LMFDB - Embedded newform 261.6.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [261,6,Mod(1,261)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(261, 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("261.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 261 = 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	261.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:	$41.8601769712$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902 $ x^7 - 3x^6 - 184x^5 + 584x^4 + 10145x^3 - 34491x^2 - 149754x + 524902
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.83960$ of defining polynomial
Character	$\chi$	$=$	261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.83960 q^{2} +2.10095 q^{4} -31.5616 q^{5} +106.304 q^{7} +174.599 q^{8} +O(q^{10})$	\(q-5.83960 q^{2} +2.10095 q^{4} -31.5616 q^{5} +106.304 q^{7} +174.599 q^{8} +184.307 q^{10} +152.796 q^{11} +325.745 q^{13} -620.776 q^{14} -1086.82 q^{16} +664.939 q^{17} -1595.33 q^{19} -66.3091 q^{20} -892.268 q^{22} -719.327 q^{23} -2128.87 q^{25} -1902.22 q^{26} +223.340 q^{28} -841.000 q^{29} +2059.61 q^{31} +759.420 q^{32} -3882.98 q^{34} -3355.13 q^{35} +14948.4 q^{37} +9316.11 q^{38} -5510.60 q^{40} -14673.9 q^{41} +10298.3 q^{43} +321.016 q^{44} +4200.58 q^{46} -4588.36 q^{47} -5506.36 q^{49} +12431.7 q^{50} +684.373 q^{52} -8952.75 q^{53} -4822.48 q^{55} +18560.6 q^{56} +4911.10 q^{58} +13734.5 q^{59} -33480.9 q^{61} -12027.3 q^{62} +30343.4 q^{64} -10281.0 q^{65} +37519.2 q^{67} +1397.00 q^{68} +19592.6 q^{70} +14763.7 q^{71} +63298.3 q^{73} -87292.9 q^{74} -3351.71 q^{76} +16242.9 q^{77} -27148.9 q^{79} +34301.6 q^{80} +85689.7 q^{82} +54499.5 q^{83} -20986.5 q^{85} -60138.1 q^{86} +26678.0 q^{88} +139829. q^{89} +34628.2 q^{91} -1511.27 q^{92} +26794.2 q^{94} +50351.2 q^{95} +89845.2 q^{97} +32154.9 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 4 q^{2} + 154 q^{4} - 32 q^{5} + 184 q^{7} - 942 q^{8}+O(q^{10}) $ 7 * q - 4 * q^2 + 154 * q^4 - 32 * q^5 + 184 * q^7 - 942 * q^8	$ 7 q - 4 q^{2} + 154 q^{4} - 32 q^{5} + 184 q^{7} - 942 q^{8} + 922 q^{10} - 1106 q^{11} + 408 q^{13} + 2008 q^{14} + 242 q^{16} + 874 q^{17} + 4288 q^{19} + 6350 q^{20} - 6114 q^{22} + 4532 q^{23} + 5527 q^{25} + 19806 q^{26} - 496 q^{28} - 5887 q^{29} + 7794 q^{31} - 7898 q^{32} + 20840 q^{34} - 7088 q^{35} + 5086 q^{37} - 23732 q^{38} + 22906 q^{40} - 19826 q^{41} + 19498 q^{43} + 6074 q^{44} - 12404 q^{46} - 14278 q^{47} + 38431 q^{49} + 41066 q^{50} - 34302 q^{52} + 58644 q^{53} - 25574 q^{55} + 79560 q^{56} + 3364 q^{58} - 12888 q^{59} + 102866 q^{61} + 42654 q^{62} - 10170 q^{64} + 149206 q^{65} + 102996 q^{67} - 85100 q^{68} + 349480 q^{70} + 51596 q^{71} - 17566 q^{73} - 12132 q^{74} + 360740 q^{76} + 94104 q^{77} + 212058 q^{79} - 142510 q^{80} + 201924 q^{82} + 122928 q^{83} - 109336 q^{85} + 63290 q^{86} + 136666 q^{88} + 66510 q^{89} + 194368 q^{91} + 110108 q^{92} + 438926 q^{94} + 131676 q^{95} - 118182 q^{97} + 29132 q^{98}+O(q^{100}) $ 7 * q - 4 * q^2 + 154 * q^4 - 32 * q^5 + 184 * q^7 - 942 * q^8 + 922 * q^10 - 1106 * q^11 + 408 * q^13 + 2008 * q^14 + 242 * q^16 + 874 * q^17 + 4288 * q^19 + 6350 * q^20 - 6114 * q^22 + 4532 * q^23 + 5527 * q^25 + 19806 * q^26 - 496 * q^28 - 5887 * q^29 + 7794 * q^31 - 7898 * q^32 + 20840 * q^34 - 7088 * q^35 + 5086 * q^37 - 23732 * q^38 + 22906 * q^40 - 19826 * q^41 + 19498 * q^43 + 6074 * q^44 - 12404 * q^46 - 14278 * q^47 + 38431 * q^49 + 41066 * q^50 - 34302 * q^52 + 58644 * q^53 - 25574 * q^55 + 79560 * q^56 + 3364 * q^58 - 12888 * q^59 + 102866 * q^61 + 42654 * q^62 - 10170 * q^64 + 149206 * q^65 + 102996 * q^67 - 85100 * q^68 + 349480 * q^70 + 51596 * q^71 - 17566 * q^73 - 12132 * q^74 + 360740 * q^76 + 94104 * q^77 + 212058 * q^79 - 142510 * q^80 + 201924 * q^82 + 122928 * q^83 - 109336 * q^85 + 63290 * q^86 + 136666 * q^88 + 66510 * q^89 + 194368 * q^91 + 110108 * q^92 + 438926 * q^94 + 131676 * q^95 - 118182 * q^97 + 29132 * 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 $

$2$	−5.83960	−1.03231	−0.516153	−	0.856497i	$-0.672636\pi$
$2$	−5.83960	−1.03231	−0.516153	+	0.856497i	$0.672636\pi$
$3$	0	0
$3$	0	0
$4$	2.10095	0.0656545
$5$	−31.5616	−0.564590	−0.282295	−	0.959328i	$-0.591096\pi$
$5$	−31.5616	−0.564590	−0.282295	+	0.959328i	$0.591096\pi$
$6$	0	0
$7$	106.304	0.819986	0.409993	−	0.912089i	$-0.365531\pi$
$7$	106.304	0.819986	0.409993	+	0.912089i	$0.365531\pi$
$8$	174.599	0.964530
$9$	0	0
$10$	184.307	0.582830
$11$	152.796	0.380741	0.190371	−	0.981712i	$-0.439031\pi$
$11$	152.796	0.380741	0.190371	+	0.981712i	$0.439031\pi$
$12$	0	0
$13$	325.745	0.534589	0.267294	−	0.963615i	$-0.413870\pi$
$13$	325.745	0.534589	0.267294	+	0.963615i	$0.413870\pi$
$14$	−620.776	−0.846476
$15$	0	0
$16$	−1086.82	−1.06134
$17$	664.939	0.558033	0.279016	−	0.960286i	$-0.409992\pi$
$17$	664.939	0.558033	0.279016	+	0.960286i	$0.409992\pi$
$18$	0	0
$19$	−1595.33	−1.01383	−0.506917	−	0.861995i	$-0.669215\pi$
$19$	−1595.33	−1.01383	−0.506917	+	0.861995i	$0.669215\pi$
$20$	−66.3091	−0.0370679
$21$	0	0
$22$	−892.268	−0.393041
$23$	−719.327	−0.283535	−0.141768	−	0.989900i	$-0.545279\pi$
$23$	−719.327	−0.283535	−0.141768	+	0.989900i	$0.545279\pi$
$24$	0	0
$25$	−2128.87	−0.681238
$26$	−1902.22	−0.551859
$27$	0	0
$28$	223.340	0.0538358
$29$	−841.000	−0.185695
$29$	−841.000	−0.185695
$30$	0	0
$31$	2059.61	0.384929	0.192464	−	0.981304i	$-0.438352\pi$
$31$	2059.61	0.384929	0.192464	+	0.981304i	$0.438352\pi$
$32$	759.420	0.131101
$33$	0	0
$34$	−3882.98	−0.576060
$35$	−3355.13	−0.462956
$36$	0	0
$37$	14948.4	1.79511	0.897556	−	0.440901i	$-0.145341\pi$
$37$	14948.4	1.79511	0.897556	+	0.440901i	$0.145341\pi$
$38$	9316.11	1.04659
$39$	0	0
$40$	−5510.60	−0.544564
$41$	−14673.9	−1.36328	−0.681641	−	0.731687i	$-0.738733\pi$
$41$	−14673.9	−1.36328	−0.681641	+	0.731687i	$0.738733\pi$
$42$	0	0
$43$	10298.3	0.849367	0.424683	−	0.905342i	$-0.360385\pi$
$43$	10298.3	0.849367	0.424683	+	0.905342i	$0.360385\pi$
$44$	321.016	0.0249974
$45$	0	0
$46$	4200.58	0.292695
$47$	−4588.36	−0.302979	−0.151490	−	0.988459i	$-0.548407\pi$
$47$	−4588.36	−0.302979	−0.151490	+	0.988459i	$0.548407\pi$
$48$	0	0
$49$	−5506.36	−0.327623
$50$	12431.7	0.703246
$51$	0	0
$52$	684.373	0.0350982
$53$	−8952.75	−0.437791	−0.218896	−	0.975748i	$-0.570245\pi$
$53$	−8952.75	−0.437791	−0.218896	+	0.975748i	$0.570245\pi$
$54$	0	0
$55$	−4822.48	−0.214963
$56$	18560.6	0.790901
$57$	0	0
$58$	4911.10	0.191694
$59$	13734.5	0.513670	0.256835	−	0.966455i	$-0.417320\pi$
$59$	13734.5	0.513670	0.256835	+	0.966455i	$0.417320\pi$
$60$	0	0
$61$	−33480.9	−1.15205	−0.576026	−	0.817431i	$-0.695397\pi$
$61$	−33480.9	−1.15205	−0.576026	+	0.817431i	$0.695397\pi$
$62$	−12027.3	−0.397364
$63$	0	0
$64$	30343.4	0.926007
$65$	−10281.0	−0.301824
$66$	0	0
$67$	37519.2	1.02110	0.510548	−	0.859849i	$-0.329442\pi$
$67$	37519.2	1.02110	0.510548	+	0.859849i	$0.329442\pi$
$68$	1397.00	0.0366374
$69$	0	0
$70$	19592.6	0.477912
$71$	14763.7	0.347576	0.173788	−	0.984783i	$-0.444399\pi$
$71$	14763.7	0.347576	0.173788	+	0.984783i	$0.444399\pi$
$72$	0	0
$73$	63298.3	1.39022	0.695112	−	0.718901i	$-0.255355\pi$
$73$	63298.3	1.39022	0.695112	+	0.718901i	$0.255355\pi$
$74$	−87292.9	−1.85310
$75$	0	0
$76$	−3351.71	−0.0665629
$77$	16242.9	0.312203
$78$	0	0
$79$	−27148.9	−0.489423	−0.244711	−	0.969596i	$-0.578693\pi$
$79$	−27148.9	−0.489423	−0.244711	+	0.969596i	$0.578693\pi$
$80$	34301.6	0.599224
$81$	0	0
$82$	85689.7	1.40732
$83$	54499.5	0.868356	0.434178	−	0.900827i	$-0.357039\pi$
$83$	54499.5	0.868356	0.434178	+	0.900827i	$0.357039\pi$
$84$	0	0
$85$	−20986.5	−0.315060
$86$	−60138.1	−0.876806
$87$	0	0
$88$	26678.0	0.367237
$89$	139829.	1.87120	0.935602	−	0.353057i	$-0.114858\pi$
$89$	139829.	1.87120	0.935602	+	0.353057i	$0.114858\pi$
$90$	0	0
$91$	34628.2	0.438355
$92$	−1511.27	−0.0186154
$93$	0	0
$94$	26794.2	0.312767
$95$	50351.2	0.572401
$96$	0	0
$97$	89845.2	0.969539	0.484770	−	0.874642i	$-0.338903\pi$
$97$	89845.2	0.969539	0.484770	+	0.874642i	$0.338903\pi$
$98$	32154.9	0.338207
$99$	0	0
$100$	−4472.64	−0.0447264
$101$	107293.	1.04657	0.523285	−	0.852158i	$-0.324706\pi$
$101$	107293.	1.04657	0.523285	+	0.852158i	$0.324706\pi$
$102$	0	0
$103$	160302.	1.48884	0.744419	−	0.667713i	$-0.232727\pi$
$103$	160302.	1.48884	0.744419	+	0.667713i	$0.232727\pi$
$104$	56874.7	0.515627
$105$	0	0
$106$	52280.5	0.451934
$107$	−79970.2	−0.675256	−0.337628	−	0.941280i	$-0.609625\pi$
$107$	−79970.2	−0.675256	−0.337628	+	0.941280i	$0.609625\pi$
$108$	0	0
$109$	−97964.0	−0.789769	−0.394885	−	0.918731i	$-0.629215\pi$
$109$	−97964.0	−0.789769	−0.394885	+	0.918731i	$0.629215\pi$
$110$	28161.3	0.221907
$111$	0	0
$112$	−115533.	−0.870287
$113$	237682.	1.75105	0.875527	−	0.483170i	$-0.160515\pi$
$113$	237682.	1.75105	0.875527	+	0.483170i	$0.160515\pi$
$114$	0	0
$115$	22703.1	0.160081
$116$	−1766.90	−0.0121917
$117$	0	0
$118$	−80204.2	−0.530264
$119$	70686.0	0.457579
$120$	0	0
$121$	−137704.	−0.855036
$122$	195515.	1.18927
$123$	0	0
$124$	4327.12	0.0252723
$125$	165820.	0.949210
$126$	0	0
$127$	194961.	1.07260	0.536299	−	0.844028i	$-0.319822\pi$
$127$	194961.	1.07260	0.536299	+	0.844028i	$0.319822\pi$
$128$	−201495.	−1.08702
$129$	0	0
$130$	60037.1	0.311574
$131$	−350750.	−1.78575	−0.892873	−	0.450309i	$-0.851314\pi$
$131$	−350750.	−1.78575	−0.892873	+	0.450309i	$0.851314\pi$
$132$	0	0
$133$	−169591.	−0.831330
$134$	−219097.	−1.05408
$135$	0	0
$136$	116097.	0.538239
$137$	−185712.	−0.845356	−0.422678	−	0.906280i	$-0.638910\pi$
$137$	−185712.	−0.845356	−0.422678	+	0.906280i	$0.638910\pi$
$138$	0	0
$139$	−316673.	−1.39019	−0.695095	−	0.718918i	$-0.744638\pi$
$139$	−316673.	−1.39019	−0.695095	+	0.718918i	$0.744638\pi$
$140$	−7048.95	−0.0303952
$141$	0	0
$142$	−86214.2	−0.358804
$143$	49772.6	0.203540
$144$	0	0
$145$	26543.3	0.104842
$146$	−369637.	−1.43514
$147$	0	0
$148$	31405.8	0.117857
$149$	326686.	1.20549	0.602747	−	0.797932i	$-0.294073\pi$
$149$	326686.	1.20549	0.602747	+	0.797932i	$0.294073\pi$
$150$	0	0
$151$	427072.	1.52426	0.762130	−	0.647424i	$-0.224154\pi$
$151$	427072.	1.52426	0.762130	+	0.647424i	$0.224154\pi$
$152$	−278543.	−0.977874
$153$	0	0
$154$	−94852.0	−0.322289
$155$	−65004.4	−0.217327
$156$	0	0
$157$	48668.5	0.157579	0.0787896	−	0.996891i	$-0.474894\pi$
$157$	48668.5	0.157579	0.0787896	+	0.996891i	$0.474894\pi$
$158$	158539.	0.505234
$159$	0	0
$160$	−23968.5	−0.0740185
$161$	−76467.7	−0.232495
$162$	0	0
$163$	−149764.	−0.441509	−0.220755	−	0.975329i	$-0.570852\pi$
$163$	−149764.	−0.441509	−0.220755	+	0.975329i	$0.570852\pi$
$164$	−30829.0	−0.0895056
$165$	0	0
$166$	−318256.	−0.896408
$167$	−371818.	−1.03167	−0.515833	−	0.856689i	$-0.672518\pi$
$167$	−371818.	−1.03167	−0.515833	+	0.856689i	$0.672518\pi$
$168$	0	0
$169$	−265183.	−0.714215
$170$	122553.	0.325238
$171$	0	0
$172$	21636.2	0.0557648
$173$	434068.	1.10266	0.551331	−	0.834287i	$-0.314120\pi$
$173$	434068.	1.10266	0.551331	+	0.834287i	$0.314120\pi$
$174$	0	0
$175$	−226308.	−0.558606
$176$	−166061.	−0.404098
$177$	0	0
$178$	−816543.	−1.93165
$179$	−341482.	−0.796590	−0.398295	−	0.917257i	$-0.630398\pi$
$179$	−341482.	−0.796590	−0.398295	+	0.917257i	$0.630398\pi$
$180$	0	0
$181$	542203.	1.23017	0.615086	−	0.788460i	$-0.289121\pi$
$181$	542203.	1.23017	0.615086	+	0.788460i	$0.289121\pi$
$182$	−202215.	−0.452517
$183$	0	0
$184$	−125593.	−0.273478
$185$	−471796.	−1.01350
$186$	0	0
$187$	101600.	0.212466
$188$	−9639.89	−0.0198920
$189$	0	0
$190$	−294031.	−0.590893
$191$	454420.	0.901310	0.450655	−	0.892698i	$-0.351190\pi$
$191$	454420.	0.901310	0.450655	+	0.892698i	$0.351190\pi$
$192$	0	0
$193$	791182.	1.52891	0.764457	−	0.644674i	$-0.223007\pi$
$193$	791182.	1.52891	0.764457	+	0.644674i	$0.223007\pi$
$194$	−524660.	−1.00086
$195$	0	0
$196$	−11568.6	−0.0215099
$197$	519219.	0.953201	0.476601	−	0.879120i	$-0.341869\pi$
$197$	519219.	0.953201	0.476601	+	0.879120i	$0.341869\pi$
$198$	0	0
$199$	−301548.	−0.539789	−0.269895	−	0.962890i	$-0.586989\pi$
$199$	−301548.	−0.539789	−0.269895	+	0.962890i	$0.586989\pi$
$200$	−371697.	−0.657074
$201$	0	0
$202$	−626548.	−1.08038
$203$	−89402.1	−0.152268
$204$	0	0
$205$	463131.	0.769696
$206$	−936103.	−1.53693
$207$	0	0
$208$	−354025.	−0.567383
$209$	−243760.	−0.386009
$210$	0	0
$211$	−473908.	−0.732804	−0.366402	−	0.930457i	$-0.619410\pi$
$211$	−473908.	−0.732804	−0.366402	+	0.930457i	$0.619410\pi$
$212$	−18809.2	−0.0287430
$213$	0	0
$214$	466994.	0.697071
$215$	−325031.	−0.479544
$216$	0	0
$217$	218945.	0.315636
$218$	572071.	0.815283
$219$	0	0
$220$	−10131.8	−0.0141133
$221$	216601.	0.298318
$222$	0	0
$223$	−135893.	−0.182993	−0.0914963	−	0.995805i	$-0.529165\pi$
$223$	−135893.	−0.182993	−0.0914963	+	0.995805i	$0.529165\pi$
$224$	80729.7	0.107501
$225$	0	0
$226$	−1.38797e6	−1.80762
$227$	1.21029e6	1.55893	0.779464	−	0.626448i	$-0.215492\pi$
$227$	1.21029e6	1.55893	0.779464	+	0.626448i	$0.215492\pi$
$228$	0	0
$229$	−577029.	−0.727125	−0.363563	−	0.931570i	$-0.618440\pi$
$229$	−577029.	−0.727125	−0.363563	+	0.931570i	$0.618440\pi$
$230$	−132577.	−0.165253
$231$	0	0
$232$	−146837.	−0.179109
$233$	966231.	1.16598	0.582990	−	0.812479i	$-0.301883\pi$
$233$	966231.	1.16598	0.582990	+	0.812479i	$0.301883\pi$
$234$	0	0
$235$	144816.	0.171059
$236$	28855.5	0.0337248
$237$	0	0
$238$	−412778.	−0.472361
$239$	−592337.	−0.670771	−0.335386	−	0.942081i	$-0.608867\pi$
$239$	−592337.	−0.670771	−0.335386	+	0.942081i	$0.608867\pi$
$240$	0	0
$241$	405276.	0.449478	0.224739	−	0.974419i	$-0.427847\pi$
$241$	405276.	0.449478	0.224739	+	0.974419i	$0.427847\pi$
$242$	804139.	0.882658
$243$	0	0
$244$	−70341.5	−0.0756374
$245$	173789.	0.184973
$246$	0	0
$247$	−519672.	−0.541985
$248$	359605.	0.371275
$249$	0	0
$250$	−968324.	−0.979875
$251$	−922052.	−0.923785	−0.461893	−	0.886936i	$-0.652829\pi$
$251$	−922052.	−0.923785	−0.461893	+	0.886936i	$0.652829\pi$
$252$	0	0
$253$	−109910.	−0.107954
$254$	−1.13849e6	−1.10725
$255$	0	0
$256$	205661.	0.196133
$257$	−461687.	−0.436028	−0.218014	−	0.975946i	$-0.569958\pi$
$257$	−461687.	−0.436028	−0.218014	+	0.975946i	$0.569958\pi$
$258$	0	0
$259$	1.58909e6	1.47197
$260$	−21599.9	−0.0198161
$261$	0	0
$262$	2.04824e6	1.84344
$263$	1.78425e6	1.59062	0.795311	−	0.606202i	$-0.207308\pi$
$263$	1.78425e6	1.59062	0.795311	+	0.606202i	$0.207308\pi$
$264$	0	0
$265$	282563.	0.247173
$266$	990344.	0.858187
$267$	0	0
$268$	78825.9	0.0670396
$269$	1.43033e6	1.20519	0.602596	−	0.798046i	$-0.294133\pi$
$269$	1.43033e6	1.20519	0.602596	+	0.798046i	$0.294133\pi$
$270$	0	0
$271$	418695.	0.346318	0.173159	−	0.984894i	$-0.444603\pi$
$271$	418695.	0.346318	0.173159	+	0.984894i	$0.444603\pi$
$272$	−722667.	−0.592265
$273$	0	0
$274$	1.08449e6	0.872666
$275$	−325282.	−0.259375
$276$	0	0
$277$	319476.	0.250172	0.125086	−	0.992146i	$-0.460079\pi$
$277$	319476.	0.250172	0.125086	+	0.992146i	$0.460079\pi$
$278$	1.84925e6	1.43510
$279$	0	0
$280$	−585802.	−0.446535
$281$	−421076.	−0.318123	−0.159061	−	0.987269i	$-0.550847\pi$
$281$	−421076.	−0.318123	−0.159061	+	0.987269i	$0.550847\pi$
$282$	0	0
$283$	−129444.	−0.0960759	−0.0480380	−	0.998846i	$-0.515297\pi$
$283$	−129444.	−0.0960759	−0.0480380	+	0.998846i	$0.515297\pi$
$284$	31017.7	0.0228199
$285$	0	0
$286$	−290652.	−0.210116
$287$	−1.55990e6	−1.11787
$288$	0	0
$289$	−977713.	−0.688600
$290$	−155002.	−0.108229
$291$	0	0
$292$	132986.	0.0912746
$293$	−1.71643e6	−1.16804	−0.584020	−	0.811740i	$-0.698521\pi$
$293$	−1.71643e6	−1.16804	−0.584020	+	0.811740i	$0.698521\pi$
$294$	0	0
$295$	−433483.	−0.290013
$296$	2.60998e6	1.73144
$297$	0	0
$298$	−1.90772e6	−1.24444
$299$	−234317.	−0.151575
$300$	0	0
$301$	1.09476e6	0.696469
$302$	−2.49393e6	−1.57350
$303$	0	0
$304$	1.73383e6	1.07603
$305$	1.05671e6	0.650437
$306$	0	0
$307$	2.88448e6	1.74671	0.873356	−	0.487083i	$-0.161939\pi$
$307$	2.88448e6	1.74671	0.873356	+	0.487083i	$0.161939\pi$
$308$	34125.4	0.0204975
$309$	0	0
$310$	379600.	0.224348
$311$	2.59382e6	1.52069	0.760343	−	0.649522i	$-0.225031\pi$
$311$	2.59382e6	1.52069	0.760343	+	0.649522i	$0.225031\pi$
$312$	0	0
$313$	1.47281e6	0.849742	0.424871	−	0.905254i	$-0.360319\pi$
$313$	1.47281e6	0.849742	0.424871	+	0.905254i	$0.360319\pi$
$314$	−284205.	−0.162670
$315$	0	0
$316$	−57038.3	−0.0321328
$317$	−661940.	−0.369973	−0.184987	−	0.982741i	$-0.559224\pi$
$317$	−661940.	−0.369973	−0.184987	+	0.982741i	$0.559224\pi$
$318$	0	0
$319$	−128501.	−0.0707019
$320$	−957685.	−0.522815
$321$	0	0
$322$	446541.	0.240006
$323$	−1.06080e6	−0.565753
$324$	0	0
$325$	−693469.	−0.364182
$326$	874564.	0.455772
$327$	0	0
$328$	−2.56204e6	−1.31493
$329$	−487763.	−0.248439
$330$	0	0
$331$	3.40774e6	1.70961	0.854805	−	0.518949i	$-0.173677\pi$
$331$	3.40774e6	1.70961	0.854805	+	0.518949i	$0.173677\pi$
$332$	114501.	0.0570115
$333$	0	0
$334$	2.17127e6	1.06499
$335$	−1.18417e6	−0.576501
$336$	0	0
$337$	−2.52588e6	−1.21154	−0.605771	−	0.795639i	$-0.707135\pi$
$337$	−2.52588e6	−1.21154	−0.605771	+	0.795639i	$0.707135\pi$
$338$	1.54856e6	0.737288
$339$	0	0
$340$	−44091.5	−0.0206851
$341$	314700.	0.146558
$342$	0	0
$343$	−2.37201e6	−1.08863
$344$	1.79807e6	0.819240
$345$	0	0
$346$	−2.53478e6	−1.13828
$347$	2.09340e6	0.933317	0.466658	−	0.884438i	$-0.345458\pi$
$347$	2.09340e6	0.933317	0.466658	+	0.884438i	$0.345458\pi$
$348$	0	0
$349$	2.90749e6	1.27777	0.638887	−	0.769300i	$-0.279395\pi$
$349$	2.90749e6	1.27777	0.638887	+	0.769300i	$0.279395\pi$
$350$	1.32155e6	0.576652
$351$	0	0
$352$	116036.	0.0499157
$353$	8176.61	0.00349250	0.00174625	−	0.999998i	$-0.499444\pi$
$353$	8176.61	0.00349250	0.00174625	+	0.999998i	$0.499444\pi$
$354$	0	0
$355$	−465966.	−0.196238
$356$	293772.	0.122853
$357$	0	0
$358$	1.99412e6	0.822324
$359$	−457001.	−0.187146	−0.0935732	−	0.995612i	$-0.529829\pi$
$359$	−457001.	−0.187146	−0.0935732	+	0.995612i	$0.529829\pi$
$360$	0	0
$361$	68986.9	0.0278611
$362$	−3.16625e6	−1.26991
$363$	0	0
$364$	72752.0	0.0287800
$365$	−1.99779e6	−0.784907
$366$	0	0
$367$	3.50049e6	1.35664	0.678318	−	0.734768i	$-0.262709\pi$
$367$	3.50049e6	1.35664	0.678318	+	0.734768i	$0.262709\pi$
$368$	781776.	0.300928
$369$	0	0
$370$	2.75510e6	1.04624
$371$	−951717.	−0.358983
$372$	0	0
$373$	−2.17554e6	−0.809645	−0.404822	−	0.914395i	$-0.632667\pi$
$373$	−2.17554e6	−0.809645	−0.404822	+	0.914395i	$0.632667\pi$
$374$	−593304.	−0.219330
$375$	0	0
$376$	−801121.	−0.292232
$377$	−273952.	−0.0992707
$378$	0	0
$379$	1.97581e6	0.706557	0.353278	−	0.935518i	$-0.385067\pi$
$379$	1.97581e6	0.706557	0.353278	+	0.935518i	$0.385067\pi$
$380$	105785.	0.0375807
$381$	0	0
$382$	−2.65363e6	−0.930428
$383$	−2.77434e6	−0.966412	−0.483206	−	0.875507i	$-0.660528\pi$
$383$	−2.77434e6	−0.966412	−0.483206	+	0.875507i	$0.660528\pi$
$384$	0	0
$385$	−512651.	−0.176267
$386$	−4.62019e6	−1.57831
$387$	0	0
$388$	188760.	0.0636547
$389$	−2.12869e6	−0.713245	−0.356622	−	0.934249i	$-0.616072\pi$
$389$	−2.12869e6	−0.713245	−0.356622	+	0.934249i	$0.616072\pi$
$390$	0	0
$391$	−478309.	−0.158222
$392$	−961402.	−0.316002
$393$	0	0
$394$	−3.03203e6	−0.983995
$395$	856861.	0.276323
$396$	0	0
$397$	2.73423e6	0.870680	0.435340	−	0.900266i	$-0.356628\pi$
$397$	2.73423e6	0.870680	0.435340	+	0.900266i	$0.356628\pi$
$398$	1.76092e6	0.557228
$399$	0	0
$400$	2.31369e6	0.723028
$401$	1.29206e6	0.401257	0.200628	−	0.979667i	$-0.435702\pi$
$401$	1.29206e6	0.401257	0.200628	+	0.979667i	$0.435702\pi$
$402$	0	0
$403$	670908.	0.205779
$404$	225417.	0.0687120
$405$	0	0
$406$	522072.	0.157187
$407$	2.28406e6	0.683473
$408$	0	0
$409$	−2.78691e6	−0.823787	−0.411894	−	0.911232i	$-0.635132\pi$
$409$	−2.78691e6	−0.823787	−0.411894	+	0.911232i	$0.635132\pi$
$410$	−2.70450e6	−0.794561
$411$	0	0
$412$	336787.	0.0977489
$413$	1.46004e6	0.421202
$414$	0	0
$415$	−1.72009e6	−0.490265
$416$	247378.	0.0700853
$417$	0	0
$418$	1.42346e6	0.398479
$419$	1.51265e6	0.420924	0.210462	−	0.977602i	$-0.432503\pi$
$419$	1.51265e6	0.420924	0.210462	+	0.977602i	$0.432503\pi$
$420$	0	0
$421$	2.55990e6	0.703911	0.351955	−	0.936017i	$-0.385517\pi$
$421$	2.55990e6	0.703911	0.351955	+	0.936017i	$0.385517\pi$
$422$	2.76743e6	0.756477
$423$	0	0
$424$	−1.56314e6	−0.422263
$425$	−1.41557e6	−0.380153
$426$	0	0
$427$	−3.55917e6	−0.944666
$428$	−168013.	−0.0443336
$429$	0	0
$430$	1.89805e6	0.495036
$431$	5.38406e6	1.39610	0.698051	−	0.716048i	$-0.254051\pi$
$431$	5.38406e6	1.39610	0.698051	+	0.716048i	$0.254051\pi$
$432$	0	0
$433$	1.90699e6	0.488797	0.244398	−	0.969675i	$-0.421410\pi$
$433$	1.90699e6	0.488797	0.244398	+	0.969675i	$0.421410\pi$
$434$	−1.27855e6	−0.325833
$435$	0	0
$436$	−205817.	−0.0518520
$437$	1.14757e6	0.287458
$438$	0	0
$439$	−3.84641e6	−0.952565	−0.476283	−	0.879292i	$-0.658016\pi$
$439$	−3.84641e6	−0.952565	−0.476283	+	0.879292i	$0.658016\pi$
$440$	−841998.	−0.207338
$441$	0	0
$442$	−1.26486e6	−0.307955
$443$	−1.43911e6	−0.348406	−0.174203	−	0.984710i	$-0.555735\pi$
$443$	−1.43911e6	−0.348406	−0.174203	+	0.984710i	$0.555735\pi$
$444$	0	0
$445$	−4.41321e6	−1.05646
$446$	793558.	0.188904
$447$	0	0
$448$	3.22564e6	0.759313
$449$	5.05146e6	1.18250	0.591249	−	0.806489i	$-0.298635\pi$
$449$	5.05146e6	1.18250	0.591249	+	0.806489i	$0.298635\pi$
$450$	0	0
$451$	−2.24211e6	−0.519058
$452$	499356.	0.114965
$453$	0	0
$454$	−7.06763e6	−1.60929
$455$	−1.09292e6	−0.247491
$456$	0	0
$457$	−3.02849e6	−0.678323	−0.339161	−	0.940728i	$-0.610143\pi$
$457$	−3.02849e6	−0.678323	−0.339161	+	0.940728i	$0.610143\pi$
$458$	3.36962e6	0.750615
$459$	0	0
$460$	47697.9	0.0105101
$461$	3.54068e6	0.775951	0.387976	−	0.921670i	$-0.373174\pi$
$461$	3.54068e6	0.775951	0.387976	+	0.921670i	$0.373174\pi$
$462$	0	0
$463$	3.78475e6	0.820512	0.410256	−	0.911970i	$-0.365439\pi$
$463$	3.78475e6	0.820512	0.410256	+	0.911970i	$0.365439\pi$
$464$	914012.	0.197087
$465$	0	0
$466$	−5.64240e6	−1.20365
$467$	3.93030e6	0.833938	0.416969	−	0.908921i	$-0.363092\pi$
$467$	3.93030e6	0.833938	0.416969	+	0.908921i	$0.363092\pi$
$468$	0	0
$469$	3.98846e6	0.837285
$470$	−845666.	−0.176585
$471$	0	0
$472$	2.39803e6	0.495450
$473$	1.57354e6	0.323389
$474$	0	0
$475$	3.39625e6	0.690663
$476$	148507.	0.0300421
$477$	0	0
$478$	3.45901e6	0.692441
$479$	3.65493e6	0.727848	0.363924	−	0.931429i	$-0.381437\pi$
$479$	3.65493e6	0.727848	0.363924	+	0.931429i	$0.381437\pi$
$480$	0	0
$481$	4.86938e6	0.959647
$482$	−2.36665e6	−0.463999
$483$	0	0
$484$	−289309.	−0.0561370
$485$	−2.83565e6	−0.547392
$486$	0	0
$487$	−3.90232e6	−0.745591	−0.372796	−	0.927913i	$-0.621601\pi$
$487$	−3.90232e6	−0.745591	−0.372796	+	0.927913i	$0.621601\pi$
$488$	−5.84571e6	−1.11119
$489$	0	0
$490$	−1.01486e6	−0.190948
$491$	−8.22645e6	−1.53996	−0.769979	−	0.638069i	$-0.779733\pi$
$491$	−8.22645e6	−1.53996	−0.769979	+	0.638069i	$0.779733\pi$
$492$	0	0
$493$	−559214.	−0.103624
$494$	3.03468e6	0.559494
$495$	0	0
$496$	−2.23841e6	−0.408542
$497$	1.56945e6	0.285007
$498$	0	0
$499$	1.95126e6	0.350803	0.175401	−	0.984497i	$-0.443878\pi$
$499$	1.95126e6	0.350803	0.175401	+	0.984497i	$0.443878\pi$
$500$	348379.	0.0623200
$501$	0	0
$502$	5.38442e6	0.953629
$503$	−6.74097e6	−1.18796	−0.593981	−	0.804479i	$-0.702445\pi$
$503$	−6.74097e6	−1.18796	−0.593981	+	0.804479i	$0.702445\pi$
$504$	0	0
$505$	−3.38633e6	−0.590883
$506$	641832.	0.111441
$507$	0	0
$508$	409601.	0.0704210
$509$	−2.57165e6	−0.439964	−0.219982	−	0.975504i	$-0.570600\pi$
$509$	−2.57165e6	−0.439964	−0.219982	+	0.975504i	$0.570600\pi$
$510$	0	0
$511$	6.72889e6	1.13996
$512$	5.24686e6	0.884554
$513$	0	0
$514$	2.69607e6	0.450114
$515$	−5.05940e6	−0.840583
$516$	0	0
$517$	−701083.	−0.115357
$518$	−9.27963e6	−1.51952
$519$	0	0
$520$	−1.79505e6	−0.291118
$521$	−1.25025e6	−0.201791	−0.100896	−	0.994897i	$-0.532171\pi$
$521$	−1.25025e6	−0.201791	−0.100896	+	0.994897i	$0.532171\pi$
$522$	0	0
$523$	−188395.	−0.0301172	−0.0150586	−	0.999887i	$-0.504793\pi$
$523$	−188395.	−0.0301172	−0.0150586	+	0.999887i	$0.504793\pi$
$524$	−736907.	−0.117242
$525$	0	0
$526$	−1.04193e7	−1.64201
$527$	1.36951e6	0.214803
$528$	0	0
$529$	−5.91891e6	−0.919608
$530$	−1.65005e6	−0.255158
$531$	0	0
$532$	−356301.	−0.0545806
$533$	−4.77995e6	−0.728795
$534$	0	0
$535$	2.52398e6	0.381243
$536$	6.55080e6	0.984878
$537$	0	0
$538$	−8.35258e6	−1.24413
$539$	−841349.	−0.124740
$540$	0	0
$541$	−7.69990e6	−1.13108	−0.565538	−	0.824722i	$-0.691331\pi$
$541$	−7.69990e6	−1.13108	−0.565538	+	0.824722i	$0.691331\pi$
$542$	−2.44501e6	−0.357506
$543$	0	0
$544$	504968.	0.0731588
$545$	3.09190e6	0.445896
$546$	0	0
$547$	1.21820e7	1.74081	0.870406	−	0.492334i	$-0.163856\pi$
$547$	1.21820e7	1.74081	0.870406	+	0.492334i	$0.163856\pi$
$548$	−390172.	−0.0555015
$549$	0	0
$550$	1.89952e6	0.267755
$551$	1.34167e6	0.188264
$552$	0	0
$553$	−2.88605e6	−0.401320
$554$	−1.86561e6	−0.258254
$555$	0	0
$556$	−665313.	−0.0912723
$557$	−5.21687e6	−0.712479	−0.356239	−	0.934395i	$-0.615941\pi$
$557$	−5.21687e6	−0.712479	−0.356239	+	0.934395i	$0.615941\pi$
$558$	0	0
$559$	3.35463e6	0.454062
$560$	3.64641e6	0.491356
$561$	0	0
$562$	2.45892e6	0.328400
$563$	782547.	0.104049	0.0520247	−	0.998646i	$-0.483433\pi$
$563$	782547.	0.104049	0.0520247	+	0.998646i	$0.483433\pi$
$564$	0	0
$565$	−7.50160e6	−0.988627
$566$	755899.	0.0991797
$567$	0	0
$568$	2.57772e6	0.335247
$569$	−1.01820e7	−1.31841	−0.659206	−	0.751962i	$-0.729107\pi$
$569$	−1.01820e7	−1.31841	−0.659206	+	0.751962i	$0.729107\pi$
$570$	0	0
$571$	−1.56900e6	−0.201388	−0.100694	−	0.994917i	$-0.532106\pi$
$571$	−1.56900e6	−0.201388	−0.100694	+	0.994917i	$0.532106\pi$
$572$	104569.	0.0133633
$573$	0	0
$574$	9.10919e6	1.15399
$575$	1.53135e6	0.193155
$576$	0	0
$577$	−1.53116e7	−1.91462	−0.957308	−	0.289069i	$-0.906654\pi$
$577$	−1.53116e7	−1.91462	−0.957308	+	0.289069i	$0.906654\pi$
$578$	5.70945e6	0.710845
$579$	0	0
$580$	55766.0	0.00688334
$581$	5.79355e6	0.712040
$582$	0	0
$583$	−1.36794e6	−0.166685
$584$	1.10518e7	1.34091
$585$	0	0
$586$	1.00233e7	1.20577
$587$	−3.65745e6	−0.438110	−0.219055	−	0.975712i	$-0.570297\pi$
$587$	−3.65745e6	−0.438110	−0.219055	+	0.975712i	$0.570297\pi$
$588$	0	0
$589$	−3.28576e6	−0.390254
$590$	2.53137e6	0.299382
$591$	0	0
$592$	−1.62462e7	−1.90523
$593$	7.18479e6	0.839029	0.419515	−	0.907749i	$-0.362200\pi$
$593$	7.18479e6	0.839029	0.419515	+	0.907749i	$0.362200\pi$
$594$	0	0
$595$	−2.23096e6	−0.258345
$596$	686350.	0.0791462
$597$	0	0
$598$	1.36832e6	0.156471
$599$	−2.35482e6	−0.268158	−0.134079	−	0.990971i	$-0.542808\pi$
$599$	−2.35482e6	−0.268158	−0.134079	+	0.990971i	$0.542808\pi$
$600$	0	0
$601$	−7.49135e6	−0.846007	−0.423003	−	0.906128i	$-0.639024\pi$
$601$	−7.49135e6	−0.846007	−0.423003	+	0.906128i	$0.639024\pi$
$602$	−6.39295e6	−0.718969
$603$	0	0
$604$	897256.	0.100075
$605$	4.34616e6	0.482745
$606$	0	0
$607$	248110.	0.0273320	0.0136660	−	0.999907i	$-0.495650\pi$
$607$	248110.	0.0273320	0.0136660	+	0.999907i	$0.495650\pi$
$608$	−1.21153e6	−0.132915
$609$	0	0
$610$	−6.17075e6	−0.671450
$611$	−1.49464e6	−0.161969
$612$	0	0
$613$	−1.97712e6	−0.212511	−0.106256	−	0.994339i	$-0.533886\pi$
$613$	−1.97712e6	−0.212511	−0.106256	+	0.994339i	$0.533886\pi$
$614$	−1.68442e7	−1.80314
$615$	0	0
$616$	2.83599e6	0.301129
$617$	−1.43605e7	−1.51865	−0.759323	−	0.650714i	$-0.774470\pi$
$617$	−1.43605e7	−1.51865	−0.759323	+	0.650714i	$0.774470\pi$
$618$	0	0
$619$	−3.63487e6	−0.381297	−0.190648	−	0.981658i	$-0.561059\pi$
$619$	−3.63487e6	−0.381297	−0.190648	+	0.981658i	$0.561059\pi$
$620$	−136571.	−0.0142685
$621$	0	0
$622$	−1.51469e7	−1.56981
$623$	1.48644e7	1.53436
$624$	0	0
$625$	1.41917e6	0.145323
$626$	−8.60065e6	−0.877193
$627$	0	0
$628$	102250.	0.0103458
$629$	9.93980e6	1.00173
$630$	0	0
$631$	−1.71093e6	−0.171065	−0.0855323	−	0.996335i	$-0.527259\pi$
$631$	−1.71093e6	−0.171065	−0.0855323	+	0.996335i	$0.527259\pi$
$632$	−4.74015e6	−0.472063
$633$	0	0
$634$	3.86547e6	0.381926
$635$	−6.15326e6	−0.605579
$636$	0	0
$637$	−1.79367e6	−0.175144
$638$	750397.	0.0729860
$639$	0	0
$640$	6.35949e6	0.613723
$641$	7.33888e6	0.705480	0.352740	−	0.935721i	$-0.385250\pi$
$641$	7.33888e6	0.705480	0.352740	+	0.935721i	$0.385250\pi$
$642$	0	0
$643$	1.65108e7	1.57485	0.787426	−	0.616409i	$-0.211413\pi$
$643$	1.65108e7	1.57485	0.787426	+	0.616409i	$0.211413\pi$
$644$	−160654.	−0.0152643
$645$	0	0
$646$	6.19464e6	0.584030
$647$	4.75302e6	0.446384	0.223192	−	0.974774i	$-0.428352\pi$
$647$	4.75302e6	0.446384	0.223192	+	0.974774i	$0.428352\pi$
$648$	0	0
$649$	2.09858e6	0.195575
$650$	4.04958e6	0.375947
$651$	0	0
$652$	−314647.	−0.0289871
$653$	−2.06957e7	−1.89932	−0.949658	−	0.313290i	$-0.898569\pi$
$653$	−2.06957e7	−1.89932	−0.949658	+	0.313290i	$0.898569\pi$
$654$	0	0
$655$	1.10702e7	1.00821
$656$	1.59478e7	1.44691
$657$	0	0
$658$	2.84834e6	0.256465
$659$	−1.75250e7	−1.57197	−0.785987	−	0.618244i	$-0.787845\pi$
$659$	−1.75250e7	−1.57197	−0.785987	+	0.618244i	$0.787845\pi$
$660$	0	0
$661$	5.06876e6	0.451230	0.225615	−	0.974217i	$-0.427561\pi$
$661$	5.06876e6	0.451230	0.225615	+	0.974217i	$0.427561\pi$
$662$	−1.98999e7	−1.76484
$663$	0	0
$664$	9.51554e6	0.837555
$665$	5.35256e6	0.469361
$666$	0	0
$667$	604954.	0.0526511
$668$	−781170.	−0.0677336
$669$	0	0
$670$	6.91505e6	0.595125
$671$	−5.11574e6	−0.438634
$672$	0	0
$673$	1.07503e7	0.914922	0.457461	−	0.889230i	$-0.348759\pi$
$673$	1.07503e7	0.914922	0.457461	+	0.889230i	$0.348759\pi$
$674$	1.47501e7	1.25068
$675$	0	0
$676$	−557135.	−0.0468914
$677$	6.95239e6	0.582992	0.291496	−	0.956572i	$-0.405847\pi$
$677$	6.95239e6	0.582992	0.291496	+	0.956572i	$0.405847\pi$
$678$	0	0
$679$	9.55094e6	0.795009
$680$	−3.66422e6	−0.303885
$681$	0	0
$682$	−1.83772e6	−0.151293
$683$	1.26571e7	1.03821	0.519103	−	0.854711i	$-0.326266\pi$
$683$	1.26571e7	1.03821	0.519103	+	0.854711i	$0.326266\pi$
$684$	0	0
$685$	5.86137e6	0.477280
$686$	1.38516e7	1.12380
$687$	0	0
$688$	−1.11924e7	−0.901470
$689$	−2.91632e6	−0.234038
$690$	0	0
$691$	−7.61067e6	−0.606356	−0.303178	−	0.952934i	$-0.598048\pi$
$691$	−7.61067e6	−0.606356	−0.303178	+	0.952934i	$0.598048\pi$
$692$	911952.	0.0723947
$693$	0	0
$694$	−1.22246e7	−0.963468
$695$	9.99470e6	0.784888
$696$	0	0
$697$	−9.75724e6	−0.760756
$698$	−1.69786e7	−1.31905
$699$	0	0
$700$	−475461.	−0.0366750
$701$	1.63413e7	1.25600	0.628002	−	0.778211i	$-0.283873\pi$
$701$	1.63413e7	1.25600	0.628002	+	0.778211i	$0.283873\pi$
$702$	0	0
$703$	−2.38477e7	−1.81995
$704$	4.63635e6	0.352569
$705$	0	0
$706$	−47748.1	−0.00360533
$707$	1.14057e7	0.858172
$708$	0	0
$709$	1.11269e7	0.831300	0.415650	−	0.909525i	$-0.363554\pi$
$709$	1.11269e7	0.831300	0.415650	+	0.909525i	$0.363554\pi$
$710$	2.72105e6	0.202577
$711$	0	0
$712$	2.44139e7	1.80483
$713$	−1.48153e6	−0.109141
$714$	0	0
$715$	−1.57090e6	−0.114917
$716$	−717434.	−0.0522997
$717$	0	0
$718$	2.66871e6	0.193192
$719$	−1.28895e7	−0.929849	−0.464924	−	0.885350i	$-0.653919\pi$
$719$	−1.28895e7	−0.929849	−0.464924	+	0.885350i	$0.653919\pi$
$720$	0	0
$721$	1.70409e7	1.22083
$722$	−402856.	−0.0287612
$723$	0	0
$724$	1.13914e6	0.0807664
$725$	1.79038e6	0.126503
$726$	0	0
$727$	−2.32214e7	−1.62949	−0.814745	−	0.579820i	$-0.803123\pi$
$727$	−2.32214e7	−1.62949	−0.814745	+	0.579820i	$0.803123\pi$
$728$	6.04603e6	0.422807
$729$	0	0
$730$	1.16663e7	0.810264
$731$	6.84776e6	0.473974
$732$	0	0
$733$	−8.72009e6	−0.599461	−0.299731	−	0.954024i	$-0.596897\pi$
$733$	−8.72009e6	−0.599461	−0.299731	+	0.954024i	$0.596897\pi$
$734$	−2.04415e7	−1.40046
$735$	0	0
$736$	−546271.	−0.0371718
$737$	5.73279e6	0.388774
$738$	0	0
$739$	2.49740e7	1.68220	0.841098	−	0.540882i	$-0.181910\pi$
$739$	2.49740e7	1.68220	0.841098	+	0.540882i	$0.181910\pi$
$740$	−991217.	−0.0665410
$741$	0	0
$742$	5.55765e6	0.370580
$743$	288063.	0.0191433	0.00957164	−	0.999954i	$-0.496953\pi$
$743$	288063.	0.0191433	0.00957164	+	0.999954i	$0.496953\pi$
$744$	0	0
$745$	−1.03107e7	−0.680611
$746$	1.27043e7	0.835801
$747$	0	0
$748$	213456.	0.0139494
$749$	−8.50119e6	−0.553701
$750$	0	0
$751$	2.19870e7	1.42255	0.711274	−	0.702915i	$-0.248119\pi$
$751$	2.19870e7	1.42255	0.711274	+	0.702915i	$0.248119\pi$
$752$	4.98670e6	0.321565
$753$	0	0
$754$	1.59977e6	0.102478
$755$	−1.34791e7	−0.860582
$756$	0	0
$757$	−6.72062e6	−0.426255	−0.213128	−	0.977024i	$-0.568365\pi$
$757$	−6.72062e6	−0.426255	−0.213128	+	0.977024i	$0.568365\pi$
$758$	−1.15379e7	−0.729382
$759$	0	0
$760$	8.79124e6	0.552098
$761$	1.10936e7	0.694400	0.347200	−	0.937791i	$-0.387132\pi$
$761$	1.10936e7	0.694400	0.347200	+	0.937791i	$0.387132\pi$
$762$	0	0
$763$	−1.04140e7	−0.647600
$764$	954712.	0.0591751
$765$	0	0
$766$	1.62010e7	0.997632
$767$	4.47396e6	0.274602
$768$	0	0
$769$	6.42928e6	0.392054	0.196027	−	0.980598i	$-0.437196\pi$
$769$	6.42928e6	0.392054	0.196027	+	0.980598i	$0.437196\pi$
$770$	2.99368e6	0.181961
$771$	0	0
$772$	1.66223e6	0.100380
$773$	1.38614e7	0.834371	0.417185	−	0.908821i	$-0.363017\pi$
$773$	1.38614e7	0.834371	0.417185	+	0.908821i	$0.363017\pi$
$774$	0	0
$775$	−4.38463e6	−0.262228
$776$	1.56868e7	0.935150
$777$	0	0
$778$	1.24307e7	0.736287
$779$	2.34097e7	1.38214
$780$	0	0
$781$	2.25584e6	0.132337
$782$	2.79313e6	0.163333
$783$	0	0
$784$	5.98440e6	0.347721
$785$	−1.53605e6	−0.0889677
$786$	0	0
$787$	2.48798e7	1.43189	0.715945	−	0.698157i	$-0.245996\pi$
$787$	2.48798e7	1.43189	0.715945	+	0.698157i	$0.245996\pi$
$788$	1.09085e6	0.0625820
$789$	0	0
$790$	−5.00372e6	−0.285250
$791$	2.52666e7	1.43584
$792$	0	0
$793$	−1.09062e7	−0.615874
$794$	−1.59668e7	−0.898808
$795$	0	0
$796$	−633537.	−0.0354396
$797$	−2.80418e7	−1.56372	−0.781861	−	0.623453i	$-0.785729\pi$
$797$	−2.80418e7	−1.56372	−0.781861	+	0.623453i	$0.785729\pi$
$798$	0	0
$799$	−3.05098e6	−0.169072
$800$	−1.61671e6	−0.0893112
$801$	0	0
$802$	−7.54513e6	−0.414220
$803$	9.67173e6	0.529316
$804$	0	0
$805$	2.41344e6	0.131264
$806$	−3.91783e6	−0.212426
$807$	0	0
$808$	1.87332e7	1.00945
$809$	−2.45320e7	−1.31784	−0.658919	−	0.752214i	$-0.728986\pi$
$809$	−2.45320e7	−1.31784	−0.658919	+	0.752214i	$0.728986\pi$
$810$	0	0
$811$	2.44306e7	1.30431	0.652156	−	0.758084i	$-0.273865\pi$
$811$	2.44306e7	1.30431	0.652156	+	0.758084i	$0.273865\pi$
$812$	−187829.	−0.00999706
$813$	0	0
$814$	−1.33380e7	−0.705553
$815$	4.72680e6	0.249272
$816$	0	0
$817$	−1.64292e7	−0.861118
$818$	1.62745e7	0.850400
$819$	0	0
$820$	973012.	0.0505340
$821$	4.22220e6	0.218615	0.109308	−	0.994008i	$-0.465137\pi$
$821$	4.22220e6	0.218615	0.109308	+	0.994008i	$0.465137\pi$
$822$	0	0
$823$	−2.11475e7	−1.08833	−0.544165	−	0.838979i	$-0.683153\pi$
$823$	−2.11475e7	−1.08833	−0.544165	+	0.838979i	$0.683153\pi$
$824$	2.79886e7	1.43603
$825$	0	0
$826$	−8.52607e6	−0.434809
$827$	−2.31990e6	−0.117952	−0.0589761	−	0.998259i	$-0.518784\pi$
$827$	−2.31990e6	−0.117952	−0.0589761	+	0.998259i	$0.518784\pi$
$828$	0	0
$829$	2.89837e7	1.46476	0.732381	−	0.680895i	$-0.238409\pi$
$829$	2.89837e7	1.46476	0.732381	+	0.680895i	$0.238409\pi$
$830$	1.00446e7	0.506103
$831$	0	0
$832$	9.88423e6	0.495033
$833$	−3.66139e6	−0.182824
$834$	0	0
$835$	1.17352e7	0.582469
$836$	−512127.	−0.0253432
$837$	0	0
$838$	−8.83328e6	−0.434522
$839$	−3.32541e7	−1.63095	−0.815473	−	0.578795i	$-0.803523\pi$
$839$	−3.32541e7	−1.63095	−0.815473	+	0.578795i	$0.803523\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	−1.49488e7	−0.726651
$843$	0	0
$844$	−995655.	−0.0481119
$845$	8.36958e6	0.403239
$846$	0	0
$847$	−1.46386e7	−0.701118
$848$	9.73000e6	0.464647
$849$	0	0
$850$	8.26635e6	0.392434
$851$	−1.07528e7	−0.508977
$852$	0	0
$853$	1.48503e7	0.698816	0.349408	−	0.936971i	$-0.386383\pi$
$853$	1.48503e7	0.698816	0.349408	+	0.936971i	$0.386383\pi$
$854$	2.07841e7	0.975184
$855$	0	0
$856$	−1.39627e7	−0.651305
$857$	−2.15523e7	−1.00240	−0.501201	−	0.865331i	$-0.667108\pi$
$857$	−2.15523e7	−1.00240	−0.501201	+	0.865331i	$0.667108\pi$
$858$	0	0
$859$	−1.84444e7	−0.852867	−0.426434	−	0.904519i	$-0.640230\pi$
$859$	−1.84444e7	−0.852867	−0.426434	+	0.904519i	$0.640230\pi$
$860$	−682872.	−0.0314843
$861$	0	0
$862$	−3.14408e7	−1.44120
$863$	2.12674e7	0.972046	0.486023	−	0.873946i	$-0.338447\pi$
$863$	2.12674e7	0.972046	0.486023	+	0.873946i	$0.338447\pi$
$864$	0	0
$865$	−1.36999e7	−0.622552
$866$	−1.11360e7	−0.504587
$867$	0	0
$868$	459993.	0.0207229
$869$	−4.14824e6	−0.186343
$870$	0	0
$871$	1.22217e7	0.545867
$872$	−1.71044e7	−0.761756
$873$	0	0
$874$	−6.70133e6	−0.296744
$875$	1.76274e7	0.778339
$876$	0	0
$877$	−2.51615e7	−1.10468	−0.552342	−	0.833618i	$-0.686266\pi$
$877$	−2.51615e7	−1.10468	−0.552342	+	0.833618i	$0.686266\pi$
$878$	2.24615e7	0.983338
$879$	0	0
$880$	5.24115e6	0.228150
$881$	−1.60794e7	−0.697961	−0.348980	−	0.937130i	$-0.613472\pi$
$881$	−1.60794e7	−0.697961	−0.348980	+	0.937130i	$0.613472\pi$
$882$	0	0
$883$	−7.09137e6	−0.306075	−0.153038	−	0.988220i	$-0.548906\pi$
$883$	−7.09137e6	−0.306075	−0.153038	+	0.988220i	$0.548906\pi$
$884$	455067.	0.0195859
$885$	0	0
$886$	8.40384e6	0.359661
$887$	1.00313e7	0.428103	0.214051	−	0.976822i	$-0.431334\pi$
$887$	1.00313e7	0.428103	0.214051	+	0.976822i	$0.431334\pi$
$888$	0	0
$889$	2.07252e7	0.879516
$890$	2.57714e7	1.09059
$891$	0	0
$892$	−285503.	−0.0120143
$893$	7.31996e6	0.307171
$894$	0	0
$895$	1.07777e7	0.449747
$896$	−2.14198e7	−0.891344
$897$	0	0
$898$	−2.94985e7	−1.22070
$899$	−1.73213e6	−0.0714795
$900$	0	0
$901$	−5.95304e6	−0.244302
$902$	1.30930e7	0.535826
$903$	0	0
$904$	4.14989e7	1.68894
$905$	−1.71128e7	−0.694543
$906$	0	0
$907$	−2.99381e7	−1.20839	−0.604193	−	0.796838i	$-0.706505\pi$
$907$	−2.99381e7	−1.20839	−0.604193	+	0.796838i	$0.706505\pi$
$908$	2.54276e6	0.102351
$909$	0	0
$910$	6.38222e6	0.255487
$911$	−2.83738e7	−1.13272	−0.566360	−	0.824158i	$-0.691649\pi$
$911$	−2.83738e7	−1.13272	−0.566360	+	0.824158i	$0.691649\pi$
$912$	0	0
$913$	8.32731e6	0.330619
$914$	1.76852e7	0.700236
$915$	0	0
$916$	−1.21231e6	−0.0477391
$917$	−3.72863e7	−1.46429
$918$	0	0
$919$	−3.91491e7	−1.52909	−0.764545	−	0.644570i	$-0.777036\pi$
$919$	−3.91491e7	−1.52909	−0.764545	+	0.644570i	$0.777036\pi$
$920$	3.96392e6	0.154403
$921$	0	0
$922$	−2.06762e7	−0.801019
$923$	4.80921e6	0.185810
$924$	0	0
$925$	−3.18233e7	−1.22290
$926$	−2.21014e7	−0.847019
$927$	0	0
$928$	−638672.	−0.0243449
$929$	−3.44851e6	−0.131097	−0.0655484	−	0.997849i	$-0.520880\pi$
$929$	−3.44851e6	−0.131097	−0.0655484	+	0.997849i	$0.520880\pi$
$930$	0	0
$931$	8.78447e6	0.332156
$932$	2.03000e6	0.0765519
$933$	0	0
$934$	−2.29514e7	−0.860879
$935$	−3.20665e6	−0.119956
$936$	0	0
$937$	−3.55183e7	−1.32161	−0.660805	−	0.750558i	$-0.729785\pi$
$937$	−3.55183e7	−1.32161	−0.660805	+	0.750558i	$0.729785\pi$
$938$	−2.32910e7	−0.864334
$939$	0	0
$940$	304250.	0.0112308
$941$	8.10962e6	0.298557	0.149278	−	0.988795i	$-0.452305\pi$
$941$	8.10962e6	0.298557	0.149278	+	0.988795i	$0.452305\pi$
$942$	0	0
$943$	1.05553e7	0.386538
$944$	−1.49269e7	−0.545180
$945$	0	0
$946$	−9.18886e6	−0.333836
$947$	−1.58835e7	−0.575535	−0.287767	−	0.957700i	$-0.592913\pi$
$947$	−1.58835e7	−0.575535	−0.287767	+	0.957700i	$0.592913\pi$
$948$	0	0
$949$	2.06191e7	0.743199
$950$	−1.98328e7	−0.712975
$951$	0	0
$952$	1.23417e7	0.441349
$953$	1.76043e7	0.627893	0.313946	−	0.949441i	$-0.398349\pi$
$953$	1.76043e7	0.627893	0.313946	+	0.949441i	$0.398349\pi$
$954$	0	0
$955$	−1.43422e7	−0.508871
$956$	−1.24447e6	−0.0440392
$957$	0	0
$958$	−2.13433e7	−0.751361
$959$	−1.97421e7	−0.693180
$960$	0	0
$961$	−2.43872e7	−0.851830
$962$	−2.84353e7	−0.990649
$963$	0	0
$964$	851464.	0.0295103
$965$	−2.49709e7	−0.863210
$966$	0	0
$967$	−5.37843e7	−1.84965	−0.924825	−	0.380393i	$-0.875789\pi$
$967$	−5.37843e7	−1.84965	−0.924825	+	0.380393i	$0.875789\pi$
$968$	−2.40430e7	−0.824708
$969$	0	0
$970$	1.65591e7	0.565076
$971$	−1.22227e7	−0.416026	−0.208013	−	0.978126i	$-0.566700\pi$
$971$	−1.22227e7	−0.416026	−0.208013	+	0.978126i	$0.566700\pi$
$972$	0	0
$973$	−3.36638e7	−1.13994
$974$	2.27880e7	0.769678
$975$	0	0
$976$	3.63875e7	1.22272
$977$	−4.33000e6	−0.145128	−0.0725640	−	0.997364i	$-0.523118\pi$
$977$	−4.33000e6	−0.145128	−0.0725640	+	0.997364i	$0.523118\pi$
$978$	0	0
$979$	2.13652e7	0.712445
$980$	365122.	0.0121443
$981$	0	0
$982$	4.80392e7	1.58971
$983$	−4.71098e7	−1.55499	−0.777494	−	0.628890i	$-0.783510\pi$
$983$	−4.71098e7	−1.55499	−0.777494	+	0.628890i	$0.783510\pi$
$984$	0	0
$985$	−1.63873e7	−0.538168
$986$	3.26559e6	0.106972
$987$	0	0
$988$	−1.09180e6	−0.0355838
$989$	−7.40786e6	−0.240825
$990$	0	0
$991$	−2.09978e7	−0.679187	−0.339594	−	0.940572i	$-0.610290\pi$
$991$	−2.09978e7	−0.679187	−0.339594	+	0.940572i	$0.610290\pi$
$992$	1.56411e6	0.0504647
$993$	0	0
$994$	−9.16495e6	−0.294215
$995$	9.51734e6	0.304760
$996$	0	0
$997$	−4.51805e7	−1.43951	−0.719753	−	0.694231i	$-0.755745\pi$
$997$	−4.51805e7	−1.43951	−0.719753	+	0.694231i	$0.755745\pi$
$998$	−1.13946e7	−0.362136
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	261.6.a.e.1.3		7
3.2	odd	2		29.6.a.b.1.5	✓	7
12.11	even	2		464.6.a.k.1.3		7
15.14	odd	2		725.6.a.b.1.3		7
87.86	odd	2		841.6.a.b.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.b.1.5	✓	7	3.2	odd	2
261.6.a.e.1.3		7	1.1	even	1	trivial
464.6.a.k.1.3		7	12.11	even	2
725.6.a.b.1.3		7	15.14	odd	2
841.6.a.b.1.3		7	87.86	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 261 = 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	261.a (trivial)

\(f(q)\)	\(=\)	\(q-5.83960 q^{2} +2.10095 q^{4} -31.5616 q^{5} +106.304 q^{7} +174.599 q^{8} +O(q^{10})\)	\(q-5.83960 q^{2} +2.10095 q^{4} -31.5616 q^{5} +106.304 q^{7} +174.599 q^{8} +184.307 q^{10} +152.796 q^{11} +325.745 q^{13} -620.776 q^{14} -1086.82 q^{16} +664.939 q^{17} -1595.33 q^{19} -66.3091 q^{20} -892.268 q^{22} -719.327 q^{23} -2128.87 q^{25} -1902.22 q^{26} +223.340 q^{28} -841.000 q^{29} +2059.61 q^{31} +759.420 q^{32} -3882.98 q^{34} -3355.13 q^{35} +14948.4 q^{37} +9316.11 q^{38} -5510.60 q^{40} -14673.9 q^{41} +10298.3 q^{43} +321.016 q^{44} +4200.58 q^{46} -4588.36 q^{47} -5506.36 q^{49} +12431.7 q^{50} +684.373 q^{52} -8952.75 q^{53} -4822.48 q^{55} +18560.6 q^{56} +4911.10 q^{58} +13734.5 q^{59} -33480.9 q^{61} -12027.3 q^{62} +30343.4 q^{64} -10281.0 q^{65} +37519.2 q^{67} +1397.00 q^{68} +19592.6 q^{70} +14763.7 q^{71} +63298.3 q^{73} -87292.9 q^{74} -3351.71 q^{76} +16242.9 q^{77} -27148.9 q^{79} +34301.6 q^{80} +85689.7 q^{82} +54499.5 q^{83} -20986.5 q^{85} -60138.1 q^{86} +26678.0 q^{88} +139829. q^{89} +34628.2 q^{91} -1511.27 q^{92} +26794.2 q^{94} +50351.2 q^{95} +89845.2 q^{97} +32154.9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 4 q^{2} + 154 q^{4} - 32 q^{5} + 184 q^{7} - 942 q^{8}+O(q^{10}) \) 7 * q - 4 * q^2 + 154 * q^4 - 32 * q^5 + 184 * q^7 - 942 * q^8	\( 7 q - 4 q^{2} + 154 q^{4} - 32 q^{5} + 184 q^{7} - 942 q^{8} + 922 q^{10} - 1106 q^{11} + 408 q^{13} + 2008 q^{14} + 242 q^{16} + 874 q^{17} + 4288 q^{19} + 6350 q^{20} - 6114 q^{22} + 4532 q^{23} + 5527 q^{25} + 19806 q^{26} - 496 q^{28} - 5887 q^{29} + 7794 q^{31} - 7898 q^{32} + 20840 q^{34} - 7088 q^{35} + 5086 q^{37} - 23732 q^{38} + 22906 q^{40} - 19826 q^{41} + 19498 q^{43} + 6074 q^{44} - 12404 q^{46} - 14278 q^{47} + 38431 q^{49} + 41066 q^{50} - 34302 q^{52} + 58644 q^{53} - 25574 q^{55} + 79560 q^{56} + 3364 q^{58} - 12888 q^{59} + 102866 q^{61} + 42654 q^{62} - 10170 q^{64} + 149206 q^{65} + 102996 q^{67} - 85100 q^{68} + 349480 q^{70} + 51596 q^{71} - 17566 q^{73} - 12132 q^{74} + 360740 q^{76} + 94104 q^{77} + 212058 q^{79} - 142510 q^{80} + 201924 q^{82} + 122928 q^{83} - 109336 q^{85} + 63290 q^{86} + 136666 q^{88} + 66510 q^{89} + 194368 q^{91} + 110108 q^{92} + 438926 q^{94} + 131676 q^{95} - 118182 q^{97} + 29132 q^{98}+O(q^{100}) \) 7 * q - 4 * q^2 + 154 * q^4 - 32 * q^5 + 184 * q^7 - 942 * q^8 + 922 * q^10 - 1106 * q^11 + 408 * q^13 + 2008 * q^14 + 242 * q^16 + 874 * q^17 + 4288 * q^19 + 6350 * q^20 - 6114 * q^22 + 4532 * q^23 + 5527 * q^25 + 19806 * q^26 - 496 * q^28 - 5887 * q^29 + 7794 * q^31 - 7898 * q^32 + 20840 * q^34 - 7088 * q^35 + 5086 * q^37 - 23732 * q^38 + 22906 * q^40 - 19826 * q^41 + 19498 * q^43 + 6074 * q^44 - 12404 * q^46 - 14278 * q^47 + 38431 * q^49 + 41066 * q^50 - 34302 * q^52 + 58644 * q^53 - 25574 * q^55 + 79560 * q^56 + 3364 * q^58 - 12888 * q^59 + 102866 * q^61 + 42654 * q^62 - 10170 * q^64 + 149206 * q^65 + 102996 * q^67 - 85100 * q^68 + 349480 * q^70 + 51596 * q^71 - 17566 * q^73 - 12132 * q^74 + 360740 * q^76 + 94104 * q^77 + 212058 * q^79 - 142510 * q^80 + 201924 * q^82 + 122928 * q^83 - 109336 * q^85 + 63290 * q^86 + 136666 * q^88 + 66510 * q^89 + 194368 * q^91 + 110108 * q^92 + 438926 * q^94 + 131676 * q^95 - 118182 * q^97 + 29132 * q^98

Introduction

L-functions

Modular forms

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Database

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