LMFDB - Embedded newform 207.6.a.f.1.5

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:	$5$
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} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.42196$ 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)

$f(q)$	$=$	$q+9.32729 q^{2} +54.9984 q^{4} -99.7124 q^{5} +125.983 q^{7} +214.513 q^{8} +O(q^{10})$	\(q+9.32729 q^{2} +54.9984 q^{4} -99.7124 q^{5} +125.983 q^{7} +214.513 q^{8} -930.047 q^{10} -177.715 q^{11} -919.618 q^{13} +1175.08 q^{14} +240.875 q^{16} -1568.24 q^{17} +447.960 q^{19} -5484.02 q^{20} -1657.60 q^{22} +529.000 q^{23} +6817.57 q^{25} -8577.55 q^{26} +6928.86 q^{28} +1298.72 q^{29} -6651.88 q^{31} -4617.70 q^{32} -14627.4 q^{34} -12562.1 q^{35} -8056.80 q^{37} +4178.25 q^{38} -21389.6 q^{40} -17941.4 q^{41} +10710.7 q^{43} -9774.04 q^{44} +4934.14 q^{46} +17886.9 q^{47} -935.293 q^{49} +63589.5 q^{50} -50577.5 q^{52} +20343.9 q^{53} +17720.4 q^{55} +27025.0 q^{56} +12113.5 q^{58} -28488.5 q^{59} +16222.3 q^{61} -62044.1 q^{62} -50778.6 q^{64} +91697.4 q^{65} +54128.0 q^{67} -86250.6 q^{68} -117170. q^{70} -42505.6 q^{71} +40936.2 q^{73} -75148.1 q^{74} +24637.1 q^{76} -22389.1 q^{77} -24760.6 q^{79} -24018.2 q^{80} -167345. q^{82} +30825.2 q^{83} +156373. q^{85} +99901.7 q^{86} -38122.1 q^{88} +66531.6 q^{89} -115856. q^{91} +29094.2 q^{92} +166836. q^{94} -44667.2 q^{95} -104506. q^{97} -8723.75 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8}+O(q^{10}) $ 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8	$ 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8} - 172 q^{10} - 1100 q^{11} - 978 q^{13} + 344 q^{14} + 1218 q^{16} - 2522 q^{17} + 2060 q^{19} - 7720 q^{20} - 2572 q^{22} + 2645 q^{23} + 12035 q^{25} - 9280 q^{26} + 8072 q^{28} - 1526 q^{29} - 7392 q^{31} + 5086 q^{32} - 15608 q^{34} - 6056 q^{35} - 8210 q^{37} + 14276 q^{38} - 37472 q^{40} - 21250 q^{41} - 4548 q^{43} + 4260 q^{44} - 4232 q^{46} - 536 q^{47} - 27979 q^{49} + 81872 q^{50} - 76380 q^{52} + 11482 q^{53} - 77064 q^{55} + 28624 q^{56} - 79680 q^{58} - 74676 q^{59} - 44618 q^{61} - 64880 q^{62} - 137382 q^{64} + 24388 q^{65} - 1412 q^{67} - 80196 q^{68} - 222304 q^{70} - 37912 q^{71} + 46546 q^{73} - 111604 q^{74} - 79548 q^{76} - 157008 q^{77} + 50544 q^{79} - 69424 q^{80} - 233720 q^{82} - 89588 q^{83} + 147892 q^{85} - 77428 q^{86} + 54484 q^{88} - 280410 q^{89} - 27416 q^{91} + 62422 q^{92} + 113632 q^{94} - 203120 q^{95} + 90074 q^{97} - 32976 q^{98}+O(q^{100}) $ 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8 - 172 * q^10 - 1100 * q^11 - 978 * q^13 + 344 * q^14 + 1218 * q^16 - 2522 * q^17 + 2060 * q^19 - 7720 * q^20 - 2572 * q^22 + 2645 * q^23 + 12035 * q^25 - 9280 * q^26 + 8072 * q^28 - 1526 * q^29 - 7392 * q^31 + 5086 * q^32 - 15608 * q^34 - 6056 * q^35 - 8210 * q^37 + 14276 * q^38 - 37472 * q^40 - 21250 * q^41 - 4548 * q^43 + 4260 * q^44 - 4232 * q^46 - 536 * q^47 - 27979 * q^49 + 81872 * q^50 - 76380 * q^52 + 11482 * q^53 - 77064 * q^55 + 28624 * q^56 - 79680 * q^58 - 74676 * q^59 - 44618 * q^61 - 64880 * q^62 - 137382 * q^64 + 24388 * q^65 - 1412 * q^67 - 80196 * q^68 - 222304 * q^70 - 37912 * q^71 + 46546 * q^73 - 111604 * q^74 - 79548 * q^76 - 157008 * q^77 + 50544 * q^79 - 69424 * q^80 - 233720 * q^82 - 89588 * q^83 + 147892 * q^85 - 77428 * q^86 + 54484 * q^88 - 280410 * q^89 - 27416 * q^91 + 62422 * q^92 + 113632 * q^94 - 203120 * q^95 + 90074 * q^97 - 32976 * 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 $

$2$	9.32729	1.64885	0.824424	−	0.565973i	$-0.191499\pi$
$2$	9.32729	1.64885	0.824424	+	0.565973i	$0.191499\pi$
$3$	0	0
$3$	0	0
$4$	54.9984	1.71870
$5$	−99.7124	−1.78371	−0.891855	−	0.452321i	$-0.850596\pi$
$5$	−99.7124	−1.78371	−0.891855	+	0.452321i	$0.850596\pi$
$6$	0	0
$7$	125.983	0.971777	0.485889	−	0.874021i	$-0.338496\pi$
$7$	125.983	0.971777	0.485889	+	0.874021i	$0.338496\pi$
$8$	214.513	1.18503
$9$	0	0
$10$	−930.047	−2.94107
$11$	−177.715	−0.442835	−0.221418	−	0.975179i	$-0.571068\pi$
$11$	−177.715	−0.442835	−0.221418	+	0.975179i	$0.571068\pi$
$12$	0	0
$13$	−919.618	−1.50921	−0.754604	−	0.656180i	$-0.772171\pi$
$13$	−919.618	−1.50921	−0.754604	+	0.656180i	$0.772171\pi$
$14$	1175.08	1.60231
$15$	0	0
$16$	240.875	0.235229
$17$	−1568.24	−1.31610	−0.658051	−	0.752973i	$-0.728619\pi$
$17$	−1568.24	−1.31610	−0.658051	+	0.752973i	$0.728619\pi$
$18$	0	0
$19$	447.960	0.284679	0.142339	−	0.989818i	$-0.454538\pi$
$19$	447.960	0.284679	0.142339	+	0.989818i	$0.454538\pi$
$20$	−5484.02	−3.06566
$21$	0	0
$22$	−1657.60	−0.730168
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	6817.57	2.18162
$26$	−8577.55	−2.48846
$27$	0	0
$28$	6928.86	1.67019
$29$	1298.72	0.286760	0.143380	−	0.989668i	$-0.454203\pi$
$29$	1298.72	0.286760	0.143380	+	0.989668i	$0.454203\pi$
$30$	0	0
$31$	−6651.88	−1.24320	−0.621599	−	0.783336i	$-0.713517\pi$
$31$	−6651.88	−1.24320	−0.621599	+	0.783336i	$0.713517\pi$
$32$	−4617.70	−0.797169
$33$	0	0
$34$	−14627.4	−2.17005
$35$	−12562.1	−1.73337
$36$	0	0
$37$	−8056.80	−0.967516	−0.483758	−	0.875202i	$-0.660728\pi$
$37$	−8056.80	−0.967516	−0.483758	+	0.875202i	$0.660728\pi$
$38$	4178.25	0.469392
$39$	0	0
$40$	−21389.6	−2.11374
$41$	−17941.4	−1.66685	−0.833427	−	0.552630i	$-0.813624\pi$
$41$	−17941.4	−1.66685	−0.833427	+	0.552630i	$0.813624\pi$
$42$	0	0
$43$	10710.7	0.883377	0.441688	−	0.897169i	$-0.354380\pi$
$43$	10710.7	0.883377	0.441688	+	0.897169i	$0.354380\pi$
$44$	−9774.04	−0.761101
$45$	0	0
$46$	4934.14	0.343809
$47$	17886.9	1.18111	0.590555	−	0.806997i	$-0.298909\pi$
$47$	17886.9	1.18111	0.590555	+	0.806997i	$0.298909\pi$
$48$	0	0
$49$	−935.293	−0.0556490
$50$	63589.5	3.59716
$51$	0	0
$52$	−50577.5	−2.59388
$53$	20343.9	0.994820	0.497410	−	0.867516i	$-0.334284\pi$
$53$	20343.9	0.994820	0.497410	+	0.867516i	$0.334284\pi$
$54$	0	0
$55$	17720.4	0.789890
$56$	27025.0	1.15158
$57$	0	0
$58$	12113.5	0.472824
$59$	−28488.5	−1.06547	−0.532734	−	0.846283i	$-0.678835\pi$
$59$	−28488.5	−1.06547	−0.532734	+	0.846283i	$0.678835\pi$
$60$	0	0
$61$	16222.3	0.558196	0.279098	−	0.960263i	$-0.409965\pi$
$61$	16222.3	0.558196	0.279098	+	0.960263i	$0.409965\pi$
$62$	−62044.1	−2.04984
$63$	0	0
$64$	−50778.6	−1.54964
$65$	91697.4	2.69199
$66$	0	0
$67$	54128.0	1.47311	0.736555	−	0.676378i	$-0.236451\pi$
$67$	54128.0	1.47311	0.736555	+	0.676378i	$0.236451\pi$
$68$	−86250.6	−2.26198
$69$	0	0
$70$	−117170.	−2.85806
$71$	−42505.6	−1.00069	−0.500346	−	0.865826i	$-0.666794\pi$
$71$	−42505.6	−1.00069	−0.500346	+	0.865826i	$0.666794\pi$
$72$	0	0
$73$	40936.2	0.899084	0.449542	−	0.893259i	$-0.351587\pi$
$73$	40936.2	0.899084	0.449542	+	0.893259i	$0.351587\pi$
$74$	−75148.1	−1.59529
$75$	0	0
$76$	24637.1	0.489277
$77$	−22389.1	−0.430337
$78$	0	0
$79$	−24760.6	−0.446369	−0.223185	−	0.974776i	$-0.571645\pi$
$79$	−24760.6	−0.446369	−0.223185	+	0.974776i	$0.571645\pi$
$80$	−24018.2	−0.419581
$81$	0	0
$82$	−167345.	−2.74839
$83$	30825.2	0.491147	0.245573	−	0.969378i	$-0.421024\pi$
$83$	30825.2	0.491147	0.245573	+	0.969378i	$0.421024\pi$
$84$	0	0
$85$	156373.	2.34755
$86$	99901.7	1.45655
$87$	0	0
$88$	−38122.1	−0.524772
$89$	66531.6	0.890334	0.445167	−	0.895448i	$-0.353144\pi$
$89$	66531.6	0.890334	0.445167	+	0.895448i	$0.353144\pi$
$90$	0	0
$91$	−115856.	−1.46661
$92$	29094.2	0.358374
$93$	0	0
$94$	166836.	1.94747
$95$	−44667.2	−0.507784
$96$	0	0
$97$	−104506.	−1.12774	−0.563872	−	0.825862i	$-0.690689\pi$
$97$	−104506.	−1.12774	−0.563872	+	0.825862i	$0.690689\pi$
$98$	−8723.75	−0.0917568
$99$	0	0
$100$	374956.	3.74956
$101$	203117.	1.98127	0.990635	−	0.136540i	$-0.0435982\pi$
$101$	203117.	1.98127	0.990635	+	0.136540i	$0.0435982\pi$
$102$	0	0
$103$	−148132.	−1.37580	−0.687901	−	0.725805i	$-0.741468\pi$
$103$	−148132.	−1.37580	−0.687901	+	0.725805i	$0.741468\pi$
$104$	−197270.	−1.78845
$105$	0	0
$106$	189754.	1.64031
$107$	43750.9	0.369426	0.184713	−	0.982793i	$-0.440864\pi$
$107$	43750.9	0.369426	0.184713	+	0.982793i	$0.440864\pi$
$108$	0	0
$109$	189052.	1.52410	0.762052	−	0.647516i	$-0.224192\pi$
$109$	189052.	1.52410	0.762052	+	0.647516i	$0.224192\pi$
$110$	165283.	1.30241
$111$	0	0
$112$	30346.1	0.228591
$113$	82382.3	0.606929	0.303465	−	0.952843i	$-0.401857\pi$
$113$	82382.3	0.606929	0.303465	+	0.952843i	$0.401857\pi$
$114$	0	0
$115$	−52747.9	−0.371929
$116$	71427.2	0.492855
$117$	0	0
$118$	−265721.	−1.75679
$119$	−197571.	−1.27896
$120$	0	0
$121$	−129468.	−0.803897
$122$	151310.	0.920381
$123$	0	0
$124$	−365843.	−2.13668
$125$	−368195.	−2.10767
$126$	0	0
$127$	−95322.4	−0.524428	−0.262214	−	0.965010i	$-0.584453\pi$
$127$	−95322.4	−0.524428	−0.262214	+	0.965010i	$0.584453\pi$
$128$	−325861.	−1.75795
$129$	0	0
$130$	855288.	4.43868
$131$	−181975.	−0.926474	−0.463237	−	0.886234i	$-0.653312\pi$
$131$	−181975.	−0.926474	−0.463237	+	0.886234i	$0.653312\pi$
$132$	0	0
$133$	56435.3	0.276644
$134$	504868.	2.42893
$135$	0	0
$136$	−336407.	−1.55962
$137$	−175024.	−0.796705	−0.398352	−	0.917232i	$-0.630418\pi$
$137$	−175024.	−0.796705	−0.398352	+	0.917232i	$0.630418\pi$
$138$	0	0
$139$	199672.	0.876556	0.438278	−	0.898839i	$-0.355588\pi$
$139$	199672.	0.876556	0.438278	+	0.898839i	$0.355588\pi$
$140$	−690894.	−2.97914
$141$	0	0
$142$	−396462.	−1.64999
$143$	163430.	0.668331
$144$	0	0
$145$	−129498.	−0.511497
$146$	381824.	1.48245
$147$	0	0
$148$	−443111.	−1.66287
$149$	−71174.8	−0.262640	−0.131320	−	0.991340i	$-0.541921\pi$
$149$	−71174.8	−0.262640	−0.131320	+	0.991340i	$0.541921\pi$
$150$	0	0
$151$	133491.	0.476440	0.238220	−	0.971211i	$-0.423436\pi$
$151$	133491.	0.476440	0.238220	+	0.971211i	$0.423436\pi$
$152$	96093.1	0.337352
$153$	0	0
$154$	−208829.	−0.709561
$155$	663276.	2.21751
$156$	0	0
$157$	−361773.	−1.17135	−0.585675	−	0.810546i	$-0.699171\pi$
$157$	−361773.	−1.17135	−0.585675	+	0.810546i	$0.699171\pi$
$158$	−230950.	−0.735995
$159$	0	0
$160$	460442.	1.42192
$161$	66645.0	0.202630
$162$	0	0
$163$	−627275.	−1.84922	−0.924611	−	0.380912i	$-0.875610\pi$
$163$	−627275.	−1.84922	−0.924611	+	0.380912i	$0.875610\pi$
$164$	−986750.	−2.86482
$165$	0	0
$166$	287516.	0.809826
$167$	−9849.85	−0.0273299	−0.0136650	−	0.999907i	$-0.504350\pi$
$167$	−9849.85	−0.0273299	−0.0136650	+	0.999907i	$0.504350\pi$
$168$	0	0
$169$	474405.	1.27771
$170$	1.45854e6	3.87075
$171$	0	0
$172$	589070.	1.51826
$173$	250617.	0.636641	0.318320	−	0.947983i	$-0.396881\pi$
$173$	250617.	0.636641	0.318320	+	0.947983i	$0.396881\pi$
$174$	0	0
$175$	858898.	2.12005
$176$	−42807.1	−0.104168
$177$	0	0
$178$	620560.	1.46802
$179$	−742331.	−1.73167	−0.865835	−	0.500330i	$-0.833212\pi$
$179$	−742331.	−1.73167	−0.865835	+	0.500330i	$0.833212\pi$
$180$	0	0
$181$	−292957.	−0.664673	−0.332336	−	0.943161i	$-0.607837\pi$
$181$	−292957.	−0.664673	−0.332336	+	0.943161i	$0.607837\pi$
$182$	−1.08062e6	−2.41822
$183$	0	0
$184$	113477.	0.247095
$185$	803363.	1.72577
$186$	0	0
$187$	278699.	0.582817
$188$	983751.	2.02997
$189$	0	0
$190$	−416624.	−0.837259
$191$	−57674.2	−0.114393	−0.0571963	−	0.998363i	$-0.518216\pi$
$191$	−57674.2	−0.114393	−0.0571963	+	0.998363i	$0.518216\pi$
$192$	0	0
$193$	−178690.	−0.345308	−0.172654	−	0.984983i	$-0.555234\pi$
$193$	−178690.	−0.345308	−0.172654	+	0.984983i	$0.555234\pi$
$194$	−974754.	−1.85948
$195$	0	0
$196$	−51439.6	−0.0956439
$197$	525992.	0.965636	0.482818	−	0.875721i	$-0.339613\pi$
$197$	525992.	0.965636	0.482818	+	0.875721i	$0.339613\pi$
$198$	0	0
$199$	−184283.	−0.329877	−0.164939	−	0.986304i	$-0.552743\pi$
$199$	−184283.	−0.329877	−0.164939	+	0.986304i	$0.552743\pi$
$200$	1.46246e6	2.58528
$201$	0	0
$202$	1.89453e6	3.26681
$203$	163616.	0.278667
$204$	0	0
$205$	1.78898e6	2.97318
$206$	−1.38167e6	−2.26849
$207$	0	0
$208$	−221513.	−0.355010
$209$	−79609.1	−0.126066
$210$	0	0
$211$	−313460.	−0.484703	−0.242352	−	0.970188i	$-0.577919\pi$
$211$	−313460.	−0.484703	−0.242352	+	0.970188i	$0.577919\pi$
$212$	1.11888e6	1.70980
$213$	0	0
$214$	408077.	0.609127
$215$	−1.06799e6	−1.57569
$216$	0	0
$217$	−838024.	−1.20811
$218$	1.76334e6	2.51302
$219$	0	0
$220$	974593.	1.35758
$221$	1.44218e6	1.98627
$222$	0	0
$223$	−1.00274e6	−1.35028	−0.675141	−	0.737689i	$-0.735917\pi$
$223$	−1.00274e6	−1.35028	−0.675141	+	0.737689i	$0.735917\pi$
$224$	−581751.	−0.774671
$225$	0	0
$226$	768404.	1.00073
$227$	1.02818e6	1.32435	0.662176	−	0.749348i	$-0.269633\pi$
$227$	1.02818e6	1.32435	0.662176	+	0.749348i	$0.269633\pi$
$228$	0	0
$229$	−83843.9	−0.105653	−0.0528266	−	0.998604i	$-0.516823\pi$
$229$	−83843.9	−0.105653	−0.0528266	+	0.998604i	$0.516823\pi$
$230$	−491995.	−0.613255
$231$	0	0
$232$	278591.	0.339819
$233$	241626.	0.291577	0.145789	−	0.989316i	$-0.453428\pi$
$233$	241626.	0.291577	0.145789	+	0.989316i	$0.453428\pi$
$234$	0	0
$235$	−1.78355e6	−2.10676
$236$	−1.56682e6	−1.83122
$237$	0	0
$238$	−1.84281e6	−2.10881
$239$	−1.32779e6	−1.50361	−0.751806	−	0.659384i	$-0.770817\pi$
$239$	−1.32779e6	−1.50361	−0.751806	+	0.659384i	$0.770817\pi$
$240$	0	0
$241$	−1.21493e6	−1.34744	−0.673721	−	0.738986i	$-0.735305\pi$
$241$	−1.21493e6	−1.34744	−0.673721	+	0.738986i	$0.735305\pi$
$242$	−1.20759e6	−1.32550
$243$	0	0
$244$	892199.	0.959372
$245$	93260.3	0.0992617
$246$	0	0
$247$	−411952.	−0.429639
$248$	−1.42691e6	−1.47322
$249$	0	0
$250$	−3.43427e6	−3.47523
$251$	−1.21766e6	−1.21995	−0.609977	−	0.792419i	$-0.708821\pi$
$251$	−1.21766e6	−1.21995	−0.609977	+	0.792419i	$0.708821\pi$
$252$	0	0
$253$	−94011.2	−0.0923376
$254$	−889100.	−0.864702
$255$	0	0
$256$	−1.41448e6	−1.34896
$257$	−1.21439e6	−1.14690	−0.573450	−	0.819241i	$-0.694395\pi$
$257$	−1.21439e6	−1.14690	−0.573450	+	0.819241i	$0.694395\pi$
$258$	0	0
$259$	−1.01502e6	−0.940210
$260$	5.04321e6	4.62672
$261$	0	0
$262$	−1.69733e6	−1.52761
$263$	−93687.8	−0.0835206	−0.0417603	−	0.999128i	$-0.513297\pi$
$263$	−93687.8	−0.0835206	−0.0417603	+	0.999128i	$0.513297\pi$
$264$	0	0
$265$	−2.02854e6	−1.77447
$266$	526388.	0.456144
$267$	0	0
$268$	2.97695e6	2.53183
$269$	323647.	0.272703	0.136352	−	0.990660i	$-0.456462\pi$
$269$	323647.	0.272703	0.136352	+	0.990660i	$0.456462\pi$
$270$	0	0
$271$	1.76235e6	1.45771	0.728853	−	0.684670i	$-0.240054\pi$
$271$	1.76235e6	1.45771	0.728853	+	0.684670i	$0.240054\pi$
$272$	−377749.	−0.309586
$273$	0	0
$274$	−1.63250e6	−1.31364
$275$	−1.21158e6	−0.966100
$276$	0	0
$277$	−1.83478e6	−1.43676	−0.718381	−	0.695650i	$-0.755117\pi$
$277$	−1.83478e6	−1.43676	−0.718381	+	0.695650i	$0.755117\pi$
$278$	1.86240e6	1.44531
$279$	0	0
$280$	−2.69472e6	−2.05409
$281$	1.71252e6	1.29381	0.646903	−	0.762572i	$-0.276064\pi$
$281$	1.71252e6	1.29381	0.646903	+	0.762572i	$0.276064\pi$
$282$	0	0
$283$	1.53094e6	1.13629	0.568147	−	0.822927i	$-0.307660\pi$
$283$	1.53094e6	1.13629	0.568147	+	0.822927i	$0.307660\pi$
$284$	−2.33774e6	−1.71989
$285$	0	0
$286$	1.52436e6	1.10198
$287$	−2.26031e6	−1.61981
$288$	0	0
$289$	1.03951e6	0.732125
$290$	−1.20787e6	−0.843381
$291$	0	0
$292$	2.25142e6	1.54526
$293$	2.02997e6	1.38140	0.690701	−	0.723141i	$-0.257302\pi$
$293$	2.02997e6	1.38140	0.690701	+	0.723141i	$0.257302\pi$
$294$	0	0
$295$	2.84066e6	1.90049
$296$	−1.72829e6	−1.14653
$297$	0	0
$298$	−663868.	−0.433053
$299$	−486478.	−0.314692
$300$	0	0
$301$	1.34936e6	0.858445
$302$	1.24511e6	0.785578
$303$	0	0
$304$	107902.	0.0669648
$305$	−1.61756e6	−0.995661
$306$	0	0
$307$	396907.	0.240349	0.120175	−	0.992753i	$-0.461655\pi$
$307$	396907.	0.240349	0.120175	+	0.992753i	$0.461655\pi$
$308$	−1.23136e6	−0.739621
$309$	0	0
$310$	6.18657e6	3.65633
$311$	−1.27993e6	−0.750385	−0.375192	−	0.926947i	$-0.622423\pi$
$311$	−1.27993e6	−0.750385	−0.375192	+	0.926947i	$0.622423\pi$
$312$	0	0
$313$	−2.41300e6	−1.39218	−0.696091	−	0.717953i	$-0.745079\pi$
$313$	−2.41300e6	−1.39218	−0.696091	+	0.717953i	$0.745079\pi$
$314$	−3.37436e6	−1.93138
$315$	0	0
$316$	−1.36180e6	−0.767175
$317$	−1.03063e6	−0.576043	−0.288022	−	0.957624i	$-0.592998\pi$
$317$	−1.03063e6	−0.576043	−0.288022	+	0.957624i	$0.592998\pi$
$318$	0	0
$319$	−230801.	−0.126988
$320$	5.06326e6	2.76411
$321$	0	0
$322$	621617.	0.334105
$323$	−702507.	−0.374666
$324$	0	0
$325$	−6.26956e6	−3.29252
$326$	−5.85078e6	−3.04909
$327$	0	0
$328$	−3.84867e6	−1.97527
$329$	2.25345e6	1.14778
$330$	0	0
$331$	231656.	0.116218	0.0581089	−	0.998310i	$-0.481493\pi$
$331$	231656.	0.116218	0.0581089	+	0.998310i	$0.481493\pi$
$332$	1.69534e6	0.844134
$333$	0	0
$334$	−91872.4	−0.0450629
$335$	−5.39723e6	−2.62760
$336$	0	0
$337$	−351073.	−0.168393	−0.0841963	−	0.996449i	$-0.526832\pi$
$337$	−351073.	−0.168393	−0.0841963	+	0.996449i	$0.526832\pi$
$338$	4.42491e6	2.10675
$339$	0	0
$340$	8.60026e6	4.03473
$341$	1.18214e6	0.550532
$342$	0	0
$343$	−2.23523e6	−1.02586
$344$	2.29758e6	1.04683
$345$	0	0
$346$	2.33757e6	1.04972
$347$	550941.	0.245630	0.122815	−	0.992430i	$-0.460808\pi$
$347$	550941.	0.245630	0.122815	+	0.992430i	$0.460808\pi$
$348$	0	0
$349$	−1.30593e6	−0.573928	−0.286964	−	0.957941i	$-0.592646\pi$
$349$	−1.30593e6	−0.573928	−0.286964	+	0.957941i	$0.592646\pi$
$350$	8.01119e6	3.49564
$351$	0	0
$352$	820634.	0.353015
$353$	−2.13442e6	−0.911682	−0.455841	−	0.890061i	$-0.650661\pi$
$353$	−2.13442e6	−0.911682	−0.455841	+	0.890061i	$0.650661\pi$
$354$	0	0
$355$	4.23834e6	1.78494
$356$	3.65913e6	1.53022
$357$	0	0
$358$	−6.92394e6	−2.85526
$359$	−1.06632e6	−0.436670	−0.218335	−	0.975874i	$-0.570063\pi$
$359$	−1.06632e6	−0.436670	−0.218335	+	0.975874i	$0.570063\pi$
$360$	0	0
$361$	−2.27543e6	−0.918958
$362$	−2.73250e6	−1.09594
$363$	0	0
$364$	−6.37191e6	−2.52067
$365$	−4.08185e6	−1.60371
$366$	0	0
$367$	1.34294e6	0.520464	0.260232	−	0.965546i	$-0.416201\pi$
$367$	1.34294e6	0.520464	0.260232	+	0.965546i	$0.416201\pi$
$368$	127423.	0.0490487
$369$	0	0
$370$	7.49320e6	2.84553
$371$	2.56299e6	0.966744
$372$	0	0
$373$	3.22658e6	1.20080	0.600400	−	0.799700i	$-0.295008\pi$
$373$	3.22658e6	1.20080	0.600400	+	0.799700i	$0.295008\pi$
$374$	2.59951e6	0.960976
$375$	0	0
$376$	3.83697e6	1.39965
$377$	−1.19432e6	−0.432781
$378$	0	0
$379$	−2.41939e6	−0.865184	−0.432592	−	0.901590i	$-0.642401\pi$
$379$	−2.41939e6	−0.865184	−0.432592	+	0.901590i	$0.642401\pi$
$380$	−2.45662e6	−0.872729
$381$	0	0
$382$	−537944.	−0.188616
$383$	−1.03280e6	−0.359764	−0.179882	−	0.983688i	$-0.557572\pi$
$383$	−1.03280e6	−0.359764	−0.179882	+	0.983688i	$0.557572\pi$
$384$	0	0
$385$	2.23247e6	0.767597
$386$	−1.66669e6	−0.569360
$387$	0	0
$388$	−5.74764e6	−1.93825
$389$	−2.56135e6	−0.858212	−0.429106	−	0.903254i	$-0.641171\pi$
$389$	−2.56135e6	−0.858212	−0.429106	+	0.903254i	$0.641171\pi$
$390$	0	0
$391$	−829598.	−0.274426
$392$	−200632.	−0.0659456
$393$	0	0
$394$	4.90608e6	1.59219
$395$	2.46894e6	0.796193
$396$	0	0
$397$	−2.07970e6	−0.662255	−0.331128	−	0.943586i	$-0.607429\pi$
$397$	−2.07970e6	−0.662255	−0.331128	+	0.943586i	$0.607429\pi$
$398$	−1.71886e6	−0.543917
$399$	0	0
$400$	1.64218e6	0.513182
$401$	1.97329e6	0.612817	0.306408	−	0.951900i	$-0.400873\pi$
$401$	1.97329e6	0.612817	0.306408	+	0.951900i	$0.400873\pi$
$402$	0	0
$403$	6.11719e6	1.87624
$404$	1.11711e7	3.40521
$405$	0	0
$406$	1.52609e6	0.459480
$407$	1.43181e6	0.428450
$408$	0	0
$409$	1.68806e6	0.498975	0.249488	−	0.968378i	$-0.419738\pi$
$409$	1.68806e6	0.498975	0.249488	+	0.968378i	$0.419738\pi$
$410$	1.66864e7	4.90233
$411$	0	0
$412$	−8.14702e6	−2.36459
$413$	−3.58907e6	−1.03540
$414$	0	0
$415$	−3.07366e6	−0.876064
$416$	4.24652e6	1.20309
$417$	0	0
$418$	−742538.	−0.207863
$419$	−1.72868e6	−0.481039	−0.240520	−	0.970644i	$-0.577318\pi$
$419$	−1.72868e6	−0.481039	−0.240520	+	0.970644i	$0.577318\pi$
$420$	0	0
$421$	932870.	0.256517	0.128258	−	0.991741i	$-0.459061\pi$
$421$	932870.	0.256517	0.128258	+	0.991741i	$0.459061\pi$
$422$	−2.92373e6	−0.799202
$423$	0	0
$424$	4.36403e6	1.17889
$425$	−1.06916e7	−2.87124
$426$	0	0
$427$	2.04373e6	0.542443
$428$	2.40623e6	0.634932
$429$	0	0
$430$	−9.96144e6	−2.59807
$431$	7.62939e6	1.97832	0.989161	−	0.146838i	$-0.0469095\pi$
$431$	7.62939e6	1.97832	0.989161	+	0.146838i	$0.0469095\pi$
$432$	0	0
$433$	4.68013e6	1.19960	0.599802	−	0.800148i	$-0.295246\pi$
$433$	4.68013e6	1.19960	0.599802	+	0.800148i	$0.295246\pi$
$434$	−7.81650e6	−1.99199
$435$	0	0
$436$	1.03976e7	2.61948
$437$	236971.	0.0593596
$438$	0	0
$439$	1.21981e6	0.302087	0.151043	−	0.988527i	$-0.451737\pi$
$439$	1.21981e6	0.302087	0.151043	+	0.988527i	$0.451737\pi$
$440$	3.80125e6	0.936041
$441$	0	0
$442$	1.34516e7	3.27506
$443$	1.82127e6	0.440926	0.220463	−	0.975395i	$-0.429243\pi$
$443$	1.82127e6	0.440926	0.220463	+	0.975395i	$0.429243\pi$
$444$	0	0
$445$	−6.63403e6	−1.58810
$446$	−9.35281e6	−2.22641
$447$	0	0
$448$	−6.39724e6	−1.50591
$449$	3.31712e6	0.776506	0.388253	−	0.921553i	$-0.373079\pi$
$449$	3.31712e6	0.776506	0.388253	+	0.921553i	$0.373079\pi$
$450$	0	0
$451$	3.18846e6	0.738142
$452$	4.53090e6	1.04313
$453$	0	0
$454$	9.59012e6	2.18366
$455$	1.15523e7	2.61602
$456$	0	0
$457$	−2.14897e6	−0.481326	−0.240663	−	0.970609i	$-0.577365\pi$
$457$	−2.14897e6	−0.481326	−0.240663	+	0.970609i	$0.577365\pi$
$458$	−782036.	−0.174206
$459$	0	0
$460$	−2.90105e6	−0.639235
$461$	6.83686e6	1.49832	0.749160	−	0.662389i	$-0.230457\pi$
$461$	6.83686e6	1.49832	0.749160	+	0.662389i	$0.230457\pi$
$462$	0	0
$463$	3.26293e6	0.707385	0.353692	−	0.935362i	$-0.384926\pi$
$463$	3.26293e6	0.707385	0.353692	+	0.935362i	$0.384926\pi$
$464$	312828.	0.0674544
$465$	0	0
$466$	2.25372e6	0.480767
$467$	4.30238e6	0.912886	0.456443	−	0.889753i	$-0.349123\pi$
$467$	4.30238e6	0.912886	0.456443	+	0.889753i	$0.349123\pi$
$468$	0	0
$469$	6.81920e6	1.43153
$470$	−1.66357e7	−3.47373
$471$	0	0
$472$	−6.11116e6	−1.26261
$473$	−1.90345e6	−0.391191
$474$	0	0
$475$	3.05400e6	0.621061
$476$	−1.08661e7	−2.19815
$477$	0	0
$478$	−1.23847e7	−2.47923
$479$	−3.15612e6	−0.628514	−0.314257	−	0.949338i	$-0.601755\pi$
$479$	−3.15612e6	−0.628514	−0.314257	+	0.949338i	$0.601755\pi$
$480$	0	0
$481$	7.40918e6	1.46018
$482$	−1.13320e7	−2.22173
$483$	0	0
$484$	−7.12055e6	−1.38166
$485$	1.04205e7	2.01157
$486$	0	0
$487$	−1.04981e6	−0.200581	−0.100290	−	0.994958i	$-0.531977\pi$
$487$	−1.04981e6	−0.200581	−0.100290	+	0.994958i	$0.531977\pi$
$488$	3.47988e6	0.661478
$489$	0	0
$490$	869867.	0.163667
$491$	−1.67278e6	−0.313138	−0.156569	−	0.987667i	$-0.550043\pi$
$491$	−1.67278e6	−0.313138	−0.156569	+	0.987667i	$0.550043\pi$
$492$	0	0
$493$	−2.03669e6	−0.377406
$494$	−3.84240e6	−0.708410
$495$	0	0
$496$	−1.60227e6	−0.292437
$497$	−5.35498e6	−0.972449
$498$	0	0
$499$	−3.07979e6	−0.553695	−0.276847	−	0.960914i	$-0.589290\pi$
$499$	−3.07979e6	−0.553695	−0.276847	+	0.960914i	$0.589290\pi$
$500$	−2.02502e7	−3.62246
$501$	0	0
$502$	−1.13575e7	−2.01152
$503$	−4.06934e6	−0.717140	−0.358570	−	0.933503i	$-0.616736\pi$
$503$	−4.06934e6	−0.717140	−0.358570	+	0.933503i	$0.616736\pi$
$504$	0	0
$505$	−2.02533e7	−3.53401
$506$	−876870.	−0.152251
$507$	0	0
$508$	−5.24258e6	−0.901334
$509$	−5.65269e6	−0.967077	−0.483538	−	0.875323i	$-0.660649\pi$
$509$	−5.65269e6	−0.967077	−0.483538	+	0.875323i	$0.660649\pi$
$510$	0	0
$511$	5.15726e6	0.873709
$512$	−2.76575e6	−0.466271
$513$	0	0
$514$	−1.13270e7	−1.89106
$515$	1.47706e7	2.45403
$516$	0	0
$517$	−3.17877e6	−0.523038
$518$	−9.46738e6	−1.55026
$519$	0	0
$520$	1.96703e7	3.19008
$521$	729056.	0.117670	0.0588351	−	0.998268i	$-0.481261\pi$
$521$	729056.	0.117670	0.0588351	+	0.998268i	$0.481261\pi$
$522$	0	0
$523$	5.74086e6	0.917747	0.458873	−	0.888502i	$-0.348253\pi$
$523$	5.74086e6	0.917747	0.458873	+	0.888502i	$0.348253\pi$
$524$	−1.00083e7	−1.59233
$525$	0	0
$526$	−873853.	−0.137713
$527$	1.04317e7	1.63618
$528$	0	0
$529$	279841.	0.0434783
$530$	−1.89208e7	−2.92583
$531$	0	0
$532$	3.10385e6	0.475468
$533$	1.64993e7	2.51563
$534$	0	0
$535$	−4.36250e6	−0.658948
$536$	1.16111e7	1.74567
$537$	0	0
$538$	3.01875e6	0.449647
$539$	166216.	0.0246434
$540$	0	0
$541$	1.12666e7	1.65500	0.827500	−	0.561466i	$-0.189762\pi$
$541$	1.12666e7	1.65500	0.827500	+	0.561466i	$0.189762\pi$
$542$	1.64380e7	2.40354
$543$	0	0
$544$	7.24165e6	1.04916
$545$	−1.88508e7	−2.71856
$546$	0	0
$547$	1.92802e6	0.275513	0.137757	−	0.990466i	$-0.456011\pi$
$547$	1.92802e6	0.275513	0.137757	+	0.990466i	$0.456011\pi$
$548$	−9.62606e6	−1.36930
$549$	0	0
$550$	−1.13008e7	−1.59295
$551$	581772.	0.0816345
$552$	0	0
$553$	−3.11942e6	−0.433771
$554$	−1.71135e7	−2.36900
$555$	0	0
$556$	1.09816e7	1.50654
$557$	1.27385e7	1.73973	0.869865	−	0.493290i	$-0.164206\pi$
$557$	1.27385e7	1.73973	0.869865	+	0.493290i	$0.164206\pi$
$558$	0	0
$559$	−9.84974e6	−1.33320
$560$	−3.02589e6	−0.407739
$561$	0	0
$562$	1.59732e7	2.13329
$563$	−1.30970e7	−1.74140	−0.870702	−	0.491810i	$-0.836335\pi$
$563$	−1.30970e7	−1.74140	−0.870702	+	0.491810i	$0.836335\pi$
$564$	0	0
$565$	−8.21454e6	−1.08259
$566$	1.42795e7	1.87358
$567$	0	0
$568$	−9.11799e6	−1.18585
$569$	−1.32454e7	−1.71509	−0.857543	−	0.514413i	$-0.828010\pi$
$569$	−1.32454e7	−1.71509	−0.857543	+	0.514413i	$0.828010\pi$
$570$	0	0
$571$	−4.22643e6	−0.542480	−0.271240	−	0.962512i	$-0.587434\pi$
$571$	−4.22643e6	−0.542480	−0.271240	+	0.962512i	$0.587434\pi$
$572$	8.98838e6	1.14866
$573$	0	0
$574$	−2.10826e7	−2.67082
$575$	3.60650e6	0.454900
$576$	0	0
$577$	−224943.	−0.0281276	−0.0140638	−	0.999901i	$-0.504477\pi$
$577$	−224943.	−0.0281276	−0.0140638	+	0.999901i	$0.504477\pi$
$578$	9.69584e6	1.20716
$579$	0	0
$580$	−7.12219e6	−0.879110
$581$	3.88346e6	0.477285
$582$	0	0
$583$	−3.61542e6	−0.440542
$584$	8.78134e6	1.06544
$585$	0	0
$586$	1.89341e7	2.27772
$587$	−4.80198e6	−0.575209	−0.287604	−	0.957749i	$-0.592859\pi$
$587$	−4.80198e6	−0.575209	−0.287604	+	0.957749i	$0.592859\pi$
$588$	0	0
$589$	−2.97978e6	−0.353912
$590$	2.64957e7	3.13361
$591$	0	0
$592$	−1.94068e6	−0.227588
$593$	9.07891e6	1.06022	0.530111	−	0.847928i	$-0.322150\pi$
$593$	9.07891e6	1.06022	0.530111	+	0.847928i	$0.322150\pi$
$594$	0	0
$595$	1.97003e7	2.28129
$596$	−3.91450e6	−0.451399
$597$	0	0
$598$	−4.53752e6	−0.518879
$599$	487665.	0.0555334	0.0277667	−	0.999614i	$-0.491160\pi$
$599$	487665.	0.0555334	0.0277667	+	0.999614i	$0.491160\pi$
$600$	0	0
$601$	−9.72337e6	−1.09807	−0.549036	−	0.835799i	$-0.685005\pi$
$601$	−9.72337e6	−1.09807	−0.549036	+	0.835799i	$0.685005\pi$
$602$	1.25859e7	1.41545
$603$	0	0
$604$	7.34177e6	0.818858
$605$	1.29096e7	1.43392
$606$	0	0
$607$	−3.30773e6	−0.364384	−0.182192	−	0.983263i	$-0.558319\pi$
$607$	−3.30773e6	−0.364384	−0.182192	+	0.983263i	$0.558319\pi$
$608$	−2.06854e6	−0.226937
$609$	0	0
$610$	−1.50875e7	−1.64169
$611$	−1.64491e7	−1.78254
$612$	0	0
$613$	−234065.	−0.0251585	−0.0125793	−	0.999921i	$-0.504004\pi$
$613$	−234065.	−0.0251585	−0.0125793	+	0.999921i	$0.504004\pi$
$614$	3.70207e6	0.396300
$615$	0	0
$616$	−4.80274e6	−0.509961
$617$	9.19095e6	0.971958	0.485979	−	0.873970i	$-0.338463\pi$
$617$	9.19095e6	0.971958	0.485979	+	0.873970i	$0.338463\pi$
$618$	0	0
$619$	7.15853e6	0.750926	0.375463	−	0.926837i	$-0.377484\pi$
$619$	7.15853e6	0.750926	0.375463	+	0.926837i	$0.377484\pi$
$620$	3.64791e7	3.81123
$621$	0	0
$622$	−1.19382e7	−1.23727
$623$	8.38185e6	0.865206
$624$	0	0
$625$	1.54088e7	1.57786
$626$	−2.25067e7	−2.29550
$627$	0	0
$628$	−1.98969e7	−2.01320
$629$	1.26350e7	1.27335
$630$	0	0
$631$	1.00474e7	1.00457	0.502283	−	0.864703i	$-0.332493\pi$
$631$	1.00474e7	1.00457	0.502283	+	0.864703i	$0.332493\pi$
$632$	−5.31147e6	−0.528960
$633$	0	0
$634$	−9.61300e6	−0.949808
$635$	9.50483e6	0.935427
$636$	0	0
$637$	860112.	0.0839859
$638$	−2.15275e6	−0.209383
$639$	0	0
$640$	3.24924e7	3.13568
$641$	−4.21517e6	−0.405201	−0.202600	−	0.979262i	$-0.564939\pi$
$641$	−4.21517e6	−0.405201	−0.202600	+	0.979262i	$0.564939\pi$
$642$	0	0
$643$	9.48561e6	0.904770	0.452385	−	0.891823i	$-0.350573\pi$
$643$	9.48561e6	0.904770	0.452385	+	0.891823i	$0.350573\pi$
$644$	3.66537e6	0.348259
$645$	0	0
$646$	−6.55249e6	−0.617768
$647$	−2.54033e6	−0.238578	−0.119289	−	0.992860i	$-0.538061\pi$
$647$	−2.54033e6	−0.238578	−0.119289	+	0.992860i	$0.538061\pi$
$648$	0	0
$649$	5.06284e6	0.471827
$650$	−5.84781e7	−5.42887
$651$	0	0
$652$	−3.44991e7	−3.17826
$653$	−9.06671e6	−0.832083	−0.416042	−	0.909346i	$-0.636583\pi$
$653$	−9.06671e6	−0.832083	−0.416042	+	0.909346i	$0.636583\pi$
$654$	0	0
$655$	1.81452e7	1.65256
$656$	−4.32164e6	−0.392093
$657$	0	0
$658$	2.10185e7	1.89251
$659$	1.63938e7	1.47050	0.735251	−	0.677795i	$-0.237064\pi$
$659$	1.63938e7	1.47050	0.735251	+	0.677795i	$0.237064\pi$
$660$	0	0
$661$	−1.06203e7	−0.945435	−0.472717	−	0.881214i	$-0.656727\pi$
$661$	−1.06203e7	−0.945435	−0.472717	+	0.881214i	$0.656727\pi$
$662$	2.16072e6	0.191626
$663$	0	0
$664$	6.61241e6	0.582022
$665$	−5.62730e6	−0.493453
$666$	0	0
$667$	687020.	0.0597936
$668$	−541726.	−0.0469719
$669$	0	0
$670$	−5.03416e7	−4.33251
$671$	−2.88294e6	−0.247189
$672$	0	0
$673$	−1.19056e7	−1.01324	−0.506620	−	0.862169i	$-0.669105\pi$
$673$	−1.19056e7	−1.01324	−0.506620	+	0.862169i	$0.669105\pi$
$674$	−3.27456e6	−0.277654
$675$	0	0
$676$	2.60915e7	2.19600
$677$	−2.17878e7	−1.82701	−0.913507	−	0.406822i	$-0.866637\pi$
$677$	−2.17878e7	−1.82701	−0.913507	+	0.406822i	$0.866637\pi$
$678$	0	0
$679$	−1.31659e7	−1.09592
$680$	3.35440e7	2.78190
$681$	0	0
$682$	1.10262e7	0.907744
$683$	−8.27503e6	−0.678762	−0.339381	−	0.940649i	$-0.610218\pi$
$683$	−8.27503e6	−0.678762	−0.339381	+	0.940649i	$0.610218\pi$
$684$	0	0
$685$	1.74521e7	1.42109
$686$	−2.08486e7	−1.69148
$687$	0	0
$688$	2.57993e6	0.207796
$689$	−1.87086e7	−1.50139
$690$	0	0
$691$	8.37140e6	0.666965	0.333483	−	0.942756i	$-0.391776\pi$
$691$	8.37140e6	0.666965	0.333483	+	0.942756i	$0.391776\pi$
$692$	1.37835e7	1.09419
$693$	0	0
$694$	5.13879e6	0.405007
$695$	−1.99098e7	−1.56352
$696$	0	0
$697$	2.81364e7	2.19375
$698$	−1.21808e7	−0.946320
$699$	0	0
$700$	4.72380e7	3.64373
$701$	−3.01421e6	−0.231675	−0.115837	−	0.993268i	$-0.536955\pi$
$701$	−3.01421e6	−0.231675	−0.115837	+	0.993268i	$0.536955\pi$
$702$	0	0
$703$	−3.60912e6	−0.275431
$704$	9.02412e6	0.686236
$705$	0	0
$706$	−1.99084e7	−1.50323
$707$	2.55893e7	1.92535
$708$	0	0
$709$	−5.74628e6	−0.429310	−0.214655	−	0.976690i	$-0.568863\pi$
$709$	−5.74628e6	−0.429310	−0.214655	+	0.976690i	$0.568863\pi$
$710$	3.95322e7	2.94310
$711$	0	0
$712$	1.42719e7	1.05507
$713$	−3.51885e6	−0.259225
$714$	0	0
$715$	−1.62960e7	−1.19211
$716$	−4.08270e7	−2.97622
$717$	0	0
$718$	−9.94592e6	−0.720002
$719$	1.64352e7	1.18564	0.592821	−	0.805334i	$-0.298014\pi$
$719$	1.64352e7	1.18564	0.592821	+	0.805334i	$0.298014\pi$
$720$	0	0
$721$	−1.86621e7	−1.33697
$722$	−2.12236e7	−1.51522
$723$	0	0
$724$	−1.61122e7	−1.14237
$725$	8.85408e6	0.625603
$726$	0	0
$727$	−4.65331e6	−0.326532	−0.163266	−	0.986582i	$-0.552203\pi$
$727$	−4.65331e6	−0.326532	−0.163266	+	0.986582i	$0.552203\pi$
$728$	−2.48526e7	−1.73798
$729$	0	0
$730$	−3.80726e7	−2.64427
$731$	−1.67969e7	−1.16261
$732$	0	0
$733$	1.76748e7	1.21505	0.607527	−	0.794299i	$-0.292162\pi$
$733$	1.76748e7	1.21505	0.607527	+	0.794299i	$0.292162\pi$
$734$	1.25260e7	0.858166
$735$	0	0
$736$	−2.44276e6	−0.166221
$737$	−9.61936e6	−0.652345
$738$	0	0
$739$	2.32944e7	1.56907	0.784533	−	0.620087i	$-0.212903\pi$
$739$	2.32944e7	1.56907	0.784533	+	0.620087i	$0.212903\pi$
$740$	4.41837e7	2.96608
$741$	0	0
$742$	2.39057e7	1.59401
$743$	−1.77107e7	−1.17696	−0.588482	−	0.808511i	$-0.700274\pi$
$743$	−1.77107e7	−1.17696	−0.588482	+	0.808511i	$0.700274\pi$
$744$	0	0
$745$	7.09701e6	0.468473
$746$	3.00953e7	1.97994
$747$	0	0
$748$	1.53280e7	1.00169
$749$	5.51186e6	0.358999
$750$	0	0
$751$	−9.59704e6	−0.620922	−0.310461	−	0.950586i	$-0.600483\pi$
$751$	−9.59704e6	−0.620922	−0.310461	+	0.950586i	$0.600483\pi$
$752$	4.30851e6	0.277832
$753$	0	0
$754$	−1.11398e7	−0.713590
$755$	−1.33107e7	−0.849832
$756$	0	0
$757$	1.81520e7	1.15129	0.575645	−	0.817700i	$-0.304751\pi$
$757$	1.81520e7	1.15129	0.575645	+	0.817700i	$0.304751\pi$
$758$	−2.25664e7	−1.42656
$759$	0	0
$760$	−9.58167e6	−0.601738
$761$	−2.18570e7	−1.36813	−0.684067	−	0.729419i	$-0.739790\pi$
$761$	−2.18570e7	−1.36813	−0.684067	+	0.729419i	$0.739790\pi$
$762$	0	0
$763$	2.38173e7	1.48109
$764$	−3.17199e6	−0.196607
$765$	0	0
$766$	−9.63319e6	−0.593196
$767$	2.61986e7	1.60801
$768$	0	0
$769$	−3.01818e6	−0.184047	−0.0920237	−	0.995757i	$-0.529334\pi$
$769$	−3.01818e6	−0.184047	−0.0920237	+	0.995757i	$0.529334\pi$
$770$	2.08229e7	1.26565
$771$	0	0
$772$	−9.82765e6	−0.593480
$773$	−7.09462e6	−0.427052	−0.213526	−	0.976937i	$-0.568495\pi$
$773$	−7.09462e6	−0.427052	−0.213526	+	0.976937i	$0.568495\pi$
$774$	0	0
$775$	−4.53497e7	−2.71219
$776$	−2.24178e7	−1.33641
$777$	0	0
$778$	−2.38904e7	−1.41506
$779$	−8.03704e6	−0.474518
$780$	0	0
$781$	7.55388e6	0.443142
$782$	−7.73790e6	−0.452487
$783$	0	0
$784$	−225289.	−0.0130903
$785$	3.60732e7	2.08935
$786$	0	0
$787$	2.85125e7	1.64096	0.820482	−	0.571672i	$-0.193705\pi$
$787$	2.85125e7	1.64096	0.820482	+	0.571672i	$0.193705\pi$
$788$	2.89287e7	1.65964
$789$	0	0
$790$	2.30286e7	1.31280
$791$	1.03788e7	0.589800
$792$	0	0
$793$	−1.49183e7	−0.842435
$794$	−1.93980e7	−1.09196
$795$	0	0
$796$	−1.01353e7	−0.566960
$797$	−7.59475e6	−0.423514	−0.211757	−	0.977322i	$-0.567919\pi$
$797$	−7.59475e6	−0.423514	−0.211757	+	0.977322i	$0.567919\pi$
$798$	0	0
$799$	−2.80509e7	−1.55446
$800$	−3.14815e7	−1.73912
$801$	0	0
$802$	1.84055e7	1.01044
$803$	−7.27497e6	−0.398146
$804$	0	0
$805$	−6.64533e6	−0.361432
$806$	5.70568e7	3.09364
$807$	0	0
$808$	4.35713e7	2.34786
$809$	−135394.	−0.00727324	−0.00363662	−	0.999993i	$-0.501158\pi$
$809$	−135394.	−0.00727324	−0.00363662	+	0.999993i	$0.501158\pi$
$810$	0	0
$811$	−2.84394e7	−1.51834	−0.759168	−	0.650895i	$-0.774394\pi$
$811$	−2.84394e7	−1.51834	−0.759168	+	0.650895i	$0.774394\pi$
$812$	8.99862e6	0.478945
$813$	0	0
$814$	1.33549e7	0.706449
$815$	6.25471e7	3.29848
$816$	0	0
$817$	4.79795e6	0.251479
$818$	1.57450e7	0.822735
$819$	0	0
$820$	9.83913e7	5.11001
$821$	9.25492e6	0.479198	0.239599	−	0.970872i	$-0.422984\pi$
$821$	9.25492e6	0.479198	0.239599	+	0.970872i	$0.422984\pi$
$822$	0	0
$823$	8.81289e6	0.453543	0.226772	−	0.973948i	$-0.427183\pi$
$823$	8.81289e6	0.453543	0.226772	+	0.973948i	$0.427183\pi$
$824$	−3.17762e7	−1.63036
$825$	0	0
$826$	−3.34763e7	−1.70721
$827$	−2.40557e7	−1.22308	−0.611539	−	0.791214i	$-0.709449\pi$
$827$	−2.40557e7	−1.22308	−0.611539	+	0.791214i	$0.709449\pi$
$828$	0	0
$829$	3.34899e7	1.69249	0.846247	−	0.532791i	$-0.178857\pi$
$829$	3.34899e7	1.69249	0.846247	+	0.532791i	$0.178857\pi$
$830$	−2.86689e7	−1.44450
$831$	0	0
$832$	4.66969e7	2.33873
$833$	1.46676e6	0.0732398
$834$	0	0
$835$	982153.	0.0487487
$836$	−4.37838e6	−0.216669
$837$	0	0
$838$	−1.61239e7	−0.793160
$839$	−6.66922e6	−0.327092	−0.163546	−	0.986536i	$-0.552293\pi$
$839$	−6.66922e6	−0.327092	−0.163546	+	0.986536i	$0.552293\pi$
$840$	0	0
$841$	−1.88245e7	−0.917769
$842$	8.70115e6	0.422957
$843$	0	0
$844$	−1.72398e7	−0.833060
$845$	−4.73040e7	−2.27906
$846$	0	0
$847$	−1.63108e7	−0.781209
$848$	4.90034e6	0.234011
$849$	0	0
$850$	−9.97235e7	−4.73424
$851$	−4.26205e6	−0.201741
$852$	0	0
$853$	2.91274e6	0.137066	0.0685330	−	0.997649i	$-0.478168\pi$
$853$	2.91274e6	0.137066	0.0685330	+	0.997649i	$0.478168\pi$
$854$	1.90625e7	0.894405
$855$	0	0
$856$	9.38512e6	0.437779
$857$	−1.52102e7	−0.707429	−0.353715	−	0.935353i	$-0.615082\pi$
$857$	−1.52102e7	−0.707429	−0.353715	+	0.935353i	$0.615082\pi$
$858$	0	0
$859$	1.06621e7	0.493016	0.246508	−	0.969141i	$-0.420717\pi$
$859$	1.06621e7	0.493016	0.246508	+	0.969141i	$0.420717\pi$
$860$	−5.87376e7	−2.70814
$861$	0	0
$862$	7.11616e7	3.26195
$863$	3.79585e7	1.73493	0.867465	−	0.497498i	$-0.165748\pi$
$863$	3.79585e7	1.73493	0.867465	+	0.497498i	$0.165748\pi$
$864$	0	0
$865$	−2.49896e7	−1.13558
$866$	4.36529e7	1.97796
$867$	0	0
$868$	−4.60900e7	−2.07638
$869$	4.40034e6	0.197668
$870$	0	0
$871$	−4.97771e7	−2.22323
$872$	4.05540e7	1.80610
$873$	0	0
$874$	2.21029e6	0.0978750
$875$	−4.63863e7	−2.04819
$876$	0	0
$877$	2.72329e7	1.19562	0.597812	−	0.801636i	$-0.296037\pi$
$877$	2.72329e7	1.19562	0.597812	+	0.801636i	$0.296037\pi$
$878$	1.13775e7	0.498095
$879$	0	0
$880$	4.26840e6	0.185805
$881$	−8.49018e6	−0.368534	−0.184267	−	0.982876i	$-0.558991\pi$
$881$	−8.49018e6	−0.368534	−0.184267	+	0.982876i	$0.558991\pi$
$882$	0	0
$883$	−2.58940e7	−1.11763	−0.558815	−	0.829292i	$-0.688744\pi$
$883$	−2.58940e7	−1.11763	−0.558815	+	0.829292i	$0.688744\pi$
$884$	7.93176e7	3.41381
$885$	0	0
$886$	1.69875e7	0.727020
$887$	1.57892e7	0.673832	0.336916	−	0.941535i	$-0.390616\pi$
$887$	1.57892e7	0.673832	0.336916	+	0.941535i	$0.390616\pi$
$888$	0	0
$889$	−1.20090e7	−0.509627
$890$	−6.18775e7	−2.61853
$891$	0	0
$892$	−5.51489e7	−2.32073
$893$	8.01261e6	0.336237
$894$	0	0
$895$	7.40196e7	3.08880
$896$	−4.10529e7	−1.70834
$897$	0	0
$898$	3.09397e7	1.28034
$899$	−8.63890e6	−0.356500
$900$	0	0
$901$	−3.19041e7	−1.30929
$902$	2.97397e7	1.21708
$903$	0	0
$904$	1.76721e7	0.719227
$905$	2.92115e7	1.18558
$906$	0	0
$907$	−1.12308e7	−0.453305	−0.226653	−	0.973976i	$-0.572778\pi$
$907$	−1.12308e7	−0.453305	−0.226653	+	0.973976i	$0.572778\pi$
$908$	5.65481e7	2.27616
$909$	0	0
$910$	1.07752e8	4.31341
$911$	−2.61794e7	−1.04511	−0.522557	−	0.852604i	$-0.675022\pi$
$911$	−2.61794e7	−1.04511	−0.522557	+	0.852604i	$0.675022\pi$
$912$	0	0
$913$	−5.47811e6	−0.217497
$914$	−2.00441e7	−0.793634
$915$	0	0
$916$	−4.61128e6	−0.181586
$917$	−2.29257e7	−0.900326
$918$	0	0
$919$	4.07119e7	1.59013	0.795065	−	0.606524i	$-0.207437\pi$
$919$	4.07119e7	1.59013	0.795065	+	0.606524i	$0.207437\pi$
$920$	−1.13151e7	−0.440746
$921$	0	0
$922$	6.37694e7	2.47050
$923$	3.90889e7	1.51025
$924$	0	0
$925$	−5.49278e7	−2.11075
$926$	3.04343e7	1.16637
$927$	0	0
$928$	−5.99707e6	−0.228596
$929$	−8.89152e6	−0.338015	−0.169008	−	0.985615i	$-0.554056\pi$
$929$	−8.89152e6	−0.338015	−0.169008	+	0.985615i	$0.554056\pi$
$930$	0	0
$931$	−418973.	−0.0158421
$932$	1.32890e7	0.501134
$933$	0	0
$934$	4.01296e7	1.50521
$935$	−2.77898e7	−1.03958
$936$	0	0
$937$	4.07318e6	0.151560	0.0757799	−	0.997125i	$-0.475855\pi$
$937$	4.07318e6	0.151560	0.0757799	+	0.997125i	$0.475855\pi$
$938$	6.36047e7	2.36038
$939$	0	0
$940$	−9.80922e7	−3.62089
$941$	1.08008e7	0.397634	0.198817	−	0.980037i	$-0.436290\pi$
$941$	1.08008e7	0.397634	0.198817	+	0.980037i	$0.436290\pi$
$942$	0	0
$943$	−9.49102e6	−0.347563
$944$	−6.86218e6	−0.250629
$945$	0	0
$946$	−1.77540e7	−0.645014
$947$	−6.93307e6	−0.251218	−0.125609	−	0.992080i	$-0.540088\pi$
$947$	−6.93307e6	−0.251218	−0.125609	+	0.992080i	$0.540088\pi$
$948$	0	0
$949$	−3.76457e7	−1.35690
$950$	2.84855e7	1.02404
$951$	0	0
$952$	−4.23816e7	−1.51560
$953$	−3.81395e7	−1.36033	−0.680163	−	0.733060i	$-0.738091\pi$
$953$	−3.81395e7	−1.36033	−0.680163	+	0.733060i	$0.738091\pi$
$954$	0	0
$955$	5.75084e6	0.204043
$956$	−7.30265e7	−2.58426
$957$	0	0
$958$	−2.94381e7	−1.03632
$959$	−2.20501e7	−0.774219
$960$	0	0
$961$	1.56184e7	0.545542
$962$	6.91076e7	2.40762
$963$	0	0
$964$	−6.68194e7	−2.31585
$965$	1.78176e7	0.615929
$966$	0	0
$967$	1.52545e6	0.0524603	0.0262302	−	0.999656i	$-0.491650\pi$
$967$	1.52545e6	0.0524603	0.0262302	+	0.999656i	$0.491650\pi$
$968$	−2.77726e7	−0.952639
$969$	0	0
$970$	9.71951e7	3.31677
$971$	4.37167e7	1.48799	0.743993	−	0.668187i	$-0.232929\pi$
$971$	4.37167e7	1.48799	0.743993	+	0.668187i	$0.232929\pi$
$972$	0	0
$973$	2.51553e7	0.851817
$974$	−9.79190e6	−0.330727
$975$	0	0
$976$	3.90754e6	0.131304
$977$	−1.59223e6	−0.0533666	−0.0266833	−	0.999644i	$-0.508495\pi$
$977$	−1.59223e6	−0.0533666	−0.0266833	+	0.999644i	$0.508495\pi$
$978$	0	0
$979$	−1.18237e7	−0.394271
$980$	5.12917e6	0.170601
$981$	0	0
$982$	−1.56025e7	−0.516316
$983$	−1.98366e7	−0.654762	−0.327381	−	0.944892i	$-0.606166\pi$
$983$	−1.98366e7	−0.654762	−0.327381	+	0.944892i	$0.606166\pi$
$984$	0	0
$985$	−5.24479e7	−1.72241
$986$	−1.89968e7	−0.622285
$987$	0	0
$988$	−2.26567e7	−0.738421
$989$	5.66595e6	0.184197
$990$	0	0
$991$	5.77793e7	1.86891	0.934454	−	0.356083i	$-0.115888\pi$
$991$	5.77793e7	1.86891	0.934454	+	0.356083i	$0.115888\pi$
$992$	3.07164e7	0.991040
$993$	0	0
$994$	−4.99475e7	−1.60342
$995$	1.83753e7	0.588405
$996$	0	0
$997$	−2.48258e7	−0.790980	−0.395490	−	0.918470i	$-0.629425\pi$
$997$	−2.48258e7	−0.790980	−0.395490	+	0.918470i	$0.629425\pi$
$998$	−2.87261e7	−0.912958
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.f.1.5		5
3.2	odd	2		69.6.a.e.1.1	✓	5
12.11	even	2		1104.6.a.r.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.1	✓	5	3.2	odd	2
207.6.a.f.1.5		5	1.1	even	1	trivial
1104.6.a.r.1.5		5	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)

\(f(q)\)	\(=\)	\(q+9.32729 q^{2} +54.9984 q^{4} -99.7124 q^{5} +125.983 q^{7} +214.513 q^{8} +O(q^{10})\)	\(q+9.32729 q^{2} +54.9984 q^{4} -99.7124 q^{5} +125.983 q^{7} +214.513 q^{8} -930.047 q^{10} -177.715 q^{11} -919.618 q^{13} +1175.08 q^{14} +240.875 q^{16} -1568.24 q^{17} +447.960 q^{19} -5484.02 q^{20} -1657.60 q^{22} +529.000 q^{23} +6817.57 q^{25} -8577.55 q^{26} +6928.86 q^{28} +1298.72 q^{29} -6651.88 q^{31} -4617.70 q^{32} -14627.4 q^{34} -12562.1 q^{35} -8056.80 q^{37} +4178.25 q^{38} -21389.6 q^{40} -17941.4 q^{41} +10710.7 q^{43} -9774.04 q^{44} +4934.14 q^{46} +17886.9 q^{47} -935.293 q^{49} +63589.5 q^{50} -50577.5 q^{52} +20343.9 q^{53} +17720.4 q^{55} +27025.0 q^{56} +12113.5 q^{58} -28488.5 q^{59} +16222.3 q^{61} -62044.1 q^{62} -50778.6 q^{64} +91697.4 q^{65} +54128.0 q^{67} -86250.6 q^{68} -117170. q^{70} -42505.6 q^{71} +40936.2 q^{73} -75148.1 q^{74} +24637.1 q^{76} -22389.1 q^{77} -24760.6 q^{79} -24018.2 q^{80} -167345. q^{82} +30825.2 q^{83} +156373. q^{85} +99901.7 q^{86} -38122.1 q^{88} +66531.6 q^{89} -115856. q^{91} +29094.2 q^{92} +166836. q^{94} -44667.2 q^{95} -104506. q^{97} -8723.75 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8}+O(q^{10}) \) 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8	\( 5 q - 8 q^{2} + 118 q^{4} - 94 q^{5} + 272 q^{7} - 258 q^{8} - 172 q^{10} - 1100 q^{11} - 978 q^{13} + 344 q^{14} + 1218 q^{16} - 2522 q^{17} + 2060 q^{19} - 7720 q^{20} - 2572 q^{22} + 2645 q^{23} + 12035 q^{25} - 9280 q^{26} + 8072 q^{28} - 1526 q^{29} - 7392 q^{31} + 5086 q^{32} - 15608 q^{34} - 6056 q^{35} - 8210 q^{37} + 14276 q^{38} - 37472 q^{40} - 21250 q^{41} - 4548 q^{43} + 4260 q^{44} - 4232 q^{46} - 536 q^{47} - 27979 q^{49} + 81872 q^{50} - 76380 q^{52} + 11482 q^{53} - 77064 q^{55} + 28624 q^{56} - 79680 q^{58} - 74676 q^{59} - 44618 q^{61} - 64880 q^{62} - 137382 q^{64} + 24388 q^{65} - 1412 q^{67} - 80196 q^{68} - 222304 q^{70} - 37912 q^{71} + 46546 q^{73} - 111604 q^{74} - 79548 q^{76} - 157008 q^{77} + 50544 q^{79} - 69424 q^{80} - 233720 q^{82} - 89588 q^{83} + 147892 q^{85} - 77428 q^{86} + 54484 q^{88} - 280410 q^{89} - 27416 q^{91} + 62422 q^{92} + 113632 q^{94} - 203120 q^{95} + 90074 q^{97} - 32976 q^{98}+O(q^{100}) \) 5 * q - 8 * q^2 + 118 * q^4 - 94 * q^5 + 272 * q^7 - 258 * q^8 - 172 * q^10 - 1100 * q^11 - 978 * q^13 + 344 * q^14 + 1218 * q^16 - 2522 * q^17 + 2060 * q^19 - 7720 * q^20 - 2572 * q^22 + 2645 * q^23 + 12035 * q^25 - 9280 * q^26 + 8072 * q^28 - 1526 * q^29 - 7392 * q^31 + 5086 * q^32 - 15608 * q^34 - 6056 * q^35 - 8210 * q^37 + 14276 * q^38 - 37472 * q^40 - 21250 * q^41 - 4548 * q^43 + 4260 * q^44 - 4232 * q^46 - 536 * q^47 - 27979 * q^49 + 81872 * q^50 - 76380 * q^52 + 11482 * q^53 - 77064 * q^55 + 28624 * q^56 - 79680 * q^58 - 74676 * q^59 - 44618 * q^61 - 64880 * q^62 - 137382 * q^64 + 24388 * q^65 - 1412 * q^67 - 80196 * q^68 - 222304 * q^70 - 37912 * q^71 + 46546 * q^73 - 111604 * q^74 - 79548 * q^76 - 157008 * q^77 + 50544 * q^79 - 69424 * q^80 - 233720 * q^82 - 89588 * q^83 + 147892 * q^85 - 77428 * q^86 + 54484 * q^88 - 280410 * q^89 - 27416 * q^91 + 62422 * q^92 + 113632 * q^94 - 203120 * q^95 + 90074 * q^97 - 32976 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more