LMFDB - Embedded newform 350.6.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [350,6,Mod(1,350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(350, 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("350.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 350 = 2 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	350.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:	$56.1343369345$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 915x^{3} - 2649x^{2} + 122688x - 432576 $ x^5 - x^4 - 915x^3 - 2649x^2 + 122688*x - 432576
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.91234$ of defining polynomial
Character	$\chi$	$=$	350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} -1.91234 q^{3} +16.0000 q^{4} -7.64937 q^{6} +49.0000 q^{7} +64.0000 q^{8} -239.343 q^{9} -780.137 q^{11} -30.5975 q^{12} +869.410 q^{13} +196.000 q^{14} +256.000 q^{16} +326.997 q^{17} -957.372 q^{18} +1768.20 q^{19} -93.7048 q^{21} -3120.55 q^{22} +1898.23 q^{23} -122.390 q^{24} +3477.64 q^{26} +922.405 q^{27} +784.000 q^{28} +4858.54 q^{29} +5436.97 q^{31} +1024.00 q^{32} +1491.89 q^{33} +1307.99 q^{34} -3829.49 q^{36} +4411.45 q^{37} +7072.79 q^{38} -1662.61 q^{39} -9886.32 q^{41} -374.819 q^{42} -15658.3 q^{43} -12482.2 q^{44} +7592.94 q^{46} +16840.3 q^{47} -489.560 q^{48} +2401.00 q^{49} -625.329 q^{51} +13910.6 q^{52} -22056.0 q^{53} +3689.62 q^{54} +3136.00 q^{56} -3381.40 q^{57} +19434.2 q^{58} +40053.7 q^{59} +21919.3 q^{61} +21747.9 q^{62} -11727.8 q^{63} +4096.00 q^{64} +5967.55 q^{66} +50458.1 q^{67} +5231.94 q^{68} -3630.07 q^{69} -28254.7 q^{71} -15317.9 q^{72} +85457.9 q^{73} +17645.8 q^{74} +28291.2 q^{76} -38226.7 q^{77} -6650.44 q^{78} +67249.4 q^{79} +56396.4 q^{81} -39545.3 q^{82} +18647.1 q^{83} -1499.28 q^{84} -62633.2 q^{86} -9291.19 q^{87} -49928.8 q^{88} -59853.7 q^{89} +42601.1 q^{91} +30371.7 q^{92} -10397.3 q^{93} +67361.3 q^{94} -1958.24 q^{96} -46061.0 q^{97} +9604.00 q^{98} +186720. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 20 q^{2} + 14 q^{3} + 80 q^{4} + 56 q^{6} + 245 q^{7} + 320 q^{8} + 655 q^{9} + 748 q^{11} + 224 q^{12} + 456 q^{13} + 980 q^{14} + 1280 q^{16} + 780 q^{17} + 2620 q^{18} - 256 q^{19} + 686 q^{21}+ \cdots + 575944 q^{99}+O(q^{100}) $ 5 * q + 20 * q^2 + 14 * q^3 + 80 * q^4 + 56 * q^6 + 245 * q^7 + 320 * q^8 + 655 * q^9 + 748 * q^11 + 224 * q^12 + 456 * q^13 + 980 * q^14 + 1280 * q^16 + 780 * q^17 + 2620 * q^18 - 256 * q^19 + 686 * q^21 + 2992 * q^22 + 3452 * q^23 + 896 * q^24 + 1824 * q^26 - 910 * q^27 + 3920 * q^28 + 7712 * q^29 + 412 * q^31 + 5120 * q^32 - 2980 * q^33 + 3120 * q^34 + 10480 * q^36 + 14400 * q^37 - 1024 * q^38 + 17668 * q^39 + 7482 * q^41 + 2744 * q^42 + 20668 * q^43 + 11968 * q^44 + 13808 * q^46 + 59202 * q^47 + 3584 * q^48 + 12005 * q^49 + 6628 * q^51 + 7296 * q^52 + 6388 * q^53 - 3640 * q^54 + 15680 * q^56 + 112616 * q^57 + 30848 * q^58 + 79596 * q^59 + 9754 * q^61 + 1648 * q^62 + 32095 * q^63 + 20480 * q^64 - 11920 * q^66 + 49820 * q^67 + 12480 * q^68 + 106152 * q^69 - 61928 * q^71 + 41920 * q^72 - 57248 * q^73 + 57600 * q^74 - 4096 * q^76 + 36652 * q^77 + 70672 * q^78 + 150912 * q^79 + 100825 * q^81 + 29928 * q^82 - 54776 * q^83 + 10976 * q^84 + 82672 * q^86 - 256362 * q^87 + 47872 * q^88 + 268950 * q^89 + 22344 * q^91 + 55232 * q^92 - 187044 * q^93 + 236808 * q^94 + 14336 * q^96 - 296156 * q^97 + 48020 * q^98 + 575944 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−1.91234	−0.122677	−0.0613384	−	0.998117i	$-0.519537\pi$
$3$	−1.91234	−0.122677	−0.0613384	+	0.998117i	$0.519537\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−7.64937	−0.0867456
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	64.0000	0.353553
$9$	−239.343	−0.984950
$10$	0	0
$11$	−780.137	−1.94397	−0.971984	−	0.235048i	$-0.924475\pi$
$11$	−780.137	−1.94397	−0.971984	+	0.235048i	$0.924475\pi$
$12$	−30.5975	−0.0613384
$13$	869.410	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$13$	869.410	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$14$	196.000	0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	326.997	0.274423	0.137212	−	0.990542i	$-0.456186\pi$
$17$	326.997	0.274423	0.137212	+	0.990542i	$0.456186\pi$
$18$	−957.372	−0.696465
$19$	1768.20	1.12369	0.561845	−	0.827242i	$-0.310092\pi$
$19$	1768.20	1.12369	0.561845	+	0.827242i	$0.310092\pi$
$20$	0	0
$21$	−93.7048	−0.0463675
$22$	−3120.55	−1.37459
$23$	1898.23	0.748221	0.374111	−	0.927384i	$-0.377948\pi$
$23$	1898.23	0.748221	0.374111	+	0.927384i	$0.377948\pi$
$24$	−122.390	−0.0433728
$25$	0	0
$26$	3477.64	1.00891
$27$	922.405	0.243507
$28$	784.000	0.188982
$29$	4858.54	1.07278	0.536390	−	0.843970i	$-0.319788\pi$
$29$	4858.54	1.07278	0.536390	+	0.843970i	$0.319788\pi$
$30$	0	0
$31$	5436.97	1.01614	0.508069	−	0.861316i	$-0.330359\pi$
$31$	5436.97	1.01614	0.508069	+	0.861316i	$0.330359\pi$
$32$	1024.00	0.176777
$33$	1491.89	0.238480
$34$	1307.99	0.194047
$35$	0	0
$36$	−3829.49	−0.492475
$37$	4411.45	0.529757	0.264879	−	0.964282i	$-0.414668\pi$
$37$	4411.45	0.529757	0.264879	+	0.964282i	$0.414668\pi$
$38$	7072.79	0.794569
$39$	−1662.61	−0.175036
$40$	0	0
$41$	−9886.32	−0.918492	−0.459246	−	0.888309i	$-0.651880\pi$
$41$	−9886.32	−0.918492	−0.459246	+	0.888309i	$0.651880\pi$
$42$	−374.819	−0.0327868
$43$	−15658.3	−1.29144	−0.645719	−	0.763575i	$-0.723442\pi$
$43$	−15658.3	−1.29144	−0.645719	+	0.763575i	$0.723442\pi$
$44$	−12482.2	−0.971984
$45$	0	0
$46$	7592.94	0.529072
$47$	16840.3	1.11200	0.556002	−	0.831181i	$-0.312335\pi$
$47$	16840.3	1.11200	0.556002	+	0.831181i	$0.312335\pi$
$48$	−489.560	−0.0306692
$49$	2401.00	0.142857
$50$	0	0
$51$	−625.329	−0.0336654
$52$	13910.6	0.713405
$53$	−22056.0	−1.07854	−0.539271	−	0.842132i	$-0.681300\pi$
$53$	−22056.0	−1.07854	−0.539271	+	0.842132i	$0.681300\pi$
$54$	3689.62	0.172186
$55$	0	0
$56$	3136.00	0.133631
$57$	−3381.40	−0.137851
$58$	19434.2	0.758570
$59$	40053.7	1.49800	0.749002	−	0.662568i	$-0.230534\pi$
$59$	40053.7	1.49800	0.749002	+	0.662568i	$0.230534\pi$
$60$	0	0
$61$	21919.3	0.754229	0.377114	−	0.926167i	$-0.376916\pi$
$61$	21919.3	0.754229	0.377114	+	0.926167i	$0.376916\pi$
$62$	21747.9	0.718518
$63$	−11727.8	−0.372276
$64$	4096.00	0.125000
$65$	0	0
$66$	5967.55	0.168631
$67$	50458.1	1.37323	0.686616	−	0.727020i	$-0.259095\pi$
$67$	50458.1	1.37323	0.686616	+	0.727020i	$0.259095\pi$
$68$	5231.94	0.137212
$69$	−3630.07	−0.0917894
$70$	0	0
$71$	−28254.7	−0.665189	−0.332594	−	0.943070i	$-0.607924\pi$
$71$	−28254.7	−0.665189	−0.332594	+	0.943070i	$0.607924\pi$
$72$	−15317.9	−0.348233
$73$	85457.9	1.87692	0.938458	−	0.345393i	$-0.112254\pi$
$73$	85457.9	1.87692	0.938458	+	0.345393i	$0.112254\pi$
$74$	17645.8	0.374595
$75$	0	0
$76$	28291.2	0.561845
$77$	−38226.7	−0.734751
$78$	−6650.44	−0.123769
$79$	67249.4	1.21233	0.606164	−	0.795339i	$-0.292707\pi$
$79$	67249.4	1.21233	0.606164	+	0.795339i	$0.292707\pi$
$80$	0	0
$81$	56396.4	0.955078
$82$	−39545.3	−0.649472
$83$	18647.1	0.297110	0.148555	−	0.988904i	$-0.452538\pi$
$83$	18647.1	0.297110	0.148555	+	0.988904i	$0.452538\pi$
$84$	−1499.28	−0.0231837
$85$	0	0
$86$	−62633.2	−0.913184
$87$	−9291.19	−0.131605
$88$	−49928.8	−0.687296
$89$	−59853.7	−0.800969	−0.400485	−	0.916303i	$-0.631158\pi$
$89$	−59853.7	−0.800969	−0.400485	+	0.916303i	$0.631158\pi$
$90$	0	0
$91$	42601.1	0.539284
$92$	30371.7	0.374111
$93$	−10397.3	−0.124657
$94$	67361.3	0.786305
$95$	0	0
$96$	−1958.24	−0.0216864
$97$	−46061.0	−0.497054	−0.248527	−	0.968625i	$-0.579947\pi$
$97$	−46061.0	−0.497054	−0.248527	+	0.968625i	$0.579947\pi$
$98$	9604.00	0.101015
$99$	186720.	1.91471
$100$	0	0
$101$	−71113.3	−0.693661	−0.346831	−	0.937928i	$-0.612742\pi$
$101$	−71113.3	−0.693661	−0.346831	+	0.937928i	$0.612742\pi$
$102$	−2501.32	−0.0238050
$103$	5051.27	0.0469146	0.0234573	−	0.999725i	$-0.492533\pi$
$103$	5051.27	0.0469146	0.0234573	+	0.999725i	$0.492533\pi$
$104$	55642.2	0.504454
$105$	0	0
$106$	−88224.0	−0.762645
$107$	−71370.8	−0.602644	−0.301322	−	0.953522i	$-0.597428\pi$
$107$	−71370.8	−0.602644	−0.301322	+	0.953522i	$0.597428\pi$
$108$	14758.5	0.121754
$109$	138106.	1.11339	0.556695	−	0.830717i	$-0.312069\pi$
$109$	138106.	1.11339	0.556695	+	0.830717i	$0.312069\pi$
$110$	0	0
$111$	−8436.20	−0.0649889
$112$	12544.0	0.0944911
$113$	−158667.	−1.16894	−0.584468	−	0.811417i	$-0.698697\pi$
$113$	−158667.	−1.16894	−0.584468	+	0.811417i	$0.698697\pi$
$114$	−13525.6	−0.0974752
$115$	0	0
$116$	77736.6	0.536390
$117$	−208087.	−1.40534
$118$	160215.	1.05925
$119$	16022.8	0.103722
$120$	0	0
$121$	447562.	2.77901
$122$	87677.4	0.533320
$123$	18906.0	0.112678
$124$	86991.5	0.508069
$125$	0	0
$126$	−46911.2	−0.263239
$127$	67468.8	0.371188	0.185594	−	0.982627i	$-0.440579\pi$
$127$	67468.8	0.371188	0.185594	+	0.982627i	$0.440579\pi$
$128$	16384.0	0.0883883
$129$	29944.0	0.158429
$130$	0	0
$131$	158205.	0.805457	0.402729	−	0.915319i	$-0.368062\pi$
$131$	158205.	0.805457	0.402729	+	0.915319i	$0.368062\pi$
$132$	23870.2	0.119240
$133$	86641.7	0.424715
$134$	201832.	0.971021
$135$	0	0
$136$	20927.8	0.0970233
$137$	200407.	0.912243	0.456122	−	0.889917i	$-0.349238\pi$
$137$	200407.	0.912243	0.456122	+	0.889917i	$0.349238\pi$
$138$	−14520.3	−0.0649049
$139$	−191630.	−0.841252	−0.420626	−	0.907234i	$-0.638190\pi$
$139$	−191630.	−0.841252	−0.420626	+	0.907234i	$0.638190\pi$
$140$	0	0
$141$	−32204.5	−0.136417
$142$	−113019.	−0.470360
$143$	−678259.	−2.77367
$144$	−61271.8	−0.246238
$145$	0	0
$146$	341831.	1.32718
$147$	−4591.53	−0.0175253
$148$	70583.2	0.264879
$149$	33501.4	0.123622	0.0618112	−	0.998088i	$-0.480312\pi$
$149$	33501.4	0.123622	0.0618112	+	0.998088i	$0.480312\pi$
$150$	0	0
$151$	−485524.	−1.73288	−0.866439	−	0.499282i	$-0.833597\pi$
$151$	−485524.	−1.73288	−0.866439	+	0.499282i	$0.833597\pi$
$152$	113165.	0.397285
$153$	−78264.3	−0.270293
$154$	−152907.	−0.519547
$155$	0	0
$156$	−26601.7	−0.0875182
$157$	−493697.	−1.59850	−0.799248	−	0.601001i	$-0.794769\pi$
$157$	−493697.	−1.59850	−0.799248	+	0.601001i	$0.794769\pi$
$158$	268997.	0.857246
$159$	42178.6	0.132312
$160$	0	0
$161$	93013.5	0.282801
$162$	225586.	0.675342
$163$	−192313.	−0.566943	−0.283472	−	0.958981i	$-0.591486\pi$
$163$	−192313.	−0.566943	−0.283472	+	0.958981i	$0.591486\pi$
$164$	−158181.	−0.459246
$165$	0	0
$166$	74588.5	0.210088
$167$	36326.5	0.100794	0.0503968	−	0.998729i	$-0.483951\pi$
$167$	36326.5	0.100794	0.0503968	+	0.998729i	$0.483951\pi$
$168$	−5997.10	−0.0163934
$169$	384581.	1.03579
$170$	0	0
$171$	−423206.	−1.10678
$172$	−250533.	−0.645719
$173$	27050.2	0.0687156	0.0343578	−	0.999410i	$-0.489061\pi$
$173$	27050.2	0.0687156	0.0343578	+	0.999410i	$0.489061\pi$
$174$	−37164.7	−0.0930589
$175$	0	0
$176$	−199715.	−0.485992
$177$	−76596.4	−0.183770
$178$	−239415.	−0.566371
$179$	123119.	0.287206	0.143603	−	0.989635i	$-0.454131\pi$
$179$	123119.	0.287206	0.143603	+	0.989635i	$0.454131\pi$
$180$	0	0
$181$	−135382.	−0.307159	−0.153580	−	0.988136i	$-0.549080\pi$
$181$	−135382.	−0.307159	−0.153580	+	0.988136i	$0.549080\pi$
$182$	170404.	0.381331
$183$	−41917.3	−0.0925263
$184$	121487.	0.264536
$185$	0	0
$186$	−41589.4	−0.0881455
$187$	−255102.	−0.533470
$188$	269445.	0.556002
$189$	45197.8	0.0920371
$190$	0	0
$191$	189708.	0.376272	0.188136	−	0.982143i	$-0.439755\pi$
$191$	189708.	0.376272	0.188136	+	0.982143i	$0.439755\pi$
$192$	−7832.95	−0.0153346
$193$	362697.	0.700891	0.350445	−	0.936583i	$-0.386030\pi$
$193$	362697.	0.700891	0.350445	+	0.936583i	$0.386030\pi$
$194$	−184244.	−0.351470
$195$	0	0
$196$	38416.0	0.0714286
$197$	828304.	1.52063	0.760316	−	0.649554i	$-0.225044\pi$
$197$	828304.	1.52063	0.760316	+	0.649554i	$0.225044\pi$
$198$	746881.	1.35391
$199$	864115.	1.54682	0.773409	−	0.633908i	$-0.218550\pi$
$199$	864115.	1.54682	0.773409	+	0.633908i	$0.218550\pi$
$200$	0	0
$201$	−96493.1	−0.168464
$202$	−284453.	−0.490493
$203$	238068.	0.405473
$204$	−10005.3	−0.0168327
$205$	0	0
$206$	20205.1	0.0331736
$207$	−454329.	−0.736961
$208$	222569.	0.356703
$209$	−1.37944e6	−2.18442
$210$	0	0
$211$	−446374.	−0.690229	−0.345114	−	0.938561i	$-0.612160\pi$
$211$	−446374.	−0.690229	−0.345114	+	0.938561i	$0.612160\pi$
$212$	−352896.	−0.539271
$213$	54032.7	0.0816032
$214$	−285483.	−0.426134
$215$	0	0
$216$	59033.9	0.0860928
$217$	266412.	0.384064
$218$	552426.	0.787286
$219$	−163425.	−0.230254
$220$	0	0
$221$	284294.	0.391550
$222$	−33744.8	−0.0459541
$223$	−615894.	−0.829362	−0.414681	−	0.909967i	$-0.636107\pi$
$223$	−615894.	−0.829362	−0.414681	+	0.909967i	$0.636107\pi$
$224$	50176.0	0.0668153
$225$	0	0
$226$	−634669.	−0.826563
$227$	143815.	0.185242	0.0926210	−	0.995701i	$-0.470475\pi$
$227$	143815.	0.185242	0.0926210	+	0.995701i	$0.470475\pi$
$228$	−54102.4	−0.0689254
$229$	−547130.	−0.689449	−0.344724	−	0.938704i	$-0.612028\pi$
$229$	−547130.	−0.689449	−0.344724	+	0.938704i	$0.612028\pi$
$230$	0	0
$231$	73102.5	0.0901369
$232$	310946.	0.379285
$233$	−624672.	−0.753811	−0.376905	−	0.926252i	$-0.623012\pi$
$233$	−624672.	−0.753811	−0.376905	+	0.926252i	$0.623012\pi$
$234$	−832349.	−0.993724
$235$	0	0
$236$	640859.	0.749002
$237$	−128604.	−0.148725
$238$	64091.3	0.0733427
$239$	−711873.	−0.806136	−0.403068	−	0.915170i	$-0.632056\pi$
$239$	−711873.	−0.806136	−0.403068	+	0.915170i	$0.632056\pi$
$240$	0	0
$241$	546788.	0.606424	0.303212	−	0.952923i	$-0.401941\pi$
$241$	546788.	0.606424	0.303212	+	0.952923i	$0.401941\pi$
$242$	1.79025e6	1.96506
$243$	−331993.	−0.360673
$244$	350709.	0.377114
$245$	0	0
$246$	75624.1	0.0796751
$247$	1.53729e6	1.60329
$248$	347966.	0.359259
$249$	−35659.7	−0.0364485
$250$	0	0
$251$	−232147.	−0.232583	−0.116292	−	0.993215i	$-0.537101\pi$
$251$	−232147.	−0.232583	−0.116292	+	0.993215i	$0.537101\pi$
$252$	−187645.	−0.186138
$253$	−1.48088e6	−1.45452
$254$	269875.	0.262469
$255$	0	0
$256$	65536.0	0.0625000
$257$	−574030.	−0.542128	−0.271064	−	0.962561i	$-0.587375\pi$
$257$	−574030.	−0.542128	−0.271064	+	0.962561i	$0.587375\pi$
$258$	119776.	0.112026
$259$	216161.	0.200229
$260$	0	0
$261$	−1.16286e6	−1.05663
$262$	632821.	0.569544
$263$	−312043.	−0.278180	−0.139090	−	0.990280i	$-0.544418\pi$
$263$	−312043.	−0.278180	−0.139090	+	0.990280i	$0.544418\pi$
$264$	95480.9	0.0843153
$265$	0	0
$266$	346567.	0.300319
$267$	114461.	0.0982603
$268$	807329.	0.686616
$269$	−2.18449e6	−1.84064	−0.920322	−	0.391161i	$-0.872074\pi$
$269$	−2.18449e6	−1.84064	−0.920322	+	0.391161i	$0.872074\pi$
$270$	0	0
$271$	951923.	0.787370	0.393685	−	0.919246i	$-0.371200\pi$
$271$	951923.	0.787370	0.393685	+	0.919246i	$0.371200\pi$
$272$	83711.1	0.0686058
$273$	−81467.8	−0.0661576
$274$	801626.	0.645053
$275$	0	0
$276$	−58081.2	−0.0458947
$277$	−1.07542e6	−0.842126	−0.421063	−	0.907031i	$-0.638343\pi$
$277$	−1.07542e6	−0.842126	−0.421063	+	0.907031i	$0.638343\pi$
$278$	−766519.	−0.594855
$279$	−1.30130e6	−1.00085
$280$	0	0
$281$	689287.	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$281$	689287.	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$282$	−128818.	−0.0964614
$283$	2.04961e6	1.52127	0.760635	−	0.649180i	$-0.224888\pi$
$283$	2.04961e6	1.52127	0.760635	+	0.649180i	$0.224888\pi$
$284$	−452075.	−0.332594
$285$	0	0
$286$	−2.71303e6	−1.96128
$287$	−484430.	−0.347157
$288$	−245087.	−0.174116
$289$	−1.31293e6	−0.924692
$290$	0	0
$291$	88084.4	0.0609770
$292$	1.36733e6	0.938458
$293$	−303846.	−0.206769	−0.103384	−	0.994641i	$-0.532967\pi$
$293$	−303846.	−0.206769	−0.103384	+	0.994641i	$0.532967\pi$
$294$	−18366.1	−0.0123922
$295$	0	0
$296$	282333.	0.187297
$297$	−719602.	−0.473370
$298$	134006.	0.0874142
$299$	1.65034e6	1.06757
$300$	0	0
$301$	−767256.	−0.488117
$302$	−1.94210e6	−1.22533
$303$	135993.	0.0850962
$304$	452659.	0.280923
$305$	0	0
$306$	−313057.	−0.191126
$307$	482206.	0.292002	0.146001	−	0.989284i	$-0.453360\pi$
$307$	482206.	0.292002	0.146001	+	0.989284i	$0.453360\pi$
$308$	−611627.	−0.367375
$309$	−9659.76	−0.00575533
$310$	0	0
$311$	−1.44282e6	−0.845886	−0.422943	−	0.906156i	$-0.639003\pi$
$311$	−1.44282e6	−0.845886	−0.422943	+	0.906156i	$0.639003\pi$
$312$	−106407.	−0.0618847
$313$	−826161.	−0.476655	−0.238327	−	0.971185i	$-0.576599\pi$
$313$	−826161.	−0.476655	−0.238327	+	0.971185i	$0.576599\pi$
$314$	−1.97479e6	−1.13031
$315$	0	0
$316$	1.07599e6	0.606164
$317$	−1.34244e6	−0.750318	−0.375159	−	0.926960i	$-0.622412\pi$
$317$	−1.34244e6	−0.750318	−0.375159	+	0.926960i	$0.622412\pi$
$318$	168715.	0.0935588
$319$	−3.79032e6	−2.08545
$320$	0	0
$321$	136485.	0.0739305
$322$	372054.	0.199971
$323$	578194.	0.308367
$324$	902342.	0.477539
$325$	0	0
$326$	−769252.	−0.400890
$327$	−264107.	−0.136587
$328$	−632725.	−0.324736
$329$	825176.	0.420298
$330$	0	0
$331$	2.62883e6	1.31884	0.659420	−	0.751775i	$-0.270802\pi$
$331$	2.62883e6	1.31884	0.659420	+	0.751775i	$0.270802\pi$
$332$	298354.	0.148555
$333$	−1.05585e6	−0.521784
$334$	145306.	0.0712718
$335$	0	0
$336$	−23988.4	−0.0115919
$337$	−3.77083e6	−1.80868	−0.904342	−	0.426809i	$-0.859638\pi$
$337$	−3.77083e6	−1.80868	−0.904342	+	0.426809i	$0.859638\pi$
$338$	1.53832e6	0.732412
$339$	303426.	0.143401
$340$	0	0
$341$	−4.24158e6	−1.97534
$342$	−1.69282e6	−0.782611
$343$	117649.	0.0539949
$344$	−1.00213e6	−0.456592
$345$	0	0
$346$	108201.	0.0485892
$347$	1.16777e6	0.520636	0.260318	−	0.965523i	$-0.416173\pi$
$347$	1.16777e6	0.520636	0.260318	+	0.965523i	$0.416173\pi$
$348$	−148659.	−0.0658026
$349$	2.40057e6	1.05499	0.527497	−	0.849557i	$-0.323131\pi$
$349$	2.40057e6	1.05499	0.527497	+	0.849557i	$0.323131\pi$
$350$	0	0
$351$	801948.	0.347439
$352$	−798860.	−0.343648
$353$	3.78830e6	1.61811	0.809054	−	0.587734i	$-0.199980\pi$
$353$	3.78830e6	1.61811	0.809054	+	0.587734i	$0.199980\pi$
$354$	−306386.	−0.129945
$355$	0	0
$356$	−957659.	−0.400485
$357$	−30641.1	−0.0127243
$358$	492477.	0.203085
$359$	3.69702e6	1.51396	0.756981	−	0.653437i	$-0.226673\pi$
$359$	3.69702e6	1.51396	0.756981	+	0.653437i	$0.226673\pi$
$360$	0	0
$361$	650423.	0.262680
$362$	−541527.	−0.217194
$363$	−855892.	−0.340920
$364$	681617.	0.269642
$365$	0	0
$366$	−167669.	−0.0654260
$367$	−1.13137e6	−0.438468	−0.219234	−	0.975672i	$-0.570356\pi$
$367$	−1.13137e6	−0.438468	−0.219234	+	0.975672i	$0.570356\pi$
$368$	485948.	0.187055
$369$	2.36622e6	0.904669
$370$	0	0
$371$	−1.08074e6	−0.407651
$372$	−166358.	−0.0623283
$373$	−3.59413e6	−1.33759	−0.668793	−	0.743449i	$-0.733189\pi$
$373$	−3.59413e6	−1.33759	−0.668793	+	0.743449i	$0.733189\pi$
$374$	−1.02041e6	−0.377220
$375$	0	0
$376$	1.07778e6	0.393153
$377$	4.22406e6	1.53065
$378$	180791.	0.0650801
$379$	−1.36485e6	−0.488076	−0.244038	−	0.969766i	$-0.578472\pi$
$379$	−1.36485e6	−0.488076	−0.244038	+	0.969766i	$0.578472\pi$
$380$	0	0
$381$	−129023.	−0.0455361
$382$	758832.	0.266065
$383$	−4.95361e6	−1.72554	−0.862770	−	0.505596i	$-0.831273\pi$
$383$	−4.95361e6	−1.72554	−0.862770	+	0.505596i	$0.831273\pi$
$384$	−31331.8	−0.0108432
$385$	0	0
$386$	1.45079e6	0.495605
$387$	3.74770e6	1.27200
$388$	−736976.	−0.248527
$389$	2.34235e6	0.784834	0.392417	−	0.919787i	$-0.371639\pi$
$389$	2.34235e6	0.784834	0.392417	+	0.919787i	$0.371639\pi$
$390$	0	0
$391$	620716.	0.205329
$392$	153664.	0.0505076
$393$	−302542.	−0.0988109
$394$	3.31321e6	1.07525
$395$	0	0
$396$	2.98752e6	0.957356
$397$	−1.54459e6	−0.491853	−0.245927	−	0.969288i	$-0.579092\pi$
$397$	−1.54459e6	−0.491853	−0.245927	+	0.969288i	$0.579092\pi$
$398$	3.45646e6	1.09376
$399$	−165688.	−0.0521027
$400$	0	0
$401$	429318.	0.133327	0.0666636	−	0.997776i	$-0.478765\pi$
$401$	429318.	0.133327	0.0666636	+	0.997776i	$0.478765\pi$
$402$	−385972.	−0.119122
$403$	4.72696e6	1.44984
$404$	−1.13781e6	−0.346831
$405$	0	0
$406$	952273.	0.286712
$407$	−3.44153e6	−1.02983
$408$	−40021.1	−0.0119025
$409$	−282665.	−0.0835533	−0.0417767	−	0.999127i	$-0.513302\pi$
$409$	−282665.	−0.0835533	−0.0417767	+	0.999127i	$0.513302\pi$
$410$	0	0
$411$	−383246.	−0.111911
$412$	80820.3	0.0234573
$413$	1.96263e6	0.566192
$414$	−1.81732e6	−0.521110
$415$	0	0
$416$	890276.	0.252227
$417$	366462.	0.103202
$418$	−5.51774e6	−1.54462
$419$	−3.15372e6	−0.877582	−0.438791	−	0.898589i	$-0.644593\pi$
$419$	−3.15372e6	−0.877582	−0.438791	+	0.898589i	$0.644593\pi$
$420$	0	0
$421$	3.80804e6	1.04712	0.523560	−	0.851989i	$-0.324604\pi$
$421$	3.80804e6	1.04712	0.523560	+	0.851989i	$0.324604\pi$
$422$	−1.78550e6	−0.488065
$423$	−4.03062e6	−1.09527
$424$	−1.41158e6	−0.381322
$425$	0	0
$426$	216131.	0.0577022
$427$	1.07405e6	0.285072
$428$	−1.14193e6	−0.301322
$429$	1.29706e6	0.340265
$430$	0	0
$431$	−1.88084e6	−0.487708	−0.243854	−	0.969812i	$-0.578412\pi$
$431$	−1.88084e6	−0.487708	−0.243854	+	0.969812i	$0.578412\pi$
$432$	236136.	0.0608768
$433$	−5.01618e6	−1.28574	−0.642871	−	0.765975i	$-0.722257\pi$
$433$	−5.01618e6	−1.28574	−0.642871	+	0.765975i	$0.722257\pi$
$434$	1.06565e6	0.271574
$435$	0	0
$436$	2.20970e6	0.556695
$437$	3.35645e6	0.840769
$438$	−653699.	−0.162814
$439$	2.64450e6	0.654911	0.327456	−	0.944867i	$-0.393809\pi$
$439$	2.64450e6	0.654911	0.327456	+	0.944867i	$0.393809\pi$
$440$	0	0
$441$	−574662.	−0.140707
$442$	1.13718e6	0.276868
$443$	7.38758e6	1.78852	0.894258	−	0.447552i	$-0.147704\pi$
$443$	7.38758e6	1.78852	0.894258	+	0.447552i	$0.147704\pi$
$444$	−134979.	−0.0324945
$445$	0	0
$446$	−2.46358e6	−0.586447
$447$	−64066.1	−0.0151656
$448$	200704.	0.0472456
$449$	4.20191e6	0.983628	0.491814	−	0.870700i	$-0.336334\pi$
$449$	4.20191e6	0.983628	0.491814	+	0.870700i	$0.336334\pi$
$450$	0	0
$451$	7.71269e6	1.78552
$452$	−2.53867e6	−0.584468
$453$	928488.	0.212584
$454$	575260.	0.130986
$455$	0	0
$456$	−216409.	−0.0487376
$457$	3.84060e6	0.860217	0.430109	−	0.902777i	$-0.358475\pi$
$457$	3.84060e6	0.860217	0.430109	+	0.902777i	$0.358475\pi$
$458$	−2.18852e6	−0.487514
$459$	301623.	0.0668241
$460$	0	0
$461$	−2.56922e6	−0.563052	−0.281526	−	0.959554i	$-0.590841\pi$
$461$	−2.56922e6	−0.563052	−0.281526	+	0.959554i	$0.590841\pi$
$462$	292410.	0.0637364
$463$	3.49589e6	0.757889	0.378944	−	0.925419i	$-0.376287\pi$
$463$	3.49589e6	0.757889	0.378944	+	0.925419i	$0.376287\pi$
$464$	1.24379e6	0.268195
$465$	0	0
$466$	−2.49869e6	−0.533025
$467$	−15357.9	−0.00325867	−0.00162933	−	0.999999i	$-0.500519\pi$
$467$	−15357.9	−0.00325867	−0.00162933	+	0.999999i	$0.500519\pi$
$468$	−3.32939e6	−0.702669
$469$	2.47245e6	0.519033
$470$	0	0
$471$	944118.	0.196098
$472$	2.56344e6	0.529624
$473$	1.22156e7	2.51051
$474$	−514415.	−0.105164
$475$	0	0
$476$	256365.	0.0518611
$477$	5.27895e6	1.06231
$478$	−2.84749e6	−0.570024
$479$	2.48585e6	0.495036	0.247518	−	0.968883i	$-0.420385\pi$
$479$	2.48585e6	0.495036	0.247518	+	0.968883i	$0.420385\pi$
$480$	0	0
$481$	3.83536e6	0.755863
$482$	2.18715e6	0.428807
$483$	−177874.	−0.0346931
$484$	7.16100e6	1.38951
$485$	0	0
$486$	−1.32797e6	−0.255034
$487$	1.01652e6	0.194219	0.0971096	−	0.995274i	$-0.469040\pi$
$487$	1.01652e6	0.194219	0.0971096	+	0.995274i	$0.469040\pi$
$488$	1.40284e6	0.266660
$489$	367768.	0.0695508
$490$	0	0
$491$	−802250.	−0.150178	−0.0750890	−	0.997177i	$-0.523924\pi$
$491$	−802250.	−0.150178	−0.0750890	+	0.997177i	$0.523924\pi$
$492$	302497.	0.0563388
$493$	1.58872e6	0.294396
$494$	6.14915e6	1.13370
$495$	0	0
$496$	1.39186e6	0.254035
$497$	−1.38448e6	−0.251418
$498$	−142639.	−0.0257730
$499$	3.00612e6	0.540450	0.270225	−	0.962797i	$-0.412902\pi$
$499$	3.00612e6	0.540450	0.270225	+	0.962797i	$0.412902\pi$
$500$	0	0
$501$	−69468.8	−0.0123650
$502$	−928587.	−0.164461
$503$	6.02092e6	1.06107	0.530534	−	0.847664i	$-0.321991\pi$
$503$	6.02092e6	1.06107	0.530534	+	0.847664i	$0.321991\pi$
$504$	−750579.	−0.131620
$505$	0	0
$506$	−5.92353e6	−1.02850
$507$	−735450.	−0.127067
$508$	1.07950e6	0.185594
$509$	8.24891e6	1.41124	0.705622	−	0.708588i	$-0.250668\pi$
$509$	8.24891e6	1.41124	0.705622	+	0.708588i	$0.250668\pi$
$510$	0	0
$511$	4.18744e6	0.709408
$512$	262144.	0.0441942
$513$	1.63099e6	0.273627
$514$	−2.29612e6	−0.383342
$515$	0	0
$516$	479104.	0.0792147
$517$	−1.31378e7	−2.16170
$518$	864644.	0.141584
$519$	−51729.2	−0.00842981
$520$	0	0
$521$	−3.69104e6	−0.595736	−0.297868	−	0.954607i	$-0.596276\pi$
$521$	−3.69104e6	−0.595736	−0.297868	+	0.954607i	$0.596276\pi$
$522$	−4.65143e6	−0.747154
$523$	−5.00116e6	−0.799496	−0.399748	−	0.916625i	$-0.630902\pi$
$523$	−5.00116e6	−0.799496	−0.399748	+	0.916625i	$0.630902\pi$
$524$	2.53128e6	0.402729
$525$	0	0
$526$	−1.24817e6	−0.196703
$527$	1.77787e6	0.278852
$528$	381923.	0.0596199
$529$	−2.83305e6	−0.440165
$530$	0	0
$531$	−9.58657e6	−1.47546
$532$	1.38627e6	0.212358
$533$	−8.59527e6	−1.31051
$534$	457843.	0.0694806
$535$	0	0
$536$	3.22932e6	0.485511
$537$	−235446.	−0.0352335
$538$	−8.73797e6	−1.30153
$539$	−1.87311e6	−0.277710
$540$	0	0
$541$	−6.07709e6	−0.892694	−0.446347	−	0.894860i	$-0.647275\pi$
$541$	−6.07709e6	−0.892694	−0.446347	+	0.894860i	$0.647275\pi$
$542$	3.80769e6	0.556754
$543$	258896.	0.0376813
$544$	334844.	0.0485116
$545$	0	0
$546$	−325871.	−0.0467805
$547$	2.13370e6	0.304905	0.152453	−	0.988311i	$-0.451283\pi$
$547$	2.13370e6	0.304905	0.152453	+	0.988311i	$0.451283\pi$
$548$	3.20651e6	0.456122
$549$	−5.24624e6	−0.742878
$550$	0	0
$551$	8.59085e6	1.20547
$552$	−232325.	−0.0324525
$553$	3.29522e6	0.458217
$554$	−4.30167e6	−0.595473
$555$	0	0
$556$	−3.06608e6	−0.420626
$557$	2.95091e6	0.403012	0.201506	−	0.979487i	$-0.435417\pi$
$557$	2.95091e6	0.403012	0.201506	+	0.979487i	$0.435417\pi$
$558$	−5.20520e6	−0.707705
$559$	−1.36135e7	−1.84264
$560$	0	0
$561$	487842.	0.0654444
$562$	2.75715e6	0.368230
$563$	1.19591e7	1.59011	0.795057	−	0.606534i	$-0.207441\pi$
$563$	1.19591e7	1.59011	0.795057	+	0.606534i	$0.207441\pi$
$564$	−515272.	−0.0682085
$565$	0	0
$566$	8.19846e6	1.07570
$567$	2.76342e6	0.360985
$568$	−1.80830e6	−0.235180
$569$	−7.83827e6	−1.01494	−0.507469	−	0.861670i	$-0.669419\pi$
$569$	−7.83827e6	−1.01494	−0.507469	+	0.861670i	$0.669419\pi$
$570$	0	0
$571$	−6.60771e6	−0.848126	−0.424063	−	0.905633i	$-0.639397\pi$
$571$	−6.60771e6	−0.848126	−0.424063	+	0.905633i	$0.639397\pi$
$572$	−1.08521e7	−1.38684
$573$	−362786.	−0.0461599
$574$	−1.93772e6	−0.245477
$575$	0	0
$576$	−980349.	−0.123119
$577$	4.77589e6	0.597193	0.298596	−	0.954380i	$-0.403482\pi$
$577$	4.77589e6	0.597193	0.298596	+	0.954380i	$0.403482\pi$
$578$	−5.25172e6	−0.653856
$579$	−693600.	−0.0859830
$580$	0	0
$581$	913709.	0.112297
$582$	352337.	0.0431173
$583$	1.72067e7	2.09665
$584$	5.46930e6	0.663590
$585$	0	0
$586$	−1.21539e6	−0.146208
$587$	7.01821e6	0.840681	0.420340	−	0.907367i	$-0.361911\pi$
$587$	7.01821e6	0.840681	0.420340	+	0.907367i	$0.361911\pi$
$588$	−73464.5	−0.00876263
$589$	9.61364e6	1.14183
$590$	0	0
$591$	−1.58400e6	−0.186546
$592$	1.12933e6	0.132439
$593$	−1.10510e7	−1.29052	−0.645261	−	0.763962i	$-0.723251\pi$
$593$	−1.10510e7	−1.29052	−0.645261	+	0.763962i	$0.723251\pi$
$594$	−2.87841e6	−0.334723
$595$	0	0
$596$	536022.	0.0618112
$597$	−1.65248e6	−0.189759
$598$	6.60137e6	0.754886
$599$	−1.64477e6	−0.187300	−0.0936501	−	0.995605i	$-0.529853\pi$
$599$	−1.64477e6	−0.187300	−0.0936501	+	0.995605i	$0.529853\pi$
$600$	0	0
$601$	1.53175e7	1.72982	0.864911	−	0.501925i	$-0.167375\pi$
$601$	1.53175e7	1.72982	0.864911	+	0.501925i	$0.167375\pi$
$602$	−3.06903e6	−0.345151
$603$	−1.20768e7	−1.35257
$604$	−7.76838e6	−0.866439
$605$	0	0
$606$	543972.	0.0601721
$607$	3.55348e6	0.391455	0.195728	−	0.980658i	$-0.437293\pi$
$607$	3.55348e6	0.391455	0.195728	+	0.980658i	$0.437293\pi$
$608$	1.81063e6	0.198642
$609$	−455268.	−0.0497421
$610$	0	0
$611$	1.46412e7	1.58662
$612$	−1.25223e6	−0.135147
$613$	−5.87140e6	−0.631088	−0.315544	−	0.948911i	$-0.602187\pi$
$613$	−5.87140e6	−0.631088	−0.315544	+	0.948911i	$0.602187\pi$
$614$	1.92882e6	0.206477
$615$	0	0
$616$	−2.44651e6	−0.259774
$617$	−1.11324e7	−1.17727	−0.588636	−	0.808398i	$-0.700335\pi$
$617$	−1.11324e7	−1.17727	−0.588636	+	0.808398i	$0.700335\pi$
$618$	−38639.0	−0.00406963
$619$	1.76747e7	1.85407	0.927034	−	0.374977i	$-0.122349\pi$
$619$	1.76747e7	1.85407	0.927034	+	0.374977i	$0.122349\pi$
$620$	0	0
$621$	1.75094e6	0.182197
$622$	−5.77129e6	−0.598132
$623$	−2.93283e6	−0.302738
$624$	−425628.	−0.0437591
$625$	0	0
$626$	−3.30464e6	−0.337046
$627$	2.63795e6	0.267977
$628$	−7.89916e6	−0.799248
$629$	1.44253e6	0.145378
$630$	0	0
$631$	−1.19809e7	−1.19789	−0.598945	−	0.800790i	$-0.704413\pi$
$631$	−1.19809e7	−1.19789	−0.598945	+	0.800790i	$0.704413\pi$
$632$	4.30396e6	0.428623
$633$	853620.	0.0846750
$634$	−5.36975e6	−0.530555
$635$	0	0
$636$	674858.	0.0661561
$637$	2.08745e6	0.203830
$638$	−1.51613e7	−1.47464
$639$	6.76257e6	0.655178
$640$	0	0
$641$	−1.88873e6	−0.181562	−0.0907809	−	0.995871i	$-0.528936\pi$
$641$	−1.88873e6	−0.181562	−0.0907809	+	0.995871i	$0.528936\pi$
$642$	545942.	0.0522768
$643$	−6.75118e6	−0.643950	−0.321975	−	0.946748i	$-0.604347\pi$
$643$	−6.75118e6	−0.643950	−0.321975	+	0.946748i	$0.604347\pi$
$644$	1.48822e6	0.141401
$645$	0	0
$646$	2.31278e6	0.218048
$647$	9.88165e6	0.928045	0.464022	−	0.885823i	$-0.346406\pi$
$647$	9.88165e6	0.928045	0.464022	+	0.885823i	$0.346406\pi$
$648$	3.60937e6	0.337671
$649$	−3.12474e7	−2.91207
$650$	0	0
$651$	−509470.	−0.0471158
$652$	−3.07701e6	−0.283472
$653$	1.45036e7	1.33104	0.665522	−	0.746378i	$-0.268209\pi$
$653$	1.45036e7	1.33104	0.665522	+	0.746378i	$0.268209\pi$
$654$	−1.05643e6	−0.0965817
$655$	0	0
$656$	−2.53090e6	−0.229623
$657$	−2.04537e7	−1.84867
$658$	3.30071e6	0.297195
$659$	2.12571e6	0.190674	0.0953370	−	0.995445i	$-0.469607\pi$
$659$	2.12571e6	0.190674	0.0953370	+	0.995445i	$0.469607\pi$
$660$	0	0
$661$	2.52280e6	0.224584	0.112292	−	0.993675i	$-0.464181\pi$
$661$	2.52280e6	0.224584	0.112292	+	0.993675i	$0.464181\pi$
$662$	1.05153e7	0.932560
$663$	−543667.	−0.0480341
$664$	1.19342e6	0.105044
$665$	0	0
$666$	−4.22340e6	−0.368957
$667$	9.22264e6	0.802677
$668$	581225.	0.0503968
$669$	1.17780e6	0.101743
$670$	0	0
$671$	−1.71001e7	−1.46620
$672$	−95953.7	−0.00819669
$673$	−1.19058e7	−1.01326	−0.506630	−	0.862164i	$-0.669109\pi$
$673$	−1.19058e7	−1.01326	−0.506630	+	0.862164i	$0.669109\pi$
$674$	−1.50833e7	−1.27893
$675$	0	0
$676$	6.15329e6	0.517894
$677$	−1.67013e7	−1.40048	−0.700241	−	0.713907i	$-0.746924\pi$
$677$	−1.67013e7	−1.40048	−0.700241	+	0.713907i	$0.746924\pi$
$678$	1.21370e6	0.101400
$679$	−2.25699e6	−0.187869
$680$	0	0
$681$	−275023.	−0.0227249
$682$	−1.69663e7	−1.39678
$683$	5.59510e6	0.458940	0.229470	−	0.973316i	$-0.426301\pi$
$683$	5.59510e6	0.458940	0.229470	+	0.973316i	$0.426301\pi$
$684$	−6.77129e6	−0.553390
$685$	0	0
$686$	470596.	0.0381802
$687$	1.04630e6	0.0845794
$688$	−4.00852e6	−0.322859
$689$	−1.91757e7	−1.53888
$690$	0	0
$691$	1.65131e6	0.131563	0.0657815	−	0.997834i	$-0.479046\pi$
$691$	1.65131e6	0.131563	0.0657815	+	0.997834i	$0.479046\pi$
$692$	432803.	0.0343578
$693$	9.14929e6	0.723693
$694$	4.67108e6	0.368145
$695$	0	0
$696$	−594636.	−0.0465295
$697$	−3.23279e6	−0.252055
$698$	9.60226e6	0.745994
$699$	1.19459e6	0.0924751
$700$	0	0
$701$	−1.74041e7	−1.33770	−0.668848	−	0.743400i	$-0.733212\pi$
$701$	−1.74041e7	−1.33770	−0.668848	+	0.743400i	$0.733212\pi$
$702$	3.20779e6	0.245676
$703$	7.80031e6	0.595283
$704$	−3.19544e6	−0.242996
$705$	0	0
$706$	1.51532e7	1.14418
$707$	−3.48455e6	−0.262179
$708$	−1.22554e6	−0.0918851
$709$	1.15661e7	0.864113	0.432057	−	0.901846i	$-0.357788\pi$
$709$	1.15661e7	0.864113	0.432057	+	0.901846i	$0.357788\pi$
$710$	0	0
$711$	−1.60957e7	−1.19408
$712$	−3.83064e6	−0.283185
$713$	1.03206e7	0.760296
$714$	−122565.	−0.00899745
$715$	0	0
$716$	1.96991e6	0.143603
$717$	1.36135e6	0.0988941
$718$	1.47881e7	1.07053
$719$	−1.14902e7	−0.828904	−0.414452	−	0.910071i	$-0.636027\pi$
$719$	−1.14902e7	−0.828904	−0.414452	+	0.910071i	$0.636027\pi$
$720$	0	0
$721$	247512.	0.0177320
$722$	2.60169e6	0.185743
$723$	−1.04565e6	−0.0743942
$724$	−2.16611e6	−0.153580
$725$	0	0
$726$	−3.42357e6	−0.241067
$727$	2.00113e7	1.40423	0.702117	−	0.712062i	$-0.252238\pi$
$727$	2.00113e7	1.40423	0.702117	+	0.712062i	$0.252238\pi$
$728$	2.72647e6	0.190666
$729$	−1.30694e7	−0.910831
$730$	0	0
$731$	−5.12021e6	−0.354400
$732$	−670676.	−0.0462632
$733$	−2.53634e7	−1.74360	−0.871800	−	0.489862i	$-0.837047\pi$
$733$	−2.53634e7	−1.74360	−0.871800	+	0.489862i	$0.837047\pi$
$734$	−4.52546e6	−0.310044
$735$	0	0
$736$	1.94379e6	0.132268
$737$	−3.93642e7	−2.66952
$738$	9.46489e6	0.639697
$739$	−9.76419e6	−0.657696	−0.328848	−	0.944383i	$-0.606660\pi$
$739$	−9.76419e6	−0.657696	−0.328848	+	0.944383i	$0.606660\pi$
$740$	0	0
$741$	−2.93982e6	−0.196687
$742$	−4.32298e6	−0.288253
$743$	6.62984e6	0.440586	0.220293	−	0.975434i	$-0.429299\pi$
$743$	6.62984e6	0.440586	0.220293	+	0.975434i	$0.429299\pi$
$744$	−665430.	−0.0440728
$745$	0	0
$746$	−1.43765e7	−0.945815
$747$	−4.46306e6	−0.292638
$748$	−4.08163e6	−0.266735
$749$	−3.49717e6	−0.227778
$750$	0	0
$751$	3.02375e6	0.195635	0.0978174	−	0.995204i	$-0.468814\pi$
$751$	3.02375e6	0.195635	0.0978174	+	0.995204i	$0.468814\pi$
$752$	4.31113e6	0.278001
$753$	443944.	0.0285326
$754$	1.68962e7	1.08234
$755$	0	0
$756$	723165.	0.0460186
$757$	4.32091e6	0.274053	0.137027	−	0.990567i	$-0.456245\pi$
$757$	4.32091e6	0.274053	0.137027	+	0.990567i	$0.456245\pi$
$758$	−5.45941e6	−0.345122
$759$	2.83195e6	0.178436
$760$	0	0
$761$	−8.52278e6	−0.533482	−0.266741	−	0.963768i	$-0.585947\pi$
$761$	−8.52278e6	−0.533482	−0.266741	+	0.963768i	$0.585947\pi$
$762$	−516094.	−0.0321989
$763$	6.76721e6	0.420822
$764$	3.03533e6	0.188136
$765$	0	0
$766$	−1.98145e7	−1.22014
$767$	3.48231e7	2.13737
$768$	−125327.	−0.00766730
$769$	1.93335e7	1.17895	0.589475	−	0.807787i	$-0.299335\pi$
$769$	1.93335e7	1.17895	0.589475	+	0.807787i	$0.299335\pi$
$770$	0	0
$771$	1.09774e6	0.0665065
$772$	5.80315e6	0.350445
$773$	−1.85205e7	−1.11482	−0.557408	−	0.830239i	$-0.688204\pi$
$773$	−1.85205e7	−1.11482	−0.557408	+	0.830239i	$0.688204\pi$
$774$	1.49908e7	0.899441
$775$	0	0
$776$	−2.94790e6	−0.175735
$777$	−413374.	−0.0245635
$778$	9.36941e6	0.554962
$779$	−1.74810e7	−1.03210
$780$	0	0
$781$	2.20425e7	1.29311
$782$	2.48286e6	0.145190
$783$	4.48154e6	0.261230
$784$	614656.	0.0357143
$785$	0	0
$786$	−1.21017e6	−0.0698699
$787$	927171.	0.0533609	0.0266804	−	0.999644i	$-0.491506\pi$
$787$	927171.	0.0533609	0.0266804	+	0.999644i	$0.491506\pi$
$788$	1.32529e7	0.760316
$789$	596733.	0.0341262
$790$	0	0
$791$	−7.77469e6	−0.441816
$792$	1.19501e7	0.676953
$793$	1.90569e7	1.07614
$794$	−6.17834e6	−0.347793
$795$	0	0
$796$	1.38258e7	0.773409
$797$	8.55917e6	0.477294	0.238647	−	0.971106i	$-0.423296\pi$
$797$	8.55917e6	0.477294	0.238647	+	0.971106i	$0.423296\pi$
$798$	−662754.	−0.0368422
$799$	5.50673e6	0.305159
$800$	0	0
$801$	1.43256e7	0.788915
$802$	1.71727e6	0.0942765
$803$	−6.66688e7	−3.64866
$804$	−1.54389e6	−0.0842318
$805$	0	0
$806$	1.89078e7	1.02519
$807$	4.17750e6	0.225804
$808$	−4.55125e6	−0.245246
$809$	2.67123e6	0.143496	0.0717481	−	0.997423i	$-0.477142\pi$
$809$	2.67123e6	0.143496	0.0717481	+	0.997423i	$0.477142\pi$
$810$	0	0
$811$	1.47660e7	0.788334	0.394167	−	0.919039i	$-0.371033\pi$
$811$	1.47660e7	0.788334	0.394167	+	0.919039i	$0.371033\pi$
$812$	3.80909e6	0.202736
$813$	−1.82040e6	−0.0965920
$814$	−1.37661e7	−0.728200
$815$	0	0
$816$	−160084.	−0.00841634
$817$	−2.76870e7	−1.45118
$818$	−1.13066e6	−0.0590811
$819$	−1.01963e7	−0.531168
$820$	0	0
$821$	−9.07572e6	−0.469919	−0.234960	−	0.972005i	$-0.575496\pi$
$821$	−9.07572e6	−0.469919	−0.234960	+	0.972005i	$0.575496\pi$
$822$	−1.53298e6	−0.0791331
$823$	3.06337e7	1.57652	0.788262	−	0.615340i	$-0.210981\pi$
$823$	3.06337e7	1.57652	0.788262	+	0.615340i	$0.210981\pi$
$824$	323281.	0.0165868
$825$	0	0
$826$	7.85053e6	0.400358
$827$	5.77395e6	0.293568	0.146784	−	0.989169i	$-0.453108\pi$
$827$	5.77395e6	0.293568	0.146784	+	0.989169i	$0.453108\pi$
$828$	−7.26926e6	−0.368480
$829$	7.67908e6	0.388082	0.194041	−	0.980993i	$-0.437841\pi$
$829$	7.67908e6	0.388082	0.194041	+	0.980993i	$0.437841\pi$
$830$	0	0
$831$	2.05656e6	0.103309
$832$	3.56110e6	0.178351
$833$	785119.	0.0392033
$834$	1.46585e6	0.0729749
$835$	0	0
$836$	−2.20710e7	−1.09221
$837$	5.01509e6	0.247437
$838$	−1.26149e7	−0.620544
$839$	1.46555e6	0.0718782	0.0359391	−	0.999354i	$-0.488558\pi$
$839$	1.46555e6	0.0718782	0.0359391	+	0.999354i	$0.488558\pi$
$840$	0	0
$841$	3.09424e6	0.150856
$842$	1.52322e7	0.740426
$843$	−1.31815e6	−0.0638847
$844$	−7.14199e6	−0.345114
$845$	0	0
$846$	−1.61225e7	−0.774471
$847$	2.19306e7	1.05037
$848$	−5.64634e6	−0.269636
$849$	−3.91956e6	−0.186624
$850$	0	0
$851$	8.37396e6	0.396376
$852$	864523.	0.0408016
$853$	−1.31222e7	−0.617496	−0.308748	−	0.951144i	$-0.599910\pi$
$853$	−1.31222e7	−0.617496	−0.308748	+	0.951144i	$0.599910\pi$
$854$	4.29619e6	0.201576
$855$	0	0
$856$	−4.56773e6	−0.213067
$857$	−3.79938e6	−0.176710	−0.0883549	−	0.996089i	$-0.528161\pi$
$857$	−3.79938e6	−0.176710	−0.0883549	+	0.996089i	$0.528161\pi$
$858$	5.18825e6	0.240604
$859$	−3.76559e6	−0.174120	−0.0870602	−	0.996203i	$-0.527747\pi$
$859$	−3.76559e6	−0.174120	−0.0870602	+	0.996203i	$0.527747\pi$
$860$	0	0
$861$	926396.	0.0425881
$862$	−7.52338e6	−0.344862
$863$	−9.96098e6	−0.455276	−0.227638	−	0.973746i	$-0.573100\pi$
$863$	−9.96098e6	−0.455276	−0.227638	+	0.973746i	$0.573100\pi$
$864$	944542.	0.0430464
$865$	0	0
$866$	−2.00647e7	−0.909156
$867$	2.51077e6	0.113438
$868$	4.26259e6	0.192032
$869$	−5.24637e7	−2.35673
$870$	0	0
$871$	4.38688e7	1.95934
$872$	8.83881e6	0.393643
$873$	1.10244e7	0.489574
$874$	1.34258e7	0.594514
$875$	0	0
$876$	−2.61479e6	−0.115127
$877$	1.25571e7	0.551304	0.275652	−	0.961258i	$-0.411106\pi$
$877$	1.25571e7	0.551304	0.275652	+	0.961258i	$0.411106\pi$
$878$	1.05780e7	0.463092
$879$	581058.	0.0253657
$880$	0	0
$881$	1.86084e7	0.807735	0.403868	−	0.914817i	$-0.367666\pi$
$881$	1.86084e7	0.807735	0.403868	+	0.914817i	$0.367666\pi$
$882$	−2.29865e6	−0.0994950
$883$	1.31567e7	0.567866	0.283933	−	0.958844i	$-0.408361\pi$
$883$	1.31567e7	0.567866	0.283933	+	0.958844i	$0.408361\pi$
$884$	4.54870e6	0.195775
$885$	0	0
$886$	2.95503e7	1.26467
$887$	6.52618e6	0.278516	0.139258	−	0.990256i	$-0.455528\pi$
$887$	6.52618e6	0.278516	0.139258	+	0.990256i	$0.455528\pi$
$888$	−539917.	−0.0229770
$889$	3.30597e6	0.140296
$890$	0	0
$891$	−4.39969e7	−1.85664
$892$	−9.85431e6	−0.414681
$893$	2.97770e7	1.24955
$894$	−256264.	−0.0107237
$895$	0	0
$896$	802816.	0.0334077
$897$	−3.15602e6	−0.130966
$898$	1.68076e7	0.695530
$899$	2.64157e7	1.09009
$900$	0	0
$901$	−7.21224e6	−0.295977
$902$	3.08507e7	1.26255
$903$	1.46726e6	0.0598807
$904$	−1.01547e7	−0.413281
$905$	0	0
$906$	3.71395e6	0.150320
$907$	−2.06271e7	−0.832570	−0.416285	−	0.909234i	$-0.636668\pi$
$907$	−2.06271e7	−0.832570	−0.416285	+	0.909234i	$0.636668\pi$
$908$	2.30104e6	0.0926210
$909$	1.70205e7	0.683222
$910$	0	0
$911$	−6.80945e6	−0.271842	−0.135921	−	0.990720i	$-0.543399\pi$
$911$	−6.80945e6	−0.271842	−0.135921	+	0.990720i	$0.543399\pi$
$912$	−865638.	−0.0344627
$913$	−1.45473e7	−0.577572
$914$	1.53624e7	0.608265
$915$	0	0
$916$	−8.75408e6	−0.344724
$917$	7.75206e6	0.304434
$918$	1.20649e6	0.0472518
$919$	2.32177e7	0.906841	0.453420	−	0.891297i	$-0.350204\pi$
$919$	2.32177e7	0.906841	0.453420	+	0.891297i	$0.350204\pi$
$920$	0	0
$921$	−922142.	−0.0358219
$922$	−1.02769e7	−0.398138
$923$	−2.45649e7	−0.949098
$924$	1.16964e6	0.0450684
$925$	0	0
$926$	1.39836e7	0.535908
$927$	−1.20899e6	−0.0462085
$928$	4.97514e6	0.189642
$929$	4.68022e7	1.77921	0.889604	−	0.456733i	$-0.150981\pi$
$929$	4.68022e7	1.77921	0.889604	+	0.456733i	$0.150981\pi$
$930$	0	0
$931$	4.24544e6	0.160527
$932$	−9.99475e6	−0.376905
$933$	2.75917e6	0.103771
$934$	−61431.7	−0.00230423
$935$	0	0
$936$	−1.33176e7	−0.496862
$937$	−1.22112e7	−0.454368	−0.227184	−	0.973852i	$-0.572952\pi$
$937$	−1.22112e7	−0.454368	−0.227184	+	0.973852i	$0.572952\pi$
$938$	9.88979e6	0.367012
$939$	1.57990e6	0.0584745
$940$	0	0
$941$	3.96195e7	1.45860	0.729298	−	0.684196i	$-0.239847\pi$
$941$	3.96195e7	1.45860	0.729298	+	0.684196i	$0.239847\pi$
$942$	3.77647e6	0.138662
$943$	−1.87666e7	−0.687235
$944$	1.02538e7	0.374501
$945$	0	0
$946$	4.88624e7	1.77520
$947$	−1.31421e7	−0.476200	−0.238100	−	0.971241i	$-0.576525\pi$
$947$	−1.31421e7	−0.476200	−0.238100	+	0.971241i	$0.576525\pi$
$948$	−2.05766e6	−0.0743623
$949$	7.42979e7	2.67800
$950$	0	0
$951$	2.56720e6	0.0920466
$952$	1.02546e6	0.0366713
$953$	−5.17474e7	−1.84568	−0.922839	−	0.385185i	$-0.874138\pi$
$953$	−5.17474e7	−1.84568	−0.922839	+	0.385185i	$0.874138\pi$
$954$	2.11158e7	0.751167
$955$	0	0
$956$	−1.13900e7	−0.403068
$957$	7.24840e6	0.255836
$958$	9.94341e6	0.350043
$959$	9.81992e6	0.344795
$960$	0	0
$961$	931510.	0.0325371
$962$	1.53414e7	0.534476
$963$	1.70821e7	0.593575
$964$	8.74861e6	0.303212
$965$	0	0
$966$	−711494.	−0.0245317
$967$	−5.60278e7	−1.92680	−0.963401	−	0.268063i	$-0.913616\pi$
$967$	−5.60278e7	−1.92680	−0.963401	+	0.268063i	$0.913616\pi$
$968$	2.86440e7	0.982529
$969$	−1.10571e6	−0.0378294
$970$	0	0
$971$	−1.07082e7	−0.364475	−0.182237	−	0.983255i	$-0.558334\pi$
$971$	−1.07082e7	−0.364475	−0.182237	+	0.983255i	$0.558334\pi$
$972$	−5.31190e6	−0.180337
$973$	−9.38986e6	−0.317963
$974$	4.06607e6	0.137334
$975$	0	0
$976$	5.61135e6	0.188557
$977$	−2.02350e7	−0.678214	−0.339107	−	0.940748i	$-0.610125\pi$
$977$	−2.02350e7	−0.678214	−0.339107	+	0.940748i	$0.610125\pi$
$978$	1.47107e6	0.0491798
$979$	4.66941e7	1.55706
$980$	0	0
$981$	−3.30548e7	−1.09663
$982$	−3.20900e6	−0.106192
$983$	1.65832e7	0.547375	0.273688	−	0.961819i	$-0.411757\pi$
$983$	1.65832e7	0.547375	0.273688	+	0.961819i	$0.411757\pi$
$984$	1.20999e6	0.0398376
$985$	0	0
$986$	6.35490e6	0.208169
$987$	−1.57802e6	−0.0515608
$988$	2.45966e7	0.801647
$989$	−2.97231e7	−0.966281
$990$	0	0
$991$	2.58044e6	0.0834660	0.0417330	−	0.999129i	$-0.486712\pi$
$991$	2.58044e6	0.0834660	0.0417330	+	0.999129i	$0.486712\pi$
$992$	5.56746e6	0.179630
$993$	−5.02721e6	−0.161791
$994$	−5.53792e6	−0.177779
$995$	0	0
$996$	−570555.	−0.0182242
$997$	−3.95431e7	−1.25989	−0.629945	−	0.776640i	$-0.716923\pi$
$997$	−3.95431e7	−1.25989	−0.629945	+	0.776640i	$0.716923\pi$
$998$	1.20245e7	0.382156
$999$	4.06914e6	0.129000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	350.6.a.z.1.3		5
5.2	odd	4		70.6.c.d.29.8	yes	10
5.3	odd	4		70.6.c.d.29.3	✓	10
5.4	even	2		350.6.a.y.1.3		5
15.2	even	4		630.6.g.h.379.4		10
15.8	even	4		630.6.g.h.379.9		10
20.3	even	4		560.6.g.d.449.6		10
20.7	even	4		560.6.g.d.449.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.c.d.29.3	✓	10	5.3	odd	4
70.6.c.d.29.8	yes	10	5.2	odd	4
350.6.a.y.1.3		5	5.4	even	2
350.6.a.z.1.3		5	1.1	even	1	trivial
560.6.g.d.449.5		10	20.7	even	4
560.6.g.d.449.6		10	20.3	even	4
630.6.g.h.379.4		10	15.2	even	4
630.6.g.h.379.9		10	15.8	even	4

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:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	350.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -1.91234 q^{3} +16.0000 q^{4} -7.64937 q^{6} +49.0000 q^{7} +64.0000 q^{8} -239.343 q^{9} -780.137 q^{11} -30.5975 q^{12} +869.410 q^{13} +196.000 q^{14} +256.000 q^{16} +326.997 q^{17} -957.372 q^{18} +1768.20 q^{19} -93.7048 q^{21} -3120.55 q^{22} +1898.23 q^{23} -122.390 q^{24} +3477.64 q^{26} +922.405 q^{27} +784.000 q^{28} +4858.54 q^{29} +5436.97 q^{31} +1024.00 q^{32} +1491.89 q^{33} +1307.99 q^{34} -3829.49 q^{36} +4411.45 q^{37} +7072.79 q^{38} -1662.61 q^{39} -9886.32 q^{41} -374.819 q^{42} -15658.3 q^{43} -12482.2 q^{44} +7592.94 q^{46} +16840.3 q^{47} -489.560 q^{48} +2401.00 q^{49} -625.329 q^{51} +13910.6 q^{52} -22056.0 q^{53} +3689.62 q^{54} +3136.00 q^{56} -3381.40 q^{57} +19434.2 q^{58} +40053.7 q^{59} +21919.3 q^{61} +21747.9 q^{62} -11727.8 q^{63} +4096.00 q^{64} +5967.55 q^{66} +50458.1 q^{67} +5231.94 q^{68} -3630.07 q^{69} -28254.7 q^{71} -15317.9 q^{72} +85457.9 q^{73} +17645.8 q^{74} +28291.2 q^{76} -38226.7 q^{77} -6650.44 q^{78} +67249.4 q^{79} +56396.4 q^{81} -39545.3 q^{82} +18647.1 q^{83} -1499.28 q^{84} -62633.2 q^{86} -9291.19 q^{87} -49928.8 q^{88} -59853.7 q^{89} +42601.1 q^{91} +30371.7 q^{92} -10397.3 q^{93} +67361.3 q^{94} -1958.24 q^{96} -46061.0 q^{97} +9604.00 q^{98} +186720. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 20 q^{2} + 14 q^{3} + 80 q^{4} + 56 q^{6} + 245 q^{7} + 320 q^{8} + 655 q^{9} + 748 q^{11} + 224 q^{12} + 456 q^{13} + 980 q^{14} + 1280 q^{16} + 780 q^{17} + 2620 q^{18} - 256 q^{19} + 686 q^{21}+ \cdots + 575944 q^{99}+O(q^{100}) \) 5 * q + 20 * q^2 + 14 * q^3 + 80 * q^4 + 56 * q^6 + 245 * q^7 + 320 * q^8 + 655 * q^9 + 748 * q^11 + 224 * q^12 + 456 * q^13 + 980 * q^14 + 1280 * q^16 + 780 * q^17 + 2620 * q^18 - 256 * q^19 + 686 * q^21 + 2992 * q^22 + 3452 * q^23 + 896 * q^24 + 1824 * q^26 - 910 * q^27 + 3920 * q^28 + 7712 * q^29 + 412 * q^31 + 5120 * q^32 - 2980 * q^33 + 3120 * q^34 + 10480 * q^36 + 14400 * q^37 - 1024 * q^38 + 17668 * q^39 + 7482 * q^41 + 2744 * q^42 + 20668 * q^43 + 11968 * q^44 + 13808 * q^46 + 59202 * q^47 + 3584 * q^48 + 12005 * q^49 + 6628 * q^51 + 7296 * q^52 + 6388 * q^53 - 3640 * q^54 + 15680 * q^56 + 112616 * q^57 + 30848 * q^58 + 79596 * q^59 + 9754 * q^61 + 1648 * q^62 + 32095 * q^63 + 20480 * q^64 - 11920 * q^66 + 49820 * q^67 + 12480 * q^68 + 106152 * q^69 - 61928 * q^71 + 41920 * q^72 - 57248 * q^73 + 57600 * q^74 - 4096 * q^76 + 36652 * q^77 + 70672 * q^78 + 150912 * q^79 + 100825 * q^81 + 29928 * q^82 - 54776 * q^83 + 10976 * q^84 + 82672 * q^86 - 256362 * q^87 + 47872 * q^88 + 268950 * q^89 + 22344 * q^91 + 55232 * q^92 - 187044 * q^93 + 236808 * q^94 + 14336 * q^96 - 296156 * q^97 + 48020 * q^98 + 575944 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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