LMFDB - Embedded newform 882.6.a.bg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, 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("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.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:	$141.458529075$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{4705}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1176 $ x^2 - x - 1176
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-33.7965$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} +77.5930 q^{5} -64.0000 q^{8} +O(q^{10})$	$q-4.00000 q^{2} +16.0000 q^{4} +77.5930 q^{5} -64.0000 q^{8} -310.372 q^{10} -618.337 q^{11} +130.372 q^{13} +256.000 q^{16} -263.895 q^{17} +91.9300 q^{19} +1241.49 q^{20} +2473.35 q^{22} +1391.69 q^{23} +2895.67 q^{25} -521.488 q^{26} +5692.02 q^{29} -6953.28 q^{31} -1024.00 q^{32} +1055.58 q^{34} +5048.02 q^{37} -367.720 q^{38} -4965.95 q^{40} +20990.1 q^{41} -21393.4 q^{43} -9893.39 q^{44} -5566.74 q^{46} -19871.5 q^{47} -11582.7 q^{50} +2085.95 q^{52} +27572.1 q^{53} -47978.6 q^{55} -22768.1 q^{58} -43536.9 q^{59} -33939.2 q^{61} +27813.1 q^{62} +4096.00 q^{64} +10116.0 q^{65} +10311.5 q^{67} -4222.32 q^{68} -76979.9 q^{71} -1986.38 q^{73} -20192.1 q^{74} +1470.88 q^{76} -5384.63 q^{79} +19863.8 q^{80} -83960.5 q^{82} +68969.3 q^{83} -20476.4 q^{85} +85573.6 q^{86} +39573.6 q^{88} +90485.6 q^{89} +22267.0 q^{92} +79486.1 q^{94} +7133.13 q^{95} -137676. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} + 18 q^{5} - 128 q^{8}+O(q^{10}) $ 2 * q - 8 * q^2 + 32 * q^4 + 18 * q^5 - 128 * q^8	$ 2 q - 8 q^{2} + 32 q^{4} + 18 q^{5} - 128 q^{8} - 72 q^{10} - 2 q^{11} - 288 q^{13} + 512 q^{16} + 1530 q^{17} - 1188 q^{19} + 288 q^{20} + 8 q^{22} - 3390 q^{23} + 3322 q^{25} + 1152 q^{26} + 3976 q^{29} - 7596 q^{31} - 2048 q^{32} - 6120 q^{34} + 2688 q^{37} + 4752 q^{38} - 1152 q^{40} + 36630 q^{41} - 23032 q^{43} - 32 q^{44} + 13560 q^{46} + 864 q^{47} - 13288 q^{50} - 4608 q^{52} + 32920 q^{53} - 84708 q^{55} - 15904 q^{58} - 26712 q^{59} - 20412 q^{61} + 30384 q^{62} + 8192 q^{64} + 35048 q^{65} - 36172 q^{67} + 24480 q^{68} - 73706 q^{71} + 74772 q^{73} - 10752 q^{74} - 19008 q^{76} - 23116 q^{79} + 4608 q^{80} - 146520 q^{82} + 147816 q^{83} - 127380 q^{85} + 92128 q^{86} + 128 q^{88} + 164646 q^{89} - 54240 q^{92} - 3456 q^{94} + 83408 q^{95} - 162036 q^{97}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 + 18 * q^5 - 128 * q^8 - 72 * q^10 - 2 * q^11 - 288 * q^13 + 512 * q^16 + 1530 * q^17 - 1188 * q^19 + 288 * q^20 + 8 * q^22 - 3390 * q^23 + 3322 * q^25 + 1152 * q^26 + 3976 * q^29 - 7596 * q^31 - 2048 * q^32 - 6120 * q^34 + 2688 * q^37 + 4752 * q^38 - 1152 * q^40 + 36630 * q^41 - 23032 * q^43 - 32 * q^44 + 13560 * q^46 + 864 * q^47 - 13288 * q^50 - 4608 * q^52 + 32920 * q^53 - 84708 * q^55 - 15904 * q^58 - 26712 * q^59 - 20412 * q^61 + 30384 * q^62 + 8192 * q^64 + 35048 * q^65 - 36172 * q^67 + 24480 * q^68 - 73706 * q^71 + 74772 * q^73 - 10752 * q^74 - 19008 * q^76 - 23116 * q^79 + 4608 * q^80 - 146520 * q^82 + 147816 * q^83 - 127380 * q^85 + 92128 * q^86 + 128 * q^88 + 164646 * q^89 - 54240 * q^92 - 3456 * q^94 + 83408 * q^95 - 162036 * q^97

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
$5$	77.5930	1.38803	0.694013	−	0.719963i	$-0.255841\pi$
$5$	77.5930	1.38803	0.694013	+	0.719963i	$0.255841\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−310.372	−0.981482
$11$	−618.337	−1.54079	−0.770395	−	0.637567i	$-0.779941\pi$
$11$	−618.337	−1.54079	−0.770395	+	0.637567i	$0.779941\pi$
$12$	0	0
$13$	130.372	0.213957	0.106978	−	0.994261i	$-0.465882\pi$
$13$	130.372	0.213957	0.106978	+	0.994261i	$0.465882\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−263.895	−0.221467	−0.110733	−	0.993850i	$-0.535320\pi$
$17$	−263.895	−0.221467	−0.110733	+	0.993850i	$0.535320\pi$
$18$	0	0
$19$	91.9300	0.0584216	0.0292108	−	0.999573i	$-0.490701\pi$
$19$	91.9300	0.0584216	0.0292108	+	0.999573i	$0.490701\pi$
$20$	1241.49	0.694013
$21$	0	0
$22$	2473.35	1.08950
$23$	1391.69	0.548557	0.274278	−	0.961650i	$-0.411561\pi$
$23$	1391.69	0.548557	0.274278	+	0.961650i	$0.411561\pi$
$24$	0	0
$25$	2895.67	0.926616
$26$	−521.488	−0.151290
$27$	0	0
$28$	0	0
$29$	5692.02	1.25682	0.628408	−	0.777884i	$-0.283707\pi$
$29$	5692.02	1.25682	0.628408	+	0.777884i	$0.283707\pi$
$30$	0	0
$31$	−6953.28	−1.29953	−0.649764	−	0.760136i	$-0.725132\pi$
$31$	−6953.28	−1.29953	−0.649764	+	0.760136i	$0.725132\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	1055.58	0.156601
$35$	0	0
$36$	0	0
$37$	5048.02	0.606201	0.303101	−	0.952959i	$-0.401978\pi$
$37$	5048.02	0.606201	0.303101	+	0.952959i	$0.401978\pi$
$38$	−367.720	−0.0413103
$39$	0	0
$40$	−4965.95	−0.490741
$41$	20990.1	1.95009	0.975047	−	0.222000i	$-0.0712584\pi$
$41$	20990.1	1.95009	0.975047	+	0.222000i	$0.0712584\pi$
$42$	0	0
$43$	−21393.4	−1.76445	−0.882223	−	0.470831i	$-0.843954\pi$
$43$	−21393.4	−1.76445	−0.882223	+	0.470831i	$0.843954\pi$
$44$	−9893.39	−0.770395
$45$	0	0
$46$	−5566.74	−0.387888
$47$	−19871.5	−1.31216	−0.656080	−	0.754692i	$-0.727787\pi$
$47$	−19871.5	−1.31216	−0.656080	+	0.754692i	$0.727787\pi$
$48$	0	0
$49$	0	0
$50$	−11582.7	−0.655216
$51$	0	0
$52$	2085.95	0.106978
$53$	27572.1	1.34828	0.674139	−	0.738604i	$-0.264515\pi$
$53$	27572.1	1.34828	0.674139	+	0.738604i	$0.264515\pi$
$54$	0	0
$55$	−47978.6	−2.13866
$56$	0	0
$57$	0	0
$58$	−22768.1	−0.888703
$59$	−43536.9	−1.62827	−0.814137	−	0.580672i	$-0.802790\pi$
$59$	−43536.9	−1.62827	−0.814137	+	0.580672i	$0.802790\pi$
$60$	0	0
$61$	−33939.2	−1.16782	−0.583911	−	0.811818i	$-0.698478\pi$
$61$	−33939.2	−1.16782	−0.583911	+	0.811818i	$0.698478\pi$
$62$	27813.1	0.918904
$63$	0	0
$64$	4096.00	0.125000
$65$	10116.0	0.296977
$66$	0	0
$67$	10311.5	0.280631	0.140315	−	0.990107i	$-0.455188\pi$
$67$	10311.5	0.280631	0.140315	+	0.990107i	$0.455188\pi$
$68$	−4222.32	−0.110733
$69$	0	0
$70$	0	0
$71$	−76979.9	−1.81231	−0.906153	−	0.422950i	$-0.860995\pi$
$71$	−76979.9	−1.81231	−0.906153	+	0.422950i	$0.860995\pi$
$72$	0	0
$73$	−1986.38	−0.0436271	−0.0218135	−	0.999762i	$-0.506944\pi$
$73$	−1986.38	−0.0436271	−0.0218135	+	0.999762i	$0.506944\pi$
$74$	−20192.1	−0.428649
$75$	0	0
$76$	1470.88	0.0292108
$77$	0	0
$78$	0	0
$79$	−5384.63	−0.0970707	−0.0485353	−	0.998821i	$-0.515455\pi$
$79$	−5384.63	−0.0970707	−0.0485353	+	0.998821i	$0.515455\pi$
$80$	19863.8	0.347006
$81$	0	0
$82$	−83960.5	−1.37892
$83$	68969.3	1.09891	0.549453	−	0.835525i	$-0.314836\pi$
$83$	68969.3	1.09891	0.549453	+	0.835525i	$0.314836\pi$
$84$	0	0
$85$	−20476.4	−0.307402
$86$	85573.6	1.24765
$87$	0	0
$88$	39573.6	0.544752
$89$	90485.6	1.21089	0.605444	−	0.795888i	$-0.292995\pi$
$89$	90485.6	1.21089	0.605444	+	0.795888i	$0.292995\pi$
$90$	0	0
$91$	0	0
$92$	22267.0	0.274278
$93$	0	0
$94$	79486.1	0.927837
$95$	7133.13	0.0810907
$96$	0	0
$97$	−137676.	−1.48569	−0.742845	−	0.669463i	$-0.766524\pi$
$97$	−137676.	−1.48569	−0.742845	+	0.669463i	$0.766524\pi$
$98$	0	0
$99$	0	0
$100$	46330.8	0.463308
$101$	−137942.	−1.34553	−0.672765	−	0.739856i	$-0.734894\pi$
$101$	−137942.	−1.34553	−0.672765	+	0.739856i	$0.734894\pi$
$102$	0	0
$103$	11512.1	0.106921	0.0534605	−	0.998570i	$-0.482975\pi$
$103$	11512.1	0.106921	0.0534605	+	0.998570i	$0.482975\pi$
$104$	−8343.81	−0.0756451
$105$	0	0
$106$	−110288.	−0.953377
$107$	98404.3	0.830911	0.415456	−	0.909613i	$-0.363622\pi$
$107$	98404.3	0.830911	0.415456	+	0.909613i	$0.363622\pi$
$108$	0	0
$109$	−167534.	−1.35063	−0.675315	−	0.737530i	$-0.735992\pi$
$109$	−167534.	−1.35063	−0.675315	+	0.737530i	$0.735992\pi$
$110$	191915.	1.51226
$111$	0	0
$112$	0	0
$113$	81313.6	0.599055	0.299528	−	0.954088i	$-0.403171\pi$
$113$	81313.6	0.599055	0.299528	+	0.954088i	$0.403171\pi$
$114$	0	0
$115$	107985.	0.761411
$116$	91072.4	0.628408
$117$	0	0
$118$	174148.	1.15136
$119$	0	0
$120$	0	0
$121$	221290.	1.37403
$122$	135757.	0.825775
$123$	0	0
$124$	−111252.	−0.649764
$125$	−17794.1	−0.101859
$126$	0	0
$127$	−90944.8	−0.500344	−0.250172	−	0.968201i	$-0.580487\pi$
$127$	−90944.8	−0.500344	−0.250172	+	0.968201i	$0.580487\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−40463.8	−0.209995
$131$	242668.	1.23547	0.617737	−	0.786385i	$-0.288050\pi$
$131$	242668.	1.23547	0.617737	+	0.786385i	$0.288050\pi$
$132$	0	0
$133$	0	0
$134$	−41246.0	−0.198436
$135$	0	0
$136$	16889.3	0.0783004
$137$	−11717.8	−0.0533391	−0.0266695	−	0.999644i	$-0.508490\pi$
$137$	−11717.8	−0.0533391	−0.0266695	+	0.999644i	$0.508490\pi$
$138$	0	0
$139$	−170260.	−0.747438	−0.373719	−	0.927542i	$-0.621918\pi$
$139$	−170260.	−0.747438	−0.373719	+	0.927542i	$0.621918\pi$
$140$	0	0
$141$	0	0
$142$	307920.	1.28149
$143$	−80613.8	−0.329662
$144$	0	0
$145$	441661.	1.74449
$146$	7945.53	0.0308490
$147$	0	0
$148$	80768.4	0.303101
$149$	169502.	0.625476	0.312738	−	0.949839i	$-0.398754\pi$
$149$	169502.	0.625476	0.312738	+	0.949839i	$0.398754\pi$
$150$	0	0
$151$	241968.	0.863605	0.431802	−	0.901968i	$-0.357878\pi$
$151$	241968.	0.863605	0.431802	+	0.901968i	$0.357878\pi$
$152$	−5883.52	−0.0206552
$153$	0	0
$154$	0	0
$155$	−539526.	−1.80378
$156$	0	0
$157$	259036.	0.838709	0.419354	−	0.907823i	$-0.362256\pi$
$157$	259036.	0.838709	0.419354	+	0.907823i	$0.362256\pi$
$158$	21538.5	0.0686393
$159$	0	0
$160$	−79455.2	−0.245371
$161$	0	0
$162$	0	0
$163$	−314858.	−0.928209	−0.464105	−	0.885780i	$-0.653624\pi$
$163$	−314858.	−0.928209	−0.464105	+	0.885780i	$0.653624\pi$
$164$	335842.	0.975047
$165$	0	0
$166$	−275877.	−0.777044
$167$	−84097.1	−0.233340	−0.116670	−	0.993171i	$-0.537222\pi$
$167$	−84097.1	−0.233340	−0.116670	+	0.993171i	$0.537222\pi$
$168$	0	0
$169$	−354296.	−0.954223
$170$	81905.6	0.217366
$171$	0	0
$172$	−342294.	−0.882223
$173$	−747559.	−1.89902	−0.949511	−	0.313733i	$-0.898420\pi$
$173$	−747559.	−1.89902	−0.949511	+	0.313733i	$0.898420\pi$
$174$	0	0
$175$	0	0
$176$	−158294.	−0.385198
$177$	0	0
$178$	−361942.	−0.856228
$179$	−90323.6	−0.210702	−0.105351	−	0.994435i	$-0.533597\pi$
$179$	−90323.6	−0.210702	−0.105351	+	0.994435i	$0.533597\pi$
$180$	0	0
$181$	−188607.	−0.427919	−0.213959	−	0.976843i	$-0.568636\pi$
$181$	−188607.	−0.427919	−0.213959	+	0.976843i	$0.568636\pi$
$182$	0	0
$183$	0	0
$184$	−89067.8	−0.193944
$185$	391691.	0.841423
$186$	0	0
$187$	163176.	0.341234
$188$	−317944.	−0.656080
$189$	0	0
$190$	−28532.5	−0.0573398
$191$	670537.	1.32996	0.664981	−	0.746861i	$-0.268440\pi$
$191$	670537.	1.32996	0.664981	+	0.746861i	$0.268440\pi$
$192$	0	0
$193$	−957035.	−1.84942	−0.924708	−	0.380677i	$-0.875691\pi$
$193$	−957035.	−1.84942	−0.924708	+	0.380677i	$0.875691\pi$
$194$	550703.	1.05054
$195$	0	0
$196$	0	0
$197$	−48400.6	−0.0888557	−0.0444279	−	0.999013i	$-0.514146\pi$
$197$	−48400.6	−0.0888557	−0.0444279	+	0.999013i	$0.514146\pi$
$198$	0	0
$199$	−761708.	−1.36350	−0.681751	−	0.731585i	$-0.738781\pi$
$199$	−761708.	−1.36350	−0.681751	+	0.731585i	$0.738781\pi$
$200$	−185323.	−0.327608
$201$	0	0
$202$	551769.	0.951434
$203$	0	0
$204$	0	0
$205$	1.62869e6	2.70678
$206$	−46048.5	−0.0756045
$207$	0	0
$208$	33375.2	0.0534892
$209$	−56843.7	−0.0900154
$210$	0	0
$211$	−698206.	−1.07964	−0.539818	−	0.841781i	$-0.681507\pi$
$211$	−698206.	−1.07964	−0.539818	+	0.841781i	$0.681507\pi$
$212$	441153.	0.674139
$213$	0	0
$214$	−393617.	−0.587543
$215$	−1.65998e6	−2.44910
$216$	0	0
$217$	0	0
$218$	670135.	0.955039
$219$	0	0
$220$	−767658.	−1.06933
$221$	−34404.5	−0.0473843
$222$	0	0
$223$	1.25402e6	1.68866	0.844332	−	0.535821i	$-0.179998\pi$
$223$	1.25402e6	1.68866	0.844332	+	0.535821i	$0.179998\pi$
$224$	0	0
$225$	0	0
$226$	−325254.	−0.423596
$227$	−799712.	−1.03008	−0.515038	−	0.857168i	$-0.672222\pi$
$227$	−799712.	−1.03008	−0.515038	+	0.857168i	$0.672222\pi$
$228$	0	0
$229$	876062.	1.10394	0.551971	−	0.833863i	$-0.313876\pi$
$229$	876062.	1.10394	0.551971	+	0.833863i	$0.313876\pi$
$230$	−431940.	−0.538399
$231$	0	0
$232$	−364289.	−0.444351
$233$	−354407.	−0.427674	−0.213837	−	0.976869i	$-0.568596\pi$
$233$	−354407.	−0.427674	−0.213837	+	0.976869i	$0.568596\pi$
$234$	0	0
$235$	−1.54189e6	−1.82131
$236$	−696591.	−0.814137
$237$	0	0
$238$	0	0
$239$	−883393.	−1.00037	−0.500183	−	0.865920i	$-0.666734\pi$
$239$	−883393.	−1.00037	−0.500183	+	0.865920i	$0.666734\pi$
$240$	0	0
$241$	618631.	0.686103	0.343051	−	0.939317i	$-0.388540\pi$
$241$	618631.	0.686103	0.343051	+	0.939317i	$0.388540\pi$
$242$	−885159.	−0.971589
$243$	0	0
$244$	−543027.	−0.583911
$245$	0	0
$246$	0	0
$247$	11985.1	0.0124997
$248$	445010.	0.459452
$249$	0	0
$250$	71176.4	0.0720254
$251$	538736.	0.539749	0.269875	−	0.962895i	$-0.413018\pi$
$251$	538736.	0.539749	0.269875	+	0.962895i	$0.413018\pi$
$252$	0	0
$253$	−860530.	−0.845211
$254$	363779.	0.353797
$255$	0	0
$256$	65536.0	0.0625000
$257$	−360248.	−0.340227	−0.170113	−	0.985425i	$-0.554413\pi$
$257$	−360248.	−0.340227	−0.170113	+	0.985425i	$0.554413\pi$
$258$	0	0
$259$	0	0
$260$	161855.	0.148489
$261$	0	0
$262$	−970671.	−0.873612
$263$	−614181.	−0.547529	−0.273764	−	0.961797i	$-0.588269\pi$
$263$	−614181.	−0.547529	−0.273764	+	0.961797i	$0.588269\pi$
$264$	0	0
$265$	2.13940e6	1.87145
$266$	0	0
$267$	0	0
$268$	164984.	0.140315
$269$	−152269.	−0.128301	−0.0641505	−	0.997940i	$-0.520434\pi$
$269$	−152269.	−0.128301	−0.0641505	+	0.997940i	$0.520434\pi$
$270$	0	0
$271$	−509710.	−0.421599	−0.210800	−	0.977529i	$-0.567607\pi$
$271$	−509710.	−0.421599	−0.210800	+	0.977529i	$0.567607\pi$
$272$	−67557.1	−0.0553667
$273$	0	0
$274$	46871.3	0.0377164
$275$	−1.79050e6	−1.42772
$276$	0	0
$277$	348739.	0.273087	0.136544	−	0.990634i	$-0.456401\pi$
$277$	348739.	0.273087	0.136544	+	0.990634i	$0.456401\pi$
$278$	681039.	0.528518
$279$	0	0
$280$	0	0
$281$	−1.45741e6	−1.10107	−0.550537	−	0.834811i	$-0.685577\pi$
$281$	−1.45741e6	−1.10107	−0.550537	+	0.834811i	$0.685577\pi$
$282$	0	0
$283$	1.96116e6	1.45562	0.727810	−	0.685779i	$-0.240538\pi$
$283$	1.96116e6	1.45562	0.727810	+	0.685779i	$0.240538\pi$
$284$	−1.23168e6	−0.906153
$285$	0	0
$286$	322455.	0.233107
$287$	0	0
$288$	0	0
$289$	−1.35022e6	−0.950952
$290$	−1.76664e6	−1.23354
$291$	0	0
$292$	−31782.1	−0.0218135
$293$	−1.37485e6	−0.935591	−0.467795	−	0.883837i	$-0.654952\pi$
$293$	−1.37485e6	−0.935591	−0.467795	+	0.883837i	$0.654952\pi$
$294$	0	0
$295$	−3.37816e6	−2.26009
$296$	−323073.	−0.214325
$297$	0	0
$298$	−678010.	−0.442278
$299$	181437.	0.117367
$300$	0	0
$301$	0	0
$302$	−967871.	−0.610661
$303$	0	0
$304$	23534.1	0.0146054
$305$	−2.63344e6	−1.62097
$306$	0	0
$307$	−2.11456e6	−1.28048	−0.640241	−	0.768174i	$-0.721166\pi$
$307$	−2.11456e6	−1.28048	−0.640241	+	0.768174i	$0.721166\pi$
$308$	0	0
$309$	0	0
$310$	2.15810e6	1.27546
$311$	373460.	0.218949	0.109475	−	0.993990i	$-0.465083\pi$
$311$	373460.	0.218949	0.109475	+	0.993990i	$0.465083\pi$
$312$	0	0
$313$	466003.	0.268861	0.134430	−	0.990923i	$-0.457080\pi$
$313$	466003.	0.268861	0.134430	+	0.990923i	$0.457080\pi$
$314$	−1.03614e6	−0.593057
$315$	0	0
$316$	−86154.1	−0.0485353
$317$	−1.77386e6	−0.991450	−0.495725	−	0.868479i	$-0.665098\pi$
$317$	−1.77386e6	−0.991450	−0.495725	+	0.868479i	$0.665098\pi$
$318$	0	0
$319$	−3.51959e6	−1.93649
$320$	317821.	0.173503
$321$	0	0
$322$	0	0
$323$	−24259.9	−0.0129385
$324$	0	0
$325$	377515.	0.198256
$326$	1.25943e6	0.656343
$327$	0	0
$328$	−1.34337e6	−0.689462
$329$	0	0
$330$	0	0
$331$	−1.77186e6	−0.888911	−0.444456	−	0.895801i	$-0.646603\pi$
$331$	−1.77186e6	−0.888911	−0.444456	+	0.895801i	$0.646603\pi$
$332$	1.10351e6	0.549453
$333$	0	0
$334$	336388.	0.164997
$335$	800100.	0.389522
$336$	0	0
$337$	−702563.	−0.336985	−0.168492	−	0.985703i	$-0.553890\pi$
$337$	−702563.	−0.336985	−0.168492	+	0.985703i	$0.553890\pi$
$338$	1.41718e6	0.674737
$339$	0	0
$340$	−327623.	−0.153701
$341$	4.29947e6	2.00230
$342$	0	0
$343$	0	0
$344$	1.36918e6	0.623826
$345$	0	0
$346$	2.99024e6	1.34281
$347$	1.76407e6	0.786490	0.393245	−	0.919434i	$-0.371353\pi$
$347$	1.76407e6	0.786490	0.393245	+	0.919434i	$0.371353\pi$
$348$	0	0
$349$	−2.00221e6	−0.879927	−0.439964	−	0.898016i	$-0.645009\pi$
$349$	−2.00221e6	−0.879927	−0.439964	+	0.898016i	$0.645009\pi$
$350$	0	0
$351$	0	0
$352$	633177.	0.272376
$353$	2.81887e6	1.20403	0.602017	−	0.798483i	$-0.294364\pi$
$353$	2.81887e6	1.20403	0.602017	+	0.798483i	$0.294364\pi$
$354$	0	0
$355$	−5.97310e6	−2.51553
$356$	1.44777e6	0.605444
$357$	0	0
$358$	361294.	0.148989
$359$	−616146.	−0.252318	−0.126159	−	0.992010i	$-0.540265\pi$
$359$	−616146.	−0.252318	−0.126159	+	0.992010i	$0.540265\pi$
$360$	0	0
$361$	−2.46765e6	−0.996587
$362$	754428.	0.302584
$363$	0	0
$364$	0	0
$365$	−154129.	−0.0605555
$366$	0	0
$367$	−1.50738e6	−0.584196	−0.292098	−	0.956388i	$-0.594353\pi$
$367$	−1.50738e6	−0.584196	−0.292098	+	0.956388i	$0.594353\pi$
$368$	356271.	0.137139
$369$	0	0
$370$	−1.56676e6	−0.594976
$371$	0	0
$372$	0	0
$373$	−525518.	−0.195576	−0.0977881	−	0.995207i	$-0.531177\pi$
$373$	−525518.	−0.195576	−0.0977881	+	0.995207i	$0.531177\pi$
$374$	−652704.	−0.241289
$375$	0	0
$376$	1.27178e6	0.463918
$377$	742080.	0.268904
$378$	0	0
$379$	−97563.3	−0.0348890	−0.0174445	−	0.999848i	$-0.505553\pi$
$379$	−97563.3	−0.0348890	−0.0174445	+	0.999848i	$0.505553\pi$
$380$	114130.	0.0405453
$381$	0	0
$382$	−2.68215e6	−0.940425
$383$	5.02105e6	1.74903	0.874516	−	0.484997i	$-0.161179\pi$
$383$	5.02105e6	1.74903	0.874516	+	0.484997i	$0.161179\pi$
$384$	0	0
$385$	0	0
$386$	3.82814e6	1.30773
$387$	0	0
$388$	−2.20281e6	−0.742845
$389$	−4.56548e6	−1.52972	−0.764862	−	0.644195i	$-0.777193\pi$
$389$	−4.56548e6	−1.52972	−0.764862	+	0.644195i	$0.777193\pi$
$390$	0	0
$391$	−367259.	−0.121487
$392$	0	0
$393$	0	0
$394$	193603.	0.0628305
$395$	−417810.	−0.134737
$396$	0	0
$397$	4.90003e6	1.56035	0.780176	−	0.625560i	$-0.215130\pi$
$397$	4.90003e6	1.56035	0.780176	+	0.625560i	$0.215130\pi$
$398$	3.04683e6	0.964141
$399$	0	0
$400$	741293.	0.231654
$401$	1.58377e6	0.491850	0.245925	−	0.969289i	$-0.420908\pi$
$401$	1.58377e6	0.491850	0.245925	+	0.969289i	$0.420908\pi$
$402$	0	0
$403$	−906513.	−0.278043
$404$	−2.20707e6	−0.672765
$405$	0	0
$406$	0	0
$407$	−3.12138e6	−0.934029
$408$	0	0
$409$	1.69804e6	0.501925	0.250962	−	0.967997i	$-0.419253\pi$
$409$	1.69804e6	0.501925	0.250962	+	0.967997i	$0.419253\pi$
$410$	−6.51475e6	−1.91398
$411$	0	0
$412$	184194.	0.0534605
$413$	0	0
$414$	0	0
$415$	5.35154e6	1.52531
$416$	−133501.	−0.0378226
$417$	0	0
$418$	227375.	0.0636505
$419$	5.55194e6	1.54493	0.772466	−	0.635056i	$-0.219023\pi$
$419$	5.55194e6	1.54493	0.772466	+	0.635056i	$0.219023\pi$
$420$	0	0
$421$	1.21525e6	0.334164	0.167082	−	0.985943i	$-0.446566\pi$
$421$	1.21525e6	0.334164	0.167082	+	0.985943i	$0.446566\pi$
$422$	2.79283e6	0.763419
$423$	0	0
$424$	−1.76461e6	−0.476688
$425$	−764154.	−0.205215
$426$	0	0
$427$	0	0
$428$	1.57447e6	0.415456
$429$	0	0
$430$	6.63991e6	1.73177
$431$	−1.32530e6	−0.343654	−0.171827	−	0.985127i	$-0.554967\pi$
$431$	−1.32530e6	−0.343654	−0.171827	+	0.985127i	$0.554967\pi$
$432$	0	0
$433$	−5.48145e6	−1.40500	−0.702499	−	0.711685i	$-0.747932\pi$
$433$	−5.48145e6	−1.40500	−0.702499	+	0.711685i	$0.747932\pi$
$434$	0	0
$435$	0	0
$436$	−2.68054e6	−0.675315
$437$	127938.	0.0320475
$438$	0	0
$439$	−6.06386e6	−1.50172	−0.750858	−	0.660464i	$-0.770360\pi$
$439$	−6.06386e6	−1.50172	−0.750858	+	0.660464i	$0.770360\pi$
$440$	3.07063e6	0.756129
$441$	0	0
$442$	137618.	0.0335058
$443$	−1.82827e6	−0.442619	−0.221309	−	0.975204i	$-0.571033\pi$
$443$	−1.82827e6	−0.442619	−0.221309	+	0.975204i	$0.571033\pi$
$444$	0	0
$445$	7.02105e6	1.68075
$446$	−5.01609e6	−1.19407
$447$	0	0
$448$	0	0
$449$	−8.20432e6	−1.92056	−0.960278	−	0.279046i	$-0.909982\pi$
$449$	−8.20432e6	−1.92056	−0.960278	+	0.279046i	$0.909982\pi$
$450$	0	0
$451$	−1.29790e7	−3.00469
$452$	1.30102e6	0.299528
$453$	0	0
$454$	3.19885e6	0.728373
$455$	0	0
$456$	0	0
$457$	−3.57078e6	−0.799784	−0.399892	−	0.916562i	$-0.630952\pi$
$457$	−3.57078e6	−0.799784	−0.399892	+	0.916562i	$0.630952\pi$
$458$	−3.50425e6	−0.780605
$459$	0	0
$460$	1.72776e6	0.380705
$461$	−5.63952e6	−1.23592	−0.617959	−	0.786211i	$-0.712040\pi$
$461$	−5.63952e6	−1.23592	−0.617959	+	0.786211i	$0.712040\pi$
$462$	0	0
$463$	796223.	0.172616	0.0863082	−	0.996268i	$-0.472493\pi$
$463$	796223.	0.172616	0.0863082	+	0.996268i	$0.472493\pi$
$464$	1.45716e6	0.314204
$465$	0	0
$466$	1.41763e6	0.302411
$467$	133452.	0.0283160	0.0141580	−	0.999900i	$-0.495493\pi$
$467$	133452.	0.0283160	0.0141580	+	0.999900i	$0.495493\pi$
$468$	0	0
$469$	0	0
$470$	6.16757e6	1.28786
$471$	0	0
$472$	2.78636e6	0.575682
$473$	1.32283e7	2.71864
$474$	0	0
$475$	266199.	0.0541344
$476$	0	0
$477$	0	0
$478$	3.53357e6	0.707366
$479$	−4.47697e6	−0.891549	−0.445775	−	0.895145i	$-0.647072\pi$
$479$	−4.47697e6	−0.891549	−0.445775	+	0.895145i	$0.647072\pi$
$480$	0	0
$481$	658121.	0.129701
$482$	−2.47452e6	−0.485148
$483$	0	0
$484$	3.54063e6	0.687017
$485$	−1.06827e7	−2.06218
$486$	0	0
$487$	4.05633e6	0.775017	0.387509	−	0.921866i	$-0.373336\pi$
$487$	4.05633e6	0.775017	0.387509	+	0.921866i	$0.373336\pi$
$488$	2.17211e6	0.412888
$489$	0	0
$490$	0	0
$491$	300928.	0.0563325	0.0281662	−	0.999603i	$-0.491033\pi$
$491$	300928.	0.0563325	0.0281662	+	0.999603i	$0.491033\pi$
$492$	0	0
$493$	−1.50210e6	−0.278343
$494$	−47940.4	−0.00883862
$495$	0	0
$496$	−1.78004e6	−0.324882
$497$	0	0
$498$	0	0
$499$	4.26638e6	0.767024	0.383512	−	0.923536i	$-0.374715\pi$
$499$	4.26638e6	0.767024	0.383512	+	0.923536i	$0.374715\pi$
$500$	−284705.	−0.0509297
$501$	0	0
$502$	−2.15495e6	−0.381660
$503$	2.34185e6	0.412704	0.206352	−	0.978478i	$-0.433841\pi$
$503$	2.34185e6	0.412704	0.206352	+	0.978478i	$0.433841\pi$
$504$	0	0
$505$	−1.07033e7	−1.86763
$506$	3.44212e6	0.597654
$507$	0	0
$508$	−1.45512e6	−0.250172
$509$	−6.29127e6	−1.07633	−0.538163	−	0.842841i	$-0.680881\pi$
$509$	−6.29127e6	−1.07633	−0.538163	+	0.842841i	$0.680881\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	1.44099e6	0.240577
$515$	893261.	0.148409
$516$	0	0
$517$	1.22873e7	2.02176
$518$	0	0
$519$	0	0
$520$	−647421.	−0.104997
$521$	1.08898e6	0.175762	0.0878812	−	0.996131i	$-0.471990\pi$
$521$	1.08898e6	0.175762	0.0878812	+	0.996131i	$0.471990\pi$
$522$	0	0
$523$	−6.04665e6	−0.966631	−0.483316	−	0.875446i	$-0.660568\pi$
$523$	−6.04665e6	−0.966631	−0.483316	+	0.875446i	$0.660568\pi$
$524$	3.88268e6	0.617737
$525$	0	0
$526$	2.45672e6	0.387161
$527$	1.83494e6	0.287802
$528$	0	0
$529$	−4.49956e6	−0.699086
$530$	−8.55760e6	−1.32331
$531$	0	0
$532$	0	0
$533$	2.73653e6	0.417236
$534$	0	0
$535$	7.63549e6	1.15333
$536$	−659936.	−0.0992179
$537$	0	0
$538$	609075.	0.0907226
$539$	0	0
$540$	0	0
$541$	−4.27361e6	−0.627772	−0.313886	−	0.949461i	$-0.601631\pi$
$541$	−4.27361e6	−0.627772	−0.313886	+	0.949461i	$0.601631\pi$
$542$	2.03884e6	0.298116
$543$	0	0
$544$	270229.	0.0391502
$545$	−1.29995e7	−1.87471
$546$	0	0
$547$	1.04021e7	1.48646	0.743230	−	0.669036i	$-0.233293\pi$
$547$	1.04021e7	1.48646	0.743230	+	0.669036i	$0.233293\pi$
$548$	−187485.	−0.0266695
$549$	0	0
$550$	7.16201e6	1.00955
$551$	523268.	0.0734252
$552$	0	0
$553$	0	0
$554$	−1.39496e6	−0.193102
$555$	0	0
$556$	−2.72416e6	−0.373719
$557$	8.19079e6	1.11863	0.559317	−	0.828954i	$-0.311063\pi$
$557$	8.19079e6	1.11863	0.559317	+	0.828954i	$0.311063\pi$
$558$	0	0
$559$	−2.78910e6	−0.377515
$560$	0	0
$561$	0	0
$562$	5.82965e6	0.778577
$563$	−5.27402e6	−0.701246	−0.350623	−	0.936517i	$-0.614030\pi$
$563$	−5.27402e6	−0.701246	−0.350623	+	0.936517i	$0.614030\pi$
$564$	0	0
$565$	6.30936e6	0.831504
$566$	−7.84466e6	−1.02928
$567$	0	0
$568$	4.92671e6	0.640747
$569$	1.27444e7	1.65020	0.825102	−	0.564984i	$-0.191118\pi$
$569$	1.27444e7	1.65020	0.825102	+	0.564984i	$0.191118\pi$
$570$	0	0
$571$	1.01607e7	1.30417	0.652084	−	0.758147i	$-0.273895\pi$
$571$	1.01607e7	1.30417	0.652084	+	0.758147i	$0.273895\pi$
$572$	−1.28982e6	−0.164831
$573$	0	0
$574$	0	0
$575$	4.02987e6	0.508301
$576$	0	0
$577$	−9.43480e6	−1.17976	−0.589879	−	0.807491i	$-0.700825\pi$
$577$	−9.43480e6	−1.17976	−0.589879	+	0.807491i	$0.700825\pi$
$578$	5.40087e6	0.672425
$579$	0	0
$580$	7.06658e6	0.872246
$581$	0	0
$582$	0	0
$583$	−1.70488e7	−2.07741
$584$	127129.	0.0154245
$585$	0	0
$586$	5.49940e6	0.661563
$587$	−8.48788e6	−1.01673	−0.508363	−	0.861143i	$-0.669749\pi$
$587$	−8.48788e6	−1.01673	−0.508363	+	0.861143i	$0.669749\pi$
$588$	0	0
$589$	−639215.	−0.0759204
$590$	1.35126e7	1.59812
$591$	0	0
$592$	1.29229e6	0.151550
$593$	3.22262e6	0.376333	0.188166	−	0.982137i	$-0.439746\pi$
$593$	3.22262e6	0.376333	0.188166	+	0.982137i	$0.439746\pi$
$594$	0	0
$595$	0	0
$596$	2.71204e6	0.312738
$597$	0	0
$598$	−725747.	−0.0829913
$599$	2.74313e6	0.312377	0.156189	−	0.987727i	$-0.450079\pi$
$599$	2.74313e6	0.312377	0.156189	+	0.987727i	$0.450079\pi$
$600$	0	0
$601$	1.28775e7	1.45427	0.727137	−	0.686492i	$-0.240851\pi$
$601$	1.28775e7	1.45427	0.727137	+	0.686492i	$0.240851\pi$
$602$	0	0
$603$	0	0
$604$	3.87148e6	0.431802
$605$	1.71705e7	1.90720
$606$	0	0
$607$	2.89143e6	0.318524	0.159262	−	0.987236i	$-0.449089\pi$
$607$	2.89143e6	0.318524	0.159262	+	0.987236i	$0.449089\pi$
$608$	−94136.3	−0.0103276
$609$	0	0
$610$	1.05338e7	1.14620
$611$	−2.59069e6	−0.280745
$612$	0	0
$613$	1.44252e7	1.55050	0.775248	−	0.631657i	$-0.217625\pi$
$613$	1.44252e7	1.55050	0.775248	+	0.631657i	$0.217625\pi$
$614$	8.45823e6	0.905438
$615$	0	0
$616$	0	0
$617$	−9.56131e6	−1.01112	−0.505562	−	0.862790i	$-0.668715\pi$
$617$	−9.56131e6	−1.01112	−0.505562	+	0.862790i	$0.668715\pi$
$618$	0	0
$619$	7.74922e6	0.812889	0.406445	−	0.913675i	$-0.366768\pi$
$619$	7.74922e6	0.812889	0.406445	+	0.913675i	$0.366768\pi$
$620$	−8.63241e6	−0.901889
$621$	0	0
$622$	−1.49384e6	−0.154820
$623$	0	0
$624$	0	0
$625$	−1.04297e7	−1.06800
$626$	−1.86401e6	−0.190113
$627$	0	0
$628$	4.14458e6	0.419354
$629$	−1.33215e6	−0.134254
$630$	0	0
$631$	−667060.	−0.0666948	−0.0333474	−	0.999444i	$-0.510617\pi$
$631$	−667060.	−0.0666948	−0.0333474	+	0.999444i	$0.510617\pi$
$632$	344616.	0.0343197
$633$	0	0
$634$	7.09544e6	0.701061
$635$	−7.05668e6	−0.694490
$636$	0	0
$637$	0	0
$638$	1.40784e7	1.36930
$639$	0	0
$640$	−1.27128e6	−0.122685
$641$	1.57468e7	1.51373	0.756864	−	0.653573i	$-0.226731\pi$
$641$	1.57468e7	1.51373	0.756864	+	0.653573i	$0.226731\pi$
$642$	0	0
$643$	1.13073e6	0.107852	0.0539262	−	0.998545i	$-0.482826\pi$
$643$	1.13073e6	0.107852	0.0539262	+	0.998545i	$0.482826\pi$
$644$	0	0
$645$	0	0
$646$	97039.5	0.00914887
$647$	2.26531e6	0.212749	0.106375	−	0.994326i	$-0.466076\pi$
$647$	2.26531e6	0.212749	0.106375	+	0.994326i	$0.466076\pi$
$648$	0	0
$649$	2.69205e7	2.50883
$650$	−1.51006e6	−0.140188
$651$	0	0
$652$	−5.03773e6	−0.464105
$653$	−1.91461e7	−1.75710	−0.878550	−	0.477651i	$-0.841488\pi$
$653$	−1.91461e7	−1.75710	−0.878550	+	0.477651i	$0.841488\pi$
$654$	0	0
$655$	1.88293e7	1.71487
$656$	5.37347e6	0.487523
$657$	0	0
$658$	0	0
$659$	−2.15578e7	−1.93371	−0.966853	−	0.255335i	$-0.917814\pi$
$659$	−2.15578e7	−1.93371	−0.966853	+	0.255335i	$0.917814\pi$
$660$	0	0
$661$	8.08195e6	0.719470	0.359735	−	0.933055i	$-0.382867\pi$
$661$	8.08195e6	0.719470	0.359735	+	0.933055i	$0.382867\pi$
$662$	7.08743e6	0.628555
$663$	0	0
$664$	−4.41404e6	−0.388522
$665$	0	0
$666$	0	0
$667$	7.92150e6	0.689434
$668$	−1.34555e6	−0.116670
$669$	0	0
$670$	−3.20040e6	−0.275434
$671$	2.09859e7	1.79937
$672$	0	0
$673$	1.12701e7	0.959159	0.479580	−	0.877498i	$-0.340789\pi$
$673$	1.12701e7	0.959159	0.479580	+	0.877498i	$0.340789\pi$
$674$	2.81025e6	0.238284
$675$	0	0
$676$	−5.66874e6	−0.477111
$677$	−8.50353e6	−0.713062	−0.356531	−	0.934283i	$-0.616041\pi$
$677$	−8.50353e6	−0.713062	−0.356531	+	0.934283i	$0.616041\pi$
$678$	0	0
$679$	0	0
$680$	1.31049e6	0.108683
$681$	0	0
$682$	−1.71979e7	−1.41584
$683$	−506305.	−0.0415298	−0.0207649	−	0.999784i	$-0.506610\pi$
$683$	−506305.	−0.0415298	−0.0207649	+	0.999784i	$0.506610\pi$
$684$	0	0
$685$	−909221.	−0.0740360
$686$	0	0
$687$	0	0
$688$	−5.47671e6	−0.441112
$689$	3.59463e6	0.288473
$690$	0	0
$691$	−5.02613e6	−0.400441	−0.200221	−	0.979751i	$-0.564166\pi$
$691$	−5.02613e6	−0.400441	−0.200221	+	0.979751i	$0.564166\pi$
$692$	−1.19609e7	−0.949511
$693$	0	0
$694$	−7.05629e6	−0.556132
$695$	−1.32110e7	−1.03746
$696$	0	0
$697$	−5.53919e6	−0.431881
$698$	8.00885e6	0.622203
$699$	0	0
$700$	0	0
$701$	1.29711e7	0.996966	0.498483	−	0.866900i	$-0.333891\pi$
$701$	1.29711e7	0.996966	0.498483	+	0.866900i	$0.333891\pi$
$702$	0	0
$703$	464065.	0.0354153
$704$	−2.53271e6	−0.192599
$705$	0	0
$706$	−1.12755e7	−0.851381
$707$	0	0
$708$	0	0
$709$	5.96629e6	0.445747	0.222874	−	0.974847i	$-0.428456\pi$
$709$	5.96629e6	0.445747	0.222874	+	0.974847i	$0.428456\pi$
$710$	2.38924e7	1.77875
$711$	0	0
$712$	−5.79108e6	−0.428114
$713$	−9.67677e6	−0.712864
$714$	0	0
$715$	−6.25507e6	−0.457580
$716$	−1.44518e6	−0.105351
$717$	0	0
$718$	2.46459e6	0.178416
$719$	1.58476e7	1.14325	0.571624	−	0.820516i	$-0.306314\pi$
$719$	1.58476e7	1.14325	0.571624	+	0.820516i	$0.306314\pi$
$720$	0	0
$721$	0	0
$722$	9.87059e6	0.704693
$723$	0	0
$724$	−3.01771e6	−0.213959
$725$	1.64822e7	1.16459
$726$	0	0
$727$	−1.52350e7	−1.06907	−0.534534	−	0.845147i	$-0.679513\pi$
$727$	−1.52350e7	−1.06907	−0.534534	+	0.845147i	$0.679513\pi$
$728$	0	0
$729$	0	0
$730$	616518.	0.0428192
$731$	5.64561e6	0.390767
$732$	0	0
$733$	−7.58714e6	−0.521576	−0.260788	−	0.965396i	$-0.583982\pi$
$733$	−7.58714e6	−0.521576	−0.260788	+	0.965396i	$0.583982\pi$
$734$	6.02954e6	0.413089
$735$	0	0
$736$	−1.42509e6	−0.0969720
$737$	−6.37598e6	−0.432393
$738$	0	0
$739$	2.82397e6	0.190217	0.0951085	−	0.995467i	$-0.469680\pi$
$739$	2.82397e6	0.190217	0.0951085	+	0.995467i	$0.469680\pi$
$740$	6.26706e6	0.420712
$741$	0	0
$742$	0	0
$743$	−8.78322e6	−0.583689	−0.291845	−	0.956466i	$-0.594269\pi$
$743$	−8.78322e6	−0.583689	−0.291845	+	0.956466i	$0.594269\pi$
$744$	0	0
$745$	1.31522e7	0.868176
$746$	2.10207e6	0.138293
$747$	0	0
$748$	2.61082e6	0.170617
$749$	0	0
$750$	0	0
$751$	−9.94029e6	−0.643131	−0.321565	−	0.946887i	$-0.604209\pi$
$751$	−9.94029e6	−0.643131	−0.321565	+	0.946887i	$0.604209\pi$
$752$	−5.08711e6	−0.328040
$753$	0	0
$754$	−2.96832e6	−0.190144
$755$	1.87750e7	1.19871
$756$	0	0
$757$	2.43882e7	1.54682	0.773411	−	0.633905i	$-0.218549\pi$
$757$	2.43882e7	1.54682	0.773411	+	0.633905i	$0.218549\pi$
$758$	390253.	0.0246702
$759$	0	0
$760$	−456520.	−0.0286699
$761$	−1.38814e7	−0.868906	−0.434453	−	0.900694i	$-0.643058\pi$
$761$	−1.38814e7	−0.868906	−0.434453	+	0.900694i	$0.643058\pi$
$762$	0	0
$763$	0	0
$764$	1.07286e7	0.664981
$765$	0	0
$766$	−2.00842e7	−1.23675
$767$	−5.67600e6	−0.348380
$768$	0	0
$769$	−5.94704e6	−0.362648	−0.181324	−	0.983423i	$-0.558038\pi$
$769$	−5.94704e6	−0.362648	−0.181324	+	0.983423i	$0.558038\pi$
$770$	0	0
$771$	0	0
$772$	−1.53126e7	−0.924708
$773$	−1.88923e7	−1.13720	−0.568599	−	0.822615i	$-0.692514\pi$
$773$	−1.88923e7	−1.13720	−0.568599	+	0.822615i	$0.692514\pi$
$774$	0	0
$775$	−2.01344e7	−1.20416
$776$	8.81125e6	0.525271
$777$	0	0
$778$	1.82619e7	1.08168
$779$	1.92962e6	0.113928
$780$	0	0
$781$	4.75995e7	2.79238
$782$	1.46904e6	0.0859044
$783$	0	0
$784$	0	0
$785$	2.00994e7	1.16415
$786$	0	0
$787$	1.71518e7	0.987125	0.493563	−	0.869710i	$-0.335694\pi$
$787$	1.71518e7	0.987125	0.493563	+	0.869710i	$0.335694\pi$
$788$	−774410.	−0.0444279
$789$	0	0
$790$	1.67124e6	0.0952732
$791$	0	0
$792$	0	0
$793$	−4.42472e6	−0.249863
$794$	−1.96001e7	−1.10334
$795$	0	0
$796$	−1.21873e7	−0.681751
$797$	1.22205e7	0.681462	0.340731	−	0.940161i	$-0.389325\pi$
$797$	1.22205e7	0.681462	0.340731	+	0.940161i	$0.389325\pi$
$798$	0	0
$799$	5.24400e6	0.290600
$800$	−2.96517e6	−0.163804
$801$	0	0
$802$	−6.33510e6	−0.347790
$803$	1.22825e6	0.0672202
$804$	0	0
$805$	0	0
$806$	3.62605e6	0.196606
$807$	0	0
$808$	8.82830e6	0.475717
$809$	−2.46850e7	−1.32605	−0.663027	−	0.748595i	$-0.730729\pi$
$809$	−2.46850e7	−1.32605	−0.663027	+	0.748595i	$0.730729\pi$
$810$	0	0
$811$	−3.18910e7	−1.70262	−0.851308	−	0.524667i	$-0.824190\pi$
$811$	−3.18910e7	−1.70262	−0.851308	+	0.524667i	$0.824190\pi$
$812$	0	0
$813$	0	0
$814$	1.24855e7	0.660458
$815$	−2.44308e7	−1.28838
$816$	0	0
$817$	−1.96670e6	−0.103082
$818$	−6.79215e6	−0.354914
$819$	0	0
$820$	2.60590e7	1.35339
$821$	1.71012e7	0.885460	0.442730	−	0.896655i	$-0.354010\pi$
$821$	1.71012e7	0.885460	0.442730	+	0.896655i	$0.354010\pi$
$822$	0	0
$823$	−295105.	−0.0151872	−0.00759358	−	0.999971i	$-0.502417\pi$
$823$	−295105.	−0.0151872	−0.00759358	+	0.999971i	$0.502417\pi$
$824$	−736776.	−0.0378022
$825$	0	0
$826$	0	0
$827$	1.30891e6	0.0665496	0.0332748	−	0.999446i	$-0.489406\pi$
$827$	1.30891e6	0.0665496	0.0332748	+	0.999446i	$0.489406\pi$
$828$	0	0
$829$	8.17046e6	0.412914	0.206457	−	0.978456i	$-0.433807\pi$
$829$	8.17046e6	0.412914	0.206457	+	0.978456i	$0.433807\pi$
$830$	−2.14061e7	−1.07856
$831$	0	0
$832$	534004.	0.0267446
$833$	0	0
$834$	0	0
$835$	−6.52535e6	−0.323882
$836$	−909500.	−0.0450077
$837$	0	0
$838$	−2.22078e7	−1.09243
$839$	2.22357e7	1.09055	0.545276	−	0.838256i	$-0.316425\pi$
$839$	2.22357e7	1.09055	0.545276	+	0.838256i	$0.316425\pi$
$840$	0	0
$841$	1.18880e7	0.579586
$842$	−4.86099e6	−0.236290
$843$	0	0
$844$	−1.11713e7	−0.539818
$845$	−2.74909e7	−1.32449
$846$	0	0
$847$	0	0
$848$	7.05845e6	0.337070
$849$	0	0
$850$	3.05662e6	0.145109
$851$	7.02526e6	0.332536
$852$	0	0
$853$	2.33553e7	1.09904	0.549518	−	0.835482i	$-0.314811\pi$
$853$	2.33553e7	1.09904	0.549518	+	0.835482i	$0.314811\pi$
$854$	0	0
$855$	0	0
$856$	−6.29788e6	−0.293772
$857$	−2.14845e7	−0.999249	−0.499625	−	0.866242i	$-0.666529\pi$
$857$	−2.14845e7	−0.999249	−0.499625	+	0.866242i	$0.666529\pi$
$858$	0	0
$859$	659584.	0.0304991	0.0152496	−	0.999884i	$-0.495146\pi$
$859$	659584.	0.0304991	0.0152496	+	0.999884i	$0.495146\pi$
$860$	−2.65596e7	−1.22455
$861$	0	0
$862$	5.30120e6	0.243000
$863$	−2.90949e6	−0.132981	−0.0664905	−	0.997787i	$-0.521180\pi$
$863$	−2.90949e6	−0.132981	−0.0664905	+	0.997787i	$0.521180\pi$
$864$	0	0
$865$	−5.80053e7	−2.63589
$866$	2.19258e7	0.993483
$867$	0	0
$868$	0	0
$869$	3.32952e6	0.149566
$870$	0	0
$871$	1.34433e6	0.0600428
$872$	1.07222e7	0.477520
$873$	0	0
$874$	−511751.	−0.0226610
$875$	0	0
$876$	0	0
$877$	−866950.	−0.0380623	−0.0190311	−	0.999819i	$-0.506058\pi$
$877$	−866950.	−0.0380623	−0.0190311	+	0.999819i	$0.506058\pi$
$878$	2.42554e7	1.06187
$879$	0	0
$880$	−1.22825e7	−0.534664
$881$	2.38015e7	1.03315	0.516577	−	0.856240i	$-0.327206\pi$
$881$	2.38015e7	1.03315	0.516577	+	0.856240i	$0.327206\pi$
$882$	0	0
$883$	−2.59761e7	−1.12117	−0.560586	−	0.828096i	$-0.689424\pi$
$883$	−2.59761e7	−1.12117	−0.560586	+	0.828096i	$0.689424\pi$
$884$	−550472.	−0.0236922
$885$	0	0
$886$	7.31306e6	0.312979
$887$	3.04811e7	1.30084	0.650418	−	0.759577i	$-0.274594\pi$
$887$	3.04811e7	1.30084	0.650418	+	0.759577i	$0.274594\pi$
$888$	0	0
$889$	0	0
$890$	−2.80842e7	−1.18847
$891$	0	0
$892$	2.00644e7	0.844332
$893$	−1.82679e6	−0.0766584
$894$	0	0
$895$	−7.00848e6	−0.292460
$896$	0	0
$897$	0	0
$898$	3.28173e7	1.35804
$899$	−3.95782e7	−1.63327
$900$	0	0
$901$	−7.27613e6	−0.298599
$902$	5.19159e7	2.12463
$903$	0	0
$904$	−5.20407e6	−0.211798
$905$	−1.46346e7	−0.593962
$906$	0	0
$907$	−1.12185e7	−0.452810	−0.226405	−	0.974033i	$-0.572697\pi$
$907$	−1.12185e7	−0.452810	−0.226405	+	0.974033i	$0.572697\pi$
$908$	−1.27954e7	−0.515038
$909$	0	0
$910$	0	0
$911$	4.12952e7	1.64855	0.824277	−	0.566187i	$-0.191582\pi$
$911$	4.12952e7	1.64855	0.824277	+	0.566187i	$0.191582\pi$
$912$	0	0
$913$	−4.26463e7	−1.69318
$914$	1.42831e7	0.565533
$915$	0	0
$916$	1.40170e7	0.551971
$917$	0	0
$918$	0	0
$919$	−2.50746e7	−0.979365	−0.489683	−	0.871901i	$-0.662887\pi$
$919$	−2.50746e7	−0.979365	−0.489683	+	0.871901i	$0.662887\pi$
$920$	−6.91104e6	−0.269199
$921$	0	0
$922$	2.25581e7	0.873926
$923$	−1.00360e7	−0.387755
$924$	0	0
$925$	1.46174e7	0.561716
$926$	−3.18489e6	−0.122058
$927$	0	0
$928$	−5.82863e6	−0.222176
$929$	−3.58109e7	−1.36137	−0.680684	−	0.732577i	$-0.738317\pi$
$929$	−3.58109e7	−1.36137	−0.680684	+	0.732577i	$0.738317\pi$
$930$	0	0
$931$	0	0
$932$	−5.67051e6	−0.213837
$933$	0	0
$934$	−533807.	−0.0200225
$935$	1.26613e7	0.473642
$936$	0	0
$937$	2.87422e7	1.06948	0.534738	−	0.845018i	$-0.320410\pi$
$937$	2.87422e7	1.06948	0.534738	+	0.845018i	$0.320410\pi$
$938$	0	0
$939$	0	0
$940$	−2.46703e7	−0.910656
$941$	6.80282e6	0.250446	0.125223	−	0.992129i	$-0.460035\pi$
$941$	6.80282e6	0.250446	0.125223	+	0.992129i	$0.460035\pi$
$942$	0	0
$943$	2.92116e7	1.06974
$944$	−1.11455e7	−0.407069
$945$	0	0
$946$	−5.29133e7	−1.92237
$947$	−4.21262e7	−1.52643	−0.763216	−	0.646143i	$-0.776381\pi$
$947$	−4.21262e7	−1.52643	−0.763216	+	0.646143i	$0.776381\pi$
$948$	0	0
$949$	−258969.	−0.00933430
$950$	−1.06480e6	−0.0382788
$951$	0	0
$952$	0	0
$953$	−1.31637e7	−0.469512	−0.234756	−	0.972054i	$-0.575429\pi$
$953$	−1.31637e7	−0.469512	−0.234756	+	0.972054i	$0.575429\pi$
$954$	0	0
$955$	5.20289e7	1.84602
$956$	−1.41343e7	−0.500183
$957$	0	0
$958$	1.79079e7	0.630421
$959$	0	0
$960$	0	0
$961$	1.97189e7	0.688771
$962$	−2.63248e6	−0.0917124
$963$	0	0
$964$	9.89810e6	0.343051
$965$	−7.42592e7	−2.56704
$966$	0	0
$967$	−3.62355e7	−1.24614	−0.623071	−	0.782165i	$-0.714115\pi$
$967$	−3.62355e7	−1.24614	−0.623071	+	0.782165i	$0.714115\pi$
$968$	−1.41625e7	−0.485795
$969$	0	0
$970$	4.27307e7	1.45818
$971$	4.02139e7	1.36876	0.684380	−	0.729125i	$-0.260073\pi$
$971$	4.02139e7	1.36876	0.684380	+	0.729125i	$0.260073\pi$
$972$	0	0
$973$	0	0
$974$	−1.62253e7	−0.548020
$975$	0	0
$976$	−8.68843e6	−0.291956
$977$	−4.98719e7	−1.67155	−0.835776	−	0.549071i	$-0.814982\pi$
$977$	−4.98719e7	−1.67155	−0.835776	+	0.549071i	$0.814982\pi$
$978$	0	0
$979$	−5.59506e7	−1.86573
$980$	0	0
$981$	0	0
$982$	−1.20371e6	−0.0398331
$983$	2.46828e7	0.814723	0.407361	−	0.913267i	$-0.366449\pi$
$983$	2.46828e7	0.814723	0.407361	+	0.913267i	$0.366449\pi$
$984$	0	0
$985$	−3.75555e6	−0.123334
$986$	6.00839e6	0.196818
$987$	0	0
$988$	191762.	0.00624985
$989$	−2.97729e7	−0.967899
$990$	0	0
$991$	−5.73487e7	−1.85498	−0.927491	−	0.373847i	$-0.878039\pi$
$991$	−5.73487e7	−1.85498	−0.927491	+	0.373847i	$0.878039\pi$
$992$	7.12016e6	0.229726
$993$	0	0
$994$	0	0
$995$	−5.91032e7	−1.89258
$996$	0	0
$997$	−5.73578e7	−1.82749	−0.913744	−	0.406291i	$-0.866822\pi$
$997$	−5.73578e7	−1.82749	−0.913744	+	0.406291i	$0.866822\pi$
$998$	−1.70655e7	−0.542368
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bg.1.2		2
3.2	odd	2		294.6.a.s.1.1	✓	2
7.6	odd	2		882.6.a.bc.1.1		2
21.2	odd	6		294.6.e.w.67.2		4
21.5	even	6		294.6.e.t.67.1		4
21.11	odd	6		294.6.e.w.79.2		4
21.17	even	6		294.6.e.t.79.1		4
21.20	even	2		294.6.a.v.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.6.a.s.1.1	✓	2	3.2	odd	2
294.6.a.v.1.2	yes	2	21.20	even	2
294.6.e.t.67.1		4	21.5	even	6
294.6.e.t.79.1		4	21.17	even	6
294.6.e.w.67.2		4	21.2	odd	6
294.6.e.w.79.2		4	21.11	odd	6
882.6.a.bc.1.1		2	7.6	odd	2
882.6.a.bg.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +77.5930 q^{5} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} +77.5930 q^{5} -64.0000 q^{8} -310.372 q^{10} -618.337 q^{11} +130.372 q^{13} +256.000 q^{16} -263.895 q^{17} +91.9300 q^{19} +1241.49 q^{20} +2473.35 q^{22} +1391.69 q^{23} +2895.67 q^{25} -521.488 q^{26} +5692.02 q^{29} -6953.28 q^{31} -1024.00 q^{32} +1055.58 q^{34} +5048.02 q^{37} -367.720 q^{38} -4965.95 q^{40} +20990.1 q^{41} -21393.4 q^{43} -9893.39 q^{44} -5566.74 q^{46} -19871.5 q^{47} -11582.7 q^{50} +2085.95 q^{52} +27572.1 q^{53} -47978.6 q^{55} -22768.1 q^{58} -43536.9 q^{59} -33939.2 q^{61} +27813.1 q^{62} +4096.00 q^{64} +10116.0 q^{65} +10311.5 q^{67} -4222.32 q^{68} -76979.9 q^{71} -1986.38 q^{73} -20192.1 q^{74} +1470.88 q^{76} -5384.63 q^{79} +19863.8 q^{80} -83960.5 q^{82} +68969.3 q^{83} -20476.4 q^{85} +85573.6 q^{86} +39573.6 q^{88} +90485.6 q^{89} +22267.0 q^{92} +79486.1 q^{94} +7133.13 q^{95} -137676. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} + 18 q^{5} - 128 q^{8}+O(q^{10}) \) 2 * q - 8 * q^2 + 32 * q^4 + 18 * q^5 - 128 * q^8	\( 2 q - 8 q^{2} + 32 q^{4} + 18 q^{5} - 128 q^{8} - 72 q^{10} - 2 q^{11} - 288 q^{13} + 512 q^{16} + 1530 q^{17} - 1188 q^{19} + 288 q^{20} + 8 q^{22} - 3390 q^{23} + 3322 q^{25} + 1152 q^{26} + 3976 q^{29} - 7596 q^{31} - 2048 q^{32} - 6120 q^{34} + 2688 q^{37} + 4752 q^{38} - 1152 q^{40} + 36630 q^{41} - 23032 q^{43} - 32 q^{44} + 13560 q^{46} + 864 q^{47} - 13288 q^{50} - 4608 q^{52} + 32920 q^{53} - 84708 q^{55} - 15904 q^{58} - 26712 q^{59} - 20412 q^{61} + 30384 q^{62} + 8192 q^{64} + 35048 q^{65} - 36172 q^{67} + 24480 q^{68} - 73706 q^{71} + 74772 q^{73} - 10752 q^{74} - 19008 q^{76} - 23116 q^{79} + 4608 q^{80} - 146520 q^{82} + 147816 q^{83} - 127380 q^{85} + 92128 q^{86} + 128 q^{88} + 164646 q^{89} - 54240 q^{92} - 3456 q^{94} + 83408 q^{95} - 162036 q^{97}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 + 18 * q^5 - 128 * q^8 - 72 * q^10 - 2 * q^11 - 288 * q^13 + 512 * q^16 + 1530 * q^17 - 1188 * q^19 + 288 * q^20 + 8 * q^22 - 3390 * q^23 + 3322 * q^25 + 1152 * q^26 + 3976 * q^29 - 7596 * q^31 - 2048 * q^32 - 6120 * q^34 + 2688 * q^37 + 4752 * q^38 - 1152 * q^40 + 36630 * q^41 - 23032 * q^43 - 32 * q^44 + 13560 * q^46 + 864 * q^47 - 13288 * q^50 - 4608 * q^52 + 32920 * q^53 - 84708 * q^55 - 15904 * q^58 - 26712 * q^59 - 20412 * q^61 + 30384 * q^62 + 8192 * q^64 + 35048 * q^65 - 36172 * q^67 + 24480 * q^68 - 73706 * q^71 + 74772 * q^73 - 10752 * q^74 - 19008 * q^76 - 23116 * q^79 + 4608 * q^80 - 146520 * q^82 + 147816 * q^83 - 127380 * q^85 + 92128 * q^86 + 128 * q^88 + 164646 * q^89 - 54240 * q^92 - 3456 * q^94 + 83408 * q^95 - 162036 * q^97

Introduction

L-functions

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