LMFDB - Embedded newform 207.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.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:	$33.1994507013$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 75x^{2} - 42x + 736 $ x^4 - 75x^2 - 42x + 736
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.50608$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.50608 q^{2} +40.3535 q^{4} +86.6918 q^{5} -64.0051 q^{7} -71.0553 q^{8} +O(q^{10})$	\(q-8.50608 q^{2} +40.3535 q^{4} +86.6918 q^{5} -64.0051 q^{7} -71.0553 q^{8} -737.408 q^{10} +285.170 q^{11} -307.400 q^{13} +544.433 q^{14} -686.909 q^{16} -2223.12 q^{17} -1802.44 q^{19} +3498.32 q^{20} -2425.68 q^{22} +529.000 q^{23} +4390.47 q^{25} +2614.77 q^{26} -2582.83 q^{28} +2359.49 q^{29} +8317.38 q^{31} +8116.67 q^{32} +18910.0 q^{34} -5548.72 q^{35} -9075.00 q^{37} +15331.7 q^{38} -6159.91 q^{40} -1543.25 q^{41} -15330.5 q^{43} +11507.6 q^{44} -4499.72 q^{46} -14725.3 q^{47} -12710.3 q^{49} -37345.7 q^{50} -12404.7 q^{52} +14163.0 q^{53} +24721.9 q^{55} +4547.90 q^{56} -20070.0 q^{58} -8408.86 q^{59} +26134.4 q^{61} -70748.4 q^{62} -47060.0 q^{64} -26649.1 q^{65} -13002.9 q^{67} -89710.5 q^{68} +47197.9 q^{70} -52490.6 q^{71} -16992.9 q^{73} +77192.7 q^{74} -72734.8 q^{76} -18252.3 q^{77} -100023. q^{79} -59549.4 q^{80} +13127.0 q^{82} +85251.7 q^{83} -192726. q^{85} +130403. q^{86} -20262.8 q^{88} +83076.2 q^{89} +19675.2 q^{91} +21347.0 q^{92} +125255. q^{94} -156257. q^{95} +31793.1 q^{97} +108115. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 62 * q^7 - 72 * q^8	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8} - 496 q^{10} + 1076 q^{11} - 396 q^{13} + 1806 q^{14} - 1982 q^{16} - 70 q^{17} - 6366 q^{19} + 5240 q^{20} - 6974 q^{22} + 2116 q^{23} + 1264 q^{25} - 2464 q^{26} - 6474 q^{28} - 3948 q^{29} + 3092 q^{31} + 3672 q^{32} + 11682 q^{34} - 1304 q^{35} - 17464 q^{37} + 12628 q^{38} - 14108 q^{40} - 18680 q^{41} - 25846 q^{43} - 20746 q^{44} - 2116 q^{46} - 18392 q^{47} + 7952 q^{49} - 69444 q^{50} + 8844 q^{52} + 26518 q^{53} - 40848 q^{55} - 54890 q^{56} + 568 q^{58} + 14520 q^{59} - 13688 q^{61} - 120136 q^{62} - 30190 q^{64} - 38324 q^{65} - 11098 q^{67} - 112138 q^{68} - 29596 q^{70} + 57496 q^{71} - 112272 q^{73} + 21226 q^{74} - 76240 q^{76} + 4792 q^{77} - 240754 q^{79} - 41200 q^{80} + 49976 q^{82} + 93268 q^{83} - 323204 q^{85} + 88224 q^{86} + 42382 q^{88} + 107582 q^{89} - 301532 q^{91} + 13754 q^{92} + 79360 q^{94} + 18640 q^{95} - 53076 q^{97} - 59664 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 62 * q^7 - 72 * q^8 - 496 * q^10 + 1076 * q^11 - 396 * q^13 + 1806 * q^14 - 1982 * q^16 - 70 * q^17 - 6366 * q^19 + 5240 * q^20 - 6974 * q^22 + 2116 * q^23 + 1264 * q^25 - 2464 * q^26 - 6474 * q^28 - 3948 * q^29 + 3092 * q^31 + 3672 * q^32 + 11682 * q^34 - 1304 * q^35 - 17464 * q^37 + 12628 * q^38 - 14108 * q^40 - 18680 * q^41 - 25846 * q^43 - 20746 * q^44 - 2116 * q^46 - 18392 * q^47 + 7952 * q^49 - 69444 * q^50 + 8844 * q^52 + 26518 * q^53 - 40848 * q^55 - 54890 * q^56 + 568 * q^58 + 14520 * q^59 - 13688 * q^61 - 120136 * q^62 - 30190 * q^64 - 38324 * q^65 - 11098 * q^67 - 112138 * q^68 - 29596 * q^70 + 57496 * q^71 - 112272 * q^73 + 21226 * q^74 - 76240 * q^76 + 4792 * q^77 - 240754 * q^79 - 41200 * q^80 + 49976 * q^82 + 93268 * q^83 - 323204 * q^85 + 88224 * q^86 + 42382 * q^88 + 107582 * q^89 - 301532 * q^91 + 13754 * q^92 + 79360 * q^94 + 18640 * q^95 - 53076 * q^97 - 59664 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−8.50608	−1.50368	−0.751839	−	0.659347i	$-0.770833\pi$
$2$	−8.50608	−1.50368	−0.751839	+	0.659347i	$0.770833\pi$
$3$	0	0
$3$	0	0
$4$	40.3535	1.26105
$5$	86.6918	1.55079	0.775395	−	0.631476i	$-0.217551\pi$
$5$	86.6918	1.55079	0.775395	+	0.631476i	$0.217551\pi$
$6$	0	0
$7$	−64.0051	−0.493707	−0.246854	−	0.969053i	$-0.579397\pi$
$7$	−64.0051	−0.493707	−0.246854	+	0.969053i	$0.579397\pi$
$8$	−71.0553	−0.392529
$9$	0	0
$10$	−737.408	−2.33189
$11$	285.170	0.710595	0.355297	−	0.934753i	$-0.384380\pi$
$11$	285.170	0.710595	0.355297	+	0.934753i	$0.384380\pi$
$12$	0	0
$13$	−307.400	−0.504482	−0.252241	−	0.967664i	$-0.581168\pi$
$13$	−307.400	−0.504482	−0.252241	+	0.967664i	$0.581168\pi$
$14$	544.433	0.742377
$15$	0	0
$16$	−686.909	−0.670809
$17$	−2223.12	−1.86569	−0.932847	−	0.360273i	$-0.882683\pi$
$17$	−2223.12	−1.86569	−0.932847	+	0.360273i	$0.882683\pi$
$18$	0	0
$19$	−1802.44	−1.14545	−0.572727	−	0.819746i	$-0.694114\pi$
$19$	−1802.44	−1.14545	−0.572727	+	0.819746i	$0.694114\pi$
$20$	3498.32	1.95562
$21$	0	0
$22$	−2425.68	−1.06851
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	4390.47	1.40495
$26$	2614.77	0.758578
$27$	0	0
$28$	−2582.83	−0.622588
$29$	2359.49	0.520982	0.260491	−	0.965476i	$-0.416116\pi$
$29$	2359.49	0.520982	0.260491	+	0.965476i	$0.416116\pi$
$30$	0	0
$31$	8317.38	1.55447	0.777235	−	0.629210i	$-0.216622\pi$
$31$	8317.38	1.55447	0.777235	+	0.629210i	$0.216622\pi$
$32$	8116.67	1.40121
$33$	0	0
$34$	18910.0	2.80540
$35$	−5548.72	−0.765637
$36$	0	0
$37$	−9075.00	−1.08979	−0.544895	−	0.838505i	$-0.683430\pi$
$37$	−9075.00	−1.08979	−0.544895	+	0.838505i	$0.683430\pi$
$38$	15331.7	1.72239
$39$	0	0
$40$	−6159.91	−0.608730
$41$	−1543.25	−0.143376	−0.0716882	−	0.997427i	$-0.522839\pi$
$41$	−1543.25	−0.143376	−0.0716882	+	0.997427i	$0.522839\pi$
$42$	0	0
$43$	−15330.5	−1.26440	−0.632202	−	0.774804i	$-0.717849\pi$
$43$	−15330.5	−1.26440	−0.632202	+	0.774804i	$0.717849\pi$
$44$	11507.6	0.896093
$45$	0	0
$46$	−4499.72	−0.313538
$47$	−14725.3	−0.972343	−0.486172	−	0.873863i	$-0.661607\pi$
$47$	−14725.3	−0.972343	−0.486172	+	0.873863i	$0.661607\pi$
$48$	0	0
$49$	−12710.3	−0.756253
$50$	−37345.7	−2.11259
$51$	0	0
$52$	−12404.7	−0.636175
$53$	14163.0	0.692572	0.346286	−	0.938129i	$-0.387443\pi$
$53$	14163.0	0.692572	0.346286	+	0.938129i	$0.387443\pi$
$54$	0	0
$55$	24721.9	1.10198
$56$	4547.90	0.193794
$57$	0	0
$58$	−20070.0	−0.783388
$59$	−8408.86	−0.314490	−0.157245	−	0.987560i	$-0.550261\pi$
$59$	−8408.86	−0.314490	−0.157245	+	0.987560i	$0.550261\pi$
$60$	0	0
$61$	26134.4	0.899267	0.449633	−	0.893213i	$-0.351555\pi$
$61$	26134.4	0.899267	0.449633	+	0.893213i	$0.351555\pi$
$62$	−70748.4	−2.33742
$63$	0	0
$64$	−47060.0	−1.43616
$65$	−26649.1	−0.782346
$66$	0	0
$67$	−13002.9	−0.353878	−0.176939	−	0.984222i	$-0.556620\pi$
$67$	−13002.9	−0.353878	−0.176939	+	0.984222i	$0.556620\pi$
$68$	−89710.5	−2.35273
$69$	0	0
$70$	47197.9	1.15127
$71$	−52490.6	−1.23576	−0.617882	−	0.786271i	$-0.712009\pi$
$71$	−52490.6	−1.23576	−0.617882	+	0.786271i	$0.712009\pi$
$72$	0	0
$73$	−16992.9	−0.373215	−0.186607	−	0.982435i	$-0.559749\pi$
$73$	−16992.9	−0.373215	−0.186607	+	0.982435i	$0.559749\pi$
$74$	77192.7	1.63869
$75$	0	0
$76$	−72734.8	−1.44447
$77$	−18252.3	−0.350826
$78$	0	0
$79$	−100023.	−1.80316	−0.901579	−	0.432614i	$-0.857591\pi$
$79$	−100023.	−1.80316	−0.901579	+	0.432614i	$0.857591\pi$
$80$	−59549.4	−1.04028
$81$	0	0
$82$	13127.0	0.215592
$83$	85251.7	1.35834	0.679169	−	0.733982i	$-0.262340\pi$
$83$	85251.7	1.35834	0.679169	+	0.733982i	$0.262340\pi$
$84$	0	0
$85$	−192726.	−2.89330
$86$	130403.	1.90126
$87$	0	0
$88$	−20262.8	−0.278929
$89$	83076.2	1.11174	0.555868	−	0.831271i	$-0.312386\pi$
$89$	83076.2	1.11174	0.555868	+	0.831271i	$0.312386\pi$
$90$	0	0
$91$	19675.2	0.249066
$92$	21347.0	0.262946
$93$	0	0
$94$	125255.	1.46209
$95$	−156257.	−1.77636
$96$	0	0
$97$	31793.1	0.343086	0.171543	−	0.985177i	$-0.445125\pi$
$97$	31793.1	0.343086	0.171543	+	0.985177i	$0.445125\pi$
$98$	108115.	1.13716
$99$	0	0
$100$	177171.	1.77171
$101$	−55155.8	−0.538007	−0.269003	−	0.963139i	$-0.586694\pi$
$101$	−55155.8	−0.538007	−0.269003	+	0.963139i	$0.586694\pi$
$102$	0	0
$103$	−125192.	−1.16274	−0.581371	−	0.813639i	$-0.697483\pi$
$103$	−125192.	−1.16274	−0.581371	+	0.813639i	$0.697483\pi$
$104$	21842.4	0.198024
$105$	0	0
$106$	−120472.	−1.04141
$107$	30718.1	0.259379	0.129690	−	0.991555i	$-0.458602\pi$
$107$	30718.1	0.259379	0.129690	+	0.991555i	$0.458602\pi$
$108$	0	0
$109$	−14066.0	−0.113398	−0.0566991	−	0.998391i	$-0.518058\pi$
$109$	−14066.0	−0.113398	−0.0566991	+	0.998391i	$0.518058\pi$
$110$	−210287.	−1.65703
$111$	0	0
$112$	43965.7	0.331184
$113$	−30310.7	−0.223305	−0.111653	−	0.993747i	$-0.535614\pi$
$113$	−30310.7	−0.223305	−0.111653	+	0.993747i	$0.535614\pi$
$114$	0	0
$115$	45860.0	0.323362
$116$	95213.4	0.656982
$117$	0	0
$118$	71526.4	0.472892
$119$	142291.	0.921107
$120$	0	0
$121$	−79729.1	−0.495055
$122$	−222302.	−1.35221
$123$	0	0
$124$	335635.	1.96026
$125$	109706.	0.627995
$126$	0	0
$127$	−143632.	−0.790211	−0.395106	−	0.918636i	$-0.629292\pi$
$127$	−143632.	−0.790211	−0.395106	+	0.918636i	$0.629292\pi$
$128$	140563.	0.758308
$129$	0	0
$130$	226679.	1.17640
$131$	269835.	1.37379	0.686895	−	0.726757i	$-0.258973\pi$
$131$	269835.	1.37379	0.686895	+	0.726757i	$0.258973\pi$
$132$	0	0
$133$	115366.	0.565519
$134$	110604.	0.532119
$135$	0	0
$136$	157964.	0.732338
$137$	−305800.	−1.39199	−0.695995	−	0.718047i	$-0.745036\pi$
$137$	−305800.	−1.39199	−0.695995	+	0.718047i	$0.745036\pi$
$138$	0	0
$139$	−175456.	−0.770251	−0.385126	−	0.922864i	$-0.625842\pi$
$139$	−175456.	−0.770251	−0.385126	+	0.922864i	$0.625842\pi$
$140$	−223910.	−0.965503
$141$	0	0
$142$	446490.	1.85819
$143$	−87661.3	−0.358482
$144$	0	0
$145$	204548.	0.807933
$146$	144543.	0.561195
$147$	0	0
$148$	−366208.	−1.37427
$149$	−507949.	−1.87436	−0.937182	−	0.348840i	$-0.886576\pi$
$149$	−507949.	−1.87436	−0.937182	+	0.348840i	$0.886576\pi$
$150$	0	0
$151$	214770.	0.766534	0.383267	−	0.923638i	$-0.374799\pi$
$151$	214770.	0.766534	0.383267	+	0.923638i	$0.374799\pi$
$152$	128073.	0.449623
$153$	0	0
$154$	155256.	0.527529
$155$	721049.	2.41066
$156$	0	0
$157$	180401.	0.584103	0.292051	−	0.956403i	$-0.405662\pi$
$157$	180401.	0.584103	0.292051	+	0.956403i	$0.405662\pi$
$158$	850808.	2.71137
$159$	0	0
$160$	703649.	2.17298
$161$	−33858.7	−0.102945
$162$	0	0
$163$	461874.	1.36161	0.680807	−	0.732463i	$-0.261629\pi$
$163$	461874.	1.36161	0.680807	+	0.732463i	$0.261629\pi$
$164$	−62275.6	−0.180804
$165$	0	0
$166$	−725158.	−2.04250
$167$	−8395.62	−0.0232949	−0.0116475	−	0.999932i	$-0.503708\pi$
$167$	−8395.62	−0.0232949	−0.0116475	+	0.999932i	$0.503708\pi$
$168$	0	0
$169$	−276798.	−0.745498
$170$	1.63935e6	4.35059
$171$	0	0
$172$	−618640.	−1.59447
$173$	−675883.	−1.71695	−0.858473	−	0.512859i	$-0.828586\pi$
$173$	−675883.	−1.71695	−0.858473	+	0.512859i	$0.828586\pi$
$174$	0	0
$175$	−281013.	−0.693635
$176$	−195886.	−0.476674
$177$	0	0
$178$	−706653.	−1.67169
$179$	−256756.	−0.598947	−0.299474	−	0.954105i	$-0.596811\pi$
$179$	−256756.	−0.598947	−0.299474	+	0.954105i	$0.596811\pi$
$180$	0	0
$181$	650703.	1.47634	0.738170	−	0.674615i	$-0.235690\pi$
$181$	650703.	1.47634	0.738170	+	0.674615i	$0.235690\pi$
$182$	−167359.	−0.374516
$183$	0	0
$184$	−37588.2	−0.0818479
$185$	−786729.	−1.69003
$186$	0	0
$187$	−633967.	−1.32575
$188$	−594217.	−1.22617
$189$	0	0
$190$	1.32914e6	2.67107
$191$	874077.	1.73367	0.866835	−	0.498595i	$-0.166151\pi$
$191$	874077.	1.73367	0.866835	+	0.498595i	$0.166151\pi$
$192$	0	0
$193$	−700334.	−1.35336	−0.676678	−	0.736279i	$-0.736581\pi$
$193$	−700334.	−1.35336	−0.676678	+	0.736279i	$0.736581\pi$
$194$	−270434.	−0.515891
$195$	0	0
$196$	−512906.	−0.953670
$197$	−1.01192e6	−1.85773	−0.928864	−	0.370420i	$-0.879214\pi$
$197$	−1.01192e6	−1.85773	−0.928864	+	0.370420i	$0.879214\pi$
$198$	0	0
$199$	751608.	1.34542	0.672712	−	0.739905i	$-0.265129\pi$
$199$	751608.	1.34542	0.672712	+	0.739905i	$0.265129\pi$
$200$	−311966.	−0.551484
$201$	0	0
$202$	469160.	0.808988
$203$	−151019.	−0.257212
$204$	0	0
$205$	−133787.	−0.222347
$206$	1.06489e6	1.74839
$207$	0	0
$208$	211156.	0.338411
$209$	−514003.	−0.813954
$210$	0	0
$211$	200532.	0.310083	0.155042	−	0.987908i	$-0.450449\pi$
$211$	200532.	0.310083	0.155042	+	0.987908i	$0.450449\pi$
$212$	571525.	0.873365
$213$	0	0
$214$	−261291.	−0.390023
$215$	−1.32903e6	−1.96083
$216$	0	0
$217$	−532355.	−0.767453
$218$	119647.	0.170514
$219$	0	0
$220$	997615.	1.38965
$221$	683387.	0.941209
$222$	0	0
$223$	227524.	0.306383	0.153192	−	0.988196i	$-0.451045\pi$
$223$	227524.	0.306383	0.153192	+	0.988196i	$0.451045\pi$
$224$	−519509.	−0.691787
$225$	0	0
$226$	257825.	0.335779
$227$	136415.	0.175711	0.0878553	−	0.996133i	$-0.471999\pi$
$227$	136415.	0.175711	0.0878553	+	0.996133i	$0.471999\pi$
$228$	0	0
$229$	−1.26300e6	−1.59153	−0.795763	−	0.605608i	$-0.792930\pi$
$229$	−1.26300e6	−1.59153	−0.795763	+	0.605608i	$0.792930\pi$
$230$	−390089.	−0.486232
$231$	0	0
$232$	−167654.	−0.204500
$233$	−449189.	−0.542050	−0.271025	−	0.962572i	$-0.587363\pi$
$233$	−449189.	−0.542050	−0.271025	+	0.962572i	$0.587363\pi$
$234$	0	0
$235$	−1.27656e6	−1.50790
$236$	−339327.	−0.396586
$237$	0	0
$238$	−1.21034e6	−1.38505
$239$	926816.	1.04954	0.524770	−	0.851244i	$-0.324151\pi$
$239$	926816.	1.04954	0.524770	+	0.851244i	$0.324151\pi$
$240$	0	0
$241$	−1.28019e6	−1.41982	−0.709910	−	0.704292i	$-0.751264\pi$
$241$	−1.28019e6	−1.41982	−0.709910	+	0.704292i	$0.751264\pi$
$242$	678182.	0.744403
$243$	0	0
$244$	1.05462e6	1.13402
$245$	−1.10188e6	−1.17279
$246$	0	0
$247$	554071.	0.577861
$248$	−590994.	−0.610174
$249$	0	0
$250$	−933170.	−0.944302
$251$	−694701.	−0.696007	−0.348003	−	0.937493i	$-0.613140\pi$
$251$	−694701.	−0.696007	−0.348003	+	0.937493i	$0.613140\pi$
$252$	0	0
$253$	150855.	0.148169
$254$	1.22175e6	1.18822
$255$	0	0
$256$	310280.	0.295906
$257$	994836.	0.939547	0.469774	−	0.882787i	$-0.344336\pi$
$257$	994836.	0.939547	0.469774	+	0.882787i	$0.344336\pi$
$258$	0	0
$259$	580847.	0.538037
$260$	−1.07538e6	−0.986574
$261$	0	0
$262$	−2.29524e6	−2.06574
$263$	−391688.	−0.349182	−0.174591	−	0.984641i	$-0.555860\pi$
$263$	−391688.	−0.349182	−0.174591	+	0.984641i	$0.555860\pi$
$264$	0	0
$265$	1.22781e6	1.07403
$266$	−981309.	−0.850358
$267$	0	0
$268$	−524713.	−0.446257
$269$	−1.20595e6	−1.01613	−0.508063	−	0.861320i	$-0.669638\pi$
$269$	−1.20595e6	−1.01613	−0.508063	+	0.861320i	$0.669638\pi$
$270$	0	0
$271$	−704840.	−0.582999	−0.291499	−	0.956571i	$-0.594154\pi$
$271$	−704840.	−0.582999	−0.291499	+	0.956571i	$0.594154\pi$
$272$	1.52708e6	1.25152
$273$	0	0
$274$	2.60116e6	2.09310
$275$	1.25203e6	0.998351
$276$	0	0
$277$	−1.05301e6	−0.824580	−0.412290	−	0.911053i	$-0.635271\pi$
$277$	−1.05301e6	−0.824580	−0.412290	+	0.911053i	$0.635271\pi$
$278$	1.49245e6	1.15821
$279$	0	0
$280$	394266.	0.300534
$281$	2.32763e6	1.75852	0.879260	−	0.476342i	$-0.158038\pi$
$281$	2.32763e6	1.75852	0.879260	+	0.476342i	$0.158038\pi$
$282$	0	0
$283$	−2.47456e6	−1.83667	−0.918335	−	0.395804i	$-0.870466\pi$
$283$	−2.47456e6	−1.83667	−0.918335	+	0.395804i	$0.870466\pi$
$284$	−2.11818e6	−1.55836
$285$	0	0
$286$	745654.	0.539042
$287$	98776.1	0.0707860
$288$	0	0
$289$	3.52240e6	2.48081
$290$	−1.73990e6	−1.21487
$291$	0	0
$292$	−685720.	−0.470641
$293$	961746.	0.654472	0.327236	−	0.944943i	$-0.393883\pi$
$293$	961746.	0.654472	0.327236	+	0.944943i	$0.393883\pi$
$294$	0	0
$295$	−728979.	−0.487708
$296$	644827.	0.427773
$297$	0	0
$298$	4.32065e6	2.81844
$299$	−162615.	−0.105192
$300$	0	0
$301$	981232.	0.624245
$302$	−1.82685e6	−1.15262
$303$	0	0
$304$	1.23811e6	0.768381
$305$	2.26564e6	1.39457
$306$	0	0
$307$	837166.	0.506951	0.253475	−	0.967342i	$-0.418426\pi$
$307$	837166.	0.506951	0.253475	+	0.967342i	$0.418426\pi$
$308$	−736545.	−0.442408
$309$	0	0
$310$	−6.13330e6	−3.62485
$311$	2.45359e6	1.43847	0.719234	−	0.694768i	$-0.244493\pi$
$311$	2.45359e6	1.43847	0.719234	+	0.694768i	$0.244493\pi$
$312$	0	0
$313$	2.20389e6	1.27154	0.635770	−	0.771879i	$-0.280683\pi$
$313$	2.20389e6	1.27154	0.635770	+	0.771879i	$0.280683\pi$
$314$	−1.53450e6	−0.878302
$315$	0	0
$316$	−4.03629e6	−2.27387
$317$	−2.16373e6	−1.20936	−0.604678	−	0.796470i	$-0.706698\pi$
$317$	−2.16373e6	−1.20936	−0.604678	+	0.796470i	$0.706698\pi$
$318$	0	0
$319$	672855.	0.370207
$320$	−4.07972e6	−2.22718
$321$	0	0
$322$	288005.	0.154796
$323$	4.00705e6	2.13707
$324$	0	0
$325$	−1.34963e6	−0.708773
$326$	−3.92874e6	−2.04743
$327$	0	0
$328$	109656.	0.0562793
$329$	942495.	0.480053
$330$	0	0
$331$	907798.	0.455427	0.227714	−	0.973728i	$-0.426875\pi$
$331$	907798.	0.455427	0.227714	+	0.973728i	$0.426875\pi$
$332$	3.44020e6	1.71293
$333$	0	0
$334$	71413.8	0.0350281
$335$	−1.12725e6	−0.548791
$336$	0	0
$337$	−2.29895e6	−1.10269	−0.551346	−	0.834277i	$-0.685886\pi$
$337$	−2.29895e6	−1.10269	−0.551346	+	0.834277i	$0.685886\pi$
$338$	2.35447e6	1.12099
$339$	0	0
$340$	−7.77717e6	−3.64858
$341$	2.37187e6	1.10460
$342$	0	0
$343$	1.88926e6	0.867075
$344$	1.08931e6	0.496315
$345$	0	0
$346$	5.74912e6	2.58173
$347$	3.67655e6	1.63914	0.819571	−	0.572977i	$-0.194212\pi$
$347$	3.67655e6	1.63914	0.819571	+	0.572977i	$0.194212\pi$
$348$	0	0
$349$	−4.10991e6	−1.80621	−0.903107	−	0.429416i	$-0.858719\pi$
$349$	−4.10991e6	−1.80621	−0.903107	+	0.429416i	$0.858719\pi$
$350$	2.39032e6	1.04300
$351$	0	0
$352$	2.31463e6	0.995692
$353$	−1.13740e6	−0.485819	−0.242910	−	0.970049i	$-0.578102\pi$
$353$	−1.13740e6	−0.485819	−0.242910	+	0.970049i	$0.578102\pi$
$354$	0	0
$355$	−4.55051e6	−1.91641
$356$	3.35241e6	1.40195
$357$	0	0
$358$	2.18399e6	0.900623
$359$	−1.64861e6	−0.675123	−0.337561	−	0.941304i	$-0.609602\pi$
$359$	−1.64861e6	−0.675123	−0.337561	+	0.941304i	$0.609602\pi$
$360$	0	0
$361$	772702.	0.312064
$362$	−5.53493e6	−2.21994
$363$	0	0
$364$	793962.	0.314084
$365$	−1.47314e6	−0.578778
$366$	0	0
$367$	781323.	0.302807	0.151403	−	0.988472i	$-0.451621\pi$
$367$	781323.	0.302807	0.151403	+	0.988472i	$0.451621\pi$
$368$	−363375.	−0.139873
$369$	0	0
$370$	6.69198e6	2.54127
$371$	−906503.	−0.341928
$372$	0	0
$373$	1.51034e6	0.562086	0.281043	−	0.959695i	$-0.409320\pi$
$373$	1.51034e6	0.562086	0.281043	+	0.959695i	$0.409320\pi$
$374$	5.39257e6	1.99350
$375$	0	0
$376$	1.04631e6	0.381673
$377$	−725306.	−0.262826
$378$	0	0
$379$	−2.11472e6	−0.756233	−0.378117	−	0.925758i	$-0.623428\pi$
$379$	−2.11472e6	−0.756233	−0.378117	+	0.925758i	$0.623428\pi$
$380$	−6.30552e6	−2.24007
$381$	0	0
$382$	−7.43497e6	−2.60688
$383$	1.57857e6	0.549878	0.274939	−	0.961462i	$-0.411342\pi$
$383$	1.57857e6	0.549878	0.274939	+	0.961462i	$0.411342\pi$
$384$	0	0
$385$	−1.58233e6	−0.544058
$386$	5.95710e6	2.03501
$387$	0	0
$388$	1.28296e6	0.432647
$389$	579997.	0.194335	0.0971677	−	0.995268i	$-0.469022\pi$
$389$	579997.	0.194335	0.0971677	+	0.995268i	$0.469022\pi$
$390$	0	0
$391$	−1.17603e6	−0.389024
$392$	903137.	0.296851
$393$	0	0
$394$	8.60751e6	2.79342
$395$	−8.67121e6	−2.79632
$396$	0	0
$397$	−642885.	−0.204719	−0.102359	−	0.994747i	$-0.532639\pi$
$397$	−642885.	−0.204719	−0.102359	+	0.994747i	$0.532639\pi$
$398$	−6.39324e6	−2.02308
$399$	0	0
$400$	−3.01585e6	−0.942455
$401$	−4.25371e6	−1.32101	−0.660506	−	0.750820i	$-0.729658\pi$
$401$	−4.25371e6	−1.32101	−0.660506	+	0.750820i	$0.729658\pi$
$402$	0	0
$403$	−2.55676e6	−0.784202
$404$	−2.22573e6	−0.678451
$405$	0	0
$406$	1.28458e6	0.386765
$407$	−2.58792e6	−0.774399
$408$	0	0
$409$	1.78257e6	0.526911	0.263456	−	0.964671i	$-0.415138\pi$
$409$	1.78257e6	0.526911	0.263456	+	0.964671i	$0.415138\pi$
$410$	1.13801e6	0.334338
$411$	0	0
$412$	−5.05193e6	−1.46627
$413$	538210.	0.155266
$414$	0	0
$415$	7.39062e6	2.10650
$416$	−2.49507e6	−0.706885
$417$	0	0
$418$	4.37215e6	1.22392
$419$	−1.47722e6	−0.411064	−0.205532	−	0.978650i	$-0.565893\pi$
$419$	−1.47722e6	−0.411064	−0.205532	+	0.978650i	$0.565893\pi$
$420$	0	0
$421$	2.38508e6	0.655840	0.327920	−	0.944705i	$-0.393652\pi$
$421$	2.38508e6	0.655840	0.327920	+	0.944705i	$0.393652\pi$
$422$	−1.70575e6	−0.466265
$423$	0	0
$424$	−1.00635e6	−0.271854
$425$	−9.76054e6	−2.62121
$426$	0	0
$427$	−1.67274e6	−0.443975
$428$	1.23958e6	0.327089
$429$	0	0
$430$	1.13048e7	2.94845
$431$	7.13658e6	1.85053	0.925267	−	0.379316i	$-0.123841\pi$
$431$	7.13658e6	1.85053	0.925267	+	0.379316i	$0.123841\pi$
$432$	0	0
$433$	5.73488e6	1.46996	0.734979	−	0.678090i	$-0.237192\pi$
$433$	5.73488e6	1.46996	0.734979	+	0.678090i	$0.237192\pi$
$434$	4.52826e6	1.15400
$435$	0	0
$436$	−567614.	−0.143000
$437$	−953492.	−0.238844
$438$	0	0
$439$	−3.39829e6	−0.841588	−0.420794	−	0.907156i	$-0.638249\pi$
$439$	−3.39829e6	−0.841588	−0.420794	+	0.907156i	$0.638249\pi$
$440$	−1.75662e6	−0.432560
$441$	0	0
$442$	−5.81295e6	−1.41527
$443$	4.85957e6	1.17649	0.588245	−	0.808683i	$-0.299819\pi$
$443$	4.85957e6	1.17649	0.588245	+	0.808683i	$0.299819\pi$
$444$	0	0
$445$	7.20203e6	1.72407
$446$	−1.93534e6	−0.460702
$447$	0	0
$448$	3.01208e6	0.709042
$449$	4.26858e6	0.999235	0.499617	−	0.866246i	$-0.333474\pi$
$449$	4.26858e6	0.999235	0.499617	+	0.866246i	$0.333474\pi$
$450$	0	0
$451$	−440089.	−0.101882
$452$	−1.22314e6	−0.281598
$453$	0	0
$454$	−1.16036e6	−0.264212
$455$	1.70568e6	0.386250
$456$	0	0
$457$	2.34262e6	0.524700	0.262350	−	0.964973i	$-0.415502\pi$
$457$	2.34262e6	0.524700	0.262350	+	0.964973i	$0.415502\pi$
$458$	1.07432e7	2.39314
$459$	0	0
$460$	1.85061e6	0.407775
$461$	1.94617e6	0.426508	0.213254	−	0.976997i	$-0.431594\pi$
$461$	1.94617e6	0.426508	0.213254	+	0.976997i	$0.431594\pi$
$462$	0	0
$463$	−7.57161e6	−1.64148	−0.820740	−	0.571302i	$-0.806439\pi$
$463$	−7.57161e6	−1.64148	−0.820740	+	0.571302i	$0.806439\pi$
$464$	−1.62075e6	−0.349479
$465$	0	0
$466$	3.82084e6	0.815068
$467$	744018.	0.157867	0.0789335	−	0.996880i	$-0.474849\pi$
$467$	744018.	0.157867	0.0789335	+	0.996880i	$0.474849\pi$
$468$	0	0
$469$	832254.	0.174712
$470$	1.08586e7	2.26740
$471$	0	0
$472$	597494.	0.123446
$473$	−4.37180e6	−0.898479
$474$	0	0
$475$	−7.91358e6	−1.60931
$476$	5.74193e6	1.16156
$477$	0	0
$478$	−7.88357e6	−1.57817
$479$	7.51659e6	1.49686	0.748432	−	0.663212i	$-0.230807\pi$
$479$	7.51659e6	1.49686	0.748432	+	0.663212i	$0.230807\pi$
$480$	0	0
$481$	2.78966e6	0.549779
$482$	1.08894e7	2.13495
$483$	0	0
$484$	−3.21735e6	−0.624287
$485$	2.75620e6	0.532055
$486$	0	0
$487$	1.74559e6	0.333518	0.166759	−	0.985998i	$-0.446670\pi$
$487$	1.74559e6	0.333518	0.166759	+	0.985998i	$0.446670\pi$
$488$	−1.85699e6	−0.352988
$489$	0	0
$490$	9.37271e6	1.76350
$491$	8.33975e6	1.56117	0.780584	−	0.625051i	$-0.214922\pi$
$491$	8.33975e6	1.56117	0.780584	+	0.625051i	$0.214922\pi$
$492$	0	0
$493$	−5.24542e6	−0.971992
$494$	−4.71298e6	−0.868916
$495$	0	0
$496$	−5.71328e6	−1.04275
$497$	3.35967e6	0.610106
$498$	0	0
$499$	2.85532e6	0.513338	0.256669	−	0.966499i	$-0.417375\pi$
$499$	2.85532e6	0.513338	0.256669	+	0.966499i	$0.417375\pi$
$500$	4.42703e6	0.791931
$501$	0	0
$502$	5.90918e6	1.04657
$503$	119267.	0.0210184	0.0105092	−	0.999945i	$-0.496655\pi$
$503$	119267.	0.0210184	0.0105092	+	0.999945i	$0.496655\pi$
$504$	0	0
$505$	−4.78156e6	−0.834336
$506$	−1.28318e6	−0.222799
$507$	0	0
$508$	−5.79607e6	−0.996492
$509$	−2.88252e6	−0.493149	−0.246575	−	0.969124i	$-0.579305\pi$
$509$	−2.88252e6	−0.493149	−0.246575	+	0.969124i	$0.579305\pi$
$510$	0	0
$511$	1.08763e6	0.184259
$512$	−7.13728e6	−1.20326
$513$	0	0
$514$	−8.46216e6	−1.41278
$515$	−1.08531e7	−1.80317
$516$	0	0
$517$	−4.19921e6	−0.690942
$518$	−4.94073e6	−0.809034
$519$	0	0
$520$	1.89356e6	0.307093
$521$	5.63013e6	0.908708	0.454354	−	0.890821i	$-0.349870\pi$
$521$	5.63013e6	0.908708	0.454354	+	0.890821i	$0.349870\pi$
$522$	0	0
$523$	1.95752e6	0.312933	0.156467	−	0.987683i	$-0.449990\pi$
$523$	1.95752e6	0.312933	0.156467	+	0.987683i	$0.449990\pi$
$524$	1.08888e7	1.73241
$525$	0	0
$526$	3.33174e6	0.525057
$527$	−1.84905e7	−2.90017
$528$	0	0
$529$	279841.	0.0434783
$530$	−1.04439e7	−1.61500
$531$	0	0
$532$	4.65540e6	0.713145
$533$	474396.	0.0723308
$534$	0	0
$535$	2.66301e6	0.402243
$536$	923927.	0.138907
$537$	0	0
$538$	1.02579e7	1.52793
$539$	−3.62461e6	−0.537389
$540$	0	0
$541$	4.40593e6	0.647208	0.323604	−	0.946193i	$-0.395105\pi$
$541$	4.40593e6	0.647208	0.323604	+	0.946193i	$0.395105\pi$
$542$	5.99543e6	0.876642
$543$	0	0
$544$	−1.80443e7	−2.61423
$545$	−1.21941e6	−0.175857
$546$	0	0
$547$	5.08750e6	0.727003	0.363502	−	0.931594i	$-0.381581\pi$
$547$	5.08750e6	0.727003	0.363502	+	0.931594i	$0.381581\pi$
$548$	−1.23401e7	−1.75536
$549$	0	0
$550$	−1.06499e7	−1.50120
$551$	−4.25284e6	−0.596760
$552$	0	0
$553$	6.40201e6	0.890233
$554$	8.95699e6	1.23990
$555$	0	0
$556$	−7.08028e6	−0.971322
$557$	−6.32135e6	−0.863320	−0.431660	−	0.902036i	$-0.642072\pi$
$557$	−6.32135e6	−0.863320	−0.431660	+	0.902036i	$0.642072\pi$
$558$	0	0
$559$	4.71260e6	0.637869
$560$	3.81147e6	0.513596
$561$	0	0
$562$	−1.97990e7	−2.64425
$563$	198801.	0.0264331	0.0132165	−	0.999913i	$-0.495793\pi$
$563$	198801.	0.0264331	0.0132165	+	0.999913i	$0.495793\pi$
$564$	0	0
$565$	−2.62769e6	−0.346300
$566$	2.10488e7	2.76176
$567$	0	0
$568$	3.72974e6	0.485073
$569$	1.26952e7	1.64384	0.821918	−	0.569605i	$-0.192904\pi$
$569$	1.26952e7	1.64384	0.821918	+	0.569605i	$0.192904\pi$
$570$	0	0
$571$	−371684.	−0.0477071	−0.0238536	−	0.999715i	$-0.507594\pi$
$571$	−371684.	−0.0477071	−0.0238536	+	0.999715i	$0.507594\pi$
$572$	−3.53744e6	−0.452063
$573$	0	0
$574$	−840198.	−0.106439
$575$	2.32256e6	0.292953
$576$	0	0
$577$	−8.62719e6	−1.07877	−0.539386	−	0.842058i	$-0.681344\pi$
$577$	−8.62719e6	−1.07877	−0.539386	+	0.842058i	$0.681344\pi$
$578$	−2.99618e7	−3.73034
$579$	0	0
$580$	8.25423e6	1.01884
$581$	−5.45654e6	−0.670621
$582$	0	0
$583$	4.03886e6	0.492138
$584$	1.20743e6	0.146498
$585$	0	0
$586$	−8.18069e6	−0.984115
$587$	−8.79998e6	−1.05411	−0.527056	−	0.849831i	$-0.676704\pi$
$587$	−8.79998e6	−1.05411	−0.527056	+	0.849831i	$0.676704\pi$
$588$	0	0
$589$	−1.49916e7	−1.78057
$590$	6.20076e6	0.733356
$591$	0	0
$592$	6.23370e6	0.731041
$593$	−6.28515e6	−0.733971	−0.366985	−	0.930227i	$-0.619610\pi$
$593$	−6.28515e6	−0.733971	−0.366985	+	0.930227i	$0.619610\pi$
$594$	0	0
$595$	1.23355e7	1.42844
$596$	−2.04975e7	−2.36366
$597$	0	0
$598$	1.38321e6	0.158174
$599$	6.18076e6	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$599$	6.18076e6	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$600$	0	0
$601$	196063.	0.0221417	0.0110708	−	0.999939i	$-0.496476\pi$
$601$	196063.	0.0221417	0.0110708	+	0.999939i	$0.496476\pi$
$602$	−8.34644e6	−0.938664
$603$	0	0
$604$	8.66672e6	0.966635
$605$	−6.91186e6	−0.767727
$606$	0	0
$607$	2.96914e6	0.327084	0.163542	−	0.986536i	$-0.447708\pi$
$607$	2.96914e6	0.327084	0.163542	+	0.986536i	$0.447708\pi$
$608$	−1.46298e7	−1.60502
$609$	0	0
$610$	−1.92717e7	−2.09699
$611$	4.52656e6	0.490530
$612$	0	0
$613$	−4.34673e6	−0.467209	−0.233605	−	0.972332i	$-0.575052\pi$
$613$	−4.34673e6	−0.467209	−0.233605	+	0.972332i	$0.575052\pi$
$614$	−7.12101e6	−0.762290
$615$	0	0
$616$	1.29692e6	0.137709
$617$	−1.11692e7	−1.18116	−0.590580	−	0.806979i	$-0.701101\pi$
$617$	−1.11692e7	−1.18116	−0.590580	+	0.806979i	$0.701101\pi$
$618$	0	0
$619$	2.31301e6	0.242633	0.121317	−	0.992614i	$-0.461288\pi$
$619$	2.31301e6	0.242633	0.121317	+	0.992614i	$0.461288\pi$
$620$	2.90968e7	3.03995
$621$	0	0
$622$	−2.08704e7	−2.16299
$623$	−5.31730e6	−0.548872
$624$	0	0
$625$	−4.20960e6	−0.431063
$626$	−1.87465e7	−1.91198
$627$	0	0
$628$	7.27980e6	0.736581
$629$	2.01748e7	2.03321
$630$	0	0
$631$	−1.11795e7	−1.11776	−0.558881	−	0.829248i	$-0.688769\pi$
$631$	−1.11795e7	−1.11776	−0.558881	+	0.829248i	$0.688769\pi$
$632$	7.10719e6	0.707791
$633$	0	0
$634$	1.84048e7	1.81848
$635$	−1.24518e7	−1.22545
$636$	0	0
$637$	3.90716e6	0.381516
$638$	−5.72336e6	−0.556672
$639$	0	0
$640$	1.21857e7	1.17598
$641$	−724583.	−0.0696536	−0.0348268	−	0.999393i	$-0.511088\pi$
$641$	−724583.	−0.0696536	−0.0348268	+	0.999393i	$0.511088\pi$
$642$	0	0
$643$	9.65287e6	0.920723	0.460361	−	0.887732i	$-0.347720\pi$
$643$	9.65287e6	0.920723	0.460361	+	0.887732i	$0.347720\pi$
$644$	−1.36632e6	−0.129818
$645$	0	0
$646$	−3.40843e7	−3.21346
$647$	1.20642e7	1.13302	0.566512	−	0.824053i	$-0.308292\pi$
$647$	1.20642e7	1.13302	0.566512	+	0.824053i	$0.308292\pi$
$648$	0	0
$649$	−2.39795e6	−0.223475
$650$	1.14801e7	1.06577
$651$	0	0
$652$	1.86382e7	1.71706
$653$	1.08359e7	0.994452	0.497226	−	0.867621i	$-0.334352\pi$
$653$	1.08359e7	0.994452	0.497226	+	0.867621i	$0.334352\pi$
$654$	0	0
$655$	2.33925e7	2.13046
$656$	1.06007e6	0.0961782
$657$	0	0
$658$	−8.01694e6	−0.721845
$659$	8.32711e6	0.746931	0.373466	−	0.927644i	$-0.378169\pi$
$659$	8.32711e6	0.746931	0.373466	+	0.927644i	$0.378169\pi$
$660$	0	0
$661$	1.42602e7	1.26946	0.634732	−	0.772732i	$-0.281110\pi$
$661$	1.42602e7	1.26946	0.634732	+	0.772732i	$0.281110\pi$
$662$	−7.72180e6	−0.684816
$663$	0	0
$664$	−6.05758e6	−0.533186
$665$	1.00013e7	0.877002
$666$	0	0
$667$	1.24817e6	0.108632
$668$	−338792.	−0.0293760
$669$	0	0
$670$	9.58846e6	0.825205
$671$	7.45276e6	0.639014
$672$	0	0
$673$	1.19925e7	1.02064	0.510319	−	0.859985i	$-0.329527\pi$
$673$	1.19925e7	1.02064	0.510319	+	0.859985i	$0.329527\pi$
$674$	1.95550e7	1.65809
$675$	0	0
$676$	−1.11698e7	−0.940107
$677$	−5.47203e6	−0.458857	−0.229428	−	0.973326i	$-0.573686\pi$
$677$	−5.47203e6	−0.458857	−0.229428	+	0.973326i	$0.573686\pi$
$678$	0	0
$679$	−2.03492e6	−0.169384
$680$	1.36942e7	1.13570
$681$	0	0
$682$	−2.01753e7	−1.66096
$683$	−1.57601e7	−1.29273	−0.646364	−	0.763029i	$-0.723711\pi$
$683$	−1.57601e7	−1.29273	−0.646364	+	0.763029i	$0.723711\pi$
$684$	0	0
$685$	−2.65104e7	−2.15868
$686$	−1.60702e7	−1.30380
$687$	0	0
$688$	1.05307e7	0.848174
$689$	−4.35370e6	−0.349390
$690$	0	0
$691$	1.53486e7	1.22285	0.611424	−	0.791303i	$-0.290597\pi$
$691$	1.53486e7	1.22285	0.611424	+	0.791303i	$0.290597\pi$
$692$	−2.72742e7	−2.16515
$693$	0	0
$694$	−3.12730e7	−2.46474
$695$	−1.52106e7	−1.19450
$696$	0	0
$697$	3.43084e6	0.267496
$698$	3.49593e7	2.71596
$699$	0	0
$700$	−1.13398e7	−0.874705
$701$	−1.48889e7	−1.14437	−0.572187	−	0.820123i	$-0.693905\pi$
$701$	−1.48889e7	−1.14437	−0.572187	+	0.820123i	$0.693905\pi$
$702$	0	0
$703$	1.63572e7	1.24830
$704$	−1.34201e7	−1.02053
$705$	0	0
$706$	9.67478e6	0.730516
$707$	3.53025e6	0.265618
$708$	0	0
$709$	2.04675e6	0.152915	0.0764573	−	0.997073i	$-0.475639\pi$
$709$	2.04675e6	0.152915	0.0764573	+	0.997073i	$0.475639\pi$
$710$	3.87070e7	2.88167
$711$	0	0
$712$	−5.90300e6	−0.436388
$713$	4.39990e6	0.324129
$714$	0	0
$715$	−7.59952e6	−0.555931
$716$	−1.03610e7	−0.755300
$717$	0	0
$718$	1.40232e7	1.01517
$719$	1.09836e7	0.792357	0.396179	−	0.918173i	$-0.370336\pi$
$719$	1.09836e7	0.792357	0.396179	+	0.918173i	$0.370336\pi$
$720$	0	0
$721$	8.01292e6	0.574054
$722$	−6.57267e6	−0.469244
$723$	0	0
$724$	2.62581e7	1.86173
$725$	1.03593e7	0.731954
$726$	0	0
$727$	−6.92842e6	−0.486181	−0.243091	−	0.970004i	$-0.578161\pi$
$727$	−6.92842e6	−0.486181	−0.243091	+	0.970004i	$0.578161\pi$
$728$	−1.39803e6	−0.0977657
$729$	0	0
$730$	1.25307e7	0.870296
$731$	3.40816e7	2.35899
$732$	0	0
$733$	3.44517e6	0.236838	0.118419	−	0.992964i	$-0.462217\pi$
$733$	3.44517e6	0.236838	0.118419	+	0.992964i	$0.462217\pi$
$734$	−6.64600e6	−0.455324
$735$	0	0
$736$	4.29372e6	0.292172
$737$	−3.70804e6	−0.251464
$738$	0	0
$739$	3.85428e6	0.259617	0.129808	−	0.991539i	$-0.458564\pi$
$739$	3.85428e6	0.259617	0.129808	+	0.991539i	$0.458564\pi$
$740$	−3.17472e7	−2.13121
$741$	0	0
$742$	7.71079e6	0.514149
$743$	−7.82591e6	−0.520071	−0.260035	−	0.965599i	$-0.583734\pi$
$743$	−7.82591e6	−0.520071	−0.260035	+	0.965599i	$0.583734\pi$
$744$	0	0
$745$	−4.40350e7	−2.90675
$746$	−1.28471e7	−0.845196
$747$	0	0
$748$	−2.55828e7	−1.67183
$749$	−1.96612e6	−0.128057
$750$	0	0
$751$	−8.96338e6	−0.579925	−0.289963	−	0.957038i	$-0.593643\pi$
$751$	−8.96338e6	−0.579925	−0.289963	+	0.957038i	$0.593643\pi$
$752$	1.01149e7	0.652257
$753$	0	0
$754$	6.16952e6	0.395205
$755$	1.86188e7	1.18873
$756$	0	0
$757$	−1.29062e7	−0.818574	−0.409287	−	0.912406i	$-0.634223\pi$
$757$	−1.29062e7	−0.818574	−0.409287	+	0.912406i	$0.634223\pi$
$758$	1.79880e7	1.13713
$759$	0	0
$760$	1.11029e7	0.697272
$761$	1.66351e7	1.04127	0.520636	−	0.853779i	$-0.325695\pi$
$761$	1.66351e7	1.04127	0.520636	+	0.853779i	$0.325695\pi$
$762$	0	0
$763$	900299.	0.0559855
$764$	3.52720e7	2.18624
$765$	0	0
$766$	−1.34274e7	−0.826840
$767$	2.58488e6	0.158655
$768$	0	0
$769$	7.10800e6	0.433443	0.216721	−	0.976233i	$-0.430464\pi$
$769$	7.10800e6	0.433443	0.216721	+	0.976233i	$0.430464\pi$
$770$	1.34594e7	0.818087
$771$	0	0
$772$	−2.82609e7	−1.70664
$773$	−1.27704e7	−0.768700	−0.384350	−	0.923187i	$-0.625574\pi$
$773$	−1.27704e7	−0.768700	−0.384350	+	0.923187i	$0.625574\pi$
$774$	0	0
$775$	3.65172e7	2.18396
$776$	−2.25906e6	−0.134671
$777$	0	0
$778$	−4.93351e6	−0.292218
$779$	2.78163e6	0.164231
$780$	0	0
$781$	−1.49687e7	−0.878128
$782$	1.00034e7	0.584967
$783$	0	0
$784$	8.73085e6	0.507302
$785$	1.56393e7	0.905821
$786$	0	0
$787$	1.38003e7	0.794240	0.397120	−	0.917767i	$-0.370010\pi$
$787$	1.38003e7	0.794240	0.397120	+	0.917767i	$0.370010\pi$
$788$	−4.08346e7	−2.34268
$789$	0	0
$790$	7.37581e7	4.20477
$791$	1.94004e6	0.110248
$792$	0	0
$793$	−8.03373e6	−0.453664
$794$	5.46844e6	0.307831
$795$	0	0
$796$	3.03300e7	1.69664
$797$	−2.14559e7	−1.19647	−0.598233	−	0.801322i	$-0.704130\pi$
$797$	−2.14559e7	−1.19647	−0.598233	+	0.801322i	$0.704130\pi$
$798$	0	0
$799$	3.27361e7	1.81409
$800$	3.56360e7	1.96863
$801$	0	0
$802$	3.61824e7	1.98638
$803$	−4.84585e6	−0.265205
$804$	0	0
$805$	−2.93527e6	−0.159646
$806$	2.17481e7	1.17919
$807$	0	0
$808$	3.91911e6	0.211183
$809$	1.09665e7	0.589113	0.294556	−	0.955634i	$-0.404828\pi$
$809$	1.09665e7	0.589113	0.294556	+	0.955634i	$0.404828\pi$
$810$	0	0
$811$	−2.58794e7	−1.38166	−0.690832	−	0.723015i	$-0.742756\pi$
$811$	−2.58794e7	−1.38166	−0.690832	+	0.723015i	$0.742756\pi$
$812$	−6.09415e6	−0.324357
$813$	0	0
$814$	2.20130e7	1.16445
$815$	4.00407e7	2.11158
$816$	0	0
$817$	2.76324e7	1.44832
$818$	−1.51627e7	−0.792305
$819$	0	0
$820$	−5.39879e6	−0.280389
$821$	2.30093e7	1.19136	0.595682	−	0.803220i	$-0.296882\pi$
$821$	2.30093e7	1.19136	0.595682	+	0.803220i	$0.296882\pi$
$822$	0	0
$823$	−7.51381e6	−0.386688	−0.193344	−	0.981131i	$-0.561933\pi$
$823$	−7.51381e6	−0.386688	−0.193344	+	0.981131i	$0.561933\pi$
$824$	8.89555e6	0.456409
$825$	0	0
$826$	−4.57806e6	−0.233470
$827$	1.08185e7	0.550051	0.275025	−	0.961437i	$-0.411314\pi$
$827$	1.08185e7	0.550051	0.275025	+	0.961437i	$0.411314\pi$
$828$	0	0
$829$	2.11950e6	0.107114	0.0535572	−	0.998565i	$-0.482944\pi$
$829$	2.11950e6	0.107114	0.0535572	+	0.998565i	$0.482944\pi$
$830$	−6.28653e7	−3.16749
$831$	0	0
$832$	1.44663e7	0.724516
$833$	2.82566e7	1.41094
$834$	0	0
$835$	−727832.	−0.0361256
$836$	−2.07418e7	−1.02643
$837$	0	0
$838$	1.25653e7	0.618108
$839$	9.53545e6	0.467667	0.233833	−	0.972277i	$-0.424873\pi$
$839$	9.53545e6	0.467667	0.233833	+	0.972277i	$0.424873\pi$
$840$	0	0
$841$	−1.49440e7	−0.728578
$842$	−2.02877e7	−0.986172
$843$	0	0
$844$	8.09218e6	0.391029
$845$	−2.39961e7	−1.15611
$846$	0	0
$847$	5.10307e6	0.244412
$848$	−9.72868e6	−0.464584
$849$	0	0
$850$	8.30240e7	3.94145
$851$	−4.80068e6	−0.227237
$852$	0	0
$853$	2.54943e7	1.19969	0.599846	−	0.800115i	$-0.295228\pi$
$853$	2.54943e7	1.19969	0.599846	+	0.800115i	$0.295228\pi$
$854$	1.42284e7	0.667595
$855$	0	0
$856$	−2.18268e6	−0.101814
$857$	−8.02146e6	−0.373079	−0.186540	−	0.982447i	$-0.559727\pi$
$857$	−8.02146e6	−0.373079	−0.186540	+	0.982447i	$0.559727\pi$
$858$	0	0
$859$	−4.80807e6	−0.222325	−0.111162	−	0.993802i	$-0.535457\pi$
$859$	−4.80807e6	−0.222325	−0.111162	+	0.993802i	$0.535457\pi$
$860$	−5.36310e7	−2.47269
$861$	0	0
$862$	−6.07044e7	−2.78261
$863$	−1.98941e7	−0.909280	−0.454640	−	0.890675i	$-0.650232\pi$
$863$	−1.98941e7	−0.909280	−0.454640	+	0.890675i	$0.650232\pi$
$864$	0	0
$865$	−5.85936e7	−2.66262
$866$	−4.87814e7	−2.21034
$867$	0	0
$868$	−2.14824e7	−0.967794
$869$	−2.85237e7	−1.28132
$870$	0	0
$871$	3.99710e6	0.178525
$872$	999467.	0.0445120
$873$	0	0
$874$	8.11049e6	0.359144
$875$	−7.02176e6	−0.310046
$876$	0	0
$877$	−1.69776e7	−0.745381	−0.372691	−	0.927956i	$-0.621565\pi$
$877$	−1.69776e7	−0.745381	−0.372691	+	0.927956i	$0.621565\pi$
$878$	2.89062e7	1.26548
$879$	0	0
$880$	−1.69817e7	−0.739221
$881$	−1.94798e7	−0.845562	−0.422781	−	0.906232i	$-0.638946\pi$
$881$	−1.94798e7	−0.845562	−0.422781	+	0.906232i	$0.638946\pi$
$882$	0	0
$883$	−4.20175e7	−1.81354	−0.906772	−	0.421621i	$-0.861461\pi$
$883$	−4.20175e7	−1.81354	−0.906772	+	0.421621i	$0.861461\pi$
$884$	2.75770e7	1.18691
$885$	0	0
$886$	−4.13359e7	−1.76906
$887$	−3.23258e7	−1.37956	−0.689779	−	0.724020i	$-0.742292\pi$
$887$	−3.23258e7	−1.37956	−0.689779	+	0.724020i	$0.742292\pi$
$888$	0	0
$889$	9.19321e6	0.390133
$890$	−6.12611e7	−2.59244
$891$	0	0
$892$	9.18139e6	0.386364
$893$	2.65415e7	1.11377
$894$	0	0
$895$	−2.22587e7	−0.928842
$896$	−8.99675e6	−0.374382
$897$	0	0
$898$	−3.63089e7	−1.50253
$899$	1.96247e7	0.809850
$900$	0	0
$901$	−3.14860e7	−1.29213
$902$	3.74344e6	0.153198
$903$	0	0
$904$	2.15373e6	0.0876538
$905$	5.64106e7	2.28949
$906$	0	0
$907$	1.37422e7	0.554676	0.277338	−	0.960772i	$-0.410548\pi$
$907$	1.37422e7	0.554676	0.277338	+	0.960772i	$0.410548\pi$
$908$	5.50483e6	0.221579
$909$	0	0
$910$	−1.45086e7	−0.580795
$911$	1.43255e7	0.571891	0.285945	−	0.958246i	$-0.407692\pi$
$911$	1.43255e7	0.571891	0.285945	+	0.958246i	$0.407692\pi$
$912$	0	0
$913$	2.43112e7	0.965228
$914$	−1.99265e7	−0.788980
$915$	0	0
$916$	−5.09663e7	−2.00699
$917$	−1.72708e7	−0.678250
$918$	0	0
$919$	4.09852e7	1.60080	0.800402	−	0.599463i	$-0.204619\pi$
$919$	4.09852e7	1.60080	0.800402	+	0.599463i	$0.204619\pi$
$920$	−3.25859e6	−0.126929
$921$	0	0
$922$	−1.65543e7	−0.641331
$923$	1.61356e7	0.623421
$924$	0	0
$925$	−3.98436e7	−1.53110
$926$	6.44047e7	2.46826
$927$	0	0
$928$	1.91512e7	0.730004
$929$	−2.88621e7	−1.09721	−0.548603	−	0.836083i	$-0.684840\pi$
$929$	−2.88621e7	−1.09721	−0.548603	+	0.836083i	$0.684840\pi$
$930$	0	0
$931$	2.29097e7	0.866253
$932$	−1.81263e7	−0.683550
$933$	0	0
$934$	−6.32868e6	−0.237381
$935$	−5.49597e7	−2.05596
$936$	0	0
$937$	−3.51671e7	−1.30854	−0.654272	−	0.756260i	$-0.727025\pi$
$937$	−3.51671e7	−1.30854	−0.654272	+	0.756260i	$0.727025\pi$
$938$	−7.07922e6	−0.262711
$939$	0	0
$940$	−5.15138e7	−1.90153
$941$	−4.23288e7	−1.55834	−0.779170	−	0.626813i	$-0.784359\pi$
$941$	−4.23288e7	−1.55834	−0.779170	+	0.626813i	$0.784359\pi$
$942$	0	0
$943$	−816381.	−0.0298960
$944$	5.77612e6	0.210963
$945$	0	0
$946$	3.71869e7	1.35102
$947$	2.17631e7	0.788579	0.394290	−	0.918986i	$-0.370991\pi$
$947$	2.17631e7	0.788579	0.394290	+	0.918986i	$0.370991\pi$
$948$	0	0
$949$	5.22360e6	0.188280
$950$	6.73136e7	2.41988
$951$	0	0
$952$	−1.01105e7	−0.361561
$953$	2.52072e7	0.899067	0.449533	−	0.893264i	$-0.351590\pi$
$953$	2.52072e7	0.899067	0.449533	+	0.893264i	$0.351590\pi$
$954$	0	0
$955$	7.57754e7	2.68856
$956$	3.74002e7	1.32352
$957$	0	0
$958$	−6.39368e7	−2.25080
$959$	1.95728e7	0.687235
$960$	0	0
$961$	4.05497e7	1.41638
$962$	−2.37291e7	−0.826690
$963$	0	0
$964$	−5.16603e7	−1.79046
$965$	−6.07133e7	−2.09877
$966$	0	0
$967$	5.15354e6	0.177231	0.0886154	−	0.996066i	$-0.471756\pi$
$967$	5.15354e6	0.177231	0.0886154	+	0.996066i	$0.471756\pi$
$968$	5.66517e6	0.194323
$969$	0	0
$970$	−2.34445e7	−0.800038
$971$	−4.87285e7	−1.65858	−0.829288	−	0.558822i	$-0.811254\pi$
$971$	−4.87285e7	−1.65858	−0.829288	+	0.558822i	$0.811254\pi$
$972$	0	0
$973$	1.12301e7	0.380279
$974$	−1.48481e7	−0.501504
$975$	0	0
$976$	−1.79520e7	−0.603237
$977$	2.08248e7	0.697981	0.348991	−	0.937126i	$-0.386524\pi$
$977$	2.08248e7	0.697981	0.348991	+	0.937126i	$0.386524\pi$
$978$	0	0
$979$	2.36908e7	0.789994
$980$	−4.44648e7	−1.47894
$981$	0	0
$982$	−7.09386e7	−2.34749
$983$	−1.68825e7	−0.557255	−0.278627	−	0.960399i	$-0.589879\pi$
$983$	−1.68825e7	−0.557255	−0.278627	+	0.960399i	$0.589879\pi$
$984$	0	0
$985$	−8.77255e7	−2.88095
$986$	4.46180e7	1.46156
$987$	0	0
$988$	2.23587e7	0.728709
$989$	−8.10984e6	−0.263646
$990$	0	0
$991$	−1.28291e6	−0.0414965	−0.0207482	−	0.999785i	$-0.506605\pi$
$991$	−1.28291e6	−0.0414965	−0.0207482	+	0.999785i	$0.506605\pi$
$992$	6.75095e7	2.17814
$993$	0	0
$994$	−2.85776e7	−0.917403
$995$	6.51583e7	2.08647
$996$	0	0
$997$	1.88209e7	0.599658	0.299829	−	0.953993i	$-0.403070\pi$
$997$	1.88209e7	0.599658	0.299829	+	0.953993i	$0.403070\pi$
$998$	−2.42876e7	−0.771894
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.e.1.1		4
3.2	odd	2		69.6.a.d.1.4	✓	4
12.11	even	2		1104.6.a.o.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.d.1.4	✓	4	3.2	odd	2
207.6.a.e.1.1		4	1.1	even	1	trivial
1104.6.a.o.1.1		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q-8.50608 q^{2} +40.3535 q^{4} +86.6918 q^{5} -64.0051 q^{7} -71.0553 q^{8} +O(q^{10})\)	\(q-8.50608 q^{2} +40.3535 q^{4} +86.6918 q^{5} -64.0051 q^{7} -71.0553 q^{8} -737.408 q^{10} +285.170 q^{11} -307.400 q^{13} +544.433 q^{14} -686.909 q^{16} -2223.12 q^{17} -1802.44 q^{19} +3498.32 q^{20} -2425.68 q^{22} +529.000 q^{23} +4390.47 q^{25} +2614.77 q^{26} -2582.83 q^{28} +2359.49 q^{29} +8317.38 q^{31} +8116.67 q^{32} +18910.0 q^{34} -5548.72 q^{35} -9075.00 q^{37} +15331.7 q^{38} -6159.91 q^{40} -1543.25 q^{41} -15330.5 q^{43} +11507.6 q^{44} -4499.72 q^{46} -14725.3 q^{47} -12710.3 q^{49} -37345.7 q^{50} -12404.7 q^{52} +14163.0 q^{53} +24721.9 q^{55} +4547.90 q^{56} -20070.0 q^{58} -8408.86 q^{59} +26134.4 q^{61} -70748.4 q^{62} -47060.0 q^{64} -26649.1 q^{65} -13002.9 q^{67} -89710.5 q^{68} +47197.9 q^{70} -52490.6 q^{71} -16992.9 q^{73} +77192.7 q^{74} -72734.8 q^{76} -18252.3 q^{77} -100023. q^{79} -59549.4 q^{80} +13127.0 q^{82} +85251.7 q^{83} -192726. q^{85} +130403. q^{86} -20262.8 q^{88} +83076.2 q^{89} +19675.2 q^{91} +21347.0 q^{92} +125255. q^{94} -156257. q^{95} +31793.1 q^{97} +108115. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 62 * q^7 - 72 * q^8	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 62 q^{7} - 72 q^{8} - 496 q^{10} + 1076 q^{11} - 396 q^{13} + 1806 q^{14} - 1982 q^{16} - 70 q^{17} - 6366 q^{19} + 5240 q^{20} - 6974 q^{22} + 2116 q^{23} + 1264 q^{25} - 2464 q^{26} - 6474 q^{28} - 3948 q^{29} + 3092 q^{31} + 3672 q^{32} + 11682 q^{34} - 1304 q^{35} - 17464 q^{37} + 12628 q^{38} - 14108 q^{40} - 18680 q^{41} - 25846 q^{43} - 20746 q^{44} - 2116 q^{46} - 18392 q^{47} + 7952 q^{49} - 69444 q^{50} + 8844 q^{52} + 26518 q^{53} - 40848 q^{55} - 54890 q^{56} + 568 q^{58} + 14520 q^{59} - 13688 q^{61} - 120136 q^{62} - 30190 q^{64} - 38324 q^{65} - 11098 q^{67} - 112138 q^{68} - 29596 q^{70} + 57496 q^{71} - 112272 q^{73} + 21226 q^{74} - 76240 q^{76} + 4792 q^{77} - 240754 q^{79} - 41200 q^{80} + 49976 q^{82} + 93268 q^{83} - 323204 q^{85} + 88224 q^{86} + 42382 q^{88} + 107582 q^{89} - 301532 q^{91} + 13754 q^{92} + 79360 q^{94} + 18640 q^{95} - 53076 q^{97} - 59664 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 62 * q^7 - 72 * q^8 - 496 * q^10 + 1076 * q^11 - 396 * q^13 + 1806 * q^14 - 1982 * q^16 - 70 * q^17 - 6366 * q^19 + 5240 * q^20 - 6974 * q^22 + 2116 * q^23 + 1264 * q^25 - 2464 * q^26 - 6474 * q^28 - 3948 * q^29 + 3092 * q^31 + 3672 * q^32 + 11682 * q^34 - 1304 * q^35 - 17464 * q^37 + 12628 * q^38 - 14108 * q^40 - 18680 * q^41 - 25846 * q^43 - 20746 * q^44 - 2116 * q^46 - 18392 * q^47 + 7952 * q^49 - 69444 * q^50 + 8844 * q^52 + 26518 * q^53 - 40848 * q^55 - 54890 * q^56 + 568 * q^58 + 14520 * q^59 - 13688 * q^61 - 120136 * q^62 - 30190 * q^64 - 38324 * q^65 - 11098 * q^67 - 112138 * q^68 - 29596 * q^70 + 57496 * q^71 - 112272 * q^73 + 21226 * q^74 - 76240 * q^76 + 4792 * q^77 - 240754 * q^79 - 41200 * q^80 + 49976 * q^82 + 93268 * q^83 - 323204 * q^85 + 88224 * q^86 + 42382 * q^88 + 107582 * q^89 - 301532 * q^91 + 13754 * q^92 + 79360 * q^94 + 18640 * q^95 - 53076 * q^97 - 59664 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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