1.91137e6	0.219487
$598$	3.51935e6	0.402448
$599$	−1.54479e7	−1.75915	−0.879573	−	0.475764i	$-0.842172\pi$
$599$	−1.54479e7	−1.75915	−0.879573	+	0.475764i	$0.842172\pi$
$600$	0	0
$601$	431785.	0.0487619	0.0243810	−	0.999703i	$-0.492239\pi$
$601$	431785.	0.0487619	0.0243810	+	0.999703i	$0.492239\pi$
$602$	5.71362e6	0.642570
$603$	1.30907e7	1.46612
$604$	1.77674e6	0.198167
$605$	0	0
$606$	198881.	0.0219995
$607$	−1.21071e7	−1.33373	−0.666863	−	0.745180i	$-0.732363\pi$
$607$	−1.21071e7	−1.33373	−0.666863	+	0.745180i	$0.732363\pi$
$608$	−1.29190e6	−0.141733
$609$	−4.93419e6	−0.539104
$610$	0	0
$611$	−1.14607e6	−0.124197
$612$	392759.	0.0423885
$613$	−9.44151e6	−1.01482	−0.507411	−	0.861704i	$-0.669398\pi$
$613$	−9.44151e6	−1.01482	−0.507411	+	0.861704i	$0.669398\pi$
$614$	145330.	0.0155573
$615$	0	0
$616$	1.12320e7	1.19263
$617$	9.98389e6	1.05581	0.527906	−	0.849303i	$-0.322977\pi$
$617$	9.98389e6	1.05581	0.527906	+	0.849303i	$0.322977\pi$
$618$	2.48157e6	0.261370
$619$	−6.44689e6	−0.676275	−0.338138	−	0.941097i	$-0.609797\pi$
$619$	−6.44689e6	−0.676275	−0.338138	+	0.941097i	$0.609797\pi$
$620$	0	0
$621$	1.08010e7	1.12392
$622$	8.79367e6	0.911369
$623$	−9.74037e6	−1.00544
$624$	1.33074e6	0.136814
$625$	0	0
$626$	2.06929e6	0.211050
$627$	−2.79194e6	−0.283620
$628$	−819528.	−0.0829211
$629$	179800.	0.0181202
$630$	0	0
$631$	4.35897e6	0.435823	0.217912	−	0.975969i	$-0.430076\pi$
$631$	4.35897e6	0.435823	0.217912	+	0.975969i	$0.430076\pi$
$632$	−1.73665e7	−1.72949
$633$	1.28003e6	0.126973
$634$	2.41176e6	0.238292
$635$	0	0
$636$	−173155.	−0.0169743
$637$	−2.99039e6	−0.291997
$638$	7.90725e6	0.769084
$639$	1.28539e7	1.24533
$640$	0	0
$641$	1.63272e7	1.56952	0.784758	−	0.619803i	$-0.212787\pi$
$641$	1.63272e7	1.56952	0.784758	+	0.619803i	$0.212787\pi$
$642$	−6.60994e6	−0.632936
$643$	−1.31929e7	−1.25838	−0.629192	−	0.777250i	$-0.716614\pi$
$643$	−1.31929e7	−1.25838	−0.629192	+	0.777250i	$0.716614\pi$
$644$	−2.08789e6	−0.198378
$645$	0	0
$646$	−4.21094e6	−0.397007
$647$	9.42830e6	0.885468	0.442734	−	0.896653i	$-0.354009\pi$
$647$	9.42830e6	0.885468	0.442734	+	0.896653i	$0.354009\pi$
$648$	−4.31388e6	−0.403581
$649$	−993180.	−0.0925584
$650$	0	0
$651$	3.80858e6	0.352217
$652$	841273.	0.0775029
$653$	−1.60701e7	−1.47481	−0.737406	−	0.675450i	$-0.763950\pi$
$653$	−1.60701e7	−1.47481	−0.737406	+	0.675450i	$0.763950\pi$
$654$	2.46432e6	0.225296
$655$	0	0
$656$	1.51400e7	1.37362
$657$	1.46672e7	1.32566
$658$	7.47365e6	0.672927
$659$	6.63639e6	0.595276	0.297638	−	0.954679i	$-0.403801\pi$
$659$	6.63639e6	0.595276	0.297638	+	0.954679i	$0.403801\pi$
$660$	0	0
$661$	2.01198e7	1.79110	0.895552	−	0.444956i	$-0.146781\pi$
$661$	2.01198e7	1.79110	0.895552	+	0.444956i	$0.146781\pi$
$662$	9.41087e6	0.834613
$663$	756618.	0.0668486
$664$	−8.69382e6	−0.765227
$665$	0	0
$666$	324812.	0.0283757
$667$	1.32169e7	1.15031
$668$	−919363.	−0.0797160
$669$	−9.78861e6	−0.845582
$670$	0	0
$671$	4.87151e6	0.417693
$672$	−1.51372e6	−0.129307
$673$	9.52533e6	0.810667	0.405334	−	0.914169i	$-0.367155\pi$
$673$	9.52533e6	0.810667	0.405334	+	0.914169i	$0.367155\pi$
$674$	−1.44525e7	−1.22544
$675$	0	0
$676$	91469.0	0.00769852
$677$	3.37825e6	0.283283	0.141641	−	0.989918i	$-0.454762\pi$
$677$	3.37825e6	0.283283	0.141641	+	0.989918i	$0.454762\pi$
$678$	−7.52265e6	−0.628487
$679$	2.64794e7	2.20412
$680$	0	0
$681$	4.87027e6	0.402425
$682$	−6.10341e6	−0.502472
$683$	8.86253e6	0.726953	0.363476	−	0.931603i	$-0.381590\pi$
$683$	8.86253e6	0.726953	0.363476	+	0.931603i	$0.381590\pi$
$684$	−692064.	−0.0565596
$685$	0	0
$686$	978195.	0.0793625
$687$	−9.75310e6	−0.788407
$688$	−5.78704e6	−0.466107
$689$	1.29529e6	0.103949
$690$	0	0
$691$	25025.8	0.00199385	0.000996925	−	1.00000i	$-0.499683\pi$
$691$	25025.8	0.00199385	0.000996925		1.00000i	$0.499683\pi$
$692$	−2.30478e6	−0.182964
$693$	1.27030e7	1.00478
$694$	6.98842e6	0.550783
$695$	0	0
$696$	4.53876e6	0.355152
$697$	8.60815e6	0.671163
$698$	−2.01048e6	−0.156193
$699$	507286.	0.0392699
$700$	0	0
$701$	−2.15506e7	−1.65640	−0.828198	−	0.560436i	$-0.810634\pi$
$701$	−2.15506e7	−1.65640	−0.828198	+	0.560436i	$0.810634\pi$
$702$	3.08568e6	0.236324
$703$	−316819.	−0.0241781
$704$	−1.02157e7	−0.776847
$705$	0	0
$706$	−1.93188e7	−1.45871
$707$	882617.	0.0664085
$708$	63399.6	0.00475339
$709$	2.07938e7	1.55352	0.776761	−	0.629796i	$-0.216861\pi$
$709$	2.07938e7	1.55352	0.776761	+	0.629796i	$0.216861\pi$
$710$	0	0
$711$	−1.96409e7	−1.45709
$712$	8.95977e6	0.662364
$713$	−1.02018e7	−0.751544
$714$	−4.93397e6	−0.362202
$715$	0	0
$716$	2.49728e6	0.182048
$717$	−5.82292e6	−0.423002
$718$	1.67208e7	1.21045
$719$	−3.65717e6	−0.263829	−0.131915	−	0.991261i	$-0.542113\pi$
$719$	−3.65717e6	−0.263829	−0.131915	+	0.991261i	$0.542113\pi$
$720$	0	0
$721$	1.10130e7	0.788982
$722$	−7.27121e6	−0.519115
$723$	−4.33900e6	−0.308705
$724$	1.07628e6	0.0763093
$725$	0	0
$726$	−1.49826e6	−0.105498
$727$	8.36880e6	0.587256	0.293628	−	0.955920i	$-0.405137\pi$
$727$	8.36880e6	0.587256	0.293628	+	0.955920i	$0.405137\pi$
$728$	5.36349e6	0.375075
$729$	396319.	0.0276202
$730$	0	0
$731$	−3.29034e6	−0.227744
$732$	−310972.	−0.0214508
$733$	−1.81111e7	−1.24504	−0.622522	−	0.782602i	$-0.713892\pi$
$733$	−1.81111e7	−1.24504	−0.622522	+	0.782602i	$0.713892\pi$
$734$	−1.83727e7	−1.25873
$735$	0	0
$736$	4.05472e6	0.275910
$737$	2.39755e7	1.62592
$738$	1.55507e7	1.05102
$739$	7.31705e6	0.492861	0.246431	−	0.969160i	$-0.420742\pi$
$739$	7.31705e6	0.492861	0.246431	+	0.969160i	$0.420742\pi$
$740$	0	0
$741$	−1.33320e6	−0.0891972
$742$	−8.44670e6	−0.563219
$743$	−1.04179e7	−0.692322	−0.346161	−	0.938175i	$-0.612515\pi$
$743$	−1.04179e7	−0.692322	−0.346161	+	0.938175i	$0.612515\pi$
$744$	−3.50336e6	−0.232034
$745$	0	0
$746$	2.50057e7	1.64510
$747$	−9.83240e6	−0.644701
$748$	719336.	0.0470087
$749$	−2.93343e7	−1.91061
$750$	0	0
$751$	−1.16729e6	−0.0755229	−0.0377615	−	0.999287i	$-0.512023\pi$
$751$	−1.16729e6	−0.0755229	−0.0377615	+	0.999287i	$0.512023\pi$
$752$	−7.56969e6	−0.488128
$753$	4.39193e6	0.282272
$754$	3.77586e6	0.241873
$755$	0	0
$756$	−1.83061e6	−0.116491
$757$	−4.75104e6	−0.301334	−0.150667	−	0.988585i	$-0.548142\pi$
$757$	−4.75104e6	−0.301334	−0.150667	+	0.988585i	$0.548142\pi$
$758$	−7.51433e6	−0.475026
$759$	8.76271e6	0.552120
$760$	0	0
$761$	−5.92209e6	−0.370692	−0.185346	−	0.982673i	$-0.559341\pi$
$761$	−5.92209e6	−0.370692	−0.185346	+	0.982673i	$0.559341\pi$
$762$	5.17909e6	0.323121
$763$	1.09364e7	0.680087
$764$	1.31139e6	0.0812828
$765$	0	0
$766$	−3.36375e7	−2.07134
$767$	−474262.	−0.0291092
$768$	2.19938e6	0.134554
$769$	−5.07027e6	−0.309183	−0.154591	−	0.987979i	$-0.549406\pi$
$769$	−5.07027e6	−0.309183	−0.154591	+	0.987979i	$0.549406\pi$
$770$	0	0
$771$	7.36006e6	0.445908
$772$	−1.08748e6	−0.0656719
$773$	−2.31839e7	−1.39552	−0.697761	−	0.716330i	$-0.745820\pi$
$773$	−2.31839e7	−1.39552	−0.697761	+	0.716330i	$0.745820\pi$
$774$	−5.94404e6	−0.356639
$775$	0	0
$776$	−2.43573e7	−1.45203
$777$	−371217.	−0.0220585
$778$	−3.25799e7	−1.92975
$779$	−1.51681e7	−0.895543
$780$	0	0
$781$	2.35419e7	1.38106
$782$	1.32163e7	0.772849
$783$	1.15883e7	0.675482
$784$	−1.97512e7	−1.14763
$785$	0	0
$786$	−1.80832e6	−0.104404
$787$	6.07650e6	0.349717	0.174858	−	0.984594i	$-0.444053\pi$
$787$	6.07650e6	0.349717	0.174858	+	0.984594i	$0.444053\pi$
$788$	2.79095e6	0.160117
$789$	9.30742e6	0.532276
$790$	0	0
$791$	−3.33848e7	−1.89718
$792$	−1.16849e7	−0.661932
$793$	2.32624e6	0.131362
$794$	9.11855e6	0.513304
$795$	0	0
$796$	867745.	0.0485411
$797$	−2.78805e7	−1.55473	−0.777363	−	0.629052i	$-0.783443\pi$
$797$	−2.78805e7	−1.55473	−0.777363	+	0.629052i	$0.783443\pi$
$798$	8.69394e6	0.483292
$799$	−4.30389e6	−0.238503
$800$	0	0
$801$	1.01332e7	0.558039
$802$	178164.	0.00978099
$803$	2.68629e7	1.47016
$804$	−1.53047e6	−0.0834999
$805$	0	0
$806$	−2.91450e6	−0.158025
$807$	2.60750e6	0.140942
$808$	−811883.	−0.0437487
$809$	6.10438e6	0.327922	0.163961	−	0.986467i	$-0.447573\pi$
$809$	6.10438e6	0.327922	0.163961	+	0.986467i	$0.447573\pi$
$810$	0	0
$811$	−2.23956e7	−1.19567	−0.597835	−	0.801619i	$-0.703972\pi$
$811$	−2.23956e7	−1.19567	−0.597835	+	0.801619i	$0.703972\pi$
$812$	−2.24007e6	−0.119226
$813$	5.28572e6	0.280465
$814$	594892.	0.0314686
$815$	0	0
$816$	4.99737e6	0.262734
$817$	5.79777e6	0.303882
$818$	−3.28943e7	−1.71885
$819$	6.06591e6	0.316000
$820$	0	0
$821$	−2.42967e7	−1.25803	−0.629014	−	0.777394i	$-0.716541\pi$
$821$	−2.42967e7	−1.25803	−0.629014	+	0.777394i	$0.716541\pi$
$822$	7.89025e6	0.407297
$823$	−3.64578e7	−1.87625	−0.938127	−	0.346293i	$-0.887440\pi$
$823$	−3.64578e7	−1.87625	−0.938127	+	0.346293i	$0.887440\pi$
$824$	−1.01304e7	−0.519767
$825$	0	0
$826$	3.09271e6	0.157721
$827$	2.81247e7	1.42996	0.714981	−	0.699143i	$-0.246435\pi$
$827$	2.81247e7	1.42996	0.714981	+	0.699143i	$0.246435\pi$
$828$	2.17210e6	0.110104
$829$	2.68734e7	1.35812	0.679058	−	0.734085i	$-0.262389\pi$
$829$	2.68734e7	1.35812	0.679058	+	0.734085i	$0.262389\pi$
$830$	0	0
$831$	−1.16224e7	−0.583841
$832$	−4.87818e6	−0.244315
$833$	−1.12299e7	−0.560743
$834$	1.44041e7	0.717087
$835$	0	0
$836$	−1.26751e6	−0.0627244
$837$	−8.94470e6	−0.441319
$838$	−7.67514e6	−0.377552
$839$	3.46774e7	1.70076	0.850378	−	0.526172i	$-0.176373\pi$
$839$	3.46774e7	1.70076	0.850378	+	0.526172i	$0.176373\pi$
$840$	0	0
$841$	−6.33089e6	−0.308656
$842$	−2.18709e7	−1.06313
$843$	8.60037e6	0.416819
$844$	581123.	0.0280810
$845$	0	0
$846$	−7.77505e6	−0.373488
$847$	−6.64914e6	−0.318462
$848$	8.55524e6	0.408548
$849$	7.03678e6	0.335046
$850$	0	0
$851$	994359.	0.0470673
$852$	−1.50280e6	−0.0709253
$853$	−1.54571e6	−0.0727368	−0.0363684	−	0.999338i	$-0.511579\pi$
$853$	−1.54571e6	−0.0727368	−0.0363684	+	0.999338i	$0.511579\pi$
$854$	−1.51696e7	−0.711754
$855$	0	0
$856$	2.69834e7	1.25867
$857$	1.27926e7	0.594987	0.297493	−	0.954724i	$-0.403849\pi$
$857$	1.27926e7	0.594987	0.297493	+	0.954724i	$0.403849\pi$
$858$	2.50336e6	0.116093
$859$	2.66940e6	0.123433	0.0617165	−	0.998094i	$-0.480343\pi$
$859$	2.66940e6	0.123433	0.0617165	+	0.998094i	$0.480343\pi$
$860$	0	0
$861$	−1.77725e7	−0.817033
$862$	7.27673e6	0.333555
$863$	3.37798e7	1.54394	0.771970	−	0.635659i	$-0.219272\pi$
$863$	3.37798e7	1.54394	0.771970	+	0.635659i	$0.219272\pi$
$864$	3.55508e6	0.162019
$865$	0	0
$866$	−6.07906e6	−0.275449
$867$	−7.17474e6	−0.324159
$868$	1.72906e6	0.0778952
$869$	−3.59721e7	−1.61591
$870$	0	0
$871$	1.14488e7	0.511344
$872$	−1.00600e7	−0.448029
$873$	−2.75473e7	−1.22333
$874$	−2.32880e7	−1.03122
$875$	0	0
$876$	−1.71479e6	−0.0755007
$877$	−3.97308e7	−1.74433	−0.872164	−	0.489214i	$-0.837284\pi$
$877$	−3.97308e7	−1.74433	−0.872164	+	0.489214i	$0.837284\pi$
$878$	−2.99596e7	−1.31160
$879$	−1.27537e7	−0.556755
$880$	0	0
$881$	−1.67058e7	−0.725149	−0.362574	−	0.931955i	$-0.618102\pi$
$881$	−1.67058e7	−0.725149	−0.362574	+	0.931955i	$0.618102\pi$
$882$	−2.02870e7	−0.878105
$883$	1.67930e7	0.724816	0.362408	−	0.932020i	$-0.381955\pi$
$883$	1.67930e7	0.724816	0.362408	+	0.932020i	$0.381955\pi$
$884$	343497.	0.0147840
$885$	0	0
$886$	−3.74293e7	−1.60187
$887$	−8.61531e6	−0.367673	−0.183837	−	0.982957i	$-0.558852\pi$
$887$	−8.61531e6	−0.367673	−0.183837	+	0.982957i	$0.558852\pi$
$888$	341468.	0.0145317
$889$	2.29843e7	0.975388
$890$	0	0
$891$	−8.93559e6	−0.377076
$892$	−4.44393e6	−0.187006
$893$	7.58371e6	0.318239
$894$	−7.44208e6	−0.311423
$895$	0	0
$896$	3.86777e7	1.60950
$897$	4.18436e6	0.173639
$898$	6.90571e6	0.285770
$899$	−1.09454e7	−0.451682
$900$	0	0
$901$	4.86425e6	0.199620
$902$	2.84811e7	1.16558
$903$	6.79326e6	0.277242
$904$	3.07093e7	1.24983
$905$	0	0
$906$	2.32202e7	0.939820
$907$	−7.34436e6	−0.296439	−0.148220	−	0.988954i	$-0.547354\pi$
$907$	−7.34436e6	−0.296439	−0.148220	+	0.988954i	$0.547354\pi$
$908$	2.21105e6	0.0889989
$909$	−918211.	−0.0368581
$910$	0	0
$911$	3.63225e7	1.45004	0.725019	−	0.688729i	$-0.241831\pi$
$911$	3.63225e7	1.45004	0.725019	+	0.688729i	$0.241831\pi$
$912$	−8.80567e6	−0.350570
$913$	−1.80080e7	−0.714971
$914$	−8.79036e6	−0.348049
$915$	0	0
$916$	−4.42781e6	−0.174361
$917$	−8.02515e6	−0.315159
$918$	1.15878e7	0.453829
$919$	2.25278e7	0.879892	0.439946	−	0.898024i	$-0.354998\pi$
$919$	2.25278e7	0.879892	0.439946	+	0.898024i	$0.354998\pi$
$920$	0	0
$921$	172791.	0.00671232
$922$	−3.35457e7	−1.29960
$923$	1.12417e7	0.434338
$924$	−1.48515e6	−0.0572255
$925$	0	0
$926$	−1.24131e7	−0.475721
$927$	−1.14571e7	−0.437901
$928$	4.35026e6	0.165823
$929$	3.93312e7	1.49520	0.747598	−	0.664152i	$-0.231207\pi$
$929$	3.93312e7	1.49520	0.747598	+	0.664152i	$0.231207\pi$
$930$	0	0
$931$	1.97878e7	0.748209
$932$	230303.	0.00868480
$933$	1.04553e7	0.393217
$934$	4.44087e7	1.66571
$935$	0	0
$936$	−5.57978e6	−0.208175
$937$	1.36354e7	0.507362	0.253681	−	0.967288i	$-0.418359\pi$
$937$	1.36354e7	0.507362	0.253681	+	0.967288i	$0.418359\pi$
$938$	−7.46584e7	−2.77058
$939$	2.46030e6	0.0910593
$940$	0	0
$941$	2.28438e7	0.840995	0.420498	−	0.907294i	$-0.361856\pi$
$941$	2.28438e7	0.840995	0.420498	+	0.907294i	$0.361856\pi$
$942$	−1.07104e7	−0.393258
$943$	4.76061e7	1.74334
$944$	−3.13245e6	−0.114407
$945$	0	0
$946$	−1.08865e7	−0.395512
$947$	6.46944e6	0.234418	0.117209	−	0.993107i	$-0.462605\pi$
$947$	6.46944e6	0.234418	0.117209	+	0.993107i	$0.462605\pi$
$948$	2.29628e6	0.0829858
$949$	1.28276e7	0.462358
$950$	0	0
$951$	2.86748e6	0.102813
$952$	2.01417e7	0.720284
$953$	−2.58068e6	−0.0920452	−0.0460226	−	0.998940i	$-0.514655\pi$
$953$	−2.58068e6	−0.0920452	−0.0460226	+	0.998940i	$0.514655\pi$
$954$	8.78734e6	0.312598
$955$	0	0
$956$	−2.64355e6	−0.0935497
$957$	9.40139e6	0.331827
$958$	1.51165e7	0.532154
$959$	3.50162e7	1.22948
$960$	0	0
$961$	−2.01807e7	−0.704899
$962$	284072.	0.00989672
$963$	3.05173e7	1.06043
$964$	−1.96986e6	−0.0682722
$965$	0	0
$966$	−2.72866e7	−0.940820
$967$	1.88591e6	0.0648568	0.0324284	−	0.999474i	$-0.489676\pi$
$967$	1.88591e6	0.0648568	0.0324284	+	0.999474i	$0.489676\pi$
$968$	6.11627e6	0.209797
$969$	−5.00663e6	−0.171292
$970$	0	0
$971$	−2.49003e7	−0.847534	−0.423767	−	0.905771i	$-0.639292\pi$
$971$	−2.49003e7	−0.847534	−0.423767	+	0.905771i	$0.639292\pi$
$972$	2.96528e6	0.100670
$973$	6.39242e7	2.16463
$974$	−2.18210e7	−0.737016
$975$	0	0
$976$	1.53645e7	0.516292
$977$	−1.01689e7	−0.340829	−0.170415	−	0.985372i	$-0.554511\pi$
$977$	−1.01689e7	−0.340829	−0.170415	+	0.985372i	$0.554511\pi$
$978$	1.09946e7	0.367562
$979$	1.85589e7	0.618864
$980$	0	0
$981$	−1.13775e7	−0.377462
$982$	−4.29143e7	−1.42011
$983$	−4.29289e6	−0.141699	−0.0708494	−	0.997487i	$-0.522571\pi$
$983$	−4.29289e6	−0.141699	−0.0708494	+	0.997487i	$0.522571\pi$
$984$	1.63482e7	0.538246
$985$	0	0
$986$	1.41796e7	0.464486
$987$	8.88586e6	0.290340
$988$	−605262.	−0.0197265
$989$	−1.81967e7	−0.591565
$990$	0	0
$991$	−2.64765e7	−0.856398	−0.428199	−	0.903684i	$-0.640852\pi$
$991$	−2.64765e7	−0.856398	−0.428199	+	0.903684i	$0.640852\pi$
$992$	−3.35786e6	−0.108339
$993$	1.11891e7	0.360100
$994$	−7.33082e7	−2.35335
$995$	0	0
$996$	1.14954e6	0.0367177
$997$	−4.21089e6	−0.134164	−0.0670820	−	0.997747i	$-0.521369\pi$
$997$	−4.21089e6	−0.134164	−0.0670820	+	0.997747i	$0.521369\pi$
$998$	5.19227e6	0.165018
$999$	871829.	0.0276387

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.g.1.5		6
5.2	odd	4		325.6.b.g.274.9		12
5.3	odd	4		325.6.b.g.274.4		12
5.4	even	2		65.6.a.d.1.2	✓	6
15.14	odd	2		585.6.a.m.1.5		6
20.19	odd	2		1040.6.a.q.1.5		6
65.64	even	2		845.6.a.h.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.d.1.2	✓	6	5.4	even	2
325.6.a.g.1.5		6	1.1	even	1	trivial
325.6.b.g.274.4		12	5.3	odd	4
325.6.b.g.274.9		12	5.2	odd	4
585.6.a.m.1.5		6	15.14	odd	2
845.6.a.h.1.5		6	65.64	even	2
1040.6.a.q.1.5		6	20.19	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q+5.93318 q^{2} +7.05430 q^{3} +3.20258 q^{4} +41.8544 q^{6} +185.746 q^{7} -170.860 q^{8} -193.237 q^{9} -353.912 q^{11} +22.5920 q^{12} -169.000 q^{13} +1102.06 q^{14} -1116.23 q^{16} -634.652 q^{17} -1146.51 q^{18} +1118.29 q^{19} +1310.31 q^{21} -2099.83 q^{22} -3509.85 q^{23} -1205.30 q^{24} -1002.71 q^{26} -3077.35 q^{27} +594.867 q^{28} -3765.67 q^{29} +2906.63 q^{31} -1155.24 q^{32} -2496.60 q^{33} -3765.50 q^{34} -618.857 q^{36} -283.305 q^{37} +6635.04 q^{38} -1192.18 q^{39} -13563.6 q^{41} +7774.29 q^{42} +5184.47 q^{43} -1133.43 q^{44} -20824.6 q^{46} +6781.50 q^{47} -7874.19 q^{48} +17694.6 q^{49} -4477.03 q^{51} -541.237 q^{52} -7664.43 q^{53} -18258.4 q^{54} -31736.6 q^{56} +7888.78 q^{57} -22342.4 q^{58} +2806.29 q^{59} -13764.7 q^{61} +17245.5 q^{62} -35893.0 q^{63} +28865.0 q^{64} -14812.8 q^{66} -67744.1 q^{67} -2032.53 q^{68} -24759.5 q^{69} -66519.0 q^{71} +33016.5 q^{72} -75902.7 q^{73} -1680.90 q^{74} +3581.43 q^{76} -65737.9 q^{77} -7073.39 q^{78} +101641. q^{79} +25248.0 q^{81} -80475.1 q^{82} +50882.7 q^{83} +4196.37 q^{84} +30760.4 q^{86} -26564.2 q^{87} +60469.5 q^{88} -52439.2 q^{89} -31391.1 q^{91} -11240.6 q^{92} +20504.2 q^{93} +40235.8 q^{94} -8149.42 q^{96} +142557. q^{97} +104985. q^{98} +68388.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 38 q^{3} + 134 q^{4} + 318 q^{6} - 220 q^{7} - 24 q^{8} + 518 q^{9} - 170 q^{11} - 2238 q^{12} - 1014 q^{13} - 1440 q^{14} + 3506 q^{16} - 728 q^{17} - 7788 q^{18} + 1218 q^{19} - 396 q^{21} - 5154 q^{22}+ \cdots - 32270 q^{99}+O(q^{100}) \) 6 * q - 38 * q^3 + 134 * q^4 + 318 * q^6 - 220 * q^7 - 24 * q^8 + 518 * q^9 - 170 * q^11 - 2238 * q^12 - 1014 * q^13 - 1440 * q^14 + 3506 * q^16 - 728 * q^17 - 7788 * q^18 + 1218 * q^19 - 396 * q^21 - 5154 * q^22 - 8954 * q^23 + 12618 * q^24 - 13112 * q^27 - 13212 * q^28 + 8364 * q^29 + 2862 * q^31 + 29672 * q^32 + 3772 * q^33 - 21484 * q^34 - 2138 * q^36 - 13840 * q^37 + 27706 * q^38 + 6422 * q^39 + 2248 * q^41 + 47028 * q^42 - 3822 * q^43 - 73482 * q^44 - 39886 * q^46 + 10860 * q^47 - 13962 * q^48 + 85902 * q^49 - 55028 * q^51 - 22646 * q^52 - 6584 * q^53 + 45504 * q^54 - 188800 * q^56 - 11692 * q^57 + 88500 * q^58 - 28874 * q^59 + 972 * q^61 - 123466 * q^62 + 125236 * q^63 + 14022 * q^64 - 158040 * q^66 - 60040 * q^67 - 3504 * q^68 + 153088 * q^69 - 74174 * q^71 - 298668 * q^72 - 48960 * q^73 + 226112 * q^74 - 93422 * q^76 - 231588 * q^77 - 53742 * q^78 + 214508 * q^79 + 116006 * q^81 - 58732 * q^82 - 123260 * q^83 - 235268 * q^84 - 304666 * q^86 - 228144 * q^87 - 127622 * q^88 + 94028 * q^89 + 37180 * q^91 - 430298 * q^92 - 192892 * q^93 - 125920 * q^94 - 385858 * q^96 - 246284 * q^97 + 4736 * q^98 - 32270 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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