LMFDB - Embedded newform 414.6.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,6,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, 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("414.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	414.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:	$66.3989014026$
Analytic rank:	$0$
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} - x^{3} - 8369x^{2} - 182616x - 370980 $ x^4 - x^3 - 8369x^2 - 182616x - 370980
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-77.4267$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} +102.561 q^{5} +126.292 q^{7} -64.0000 q^{8} +O(q^{10})$	\(q-4.00000 q^{2} +16.0000 q^{4} +102.561 q^{5} +126.292 q^{7} -64.0000 q^{8} -410.245 q^{10} +90.5883 q^{11} +1149.78 q^{13} -505.168 q^{14} +256.000 q^{16} -714.399 q^{17} +2072.71 q^{19} +1640.98 q^{20} -362.353 q^{22} -529.000 q^{23} +7393.80 q^{25} -4599.12 q^{26} +2020.67 q^{28} +3746.42 q^{29} +3106.00 q^{31} -1024.00 q^{32} +2857.60 q^{34} +12952.7 q^{35} -11515.3 q^{37} -8290.85 q^{38} -6563.92 q^{40} +15719.4 q^{41} +5210.24 q^{43} +1449.41 q^{44} +2116.00 q^{46} -14937.4 q^{47} -857.301 q^{49} -29575.2 q^{50} +18396.5 q^{52} -29596.2 q^{53} +9290.84 q^{55} -8082.70 q^{56} -14985.7 q^{58} -36544.0 q^{59} -30666.3 q^{61} -12424.0 q^{62} +4096.00 q^{64} +117923. q^{65} -50394.1 q^{67} -11430.4 q^{68} -51810.7 q^{70} -75712.7 q^{71} -8603.49 q^{73} +46061.1 q^{74} +33163.4 q^{76} +11440.6 q^{77} +31180.1 q^{79} +26255.7 q^{80} -62877.4 q^{82} +47887.8 q^{83} -73269.6 q^{85} -20841.0 q^{86} -5797.65 q^{88} +122047. q^{89} +145208. q^{91} -8464.00 q^{92} +59749.7 q^{94} +212580. q^{95} +68389.3 q^{97} +3429.20 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8}+O(q^{10}) $ 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8	$ 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8} + 216 q^{10} + 6 q^{11} + 1248 q^{13} - 1392 q^{14} + 1024 q^{16} + 56 q^{17} + 1530 q^{19} - 864 q^{20} - 24 q^{22} - 2116 q^{23} + 8852 q^{25} - 4992 q^{26} + 5568 q^{28} - 2292 q^{29} + 1556 q^{31} - 4096 q^{32} - 224 q^{34} - 5688 q^{35} - 9586 q^{37} - 6120 q^{38} + 3456 q^{40} + 11768 q^{41} + 13758 q^{43} + 96 q^{44} + 8464 q^{46} - 10636 q^{47} + 11360 q^{49} - 35408 q^{50} + 19968 q^{52} - 26686 q^{53} - 28900 q^{55} - 22272 q^{56} + 9168 q^{58} + 9108 q^{59} - 37878 q^{61} - 6224 q^{62} + 16384 q^{64} + 72212 q^{65} - 23302 q^{67} + 896 q^{68} + 22752 q^{70} - 31728 q^{71} - 28340 q^{73} + 38344 q^{74} + 24480 q^{76} + 121276 q^{77} - 26668 q^{79} - 13824 q^{80} - 47072 q^{82} + 119026 q^{83} - 217876 q^{85} - 55032 q^{86} - 384 q^{88} + 148236 q^{89} + 89008 q^{91} - 33856 q^{92} + 42544 q^{94} + 399292 q^{95} - 16092 q^{97} - 45440 q^{98}+O(q^{100}) $ 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8 + 216 * q^10 + 6 * q^11 + 1248 * q^13 - 1392 * q^14 + 1024 * q^16 + 56 * q^17 + 1530 * q^19 - 864 * q^20 - 24 * q^22 - 2116 * q^23 + 8852 * q^25 - 4992 * q^26 + 5568 * q^28 - 2292 * q^29 + 1556 * q^31 - 4096 * q^32 - 224 * q^34 - 5688 * q^35 - 9586 * q^37 - 6120 * q^38 + 3456 * q^40 + 11768 * q^41 + 13758 * q^43 + 96 * q^44 + 8464 * q^46 - 10636 * q^47 + 11360 * q^49 - 35408 * q^50 + 19968 * q^52 - 26686 * q^53 - 28900 * q^55 - 22272 * q^56 + 9168 * q^58 + 9108 * q^59 - 37878 * q^61 - 6224 * q^62 + 16384 * q^64 + 72212 * q^65 - 23302 * q^67 + 896 * q^68 + 22752 * q^70 - 31728 * q^71 - 28340 * q^73 + 38344 * q^74 + 24480 * q^76 + 121276 * q^77 - 26668 * q^79 - 13824 * q^80 - 47072 * q^82 + 119026 * q^83 - 217876 * q^85 - 55032 * q^86 - 384 * q^88 + 148236 * q^89 + 89008 * q^91 - 33856 * q^92 + 42544 * q^94 + 399292 * q^95 - 16092 * q^97 - 45440 * 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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	102.561	1.83467	0.917335	−	0.398115i	$-0.130336\pi$
$5$	102.561	1.83467	0.917335	+	0.398115i	$0.130336\pi$
$6$	0	0
$7$	126.292	0.974162	0.487081	−	0.873357i	$-0.338062\pi$
$7$	126.292	0.974162	0.487081	+	0.873357i	$0.338062\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−410.245	−1.29731
$11$	90.5883	0.225731	0.112865	−	0.993610i	$-0.463997\pi$
$11$	90.5883	0.225731	0.112865	+	0.993610i	$0.463997\pi$
$12$	0	0
$13$	1149.78	1.88693	0.943467	−	0.331466i	$-0.107543\pi$
$13$	1149.78	1.88693	0.943467	+	0.331466i	$0.107543\pi$
$14$	−505.168	−0.688836
$15$	0	0
$16$	256.000	0.250000
$17$	−714.399	−0.599540	−0.299770	−	0.954011i	$-0.596910\pi$
$17$	−714.399	−0.599540	−0.299770	+	0.954011i	$0.596910\pi$
$18$	0	0
$19$	2072.71	1.31721	0.658605	−	0.752489i	$-0.271147\pi$
$19$	2072.71	1.31721	0.658605	+	0.752489i	$0.271147\pi$
$20$	1640.98	0.917335
$21$	0	0
$22$	−362.353	−0.159616
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	7393.80	2.36602
$26$	−4599.12	−1.33426
$27$	0	0
$28$	2020.67	0.487081
$29$	3746.42	0.827221	0.413611	−	0.910454i	$-0.364267\pi$
$29$	3746.42	0.827221	0.413611	+	0.910454i	$0.364267\pi$
$30$	0	0
$31$	3106.00	0.580493	0.290246	−	0.956952i	$-0.406263\pi$
$31$	3106.00	0.580493	0.290246	+	0.956952i	$0.406263\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	2857.60	0.423939
$35$	12952.7	1.78727
$36$	0	0
$37$	−11515.3	−1.38283	−0.691417	−	0.722456i	$-0.743013\pi$
$37$	−11515.3	−1.38283	−0.691417	+	0.722456i	$0.743013\pi$
$38$	−8290.85	−0.931409
$39$	0	0
$40$	−6563.92	−0.648654
$41$	15719.4	1.46041	0.730206	−	0.683227i	$-0.239424\pi$
$41$	15719.4	1.46041	0.730206	+	0.683227i	$0.239424\pi$
$42$	0	0
$43$	5210.24	0.429721	0.214860	−	0.976645i	$-0.431070\pi$
$43$	5210.24	0.429721	0.214860	+	0.976645i	$0.431070\pi$
$44$	1449.41	0.112865
$45$	0	0
$46$	2116.00	0.147442
$47$	−14937.4	−0.986350	−0.493175	−	0.869930i	$-0.664164\pi$
$47$	−14937.4	−0.986350	−0.493175	+	0.869930i	$0.664164\pi$
$48$	0	0
$49$	−857.301	−0.0510086
$50$	−29575.2	−1.67303
$51$	0	0
$52$	18396.5	0.943467
$53$	−29596.2	−1.44726	−0.723630	−	0.690188i	$-0.757528\pi$
$53$	−29596.2	−1.44726	−0.723630	+	0.690188i	$0.757528\pi$
$54$	0	0
$55$	9290.84	0.414141
$56$	−8082.70	−0.344418
$57$	0	0
$58$	−14985.7	−0.584934
$59$	−36544.0	−1.36674	−0.683370	−	0.730073i	$-0.739486\pi$
$59$	−36544.0	−1.36674	−0.683370	+	0.730073i	$0.739486\pi$
$60$	0	0
$61$	−30666.3	−1.05521	−0.527603	−	0.849491i	$-0.676909\pi$
$61$	−30666.3	−1.05521	−0.527603	+	0.849491i	$0.676909\pi$
$62$	−12424.0	−0.410470
$63$	0	0
$64$	4096.00	0.125000
$65$	117923.	3.46190
$66$	0	0
$67$	−50394.1	−1.37149	−0.685745	−	0.727842i	$-0.740523\pi$
$67$	−50394.1	−1.37149	−0.685745	+	0.727842i	$0.740523\pi$
$68$	−11430.4	−0.299770
$69$	0	0
$70$	−51810.7	−1.26379
$71$	−75712.7	−1.78247	−0.891236	−	0.453540i	$-0.850161\pi$
$71$	−75712.7	−1.78247	−0.891236	+	0.453540i	$0.850161\pi$
$72$	0	0
$73$	−8603.49	−0.188959	−0.0944794	−	0.995527i	$-0.530119\pi$
$73$	−8603.49	−0.188959	−0.0944794	+	0.995527i	$0.530119\pi$
$74$	46061.1	0.977811
$75$	0	0
$76$	33163.4	0.658605
$77$	11440.6	0.219898
$78$	0	0
$79$	31180.1	0.562096	0.281048	−	0.959694i	$-0.409318\pi$
$79$	31180.1	0.562096	0.281048	+	0.959694i	$0.409318\pi$
$80$	26255.7	0.458668
$81$	0	0
$82$	−62877.4	−1.03267
$83$	47887.8	0.763009	0.381504	−	0.924367i	$-0.375406\pi$
$83$	47887.8	0.763009	0.381504	+	0.924367i	$0.375406\pi$
$84$	0	0
$85$	−73269.6	−1.09996
$86$	−20841.0	−0.303859
$87$	0	0
$88$	−5797.65	−0.0798078
$89$	122047.	1.63325	0.816626	−	0.577167i	$-0.195842\pi$
$89$	122047.	1.63325	0.816626	+	0.577167i	$0.195842\pi$
$90$	0	0
$91$	145208.	1.83818
$92$	−8464.00	−0.104257
$93$	0	0
$94$	59749.7	0.697455
$95$	212580.	2.41665
$96$	0	0
$97$	68389.3	0.738004	0.369002	−	0.929429i	$-0.379700\pi$
$97$	68389.3	0.738004	0.369002	+	0.929429i	$0.379700\pi$
$98$	3429.20	0.0360685
$99$	0	0
$100$	118301.	1.18301
$101$	−93191.9	−0.909023	−0.454511	−	0.890741i	$-0.650186\pi$
$101$	−93191.9	−0.909023	−0.454511	+	0.890741i	$0.650186\pi$
$102$	0	0
$103$	−103837.	−0.964409	−0.482204	−	0.876059i	$-0.660164\pi$
$103$	−103837.	−0.964409	−0.482204	+	0.876059i	$0.660164\pi$
$104$	−73586.0	−0.667132
$105$	0	0
$106$	118385.	1.02337
$107$	151252.	1.27715	0.638576	−	0.769559i	$-0.279524\pi$
$107$	151252.	1.27715	0.638576	+	0.769559i	$0.279524\pi$
$108$	0	0
$109$	11831.8	0.0953862	0.0476931	−	0.998862i	$-0.484813\pi$
$109$	11831.8	0.0953862	0.0476931	+	0.998862i	$0.484813\pi$
$110$	−37163.4	−0.292842
$111$	0	0
$112$	32330.8	0.243540
$113$	−62621.6	−0.461347	−0.230674	−	0.973031i	$-0.574093\pi$
$113$	−62621.6	−0.461347	−0.230674	+	0.973031i	$0.574093\pi$
$114$	0	0
$115$	−54254.9	−0.382555
$116$	59942.8	0.413611
$117$	0	0
$118$	146176.	0.966430
$119$	−90222.9	−0.584049
$120$	0	0
$121$	−152845.	−0.949046
$122$	122665.	0.746143
$123$	0	0
$124$	49695.9	0.290246
$125$	437814.	2.50619
$126$	0	0
$127$	−229747.	−1.26398	−0.631990	−	0.774976i	$-0.717762\pi$
$127$	−229747.	−1.26398	−0.631990	+	0.774976i	$0.717762\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−471692.	−2.44794
$131$	−19533.1	−0.0994472	−0.0497236	−	0.998763i	$-0.515834\pi$
$131$	−19533.1	−0.0994472	−0.0497236	+	0.998763i	$0.515834\pi$
$132$	0	0
$133$	261767.	1.28318
$134$	201576.	0.969790
$135$	0	0
$136$	45721.5	0.211970
$137$	−22459.8	−0.102236	−0.0511181	−	0.998693i	$-0.516278\pi$
$137$	−22459.8	−0.102236	−0.0511181	+	0.998693i	$0.516278\pi$
$138$	0	0
$139$	100788.	0.442456	0.221228	−	0.975222i	$-0.428994\pi$
$139$	100788.	0.442456	0.221228	+	0.975222i	$0.428994\pi$
$140$	207243.	0.893633
$141$	0	0
$142$	302851.	1.26040
$143$	104157.	0.425939
$144$	0	0
$145$	384238.	1.51768
$146$	34413.9	0.133614
$147$	0	0
$148$	−184244.	−0.691417
$149$	84542.7	0.311968	0.155984	−	0.987760i	$-0.450145\pi$
$149$	84542.7	0.311968	0.155984	+	0.987760i	$0.450145\pi$
$150$	0	0
$151$	−24889.3	−0.0888321	−0.0444161	−	0.999013i	$-0.514143\pi$
$151$	−24889.3	−0.0888321	−0.0444161	+	0.999013i	$0.514143\pi$
$152$	−132654.	−0.465704
$153$	0	0
$154$	−45762.3	−0.155491
$155$	318555.	1.06501
$156$	0	0
$157$	112479.	0.364187	0.182093	−	0.983281i	$-0.441713\pi$
$157$	112479.	0.364187	0.182093	+	0.983281i	$0.441713\pi$
$158$	−124721.	−0.397462
$159$	0	0
$160$	−105023.	−0.324327
$161$	−66808.5	−0.203127
$162$	0	0
$163$	303309.	0.894161	0.447080	−	0.894494i	$-0.352464\pi$
$163$	303309.	0.894161	0.447080	+	0.894494i	$0.352464\pi$
$164$	251510.	0.730206
$165$	0	0
$166$	−191551.	−0.539529
$167$	−414178.	−1.14920	−0.574601	−	0.818434i	$-0.694843\pi$
$167$	−414178.	−1.14920	−0.574601	+	0.818434i	$0.694843\pi$
$168$	0	0
$169$	950704.	2.56052
$170$	293078.	0.777789
$171$	0	0
$172$	83363.8	0.214860
$173$	735126.	1.86744	0.933719	−	0.358006i	$-0.116543\pi$
$173$	735126.	1.86744	0.933719	+	0.358006i	$0.116543\pi$
$174$	0	0
$175$	933779.	2.30488
$176$	23190.6	0.0564326
$177$	0	0
$178$	−488189.	−1.15488
$179$	13360.3	0.0311661	0.0155831	−	0.999879i	$-0.495040\pi$
$179$	13360.3	0.0311661	0.0155831	+	0.999879i	$0.495040\pi$
$180$	0	0
$181$	−581129.	−1.31849	−0.659244	−	0.751929i	$-0.729123\pi$
$181$	−581129.	−1.31849	−0.659244	+	0.751929i	$0.729123\pi$
$182$	−580833.	−1.29979
$183$	0	0
$184$	33856.0	0.0737210
$185$	−1.18102e6	−2.53704
$186$	0	0
$187$	−64716.2	−0.135335
$188$	−238999.	−0.493175
$189$	0	0
$190$	−850320.	−1.70883
$191$	−838828.	−1.66375	−0.831877	−	0.554959i	$-0.812734\pi$
$191$	−838828.	−1.66375	−0.831877	+	0.554959i	$0.812734\pi$
$192$	0	0
$193$	−64900.9	−0.125417	−0.0627086	−	0.998032i	$-0.519974\pi$
$193$	−64900.9	−0.125417	−0.0627086	+	0.998032i	$0.519974\pi$
$194$	−273557.	−0.521848
$195$	0	0
$196$	−13716.8	−0.0255043
$197$	580682.	1.06604	0.533019	−	0.846103i	$-0.321057\pi$
$197$	580682.	1.06604	0.533019	+	0.846103i	$0.321057\pi$
$198$	0	0
$199$	−516699.	−0.924921	−0.462461	−	0.886640i	$-0.653033\pi$
$199$	−516699.	−0.924921	−0.462461	+	0.886640i	$0.653033\pi$
$200$	−473203.	−0.836513
$201$	0	0
$202$	372768.	0.642776
$203$	473144.	0.805848
$204$	0	0
$205$	1.61220e6	2.67937
$206$	415350.	0.681940
$207$	0	0
$208$	294344.	0.471734
$209$	187764.	0.297335
$210$	0	0
$211$	−77539.1	−0.119899	−0.0599493	−	0.998201i	$-0.519094\pi$
$211$	−77539.1	−0.119899	−0.0599493	+	0.998201i	$0.519094\pi$
$212$	−473540.	−0.723630
$213$	0	0
$214$	−605009.	−0.903082
$215$	534368.	0.788397
$216$	0	0
$217$	392263.	0.565494
$218$	−47327.3	−0.0674483
$219$	0	0
$220$	148654.	0.207071
$221$	−821402.	−1.13129
$222$	0	0
$223$	430698.	0.579976	0.289988	−	0.957030i	$-0.406349\pi$
$223$	430698.	0.579976	0.289988	+	0.957030i	$0.406349\pi$
$224$	−129323.	−0.172209
$225$	0	0
$226$	250486.	0.326222
$227$	734612.	0.946223	0.473111	−	0.881003i	$-0.343131\pi$
$227$	734612.	0.946223	0.473111	+	0.881003i	$0.343131\pi$
$228$	0	0
$229$	70048.0	0.0882687	0.0441344	−	0.999026i	$-0.485947\pi$
$229$	70048.0	0.0882687	0.0441344	+	0.999026i	$0.485947\pi$
$230$	217020.	0.270507
$231$	0	0
$232$	−239771.	−0.292467
$233$	448068.	0.540698	0.270349	−	0.962762i	$-0.412861\pi$
$233$	448068.	0.540698	0.270349	+	0.962762i	$0.412861\pi$
$234$	0	0
$235$	−1.53200e6	−1.80963
$236$	−584703.	−0.683370
$237$	0	0
$238$	360892.	0.412985
$239$	−928167.	−1.05107	−0.525535	−	0.850772i	$-0.676135\pi$
$239$	−928167.	−1.05107	−0.525535	+	0.850772i	$0.676135\pi$
$240$	0	0
$241$	584553.	0.648308	0.324154	−	0.946004i	$-0.394920\pi$
$241$	584553.	0.648308	0.324154	+	0.946004i	$0.394920\pi$
$242$	611379.	0.671077
$243$	0	0
$244$	−490661.	−0.527603
$245$	−87925.9	−0.0935840
$246$	0	0
$247$	2.38317e6	2.48549
$248$	−198784.	−0.205235
$249$	0	0
$250$	−1.75125e6	−1.77215
$251$	640483.	0.641687	0.320843	−	0.947132i	$-0.396034\pi$
$251$	640483.	0.641687	0.320843	+	0.947132i	$0.396034\pi$
$252$	0	0
$253$	−47921.2	−0.0470681
$254$	918988.	0.893769
$255$	0	0
$256$	65536.0	0.0625000
$257$	−232779.	−0.219842	−0.109921	−	0.993940i	$-0.535060\pi$
$257$	−232779.	−0.219842	−0.109921	+	0.993940i	$0.535060\pi$
$258$	0	0
$259$	−1.45429e6	−1.34710
$260$	1.88677e6	1.73095
$261$	0	0
$262$	78132.4	0.0703198
$263$	−1.68325e6	−1.50058	−0.750289	−	0.661110i	$-0.770086\pi$
$263$	−1.68325e6	−1.50058	−0.750289	+	0.661110i	$0.770086\pi$
$264$	0	0
$265$	−3.03543e6	−2.65525
$266$	−1.04707e6	−0.907343
$267$	0	0
$268$	−806306.	−0.685745
$269$	193031.	0.162647	0.0813235	−	0.996688i	$-0.474085\pi$
$269$	193031.	0.162647	0.0813235	+	0.996688i	$0.474085\pi$
$270$	0	0
$271$	1.07272e6	0.887285	0.443643	−	0.896204i	$-0.353686\pi$
$271$	1.07272e6	0.887285	0.443643	+	0.896204i	$0.353686\pi$
$272$	−182886.	−0.149885
$273$	0	0
$274$	89839.2	0.0722919
$275$	669792.	0.534082
$276$	0	0
$277$	−646094.	−0.505937	−0.252969	−	0.967475i	$-0.581407\pi$
$277$	−646094.	−0.505937	−0.252969	+	0.967475i	$0.581407\pi$
$278$	−403150.	−0.312864
$279$	0	0
$280$	−828971.	−0.631894
$281$	376738.	0.284626	0.142313	−	0.989822i	$-0.454546\pi$
$281$	376738.	0.284626	0.142313	+	0.989822i	$0.454546\pi$
$282$	0	0
$283$	238880.	0.177302	0.0886509	−	0.996063i	$-0.471744\pi$
$283$	238880.	0.177302	0.0886509	+	0.996063i	$0.471744\pi$
$284$	−1.21140e6	−0.891236
$285$	0	0
$286$	−416627.	−0.301184
$287$	1.98523e6	1.42268
$288$	0	0
$289$	−909491.	−0.640551
$290$	−1.53695e6	−1.07316
$291$	0	0
$292$	−137656.	−0.0944794
$293$	2.25101e6	1.53182	0.765912	−	0.642946i	$-0.222288\pi$
$293$	2.25101e6	1.53182	0.765912	+	0.642946i	$0.222288\pi$
$294$	0	0
$295$	−3.74799e6	−2.50752
$296$	736977.	0.488905
$297$	0	0
$298$	−338171.	−0.220595
$299$	−608234.	−0.393453
$300$	0	0
$301$	658012.	0.418618
$302$	99557.1	0.0628138
$303$	0	0
$304$	530615.	0.329303
$305$	−3.14517e6	−1.93595
$306$	0	0
$307$	−2.76566e6	−1.67476	−0.837382	−	0.546618i	$-0.815915\pi$
$307$	−2.76566e6	−1.67476	−0.837382	+	0.546618i	$0.815915\pi$
$308$	183049.	0.109949
$309$	0	0
$310$	−1.27422e6	−0.753078
$311$	1.38972e6	0.814753	0.407377	−	0.913260i	$-0.366444\pi$
$311$	1.38972e6	0.814753	0.407377	+	0.913260i	$0.366444\pi$
$312$	0	0
$313$	−2.59777e6	−1.49879	−0.749393	−	0.662125i	$-0.769655\pi$
$313$	−2.59777e6	−1.49879	−0.749393	+	0.662125i	$0.769655\pi$
$314$	−449918.	−0.257519
$315$	0	0
$316$	498882.	0.281048
$317$	−1.77867e6	−0.994140	−0.497070	−	0.867710i	$-0.665591\pi$
$317$	−1.77867e6	−0.994140	−0.497070	+	0.867710i	$0.665591\pi$
$318$	0	0
$319$	339382.	0.186729
$320$	420091.	0.229334
$321$	0	0
$322$	267234.	0.143632
$323$	−1.48074e6	−0.789721
$324$	0	0
$325$	8.50126e6	4.46452
$326$	−1.21323e6	−0.632267
$327$	0	0
$328$	−1.00604e6	−0.516333
$329$	−1.88648e6	−0.960865
$330$	0	0
$331$	491570.	0.246613	0.123306	−	0.992369i	$-0.460650\pi$
$331$	491570.	0.246613	0.123306	+	0.992369i	$0.460650\pi$
$332$	766204.	0.381504
$333$	0	0
$334$	1.65671e6	0.812608
$335$	−5.16848e6	−2.51623
$336$	0	0
$337$	3.30544e6	1.58546	0.792730	−	0.609573i	$-0.208659\pi$
$337$	3.30544e6	1.58546	0.792730	+	0.609573i	$0.208659\pi$
$338$	−3.80281e6	−1.81056
$339$	0	0
$340$	−1.17231e6	−0.549980
$341$	281367.	0.131035
$342$	0	0
$343$	−2.23086e6	−1.02385
$344$	−333455.	−0.151929
$345$	0	0
$346$	−2.94050e6	−1.32048
$347$	−212149.	−0.0945840	−0.0472920	−	0.998881i	$-0.515059\pi$
$347$	−212149.	−0.0945840	−0.0472920	+	0.998881i	$0.515059\pi$
$348$	0	0
$349$	3.89733e6	1.71279	0.856395	−	0.516322i	$-0.172699\pi$
$349$	3.89733e6	1.71279	0.856395	+	0.516322i	$0.172699\pi$
$350$	−3.73512e6	−1.62980
$351$	0	0
$352$	−92762.4	−0.0399039
$353$	2.17818e6	0.930374	0.465187	−	0.885212i	$-0.345987\pi$
$353$	2.17818e6	0.930374	0.465187	+	0.885212i	$0.345987\pi$
$354$	0	0
$355$	−7.76518e6	−3.27025
$356$	1.95276e6	0.816626
$357$	0	0
$358$	−53441.1	−0.0220378
$359$	−398375.	−0.163138	−0.0815691	−	0.996668i	$-0.525993\pi$
$359$	−398375.	−0.163138	−0.0815691	+	0.996668i	$0.525993\pi$
$360$	0	0
$361$	1.82004e6	0.735044
$362$	2.32452e6	0.932311
$363$	0	0
$364$	2.32333e6	0.919090
$365$	−882384.	−0.346677
$366$	0	0
$367$	−2.97637e6	−1.15351	−0.576755	−	0.816917i	$-0.695681\pi$
$367$	−2.97637e6	−1.15351	−0.576755	+	0.816917i	$0.695681\pi$
$368$	−135424.	−0.0521286
$369$	0	0
$370$	4.72408e6	1.79396
$371$	−3.73777e6	−1.40987
$372$	0	0
$373$	2.83201e6	1.05396	0.526978	−	0.849879i	$-0.323325\pi$
$373$	2.83201e6	1.05396	0.526978	+	0.849879i	$0.323325\pi$
$374$	258865.	0.0956960
$375$	0	0
$376$	955995.	0.348727
$377$	4.30757e6	1.56091
$378$	0	0
$379$	681684.	0.243773	0.121886	−	0.992544i	$-0.461106\pi$
$379$	681684.	0.243773	0.121886	+	0.992544i	$0.461106\pi$
$380$	3.40128e6	1.20832
$381$	0	0
$382$	3.35531e6	1.17645
$383$	148147.	0.0516054	0.0258027	−	0.999667i	$-0.491786\pi$
$383$	148147.	0.0516054	0.0258027	+	0.999667i	$0.491786\pi$
$384$	0	0
$385$	1.17336e6	0.403441
$386$	259604.	0.0886834
$387$	0	0
$388$	1.09423e6	0.369002
$389$	1.20984e6	0.405373	0.202687	−	0.979244i	$-0.435033\pi$
$389$	1.20984e6	0.405373	0.202687	+	0.979244i	$0.435033\pi$
$390$	0	0
$391$	377917.	0.125013
$392$	54867.3	0.0180343
$393$	0	0
$394$	−2.32273e6	−0.753803
$395$	3.19787e6	1.03126
$396$	0	0
$397$	−4.48036e6	−1.42671	−0.713357	−	0.700801i	$-0.752826\pi$
$397$	−4.48036e6	−1.42671	−0.713357	+	0.700801i	$0.752826\pi$
$398$	2.06680e6	0.654018
$399$	0	0
$400$	1.89281e6	0.591504
$401$	−2.08714e6	−0.648171	−0.324086	−	0.946028i	$-0.605057\pi$
$401$	−2.08714e6	−0.648171	−0.324086	+	0.946028i	$0.605057\pi$
$402$	0	0
$403$	3.57122e6	1.09535
$404$	−1.49107e6	−0.454511
$405$	0	0
$406$	−1.89257e6	−0.569820
$407$	−1.04315e6	−0.312148
$408$	0	0
$409$	−4.39203e6	−1.29825	−0.649123	−	0.760684i	$-0.724864\pi$
$409$	−4.39203e6	−1.29825	−0.649123	+	0.760684i	$0.724864\pi$
$410$	−6.44879e6	−1.89460
$411$	0	0
$412$	−1.66140e6	−0.482204
$413$	−4.61521e6	−1.33143
$414$	0	0
$415$	4.91143e6	1.39987
$416$	−1.17738e6	−0.333566
$417$	0	0
$418$	−751054.	−0.210247
$419$	5.63480e6	1.56799	0.783995	−	0.620767i	$-0.213179\pi$
$419$	5.63480e6	1.56799	0.783995	+	0.620767i	$0.213179\pi$
$420$	0	0
$421$	3.79377e6	1.04320	0.521598	−	0.853192i	$-0.325336\pi$
$421$	3.79377e6	1.04320	0.521598	+	0.853192i	$0.325336\pi$
$422$	310156.	0.0847812
$423$	0	0
$424$	1.89416e6	0.511684
$425$	−5.28212e6	−1.41852
$426$	0	0
$427$	−3.87291e6	−1.02794
$428$	2.42004e6	0.638576
$429$	0	0
$430$	−2.13747e6	−0.557481
$431$	5.86627e6	1.52114	0.760569	−	0.649257i	$-0.224920\pi$
$431$	5.86627e6	1.52114	0.760569	+	0.649257i	$0.224920\pi$
$432$	0	0
$433$	1.52396e6	0.390619	0.195310	−	0.980742i	$-0.437429\pi$
$433$	1.52396e6	0.390619	0.195310	+	0.980742i	$0.437429\pi$
$434$	−1.56905e6	−0.399865
$435$	0	0
$436$	189309.	0.0476931
$437$	−1.09647e6	−0.274657
$438$	0	0
$439$	−140045.	−0.0346822	−0.0173411	−	0.999850i	$-0.505520\pi$
$439$	−140045.	−0.0346822	−0.0173411	+	0.999850i	$0.505520\pi$
$440$	−594614.	−0.146421
$441$	0	0
$442$	3.28561e6	0.799945
$443$	4.48550e6	1.08593	0.542964	−	0.839756i	$-0.317302\pi$
$443$	4.48550e6	1.08593	0.542964	+	0.839756i	$0.317302\pi$
$444$	0	0
$445$	1.25173e7	2.99648
$446$	−1.72279e6	−0.410105
$447$	0	0
$448$	517293.	0.121770
$449$	231227.	0.0541282	0.0270641	−	0.999634i	$-0.491384\pi$
$449$	231227.	0.0541282	0.0270641	+	0.999634i	$0.491384\pi$
$450$	0	0
$451$	1.42399e6	0.329659
$452$	−1.00195e6	−0.230674
$453$	0	0
$454$	−2.93845e6	−0.669081
$455$	1.48927e7	3.37245
$456$	0	0
$457$	−1.38930e6	−0.311176	−0.155588	−	0.987822i	$-0.549727\pi$
$457$	−1.38930e6	−0.311176	−0.155588	+	0.987822i	$0.549727\pi$
$458$	−280192.	−0.0624154
$459$	0	0
$460$	−868078.	−0.191278
$461$	−330967.	−0.0725324	−0.0362662	−	0.999342i	$-0.511546\pi$
$461$	−330967.	−0.0725324	−0.0362662	+	0.999342i	$0.511546\pi$
$462$	0	0
$463$	1.33825e6	0.290124	0.145062	−	0.989423i	$-0.453662\pi$
$463$	1.33825e6	0.290124	0.145062	+	0.989423i	$0.453662\pi$
$464$	959084.	0.206805
$465$	0	0
$466$	−1.79227e6	−0.382331
$467$	2.07392e6	0.440047	0.220024	−	0.975495i	$-0.429387\pi$
$467$	2.07392e6	0.440047	0.220024	+	0.975495i	$0.429387\pi$
$468$	0	0
$469$	−6.36438e6	−1.33605
$470$	6.12800e6	1.27960
$471$	0	0
$472$	2.33881e6	0.483215
$473$	471987.	0.0970011
$474$	0	0
$475$	1.53252e7	3.11654
$476$	−1.44357e6	−0.292025
$477$	0	0
$478$	3.71267e6	0.743219
$479$	−5.74666e6	−1.14440	−0.572198	−	0.820115i	$-0.693909\pi$
$479$	−5.74666e6	−1.14440	−0.572198	+	0.820115i	$0.693909\pi$
$480$	0	0
$481$	−1.32400e7	−2.60932
$482$	−2.33821e6	−0.458423
$483$	0	0
$484$	−2.44552e6	−0.474523
$485$	7.01409e6	1.35399
$486$	0	0
$487$	−2.91443e6	−0.556841	−0.278421	−	0.960459i	$-0.589811\pi$
$487$	−2.91443e6	−0.556841	−0.278421	+	0.960459i	$0.589811\pi$
$488$	1.96264e6	0.373071
$489$	0	0
$490$	351703.	0.0661738
$491$	9.14846e6	1.71255	0.856277	−	0.516518i	$-0.172772\pi$
$491$	9.14846e6	1.71255	0.856277	+	0.516518i	$0.172772\pi$
$492$	0	0
$493$	−2.67644e6	−0.495953
$494$	−9.53267e6	−1.75751
$495$	0	0
$496$	795135.	0.145123
$497$	−9.56191e6	−1.73642
$498$	0	0
$499$	−5.38360e6	−0.967880	−0.483940	−	0.875101i	$-0.660795\pi$
$499$	−5.38360e6	−0.967880	−0.483940	+	0.875101i	$0.660795\pi$
$500$	7.00502e6	1.25310
$501$	0	0
$502$	−2.56193e6	−0.453741
$503$	−2.03879e6	−0.359297	−0.179648	−	0.983731i	$-0.557496\pi$
$503$	−2.03879e6	−0.359297	−0.179648	+	0.983731i	$0.557496\pi$
$504$	0	0
$505$	−9.55787e6	−1.66776
$506$	191685.	0.0332822
$507$	0	0
$508$	−3.67595e6	−0.631990
$509$	−6.82539e6	−1.16770	−0.583852	−	0.811860i	$-0.698455\pi$
$509$	−6.82539e6	−1.16770	−0.583852	+	0.811860i	$0.698455\pi$
$510$	0	0
$511$	−1.08655e6	−0.184077
$512$	−262144.	−0.0441942
$513$	0	0
$514$	931116.	0.155452
$515$	−1.06497e7	−1.76937
$516$	0	0
$517$	−1.35316e6	−0.222649
$518$	5.81715e6	0.952546
$519$	0	0
$520$	−7.54707e6	−1.22397
$521$	−1.30318e6	−0.210335	−0.105167	−	0.994455i	$-0.533538\pi$
$521$	−1.30318e6	−0.210335	−0.105167	+	0.994455i	$0.533538\pi$
$522$	0	0
$523$	−1.10662e7	−1.76906	−0.884530	−	0.466483i	$-0.845521\pi$
$523$	−1.10662e7	−1.76906	−0.884530	+	0.466483i	$0.845521\pi$
$524$	−312529.	−0.0497236
$525$	0	0
$526$	6.73299e6	1.06107
$527$	−2.21892e6	−0.348029
$528$	0	0
$529$	279841.	0.0434783
$530$	1.21417e7	1.87754
$531$	0	0
$532$	4.18828e6	0.641588
$533$	1.80738e7	2.75570
$534$	0	0
$535$	1.55126e7	2.34315
$536$	3.22522e6	0.484895
$537$	0	0
$538$	−772123.	−0.115009
$539$	−77661.4	−0.0115142
$540$	0	0
$541$	4.00655e6	0.588541	0.294271	−	0.955722i	$-0.404923\pi$
$541$	4.00655e6	0.588541	0.294271	+	0.955722i	$0.404923\pi$
$542$	−4.29088e6	−0.627406
$543$	0	0
$544$	731544.	0.105985
$545$	1.21349e6	0.175002
$546$	0	0
$547$	−1.12439e7	−1.60675	−0.803377	−	0.595471i	$-0.796965\pi$
$547$	−1.12439e7	−1.60675	−0.803377	+	0.595471i	$0.796965\pi$
$548$	−359357.	−0.0511181
$549$	0	0
$550$	−2.67917e6	−0.377653
$551$	7.76526e6	1.08962
$552$	0	0
$553$	3.93781e6	0.547572
$554$	2.58438e6	0.357751
$555$	0	0
$556$	1.61260e6	0.221228
$557$	−1.13866e7	−1.55510	−0.777548	−	0.628824i	$-0.783537\pi$
$557$	−1.13866e7	−1.55510	−0.777548	+	0.628824i	$0.783537\pi$
$558$	0	0
$559$	5.99063e6	0.810855
$560$	3.31588e6	0.446817
$561$	0	0
$562$	−1.50695e6	−0.201261
$563$	−502978.	−0.0668772	−0.0334386	−	0.999441i	$-0.510646\pi$
$563$	−502978.	−0.0668772	−0.0334386	+	0.999441i	$0.510646\pi$
$564$	0	0
$565$	−6.42255e6	−0.846420
$566$	−955518.	−0.125371
$567$	0	0
$568$	4.84561e6	0.630199
$569$	−2.86146e6	−0.370516	−0.185258	−	0.982690i	$-0.559312\pi$
$569$	−2.86146e6	−0.370516	−0.185258	+	0.982690i	$0.559312\pi$
$570$	0	0
$571$	7.15355e6	0.918187	0.459094	−	0.888388i	$-0.348174\pi$
$571$	7.15355e6	0.918187	0.459094	+	0.888388i	$0.348174\pi$
$572$	1.66651e6	0.212969
$573$	0	0
$574$	−7.94092e6	−1.00598
$575$	−3.91132e6	−0.493349
$576$	0	0
$577$	9.80232e6	1.22571	0.612857	−	0.790194i	$-0.290020\pi$
$577$	9.80232e6	1.22571	0.612857	+	0.790194i	$0.290020\pi$
$578$	3.63796e6	0.452938
$579$	0	0
$580$	6.14780e6	0.758840
$581$	6.04785e6	0.743294
$582$	0	0
$583$	−2.68107e6	−0.326691
$584$	550623.	0.0668071
$585$	0	0
$586$	−9.00405e6	−1.08316
$587$	9.14289e6	1.09519	0.547594	−	0.836744i	$-0.315544\pi$
$587$	9.14289e6	1.09519	0.547594	+	0.836744i	$0.315544\pi$
$588$	0	0
$589$	6.43784e6	0.764631
$590$	1.49920e7	1.77308
$591$	0	0
$592$	−2.94791e6	−0.345708
$593$	9.03754e6	1.05539	0.527696	−	0.849433i	$-0.323056\pi$
$593$	9.03754e6	1.05539	0.527696	+	0.849433i	$0.323056\pi$
$594$	0	0
$595$	−9.25338e6	−1.07154
$596$	1.35268e6	0.155984
$597$	0	0
$598$	2.43294e6	0.278213
$599$	3.80735e6	0.433566	0.216783	−	0.976220i	$-0.430444\pi$
$599$	3.80735e6	0.433566	0.216783	+	0.976220i	$0.430444\pi$
$600$	0	0
$601$	−1.87780e6	−0.212063	−0.106031	−	0.994363i	$-0.533814\pi$
$601$	−1.87780e6	−0.212063	−0.106031	+	0.994363i	$0.533814\pi$
$602$	−2.63205e6	−0.296007
$603$	0	0
$604$	−398228.	−0.0444161
$605$	−1.56759e7	−1.74119
$606$	0	0
$607$	4.54959e6	0.501187	0.250594	−	0.968092i	$-0.419374\pi$
$607$	4.54959e6	0.501187	0.250594	+	0.968092i	$0.419374\pi$
$608$	−2.12246e6	−0.232852
$609$	0	0
$610$	1.25807e7	1.36893
$611$	−1.71748e7	−1.86118
$612$	0	0
$613$	−1.34888e7	−1.44985	−0.724923	−	0.688830i	$-0.758125\pi$
$613$	−1.34888e7	−1.44985	−0.724923	+	0.688830i	$0.758125\pi$
$614$	1.10627e7	1.18424
$615$	0	0
$616$	−732197.	−0.0777457
$617$	−6.99271e6	−0.739491	−0.369745	−	0.929133i	$-0.620555\pi$
$617$	−6.99271e6	−0.739491	−0.369745	+	0.929133i	$0.620555\pi$
$618$	0	0
$619$	1.60036e7	1.67877	0.839386	−	0.543535i	$-0.182915\pi$
$619$	1.60036e7	1.67877	0.839386	+	0.543535i	$0.182915\pi$
$620$	5.09688e6	0.532507
$621$	0	0
$622$	−5.55888e6	−0.576117
$623$	1.54136e7	1.59105
$624$	0	0
$625$	2.17971e7	2.23202
$626$	1.03911e7	1.05980
$627$	0	0
$628$	1.79967e6	0.182093
$629$	8.22650e6	0.829065
$630$	0	0
$631$	−5.91454e6	−0.591354	−0.295677	−	0.955288i	$-0.595545\pi$
$631$	−5.91454e6	−0.591354	−0.295677	+	0.955288i	$0.595545\pi$
$632$	−1.99553e6	−0.198731
$633$	0	0
$634$	7.11469e6	0.702963
$635$	−2.35631e7	−2.31899
$636$	0	0
$637$	−985709.	−0.0962498
$638$	−1.35753e6	−0.132037
$639$	0	0
$640$	−1.68036e6	−0.162164
$641$	3.74995e6	0.360480	0.180240	−	0.983623i	$-0.442313\pi$
$641$	3.74995e6	0.360480	0.180240	+	0.983623i	$0.442313\pi$
$642$	0	0
$643$	−1.53319e7	−1.46240	−0.731202	−	0.682161i	$-0.761040\pi$
$643$	−1.53319e7	−1.46240	−0.731202	+	0.682161i	$0.761040\pi$
$644$	−1.06894e6	−0.101563
$645$	0	0
$646$	5.92298e6	0.558417
$647$	−1.31456e7	−1.23458	−0.617289	−	0.786736i	$-0.711769\pi$
$647$	−1.31456e7	−1.23458	−0.617289	+	0.786736i	$0.711769\pi$
$648$	0	0
$649$	−3.31045e6	−0.308515
$650$	−3.40050e7	−3.15689
$651$	0	0
$652$	4.85294e6	0.447080
$653$	3.77381e6	0.346335	0.173168	−	0.984892i	$-0.444600\pi$
$653$	3.77381e6	0.346335	0.173168	+	0.984892i	$0.444600\pi$
$654$	0	0
$655$	−2.00334e6	−0.182453
$656$	4.02416e6	0.365103
$657$	0	0
$658$	7.54592e6	0.679434
$659$	−173359.	−0.0155501	−0.00777503	−	0.999970i	$-0.502475\pi$
$659$	−173359.	−0.0155501	−0.00777503	+	0.999970i	$0.502475\pi$
$660$	0	0
$661$	−4.23487e6	−0.376996	−0.188498	−	0.982074i	$-0.560362\pi$
$661$	−4.23487e6	−0.376996	−0.188498	+	0.982074i	$0.560362\pi$
$662$	−1.96628e6	−0.174382
$663$	0	0
$664$	−3.06482e6	−0.269764
$665$	2.68472e7	2.35421
$666$	0	0
$667$	−1.98186e6	−0.172488
$668$	−6.62685e6	−0.574601
$669$	0	0
$670$	2.06739e7	1.77925
$671$	−2.77801e6	−0.238192
$672$	0	0
$673$	2.95842e6	0.251781	0.125890	−	0.992044i	$-0.459821\pi$
$673$	2.95842e6	0.251781	0.125890	+	0.992044i	$0.459821\pi$
$674$	−1.32218e7	−1.12109
$675$	0	0
$676$	1.52113e7	1.28026
$677$	−2.12051e7	−1.77815	−0.889074	−	0.457764i	$-0.848650\pi$
$677$	−2.12051e7	−1.77815	−0.889074	+	0.457764i	$0.848650\pi$
$678$	0	0
$679$	8.63703e6	0.718935
$680$	4.68926e6	0.388894
$681$	0	0
$682$	−1.12547e6	−0.0926557
$683$	−1.16899e7	−0.958869	−0.479435	−	0.877578i	$-0.659158\pi$
$683$	−1.16899e7	−0.958869	−0.479435	+	0.877578i	$0.659158\pi$
$684$	0	0
$685$	−2.30350e6	−0.187570
$686$	8.92345e6	0.723973
$687$	0	0
$688$	1.33382e6	0.107430
$689$	−3.40292e7	−2.73089
$690$	0	0
$691$	−1.95865e7	−1.56049	−0.780247	−	0.625472i	$-0.784906\pi$
$691$	−1.95865e7	−1.56049	−0.780247	+	0.625472i	$0.784906\pi$
$692$	1.17620e7	0.933719
$693$	0	0
$694$	848597.	0.0668810
$695$	1.03369e7	0.811761
$696$	0	0
$697$	−1.12299e7	−0.875576
$698$	−1.55893e7	−1.21112
$699$	0	0
$700$	1.49405e7	1.15244
$701$	1.91397e7	1.47109	0.735546	−	0.677475i	$-0.236926\pi$
$701$	1.91397e7	1.47109	0.735546	+	0.677475i	$0.236926\pi$
$702$	0	0
$703$	−2.38679e7	−1.82148
$704$	371050.	0.0282163
$705$	0	0
$706$	−8.71273e6	−0.657874
$707$	−1.17694e7	−0.885535
$708$	0	0
$709$	1.71565e7	1.28178	0.640890	−	0.767633i	$-0.278566\pi$
$709$	1.71565e7	1.28178	0.640890	+	0.767633i	$0.278566\pi$
$710$	3.10607e7	2.31242
$711$	0	0
$712$	−7.81103e6	−0.577442
$713$	−1.64307e6	−0.121041
$714$	0	0
$715$	1.06824e7	0.781457
$716$	213764.	0.0155831
$717$	0	0
$718$	1.59350e6	0.115356
$719$	−1.92589e7	−1.38935	−0.694673	−	0.719326i	$-0.744451\pi$
$719$	−1.92589e7	−1.38935	−0.694673	+	0.719326i	$0.744451\pi$
$720$	0	0
$721$	−1.31139e7	−0.939490
$722$	−7.28016e6	−0.519754
$723$	0	0
$724$	−9.29806e6	−0.659244
$725$	2.77003e7	1.95722
$726$	0	0
$727$	−9.40978e6	−0.660303	−0.330151	−	0.943928i	$-0.607100\pi$
$727$	−9.40978e6	−0.660303	−0.330151	+	0.943928i	$0.607100\pi$
$728$	−9.29333e6	−0.649895
$729$	0	0
$730$	3.52954e6	0.245138
$731$	−3.72219e6	−0.257635
$732$	0	0
$733$	−1.49448e7	−1.02738	−0.513688	−	0.857977i	$-0.671721\pi$
$733$	−1.49448e7	−1.02738	−0.513688	+	0.857977i	$0.671721\pi$
$734$	1.19055e7	0.815655
$735$	0	0
$736$	541696.	0.0368605
$737$	−4.56511e6	−0.309587
$738$	0	0
$739$	−2.77526e7	−1.86936	−0.934679	−	0.355493i	$-0.884313\pi$
$739$	−2.77526e7	−1.86936	−0.934679	+	0.355493i	$0.884313\pi$
$740$	−1.88963e7	−1.26852
$741$	0	0
$742$	1.49511e7	0.996926
$743$	5.49249e6	0.365004	0.182502	−	0.983206i	$-0.441580\pi$
$743$	5.49249e6	0.365004	0.182502	+	0.983206i	$0.441580\pi$
$744$	0	0
$745$	8.67080e6	0.572359
$746$	−1.13280e7	−0.745260
$747$	0	0
$748$	−1.03546e6	−0.0676673
$749$	1.91020e7	1.24415
$750$	0	0
$751$	5.90981e6	0.382361	0.191181	−	0.981555i	$-0.438768\pi$
$751$	5.90981e6	0.382361	0.191181	+	0.981555i	$0.438768\pi$
$752$	−3.82398e6	−0.246588
$753$	0	0
$754$	−1.72303e7	−1.10373
$755$	−2.55267e6	−0.162978
$756$	0	0
$757$	1.21520e7	0.770737	0.385369	−	0.922763i	$-0.374074\pi$
$757$	1.21520e7	0.770737	0.385369	+	0.922763i	$0.374074\pi$
$758$	−2.72674e6	−0.172373
$759$	0	0
$760$	−1.36051e7	−0.854414
$761$	1.65759e7	1.03757	0.518783	−	0.854906i	$-0.326385\pi$
$761$	1.65759e7	1.03757	0.518783	+	0.854906i	$0.326385\pi$
$762$	0	0
$763$	1.49427e6	0.0929217
$764$	−1.34212e7	−0.831877
$765$	0	0
$766$	−592587.	−0.0364905
$767$	−4.20176e7	−2.57895
$768$	0	0
$769$	2.49448e7	1.52112	0.760562	−	0.649266i	$-0.224924\pi$
$769$	2.49448e7	1.52112	0.760562	+	0.649266i	$0.224924\pi$
$770$	−4.69344e6	−0.285276
$771$	0	0
$772$	−1.03841e6	−0.0627086
$773$	−1.49304e7	−0.898715	−0.449357	−	0.893352i	$-0.648347\pi$
$773$	−1.49304e7	−0.898715	−0.449357	+	0.893352i	$0.648347\pi$
$774$	0	0
$775$	2.29651e7	1.37346
$776$	−4.37691e6	−0.260924
$777$	0	0
$778$	−4.83937e6	−0.286642
$779$	3.25817e7	1.92367
$780$	0	0
$781$	−6.85868e6	−0.402358
$782$	−1.51167e6	−0.0883974
$783$	0	0
$784$	−219469.	−0.0127521
$785$	1.15360e7	0.668163
$786$	0	0
$787$	227162.	0.0130737	0.00653684	−	0.999979i	$-0.497919\pi$
$787$	227162.	0.0130737	0.00653684	+	0.999979i	$0.497919\pi$
$788$	9.29092e6	0.533019
$789$	0	0
$790$	−1.27915e7	−0.729211
$791$	−7.90861e6	−0.449427
$792$	0	0
$793$	−3.52595e7	−1.99110
$794$	1.79215e7	1.00884
$795$	0	0
$796$	−8.26718e6	−0.462461
$797$	2.65443e6	0.148022	0.0740108	−	0.997257i	$-0.476420\pi$
$797$	2.65443e6	0.148022	0.0740108	+	0.997257i	$0.476420\pi$
$798$	0	0
$799$	1.06713e7	0.591357
$800$	−7.57125e6	−0.418257
$801$	0	0
$802$	8.34855e6	0.458326
$803$	−779375.	−0.0426538
$804$	0	0
$805$	−6.85196e6	−0.372671
$806$	−1.42849e7	−0.774531
$807$	0	0
$808$	5.96428e6	0.321388
$809$	−7.38114e6	−0.396508	−0.198254	−	0.980151i	$-0.563527\pi$
$809$	−7.38114e6	−0.396508	−0.198254	+	0.980151i	$0.563527\pi$
$810$	0	0
$811$	−3.31527e6	−0.176997	−0.0884987	−	0.996076i	$-0.528207\pi$
$811$	−3.31527e6	−0.176997	−0.0884987	+	0.996076i	$0.528207\pi$
$812$	7.57030e6	0.402924
$813$	0	0
$814$	4.17259e6	0.220722
$815$	3.11077e7	1.64049
$816$	0	0
$817$	1.07993e7	0.566033
$818$	1.75681e7	0.917998
$819$	0	0
$820$	2.57951e7	1.33969
$821$	−2.63736e6	−0.136556	−0.0682780	−	0.997666i	$-0.521750\pi$
$821$	−2.63736e6	−0.136556	−0.0682780	+	0.997666i	$0.521750\pi$
$822$	0	0
$823$	2.39962e6	0.123493	0.0617467	−	0.998092i	$-0.480333\pi$
$823$	2.39962e6	0.123493	0.0617467	+	0.998092i	$0.480333\pi$
$824$	6.64560e6	0.340970
$825$	0	0
$826$	1.84609e7	0.941460
$827$	−8.25570e6	−0.419749	−0.209875	−	0.977728i	$-0.567306\pi$
$827$	−8.25570e6	−0.419749	−0.209875	+	0.977728i	$0.567306\pi$
$828$	0	0
$829$	2.99296e7	1.51256	0.756282	−	0.654245i	$-0.227014\pi$
$829$	2.99296e7	1.51256	0.756282	+	0.654245i	$0.227014\pi$
$830$	−1.96457e7	−0.989857
$831$	0	0
$832$	4.70950e6	0.235867
$833$	612455.	0.0305817
$834$	0	0
$835$	−4.24786e7	−2.10841
$836$	3.00422e6	0.148667
$837$	0	0
$838$	−2.25392e7	−1.10874
$839$	−1.33410e7	−0.654309	−0.327155	−	0.944971i	$-0.606090\pi$
$839$	−1.33410e7	−0.654309	−0.327155	+	0.944971i	$0.606090\pi$
$840$	0	0
$841$	−6.47546e6	−0.315705
$842$	−1.51751e7	−0.737650
$843$	0	0
$844$	−1.24062e6	−0.0599493
$845$	9.75053e7	4.69771
$846$	0	0
$847$	−1.93031e7	−0.924524
$848$	−7.57663e6	−0.361815
$849$	0	0
$850$	2.11285e7	1.00305
$851$	6.09158e6	0.288341
$852$	0	0
$853$	−3.28337e7	−1.54507	−0.772533	−	0.634974i	$-0.781011\pi$
$853$	−3.28337e7	−1.54507	−0.772533	+	0.634974i	$0.781011\pi$
$854$	1.54917e7	0.726864
$855$	0	0
$856$	−9.68014e6	−0.451541
$857$	9.91548e6	0.461171	0.230585	−	0.973052i	$-0.425936\pi$
$857$	9.91548e6	0.461171	0.230585	+	0.973052i	$0.425936\pi$
$858$	0	0
$859$	2.44468e7	1.13042	0.565209	−	0.824947i	$-0.308795\pi$
$859$	2.44468e7	1.13042	0.565209	+	0.824947i	$0.308795\pi$
$860$	8.54990e6	0.394198
$861$	0	0
$862$	−2.34651e7	−1.07561
$863$	−1.88703e7	−0.862484	−0.431242	−	0.902236i	$-0.641924\pi$
$863$	−1.88703e7	−0.862484	−0.431242	+	0.902236i	$0.641924\pi$
$864$	0	0
$865$	7.53954e7	3.42614
$866$	−6.09584e6	−0.276210
$867$	0	0
$868$	6.27621e6	0.282747
$869$	2.82456e6	0.126882
$870$	0	0
$871$	−5.79422e7	−2.58791
$872$	−757237.	−0.0337241
$873$	0	0
$874$	4.38586e6	0.194212
$875$	5.52924e7	2.44144
$876$	0	0
$877$	3.79461e7	1.66597	0.832987	−	0.553292i	$-0.186629\pi$
$877$	3.79461e7	1.66597	0.832987	+	0.553292i	$0.186629\pi$
$878$	560181.	0.0245240
$879$	0	0
$880$	2.37846e6	0.103535
$881$	2.74586e7	1.19190	0.595949	−	0.803022i	$-0.296776\pi$
$881$	2.74586e7	1.19190	0.595949	+	0.803022i	$0.296776\pi$
$882$	0	0
$883$	2.96084e7	1.27795	0.638974	−	0.769228i	$-0.279359\pi$
$883$	2.96084e7	1.27795	0.638974	+	0.769228i	$0.279359\pi$
$884$	−1.31424e7	−0.565647
$885$	0	0
$886$	−1.79420e7	−0.767867
$887$	−4.12029e7	−1.75840	−0.879202	−	0.476449i	$-0.841924\pi$
$887$	−4.12029e7	−1.75840	−0.879202	+	0.476449i	$0.841924\pi$
$888$	0	0
$889$	−2.90152e7	−1.23132
$890$	−5.00693e7	−2.11883
$891$	0	0
$892$	6.89116e6	0.289988
$893$	−3.09610e7	−1.29923
$894$	0	0
$895$	1.37025e6	0.0571795
$896$	−2.06917e6	−0.0861046
$897$	0	0
$898$	−924909.	−0.0382744
$899$	1.16364e7	0.480196
$900$	0	0
$901$	2.11435e7	0.867691
$902$	−5.69596e6	−0.233104
$903$	0	0
$904$	4.00778e6	0.163111
$905$	−5.96013e7	−2.41899
$906$	0	0
$907$	1.75022e7	0.706440	0.353220	−	0.935540i	$-0.385087\pi$
$907$	1.75022e7	0.706440	0.353220	+	0.935540i	$0.385087\pi$
$908$	1.17538e7	0.473111
$909$	0	0
$910$	−5.95710e7	−2.38469
$911$	−9.14766e6	−0.365186	−0.182593	−	0.983189i	$-0.558449\pi$
$911$	−9.14766e6	−0.365186	−0.182593	+	0.983189i	$0.558449\pi$
$912$	0	0
$913$	4.33807e6	0.172234
$914$	5.55721e6	0.220035
$915$	0	0
$916$	1.12077e6	0.0441344
$917$	−2.46688e6	−0.0968777
$918$	0	0
$919$	−2.46569e7	−0.963051	−0.481526	−	0.876432i	$-0.659917\pi$
$919$	−2.46569e7	−0.963051	−0.481526	+	0.876432i	$0.659917\pi$
$920$	3.47231e6	0.135254
$921$	0	0
$922$	1.32387e6	0.0512881
$923$	−8.70530e7	−3.36341
$924$	0	0
$925$	−8.51417e7	−3.27181
$926$	−5.35299e6	−0.205149
$927$	0	0
$928$	−3.83634e6	−0.146233
$929$	−3.70756e7	−1.40945	−0.704723	−	0.709483i	$-0.748929\pi$
$929$	−3.70756e7	−1.40945	−0.704723	+	0.709483i	$0.748929\pi$
$930$	0	0
$931$	−1.77694e6	−0.0671890
$932$	7.16909e6	0.270349
$933$	0	0
$934$	−8.29568e6	−0.311161
$935$	−6.63737e6	−0.248294
$936$	0	0
$937$	2.93182e7	1.09091	0.545455	−	0.838140i	$-0.316357\pi$
$937$	2.93182e7	1.09091	0.545455	+	0.838140i	$0.316357\pi$
$938$	2.54575e7	0.944732
$939$	0	0
$940$	−2.45120e7	−0.904814
$941$	1.12790e7	0.415239	0.207620	−	0.978210i	$-0.433428\pi$
$941$	1.12790e7	0.415239	0.207620	+	0.978210i	$0.433428\pi$
$942$	0	0
$943$	−8.31554e6	−0.304517
$944$	−9.35525e6	−0.341685
$945$	0	0
$946$	−1.88795e6	−0.0685902
$947$	−1.16239e7	−0.421191	−0.210595	−	0.977573i	$-0.567540\pi$
$947$	−1.16239e7	−0.421191	−0.210595	+	0.977573i	$0.567540\pi$
$948$	0	0
$949$	−9.89213e6	−0.356553
$950$	−6.13009e7	−2.20373
$951$	0	0
$952$	5.77427e6	0.206493
$953$	−2.66642e7	−0.951034	−0.475517	−	0.879706i	$-0.657739\pi$
$953$	−2.66642e7	−0.951034	−0.475517	+	0.879706i	$0.657739\pi$
$954$	0	0
$955$	−8.60312e7	−3.05244
$956$	−1.48507e7	−0.525535
$957$	0	0
$958$	2.29866e7	0.809210
$959$	−2.83650e6	−0.0995946
$960$	0	0
$961$	−1.89819e7	−0.663028
$962$	5.29602e7	1.84506
$963$	0	0
$964$	9.35285e6	0.324154
$965$	−6.65631e6	−0.230099
$966$	0	0
$967$	3.95144e7	1.35890	0.679452	−	0.733720i	$-0.262217\pi$
$967$	3.95144e7	1.35890	0.679452	+	0.733720i	$0.262217\pi$
$968$	9.78206e6	0.335538
$969$	0	0
$970$	−2.80563e7	−0.957419
$971$	−2.45153e7	−0.834430	−0.417215	−	0.908808i	$-0.636994\pi$
$971$	−2.45153e7	−0.834430	−0.417215	+	0.908808i	$0.636994\pi$
$972$	0	0
$973$	1.27287e7	0.431024
$974$	1.16577e7	0.393746
$975$	0	0
$976$	−7.85057e6	−0.263801
$977$	−2.39782e7	−0.803673	−0.401837	−	0.915711i	$-0.631628\pi$
$977$	−2.39782e7	−0.803673	−0.401837	+	0.915711i	$0.631628\pi$
$978$	0	0
$979$	1.10561e7	0.368675
$980$	−1.40681e6	−0.0467920
$981$	0	0
$982$	−3.65938e7	−1.21096
$983$	9.53736e6	0.314807	0.157403	−	0.987534i	$-0.449688\pi$
$983$	9.53736e6	0.314807	0.157403	+	0.987534i	$0.449688\pi$
$984$	0	0
$985$	5.95555e7	1.95583
$986$	1.07058e7	0.350692
$987$	0	0
$988$	3.81307e7	1.24274
$989$	−2.75622e6	−0.0896030
$990$	0	0
$991$	1.66472e7	0.538465	0.269232	−	0.963075i	$-0.413230\pi$
$991$	1.66472e7	0.538465	0.269232	+	0.963075i	$0.413230\pi$
$992$	−3.18054e6	−0.102618
$993$	0	0
$994$	3.82476e7	1.22783
$995$	−5.29933e7	−1.69693
$996$	0	0
$997$	5.79868e7	1.84753	0.923765	−	0.382960i	$-0.125095\pi$
$997$	5.79868e7	1.84753	0.923765	+	0.382960i	$0.125095\pi$
$998$	2.15344e7	0.684394
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.6.a.o.1.4		4
3.2	odd	2		138.6.a.i.1.1	✓	4
12.11	even	2		1104.6.a.m.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.6.a.i.1.1	✓	4	3.2	odd	2
414.6.a.o.1.4		4	1.1	even	1	trivial
1104.6.a.m.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:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +102.561 q^{5} +126.292 q^{7} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} +102.561 q^{5} +126.292 q^{7} -64.0000 q^{8} -410.245 q^{10} +90.5883 q^{11} +1149.78 q^{13} -505.168 q^{14} +256.000 q^{16} -714.399 q^{17} +2072.71 q^{19} +1640.98 q^{20} -362.353 q^{22} -529.000 q^{23} +7393.80 q^{25} -4599.12 q^{26} +2020.67 q^{28} +3746.42 q^{29} +3106.00 q^{31} -1024.00 q^{32} +2857.60 q^{34} +12952.7 q^{35} -11515.3 q^{37} -8290.85 q^{38} -6563.92 q^{40} +15719.4 q^{41} +5210.24 q^{43} +1449.41 q^{44} +2116.00 q^{46} -14937.4 q^{47} -857.301 q^{49} -29575.2 q^{50} +18396.5 q^{52} -29596.2 q^{53} +9290.84 q^{55} -8082.70 q^{56} -14985.7 q^{58} -36544.0 q^{59} -30666.3 q^{61} -12424.0 q^{62} +4096.00 q^{64} +117923. q^{65} -50394.1 q^{67} -11430.4 q^{68} -51810.7 q^{70} -75712.7 q^{71} -8603.49 q^{73} +46061.1 q^{74} +33163.4 q^{76} +11440.6 q^{77} +31180.1 q^{79} +26255.7 q^{80} -62877.4 q^{82} +47887.8 q^{83} -73269.6 q^{85} -20841.0 q^{86} -5797.65 q^{88} +122047. q^{89} +145208. q^{91} -8464.00 q^{92} +59749.7 q^{94} +212580. q^{95} +68389.3 q^{97} +3429.20 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8}+O(q^{10}) \) 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8	\( 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8} + 216 q^{10} + 6 q^{11} + 1248 q^{13} - 1392 q^{14} + 1024 q^{16} + 56 q^{17} + 1530 q^{19} - 864 q^{20} - 24 q^{22} - 2116 q^{23} + 8852 q^{25} - 4992 q^{26} + 5568 q^{28} - 2292 q^{29} + 1556 q^{31} - 4096 q^{32} - 224 q^{34} - 5688 q^{35} - 9586 q^{37} - 6120 q^{38} + 3456 q^{40} + 11768 q^{41} + 13758 q^{43} + 96 q^{44} + 8464 q^{46} - 10636 q^{47} + 11360 q^{49} - 35408 q^{50} + 19968 q^{52} - 26686 q^{53} - 28900 q^{55} - 22272 q^{56} + 9168 q^{58} + 9108 q^{59} - 37878 q^{61} - 6224 q^{62} + 16384 q^{64} + 72212 q^{65} - 23302 q^{67} + 896 q^{68} + 22752 q^{70} - 31728 q^{71} - 28340 q^{73} + 38344 q^{74} + 24480 q^{76} + 121276 q^{77} - 26668 q^{79} - 13824 q^{80} - 47072 q^{82} + 119026 q^{83} - 217876 q^{85} - 55032 q^{86} - 384 q^{88} + 148236 q^{89} + 89008 q^{91} - 33856 q^{92} + 42544 q^{94} + 399292 q^{95} - 16092 q^{97} - 45440 q^{98}+O(q^{100}) \) 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8 + 216 * q^10 + 6 * q^11 + 1248 * q^13 - 1392 * q^14 + 1024 * q^16 + 56 * q^17 + 1530 * q^19 - 864 * q^20 - 24 * q^22 - 2116 * q^23 + 8852 * q^25 - 4992 * q^26 + 5568 * q^28 - 2292 * q^29 + 1556 * q^31 - 4096 * q^32 - 224 * q^34 - 5688 * q^35 - 9586 * q^37 - 6120 * q^38 + 3456 * q^40 + 11768 * q^41 + 13758 * q^43 + 96 * q^44 + 8464 * q^46 - 10636 * q^47 + 11360 * q^49 - 35408 * q^50 + 19968 * q^52 - 26686 * q^53 - 28900 * q^55 - 22272 * q^56 + 9168 * q^58 + 9108 * q^59 - 37878 * q^61 - 6224 * q^62 + 16384 * q^64 + 72212 * q^65 - 23302 * q^67 + 896 * q^68 + 22752 * q^70 - 31728 * q^71 - 28340 * q^73 + 38344 * q^74 + 24480 * q^76 + 121276 * q^77 - 26668 * q^79 - 13824 * q^80 - 47072 * q^82 + 119026 * q^83 - 217876 * q^85 - 55032 * q^86 - 384 * q^88 + 148236 * q^89 + 89008 * q^91 - 33856 * q^92 + 42544 * q^94 + 399292 * q^95 - 16092 * q^97 - 45440 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more