LMFDB - Embedded newform 1104.6.a.u.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 882x^{4} + 3882x^{3} + 177571x^{2} - 926456x - 1499874 $ x^6 - 2x^5 - 882x^4 + 3882x^3 + 177571x^2 - 926456*x - 1499874
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{3}\cdot 5 $
Twist minimal:	no (minimal twist has level 552)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.30472$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.00000 q^{3} +54.5639 q^{5} -101.630 q^{7} +81.0000 q^{9} +O(q^{10})$	$q+9.00000 q^{3} +54.5639 q^{5} -101.630 q^{7} +81.0000 q^{9} +47.2374 q^{11} +346.934 q^{13} +491.075 q^{15} -2088.79 q^{17} +655.254 q^{19} -914.672 q^{21} +529.000 q^{23} -147.779 q^{25} +729.000 q^{27} -3096.35 q^{29} +5702.36 q^{31} +425.136 q^{33} -5545.34 q^{35} +9196.94 q^{37} +3122.41 q^{39} -16791.9 q^{41} -15115.5 q^{43} +4419.68 q^{45} -6384.59 q^{47} -6478.31 q^{49} -18799.1 q^{51} -4179.63 q^{53} +2577.46 q^{55} +5897.29 q^{57} +11304.0 q^{59} -3907.89 q^{61} -8232.04 q^{63} +18930.1 q^{65} +32512.7 q^{67} +4761.00 q^{69} +4079.07 q^{71} -1902.15 q^{73} -1330.01 q^{75} -4800.74 q^{77} -17446.3 q^{79} +6561.00 q^{81} -82727.3 q^{83} -113973. q^{85} -27867.1 q^{87} +57311.9 q^{89} -35259.0 q^{91} +51321.2 q^{93} +35753.2 q^{95} +796.998 q^{97} +3826.23 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 54 q^{3} - 70 q^{5} - 144 q^{7} + 486 q^{9}+O(q^{10}) $ 6 * q + 54 * q^3 - 70 * q^5 - 144 * q^7 + 486 * q^9	$ 6 q + 54 q^{3} - 70 q^{5} - 144 q^{7} + 486 q^{9} + 118 q^{11} - 692 q^{13} - 630 q^{15} - 1200 q^{17} - 470 q^{19} - 1296 q^{21} + 3174 q^{23} + 6586 q^{25} + 4374 q^{27} + 2728 q^{29} - 6540 q^{31} + 1062 q^{33} - 6208 q^{35} + 7246 q^{37} - 6228 q^{39} + 13396 q^{41} - 13394 q^{43} - 5670 q^{45} - 35996 q^{47} - 2334 q^{49} - 10800 q^{51} + 6274 q^{53} - 44164 q^{55} - 4230 q^{57} - 3756 q^{59} - 4526 q^{61} - 11664 q^{63} - 33828 q^{65} - 9646 q^{67} + 28566 q^{69} + 4912 q^{71} - 29352 q^{73} + 59274 q^{75} - 146060 q^{77} - 17608 q^{79} + 39366 q^{81} + 175218 q^{83} - 99652 q^{85} + 24552 q^{87} - 157740 q^{89} + 62512 q^{91} - 58860 q^{93} + 215644 q^{95} - 150008 q^{97} + 9558 q^{99}+O(q^{100}) $ 6 * q + 54 * q^3 - 70 * q^5 - 144 * q^7 + 486 * q^9 + 118 * q^11 - 692 * q^13 - 630 * q^15 - 1200 * q^17 - 470 * q^19 - 1296 * q^21 + 3174 * q^23 + 6586 * q^25 + 4374 * q^27 + 2728 * q^29 - 6540 * q^31 + 1062 * q^33 - 6208 * q^35 + 7246 * q^37 - 6228 * q^39 + 13396 * q^41 - 13394 * q^43 - 5670 * q^45 - 35996 * q^47 - 2334 * q^49 - 10800 * q^51 + 6274 * q^53 - 44164 * q^55 - 4230 * q^57 - 3756 * q^59 - 4526 * q^61 - 11664 * q^63 - 33828 * q^65 - 9646 * q^67 + 28566 * q^69 + 4912 * q^71 - 29352 * q^73 + 59274 * q^75 - 146060 * q^77 - 17608 * q^79 + 39366 * q^81 + 175218 * q^83 - 99652 * q^85 + 24552 * q^87 - 157740 * q^89 + 62512 * q^91 - 58860 * q^93 + 215644 * q^95 - 150008 * q^97 + 9558 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	54.5639	0.976069	0.488035	−	0.872824i	$-0.337714\pi$
$5$	54.5639	0.976069	0.488035	+	0.872824i	$0.337714\pi$
$6$	0	0
$7$	−101.630	−0.783931	−0.391965	−	0.919980i	$-0.628205\pi$
$7$	−101.630	−0.783931	−0.391965	+	0.919980i	$0.628205\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	47.2374	0.117707	0.0588537	−	0.998267i	$-0.481255\pi$
$11$	47.2374	0.117707	0.0588537	+	0.998267i	$0.481255\pi$
$12$	0	0
$13$	346.934	0.569362	0.284681	−	0.958622i	$-0.408112\pi$
$13$	346.934	0.569362	0.284681	+	0.958622i	$0.408112\pi$
$14$	0	0
$15$	491.075	0.563534
$16$	0	0
$17$	−2088.79	−1.75297	−0.876483	−	0.481434i	$-0.840116\pi$
$17$	−2088.79	−1.75297	−0.876483	+	0.481434i	$0.840116\pi$
$18$	0	0
$19$	655.254	0.416414	0.208207	−	0.978085i	$-0.433237\pi$
$19$	655.254	0.416414	0.208207	+	0.978085i	$0.433237\pi$
$20$	0	0
$21$	−914.672	−0.452603
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−147.779	−0.0472892
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−3096.35	−0.683683	−0.341841	−	0.939758i	$-0.611051\pi$
$29$	−3096.35	−0.683683	−0.341841	+	0.939758i	$0.611051\pi$
$30$	0	0
$31$	5702.36	1.06574	0.532869	−	0.846198i	$-0.321114\pi$
$31$	5702.36	1.06574	0.532869	+	0.846198i	$0.321114\pi$
$32$	0	0
$33$	425.136	0.0679584
$34$	0	0
$35$	−5545.34	−0.765170
$36$	0	0
$37$	9196.94	1.10443	0.552216	−	0.833701i	$-0.313782\pi$
$37$	9196.94	1.10443	0.552216	+	0.833701i	$0.313782\pi$
$38$	0	0
$39$	3122.41	0.328722
$40$	0	0
$41$	−16791.9	−1.56006	−0.780029	−	0.625744i	$-0.784796\pi$
$41$	−16791.9	−1.56006	−0.780029	+	0.625744i	$0.784796\pi$
$42$	0	0
$43$	−15115.5	−1.24667	−0.623334	−	0.781956i	$-0.714222\pi$
$43$	−15115.5	−1.24667	−0.623334	+	0.781956i	$0.714222\pi$
$44$	0	0
$45$	4419.68	0.325356
$46$	0	0
$47$	−6384.59	−0.421588	−0.210794	−	0.977530i	$-0.567605\pi$
$47$	−6384.59	−0.421588	−0.210794	+	0.977530i	$0.567605\pi$
$48$	0	0
$49$	−6478.31	−0.385453
$50$	0	0
$51$	−18799.1	−1.01207
$52$	0	0
$53$	−4179.63	−0.204384	−0.102192	−	0.994765i	$-0.532586\pi$
$53$	−4179.63	−0.204384	−0.102192	+	0.994765i	$0.532586\pi$
$54$	0	0
$55$	2577.46	0.114891
$56$	0	0
$57$	5897.29	0.240417
$58$	0	0
$59$	11304.0	0.422769	0.211384	−	0.977403i	$-0.432203\pi$
$59$	11304.0	0.422769	0.211384	+	0.977403i	$0.432203\pi$
$60$	0	0
$61$	−3907.89	−0.134468	−0.0672338	−	0.997737i	$-0.521417\pi$
$61$	−3907.89	−0.134468	−0.0672338	+	0.997737i	$0.521417\pi$
$62$	0	0
$63$	−8232.04	−0.261310
$64$	0	0
$65$	18930.1	0.555737
$66$	0	0
$67$	32512.7	0.884842	0.442421	−	0.896807i	$-0.354120\pi$
$67$	32512.7	0.884842	0.442421	+	0.896807i	$0.354120\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	4079.07	0.0960318	0.0480159	−	0.998847i	$-0.484710\pi$
$71$	4079.07	0.0960318	0.0480159	+	0.998847i	$0.484710\pi$
$72$	0	0
$73$	−1902.15	−0.0417770	−0.0208885	−	0.999782i	$-0.506650\pi$
$73$	−1902.15	−0.0417770	−0.0208885	+	0.999782i	$0.506650\pi$
$74$	0	0
$75$	−1330.01	−0.0273024
$76$	0	0
$77$	−4800.74	−0.0922745
$78$	0	0
$79$	−17446.3	−0.314511	−0.157255	−	0.987558i	$-0.550265\pi$
$79$	−17446.3	−0.314511	−0.157255	+	0.987558i	$0.550265\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−82727.3	−1.31812	−0.659058	−	0.752092i	$-0.729045\pi$
$83$	−82727.3	−1.31812	−0.659058	+	0.752092i	$0.729045\pi$
$84$	0	0
$85$	−113973.	−1.71102
$86$	0	0
$87$	−27867.1	−0.394724
$88$	0	0
$89$	57311.9	0.766955	0.383477	−	0.923550i	$-0.374726\pi$
$89$	57311.9	0.766955	0.383477	+	0.923550i	$0.374726\pi$
$90$	0	0
$91$	−35259.0	−0.446341
$92$	0	0
$93$	51321.2	0.615304
$94$	0	0
$95$	35753.2	0.406449
$96$	0	0
$97$	796.998	0.00860059	0.00430029	−	0.999991i	$-0.498631\pi$
$97$	796.998	0.00860059	0.00430029	+	0.999991i	$0.498631\pi$
$98$	0	0
$99$	3826.23	0.0392358
$100$	0	0
$101$	−106605.	−1.03986	−0.519929	−	0.854209i	$-0.674042\pi$
$101$	−106605.	−1.03986	−0.519929	+	0.854209i	$0.674042\pi$
$102$	0	0
$103$	−16813.5	−0.156159	−0.0780794	−	0.996947i	$-0.524879\pi$
$103$	−16813.5	−0.156159	−0.0780794	+	0.996947i	$0.524879\pi$
$104$	0	0
$105$	−49908.1	−0.441771
$106$	0	0
$107$	−2278.20	−0.0192368	−0.00961841	−	0.999954i	$-0.503062\pi$
$107$	−2278.20	−0.0192368	−0.00961841	+	0.999954i	$0.503062\pi$
$108$	0	0
$109$	−175163.	−1.41213	−0.706066	−	0.708146i	$-0.749532\pi$
$109$	−175163.	−1.41213	−0.706066	+	0.708146i	$0.749532\pi$
$110$	0	0
$111$	82772.4	0.637644
$112$	0	0
$113$	−191233.	−1.40886	−0.704429	−	0.709775i	$-0.748797\pi$
$113$	−191233.	−1.40886	−0.704429	+	0.709775i	$0.748797\pi$
$114$	0	0
$115$	28864.3	0.203524
$116$	0	0
$117$	28101.7	0.189787
$118$	0	0
$119$	212285.	1.37420
$120$	0	0
$121$	−158820.	−0.986145
$122$	0	0
$123$	−151127.	−0.900700
$124$	0	0
$125$	−178576.	−1.02223
$126$	0	0
$127$	50151.5	0.275915	0.137957	−	0.990438i	$-0.455946\pi$
$127$	50151.5	0.275915	0.137957	+	0.990438i	$0.455946\pi$
$128$	0	0
$129$	−136039.	−0.719764
$130$	0	0
$131$	−45626.0	−0.232292	−0.116146	−	0.993232i	$-0.537054\pi$
$131$	−45626.0	−0.232292	−0.116146	+	0.993232i	$0.537054\pi$
$132$	0	0
$133$	−66593.6	−0.326440
$134$	0	0
$135$	39777.1	0.187845
$136$	0	0
$137$	198119.	0.901830	0.450915	−	0.892567i	$-0.351098\pi$
$137$	198119.	0.901830	0.450915	+	0.892567i	$0.351098\pi$
$138$	0	0
$139$	11174.1	0.0490543	0.0245271	−	0.999699i	$-0.492192\pi$
$139$	11174.1	0.0490543	0.0245271	+	0.999699i	$0.492192\pi$
$140$	0	0
$141$	−57461.3	−0.243404
$142$	0	0
$143$	16388.3	0.0670182
$144$	0	0
$145$	−168949.	−0.667322
$146$	0	0
$147$	−58304.8	−0.222541
$148$	0	0
$149$	41592.8	0.153480	0.0767402	−	0.997051i	$-0.475549\pi$
$149$	41592.8	0.153480	0.0767402	+	0.997051i	$0.475549\pi$
$150$	0	0
$151$	130094.	0.464317	0.232158	−	0.972678i	$-0.425421\pi$
$151$	130094.	0.464317	0.232158	+	0.972678i	$0.425421\pi$
$152$	0	0
$153$	−169192.	−0.584322
$154$	0	0
$155$	311143.	1.04023
$156$	0	0
$157$	−352220.	−1.14042	−0.570209	−	0.821499i	$-0.693138\pi$
$157$	−352220.	−1.14042	−0.570209	+	0.821499i	$0.693138\pi$
$158$	0	0
$159$	−37616.6	−0.118001
$160$	0	0
$161$	−53762.4	−0.163461
$162$	0	0
$163$	−147069.	−0.433564	−0.216782	−	0.976220i	$-0.569556\pi$
$163$	−147069.	−0.433564	−0.216782	+	0.976220i	$0.569556\pi$
$164$	0	0
$165$	23197.1	0.0663321
$166$	0	0
$167$	287767.	0.798454	0.399227	−	0.916852i	$-0.369278\pi$
$167$	287767.	0.798454	0.399227	+	0.916852i	$0.369278\pi$
$168$	0	0
$169$	−250930.	−0.675826
$170$	0	0
$171$	53075.6	0.138805
$172$	0	0
$173$	200425.	0.509140	0.254570	−	0.967054i	$-0.418066\pi$
$173$	200425.	0.509140	0.254570	+	0.967054i	$0.418066\pi$
$174$	0	0
$175$	15018.8	0.0370715
$176$	0	0
$177$	101736.	0.244086
$178$	0	0
$179$	589538.	1.37524	0.687621	−	0.726070i	$-0.258655\pi$
$179$	589538.	1.37524	0.687621	+	0.726070i	$0.258655\pi$
$180$	0	0
$181$	63147.2	0.143271	0.0716354	−	0.997431i	$-0.477178\pi$
$181$	63147.2	0.143271	0.0716354	+	0.997431i	$0.477178\pi$
$182$	0	0
$183$	−35171.0	−0.0776349
$184$	0	0
$185$	501821.	1.07800
$186$	0	0
$187$	−98669.1	−0.206337
$188$	0	0
$189$	−74088.4	−0.150868
$190$	0	0
$191$	437404.	0.867561	0.433780	−	0.901019i	$-0.357179\pi$
$191$	437404.	0.867561	0.433780	+	0.901019i	$0.357179\pi$
$192$	0	0
$193$	115797.	0.223772	0.111886	−	0.993721i	$-0.464311\pi$
$193$	115797.	0.223772	0.111886	+	0.993721i	$0.464311\pi$
$194$	0	0
$195$	170371.	0.320855
$196$	0	0
$197$	−674946.	−1.23909	−0.619546	−	0.784960i	$-0.712683\pi$
$197$	−674946.	−1.23909	−0.619546	+	0.784960i	$0.712683\pi$
$198$	0	0
$199$	−124153.	−0.222240	−0.111120	−	0.993807i	$-0.535444\pi$
$199$	−124153.	−0.222240	−0.111120	+	0.993807i	$0.535444\pi$
$200$	0	0
$201$	292614.	0.510864
$202$	0	0
$203$	314682.	0.535960
$204$	0	0
$205$	−916233.	−1.52272
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	30952.5	0.0490151
$210$	0	0
$211$	135958.	0.210232	0.105116	−	0.994460i	$-0.466479\pi$
$211$	135958.	0.210232	0.105116	+	0.994460i	$0.466479\pi$
$212$	0	0
$213$	36711.6	0.0554440
$214$	0	0
$215$	−824760.	−1.21683
$216$	0	0
$217$	−579532.	−0.835464
$218$	0	0
$219$	−17119.4	−0.0241200
$220$	0	0
$221$	−724674.	−0.998072
$222$	0	0
$223$	−567236.	−0.763839	−0.381919	−	0.924196i	$-0.624737\pi$
$223$	−567236.	−0.763839	−0.381919	+	0.924196i	$0.624737\pi$
$224$	0	0
$225$	−11970.1	−0.0157631
$226$	0	0
$227$	1.08595e6	1.39876	0.699382	−	0.714748i	$-0.253459\pi$
$227$	1.08595e6	1.39876	0.699382	+	0.714748i	$0.253459\pi$
$228$	0	0
$229$	886602.	1.11722	0.558612	−	0.829429i	$-0.311334\pi$
$229$	886602.	1.11722	0.558612	+	0.829429i	$0.311334\pi$
$230$	0	0
$231$	−43206.7	−0.0532747
$232$	0	0
$233$	220333.	0.265882	0.132941	−	0.991124i	$-0.457558\pi$
$233$	220333.	0.265882	0.132941	+	0.991124i	$0.457558\pi$
$234$	0	0
$235$	−348368.	−0.411499
$236$	0	0
$237$	−157017.	−0.181583
$238$	0	0
$239$	−1.44167e6	−1.63257	−0.816285	−	0.577649i	$-0.803970\pi$
$239$	−1.44167e6	−1.63257	−0.816285	+	0.577649i	$0.803970\pi$
$240$	0	0
$241$	983695.	1.09098	0.545491	−	0.838116i	$-0.316343\pi$
$241$	983695.	1.09098	0.545491	+	0.838116i	$0.316343\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	−353482.	−0.376229
$246$	0	0
$247$	227330.	0.237091
$248$	0	0
$249$	−744546.	−0.761015
$250$	0	0
$251$	−1.87361e6	−1.87713	−0.938566	−	0.345100i	$-0.887845\pi$
$251$	−1.87361e6	−1.87713	−0.938566	+	0.345100i	$0.887845\pi$
$252$	0	0
$253$	24988.6	0.0245437
$254$	0	0
$255$	−1.02576e6	−0.987855
$256$	0	0
$257$	274903.	0.259625	0.129813	−	0.991539i	$-0.458562\pi$
$257$	274903.	0.259625	0.129813	+	0.991539i	$0.458562\pi$
$258$	0	0
$259$	−934686.	−0.865798
$260$	0	0
$261$	−250804.	−0.227894
$262$	0	0
$263$	1.36474e6	1.21663	0.608316	−	0.793695i	$-0.291845\pi$
$263$	1.36474e6	1.21663	0.608316	+	0.793695i	$0.291845\pi$
$264$	0	0
$265$	−228057.	−0.199493
$266$	0	0
$267$	515807.	0.442802
$268$	0	0
$269$	−754763.	−0.635960	−0.317980	−	0.948097i	$-0.603005\pi$
$269$	−754763.	−0.635960	−0.317980	+	0.948097i	$0.603005\pi$
$270$	0	0
$271$	541736.	0.448089	0.224045	−	0.974579i	$-0.428074\pi$
$271$	541736.	0.448089	0.224045	+	0.974579i	$0.428074\pi$
$272$	0	0
$273$	−317331.	−0.257695
$274$	0	0
$275$	−6980.68	−0.00556629
$276$	0	0
$277$	−730028.	−0.571663	−0.285831	−	0.958280i	$-0.592270\pi$
$277$	−730028.	−0.571663	−0.285831	+	0.958280i	$0.592270\pi$
$278$	0	0
$279$	461891.	0.355246
$280$	0	0
$281$	−2.15313e6	−1.62669	−0.813346	−	0.581781i	$-0.802356\pi$
$281$	−2.15313e6	−1.62669	−0.813346	+	0.581781i	$0.802356\pi$
$282$	0	0
$283$	−1.72440e6	−1.27989	−0.639943	−	0.768423i	$-0.721042\pi$
$283$	−1.72440e6	−1.27989	−0.639943	+	0.768423i	$0.721042\pi$
$284$	0	0
$285$	321779.	0.234664
$286$	0	0
$287$	1.70657e6	1.22298
$288$	0	0
$289$	2.94320e6	2.07289
$290$	0	0
$291$	7172.98	0.00496555
$292$	0	0
$293$	−2.22313e6	−1.51285	−0.756425	−	0.654080i	$-0.773056\pi$
$293$	−2.22313e6	−1.51285	−0.756425	+	0.654080i	$0.773056\pi$
$294$	0	0
$295$	616792.	0.412652
$296$	0	0
$297$	34436.0	0.0226528
$298$	0	0
$299$	183528.	0.118720
$300$	0	0
$301$	1.53619e6	0.977301
$302$	0	0
$303$	−959445.	−0.600363
$304$	0	0
$305$	−213230.	−0.131250
$306$	0	0
$307$	−2.41373e6	−1.46165	−0.730825	−	0.682565i	$-0.760864\pi$
$307$	−2.41373e6	−1.46165	−0.730825	+	0.682565i	$0.760864\pi$
$308$	0	0
$309$	−151322.	−0.0901583
$310$	0	0
$311$	71631.7	0.0419956	0.0209978	−	0.999780i	$-0.493316\pi$
$311$	71631.7	0.0419956	0.0209978	+	0.999780i	$0.493316\pi$
$312$	0	0
$313$	−1.36257e6	−0.786137	−0.393068	−	0.919509i	$-0.628586\pi$
$313$	−1.36257e6	−0.786137	−0.393068	+	0.919509i	$0.628586\pi$
$314$	0	0
$315$	−449173.	−0.255057
$316$	0	0
$317$	−3.16634e6	−1.76974	−0.884870	−	0.465838i	$-0.845753\pi$
$317$	−3.16634e6	−1.76974	−0.884870	+	0.465838i	$0.845753\pi$
$318$	0	0
$319$	−146263.	−0.0804746
$320$	0	0
$321$	−20503.8	−0.0111064
$322$	0	0
$323$	−1.36869e6	−0.729960
$324$	0	0
$325$	−51269.5	−0.0269247
$326$	0	0
$327$	−1.57646e6	−0.815295
$328$	0	0
$329$	648867.	0.330496
$330$	0	0
$331$	167455.	0.0840097	0.0420048	−	0.999117i	$-0.486626\pi$
$331$	167455.	0.0840097	0.0420048	+	0.999117i	$0.486626\pi$
$332$	0	0
$333$	744952.	0.368144
$334$	0	0
$335$	1.77402e6	0.863667
$336$	0	0
$337$	−108256.	−0.0519250	−0.0259625	−	0.999663i	$-0.508265\pi$
$337$	−108256.	−0.0519250	−0.0259625	+	0.999663i	$0.508265\pi$
$338$	0	0
$339$	−1.72110e6	−0.813404
$340$	0	0
$341$	269364.	0.125445
$342$	0	0
$343$	2.36649e6	1.08610
$344$	0	0
$345$	259779.	0.117505
$346$	0	0
$347$	−1.48871e6	−0.663721	−0.331861	−	0.943328i	$-0.607676\pi$
$347$	−1.48871e6	−0.663721	−0.331861	+	0.943328i	$0.607676\pi$
$348$	0	0
$349$	−3.75870e6	−1.65186	−0.825932	−	0.563769i	$-0.809351\pi$
$349$	−3.75870e6	−1.65186	−0.825932	+	0.563769i	$0.809351\pi$
$350$	0	0
$351$	252915.	0.109574
$352$	0	0
$353$	656280.	0.280319	0.140159	−	0.990129i	$-0.455239\pi$
$353$	656280.	0.280319	0.140159	+	0.990129i	$0.455239\pi$
$354$	0	0
$355$	222570.	0.0937336
$356$	0	0
$357$	1.91056e6	0.793396
$358$	0	0
$359$	−2.48344e6	−1.01699	−0.508496	−	0.861064i	$-0.669798\pi$
$359$	−2.48344e6	−1.01699	−0.508496	+	0.861064i	$0.669798\pi$
$360$	0	0
$361$	−2.04674e6	−0.826599
$362$	0	0
$363$	−1.42938e6	−0.569351
$364$	0	0
$365$	−103789.	−0.0407773
$366$	0	0
$367$	−2.13049e6	−0.825685	−0.412842	−	0.910802i	$-0.635464\pi$
$367$	−2.13049e6	−0.825685	−0.412842	+	0.910802i	$0.635464\pi$
$368$	0	0
$369$	−1.36015e6	−0.520019
$370$	0	0
$371$	424776.	0.160223
$372$	0	0
$373$	4.94180e6	1.83913	0.919567	−	0.392933i	$-0.128539\pi$
$373$	4.94180e6	1.83913	0.919567	+	0.392933i	$0.128539\pi$
$374$	0	0
$375$	−1.60718e6	−0.590183
$376$	0	0
$377$	−1.07423e6	−0.389263
$378$	0	0
$379$	−3.43517e6	−1.22843	−0.614215	−	0.789139i	$-0.710527\pi$
$379$	−3.43517e6	−1.22843	−0.614215	+	0.789139i	$0.710527\pi$
$380$	0	0
$381$	451364.	0.159299
$382$	0	0
$383$	1.98312e6	0.690798	0.345399	−	0.938456i	$-0.387744\pi$
$383$	1.98312e6	0.690798	0.345399	+	0.938456i	$0.387744\pi$
$384$	0	0
$385$	−261947.	−0.0900663
$386$	0	0
$387$	−1.22435e6	−0.415556
$388$	0	0
$389$	−1.80609e6	−0.605154	−0.302577	−	0.953125i	$-0.597847\pi$
$389$	−1.80609e6	−0.605154	−0.302577	+	0.953125i	$0.597847\pi$
$390$	0	0
$391$	−1.10497e6	−0.365519
$392$	0	0
$393$	−410634.	−0.134114
$394$	0	0
$395$	−951939.	−0.306984
$396$	0	0
$397$	−443256.	−0.141149	−0.0705745	−	0.997507i	$-0.522483\pi$
$397$	−443256.	−0.141149	−0.0705745	+	0.997507i	$0.522483\pi$
$398$	0	0
$399$	−599342.	−0.188470
$400$	0	0
$401$	1.55464e6	0.482801	0.241400	−	0.970426i	$-0.422393\pi$
$401$	1.55464e6	0.482801	0.241400	+	0.970426i	$0.422393\pi$
$402$	0	0
$403$	1.97834e6	0.606791
$404$	0	0
$405$	357994.	0.108452
$406$	0	0
$407$	434439.	0.130000
$408$	0	0
$409$	−2.88254e6	−0.852054	−0.426027	−	0.904710i	$-0.640087\pi$
$409$	−2.88254e6	−0.852054	−0.426027	+	0.904710i	$0.640087\pi$
$410$	0	0
$411$	1.78307e6	0.520672
$412$	0	0
$413$	−1.14883e6	−0.331421
$414$	0	0
$415$	−4.51393e6	−1.28657
$416$	0	0
$417$	100567.	0.0283215
$418$	0	0
$419$	−924984.	−0.257394	−0.128697	−	0.991684i	$-0.541080\pi$
$419$	−924984.	−0.257394	−0.128697	+	0.991684i	$0.541080\pi$
$420$	0	0
$421$	1.87481e6	0.515527	0.257763	−	0.966208i	$-0.417014\pi$
$421$	1.87481e6	0.515527	0.257763	+	0.966208i	$0.417014\pi$
$422$	0	0
$423$	−517152.	−0.140529
$424$	0	0
$425$	308680.	0.0828964
$426$	0	0
$427$	397159.	0.105413
$428$	0	0
$429$	147494.	0.0386930
$430$	0	0
$431$	−3.05096e6	−0.791121	−0.395561	−	0.918440i	$-0.629450\pi$
$431$	−3.05096e6	−0.791121	−0.395561	+	0.918440i	$0.629450\pi$
$432$	0	0
$433$	6.71772e6	1.72188	0.860939	−	0.508709i	$-0.169877\pi$
$433$	6.71772e6	1.72188	0.860939	+	0.508709i	$0.169877\pi$
$434$	0	0
$435$	−1.52054e6	−0.385278
$436$	0	0
$437$	346629.	0.0868284
$438$	0	0
$439$	2.08362e6	0.516009	0.258004	−	0.966144i	$-0.416935\pi$
$439$	2.08362e6	0.516009	0.258004	+	0.966144i	$0.416935\pi$
$440$	0	0
$441$	−524743.	−0.128484
$442$	0	0
$443$	−3.94396e6	−0.954823	−0.477411	−	0.878680i	$-0.658425\pi$
$443$	−3.94396e6	−0.954823	−0.477411	+	0.878680i	$0.658425\pi$
$444$	0	0
$445$	3.12716e6	0.748601
$446$	0	0
$447$	374335.	0.0886119
$448$	0	0
$449$	−2.15958e6	−0.505537	−0.252769	−	0.967527i	$-0.581341\pi$
$449$	−2.15958e6	−0.505537	−0.252769	+	0.967527i	$0.581341\pi$
$450$	0	0
$451$	−793206.	−0.183630
$452$	0	0
$453$	1.17085e6	0.268074
$454$	0	0
$455$	−1.92387e6	−0.435659
$456$	0	0
$457$	6.66198e6	1.49215	0.746076	−	0.665861i	$-0.231935\pi$
$457$	6.66198e6	1.49215	0.746076	+	0.665861i	$0.231935\pi$
$458$	0	0
$459$	−1.52273e6	−0.337358
$460$	0	0
$461$	3.02524e6	0.662991	0.331495	−	0.943457i	$-0.392447\pi$
$461$	3.02524e6	0.662991	0.331495	+	0.943457i	$0.392447\pi$
$462$	0	0
$463$	4.93239e6	1.06931	0.534657	−	0.845069i	$-0.320441\pi$
$463$	4.93239e6	1.06931	0.534657	+	0.845069i	$0.320441\pi$
$464$	0	0
$465$	2.80029e6	0.600579
$466$	0	0
$467$	4.62763e6	0.981898	0.490949	−	0.871188i	$-0.336650\pi$
$467$	4.62763e6	0.981898	0.490949	+	0.871188i	$0.336650\pi$
$468$	0	0
$469$	−3.30427e6	−0.693655
$470$	0	0
$471$	−3.16998e6	−0.658421
$472$	0	0
$473$	−714015.	−0.146742
$474$	0	0
$475$	−96832.7	−0.0196919
$476$	0	0
$477$	−338550.	−0.0681281
$478$	0	0
$479$	−2.86608e6	−0.570755	−0.285377	−	0.958415i	$-0.592119\pi$
$479$	−2.86608e6	−0.570755	−0.285377	+	0.958415i	$0.592119\pi$
$480$	0	0
$481$	3.19073e6	0.628822
$482$	0	0
$483$	−483861.	−0.0943742
$484$	0	0
$485$	43487.3	0.00839477
$486$	0	0
$487$	−4.44248e6	−0.848796	−0.424398	−	0.905476i	$-0.639514\pi$
$487$	−4.44248e6	−0.848796	−0.424398	+	0.905476i	$0.639514\pi$
$488$	0	0
$489$	−1.32362e6	−0.250318
$490$	0	0
$491$	3.07944e6	0.576458	0.288229	−	0.957561i	$-0.406933\pi$
$491$	3.07944e6	0.576458	0.288229	+	0.957561i	$0.406933\pi$
$492$	0	0
$493$	6.46763e6	1.19847
$494$	0	0
$495$	208774.	0.0382969
$496$	0	0
$497$	−414556.	−0.0752822
$498$	0	0
$499$	4.46439e6	0.802621	0.401311	−	0.915942i	$-0.368555\pi$
$499$	4.46439e6	0.802621	0.401311	+	0.915942i	$0.368555\pi$
$500$	0	0
$501$	2.58990e6	0.460988
$502$	0	0
$503$	−1.17377e6	−0.206853	−0.103427	−	0.994637i	$-0.532981\pi$
$503$	−1.17377e6	−0.206853	−0.103427	+	0.994637i	$0.532981\pi$
$504$	0	0
$505$	−5.81679e6	−1.01497
$506$	0	0
$507$	−2.25837e6	−0.390189
$508$	0	0
$509$	−7.30781e6	−1.25024	−0.625119	−	0.780530i	$-0.714949\pi$
$509$	−7.30781e6	−1.25024	−0.625119	+	0.780530i	$0.714949\pi$
$510$	0	0
$511$	193316.	0.0327503
$512$	0	0
$513$	477680.	0.0801390
$514$	0	0
$515$	−917413.	−0.152422
$516$	0	0
$517$	−301591.	−0.0496241
$518$	0	0
$519$	1.80383e6	0.293952
$520$	0	0
$521$	1.48550e6	0.239761	0.119880	−	0.992788i	$-0.461749\pi$
$521$	1.48550e6	0.239761	0.119880	+	0.992788i	$0.461749\pi$
$522$	0	0
$523$	−2.72619e6	−0.435814	−0.217907	−	0.975970i	$-0.569923\pi$
$523$	−2.72619e6	−0.435814	−0.217907	+	0.975970i	$0.569923\pi$
$524$	0	0
$525$	135169.	0.0214032
$526$	0	0
$527$	−1.19110e7	−1.86820
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	915626.	0.140923
$532$	0	0
$533$	−5.82569e6	−0.888238
$534$	0	0
$535$	−124308.	−0.0187765
$536$	0	0
$537$	5.30584e6	0.793997
$538$	0	0
$539$	−306018.	−0.0453707
$540$	0	0
$541$	−3.08074e6	−0.452545	−0.226272	−	0.974064i	$-0.572654\pi$
$541$	−3.08074e6	−0.452545	−0.226272	+	0.974064i	$0.572654\pi$
$542$	0	0
$543$	568325.	0.0827174
$544$	0	0
$545$	−9.55757e6	−1.37834
$546$	0	0
$547$	−1.13704e7	−1.62482	−0.812412	−	0.583084i	$-0.801846\pi$
$547$	−1.13704e7	−1.62482	−0.812412	+	0.583084i	$0.801846\pi$
$548$	0	0
$549$	−316539.	−0.0448225
$550$	0	0
$551$	−2.02889e6	−0.284695
$552$	0	0
$553$	1.77307e6	0.246555
$554$	0	0
$555$	4.51639e6	0.622385
$556$	0	0
$557$	−7.64168e6	−1.04364	−0.521820	−	0.853056i	$-0.674747\pi$
$557$	−7.64168e6	−1.04364	−0.521820	+	0.853056i	$0.674747\pi$
$558$	0	0
$559$	−5.24408e6	−0.709806
$560$	0	0
$561$	−888022.	−0.119129
$562$	0	0
$563$	5.81027e6	0.772548	0.386274	−	0.922384i	$-0.373762\pi$
$563$	5.81027e6	0.772548	0.386274	+	0.922384i	$0.373762\pi$
$564$	0	0
$565$	−1.04344e7	−1.37514
$566$	0	0
$567$	−666796.	−0.0871034
$568$	0	0
$569$	−8.07512e6	−1.04561	−0.522803	−	0.852453i	$-0.675114\pi$
$569$	−8.07512e6	−1.04561	−0.522803	+	0.852453i	$0.675114\pi$
$570$	0	0
$571$	1.17797e7	1.51198	0.755989	−	0.654584i	$-0.227156\pi$
$571$	1.17797e7	1.51198	0.755989	+	0.654584i	$0.227156\pi$
$572$	0	0
$573$	3.93664e6	0.500886
$574$	0	0
$575$	−78175.0	−0.00986049
$576$	0	0
$577$	1.06619e7	1.33320	0.666602	−	0.745414i	$-0.267748\pi$
$577$	1.06619e7	1.33320	0.666602	+	0.745414i	$0.267748\pi$
$578$	0	0
$579$	1.04218e6	0.129195
$580$	0	0
$581$	8.40759e6	1.03331
$582$	0	0
$583$	−197434.	−0.0240576
$584$	0	0
$585$	1.53334e6	0.185246
$586$	0	0
$587$	4.91414e6	0.588643	0.294322	−	0.955706i	$-0.404906\pi$
$587$	4.91414e6	0.588643	0.294322	+	0.955706i	$0.404906\pi$
$588$	0	0
$589$	3.73649e6	0.443788
$590$	0	0
$591$	−6.07451e6	−0.715390
$592$	0	0
$593$	−1.58488e7	−1.85080	−0.925398	−	0.378997i	$-0.876269\pi$
$593$	−1.58488e7	−1.85080	−0.925398	+	0.378997i	$0.876269\pi$
$594$	0	0
$595$	1.15831e7	1.34132
$596$	0	0
$597$	−1.11737e6	−0.128311
$598$	0	0
$599$	2.95394e6	0.336384	0.168192	−	0.985754i	$-0.446207\pi$
$599$	2.95394e6	0.336384	0.168192	+	0.985754i	$0.446207\pi$
$600$	0	0
$601$	−1.87544e6	−0.211796	−0.105898	−	0.994377i	$-0.533772\pi$
$601$	−1.87544e6	−0.211796	−0.105898	+	0.994377i	$0.533772\pi$
$602$	0	0
$603$	2.63353e6	0.294947
$604$	0	0
$605$	−8.66582e6	−0.962546
$606$	0	0
$607$	−4.85492e6	−0.534823	−0.267411	−	0.963582i	$-0.586168\pi$
$607$	−4.85492e6	−0.534823	−0.267411	+	0.963582i	$0.586168\pi$
$608$	0	0
$609$	2.83214e6	0.309437
$610$	0	0
$611$	−2.21503e6	−0.240036
$612$	0	0
$613$	4.38703e6	0.471541	0.235770	−	0.971809i	$-0.424239\pi$
$613$	4.38703e6	0.471541	0.235770	+	0.971809i	$0.424239\pi$
$614$	0	0
$615$	−8.24609e6	−0.879145
$616$	0	0
$617$	7.47992e6	0.791014	0.395507	−	0.918463i	$-0.370569\pi$
$617$	7.47992e6	0.791014	0.395507	+	0.918463i	$0.370569\pi$
$618$	0	0
$619$	−1.70115e7	−1.78449	−0.892246	−	0.451549i	$-0.850872\pi$
$619$	−1.70115e7	−1.78449	−0.892246	+	0.451549i	$0.850872\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	−5.82462e6	−0.601239
$624$	0	0
$625$	−9.28198e6	−0.950474
$626$	0	0
$627$	278572.	0.0282989
$628$	0	0
$629$	−1.92105e7	−1.93603
$630$	0	0
$631$	−2.93235e6	−0.293185	−0.146593	−	0.989197i	$-0.546831\pi$
$631$	−2.93235e6	−0.293185	−0.146593	+	0.989197i	$0.546831\pi$
$632$	0	0
$633$	1.22362e6	0.121377
$634$	0	0
$635$	2.73646e6	0.269312
$636$	0	0
$637$	−2.24755e6	−0.219462
$638$	0	0
$639$	330404.	0.0320106
$640$	0	0
$641$	−516075.	−0.0496099	−0.0248049	−	0.999692i	$-0.507896\pi$
$641$	−516075.	−0.0496099	−0.0248049	+	0.999692i	$0.507896\pi$
$642$	0	0
$643$	1.84195e7	1.75692	0.878458	−	0.477820i	$-0.158573\pi$
$643$	1.84195e7	1.75692	0.878458	+	0.477820i	$0.158573\pi$
$644$	0	0
$645$	−7.42284e6	−0.702539
$646$	0	0
$647$	1.91829e7	1.80158	0.900792	−	0.434251i	$-0.142987\pi$
$647$	1.91829e7	1.80158	0.900792	+	0.434251i	$0.142987\pi$
$648$	0	0
$649$	533972.	0.0497630
$650$	0	0
$651$	−5.21578e6	−0.482355
$652$	0	0
$653$	240221.	0.0220459	0.0110230	−	0.999939i	$-0.496491\pi$
$653$	240221.	0.0220459	0.0110230	+	0.999939i	$0.496491\pi$
$654$	0	0
$655$	−2.48953e6	−0.226733
$656$	0	0
$657$	−154074.	−0.0139257
$658$	0	0
$659$	1.14851e7	1.03020	0.515101	−	0.857129i	$-0.327754\pi$
$659$	1.14851e7	1.03020	0.515101	+	0.857129i	$0.327754\pi$
$660$	0	0
$661$	1.21290e7	1.07974	0.539872	−	0.841747i	$-0.318473\pi$
$661$	1.21290e7	1.07974	0.539872	+	0.841747i	$0.318473\pi$
$662$	0	0
$663$	−6.52207e6	−0.576237
$664$	0	0
$665$	−3.63361e6	−0.318628
$666$	0	0
$667$	−1.63797e6	−0.142558
$668$	0	0
$669$	−5.10512e6	−0.441002
$670$	0	0
$671$	−184598.	−0.0158278
$672$	0	0
$673$	216963.	0.0184649	0.00923247	−	0.999957i	$-0.497061\pi$
$673$	216963.	0.0184649	0.00923247	+	0.999957i	$0.497061\pi$
$674$	0	0
$675$	−107731.	−0.00910082
$676$	0	0
$677$	1.73783e7	1.45725	0.728627	−	0.684911i	$-0.240159\pi$
$677$	1.73783e7	1.45725	0.728627	+	0.684911i	$0.240159\pi$
$678$	0	0
$679$	−80999.1	−0.00674226
$680$	0	0
$681$	9.77353e6	0.807577
$682$	0	0
$683$	2.20245e7	1.80657	0.903283	−	0.429046i	$-0.141150\pi$
$683$	2.20245e7	1.80657	0.903283	+	0.429046i	$0.141150\pi$
$684$	0	0
$685$	1.08101e7	0.880248
$686$	0	0
$687$	7.97942e6	0.645029
$688$	0	0
$689$	−1.45006e6	−0.116369
$690$	0	0
$691$	7.22314e6	0.575481	0.287740	−	0.957708i	$-0.407096\pi$
$691$	7.22314e6	0.575481	0.287740	+	0.957708i	$0.407096\pi$
$692$	0	0
$693$	−388860.	−0.0307582
$694$	0	0
$695$	609705.	0.0478804
$696$	0	0
$697$	3.50748e7	2.73473
$698$	0	0
$699$	1.98300e6	0.153507
$700$	0	0
$701$	−26729.6	−0.00205446	−0.00102723	−	0.999999i	$-0.500327\pi$
$701$	−26729.6	−0.00205446	−0.00102723	+	0.999999i	$0.500327\pi$
$702$	0	0
$703$	6.02633e6	0.459901
$704$	0	0
$705$	−3.13532e6	−0.237579
$706$	0	0
$707$	1.08343e7	0.815177
$708$	0	0
$709$	8.40889e6	0.628237	0.314118	−	0.949384i	$-0.398291\pi$
$709$	8.40889e6	0.628237	0.314118	+	0.949384i	$0.398291\pi$
$710$	0	0
$711$	−1.41315e6	−0.104837
$712$	0	0
$713$	3.01655e6	0.222222
$714$	0	0
$715$	894208.	0.0654144
$716$	0	0
$717$	−1.29750e7	−0.942565
$718$	0	0
$719$	−2.05917e6	−0.148549	−0.0742745	−	0.997238i	$-0.523664\pi$
$719$	−2.05917e6	−0.148549	−0.0742745	+	0.997238i	$0.523664\pi$
$720$	0	0
$721$	1.70876e6	0.122418
$722$	0	0
$723$	8.85326e6	0.629879
$724$	0	0
$725$	457575.	0.0323308
$726$	0	0
$727$	−4.23128e6	−0.296918	−0.148459	−	0.988919i	$-0.547431\pi$
$727$	−4.23128e6	−0.296918	−0.148459	+	0.988919i	$0.547431\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	3.15731e7	2.18537
$732$	0	0
$733$	−4.03260e6	−0.277221	−0.138610	−	0.990347i	$-0.544264\pi$
$733$	−4.03260e6	−0.277221	−0.138610	+	0.990347i	$0.544264\pi$
$734$	0	0
$735$	−3.18134e6	−0.217216
$736$	0	0
$737$	1.53581e6	0.104153
$738$	0	0
$739$	6.54455e6	0.440827	0.220414	−	0.975406i	$-0.429259\pi$
$739$	6.54455e6	0.440827	0.220414	+	0.975406i	$0.429259\pi$
$740$	0	0
$741$	2.04597e6	0.136884
$742$	0	0
$743$	6.76099e6	0.449302	0.224651	−	0.974439i	$-0.427876\pi$
$743$	6.76099e6	0.449302	0.224651	+	0.974439i	$0.427876\pi$
$744$	0	0
$745$	2.26947e6	0.149807
$746$	0	0
$747$	−6.70091e6	−0.439372
$748$	0	0
$749$	231534.	0.0150803
$750$	0	0
$751$	1.52676e7	0.987806	0.493903	−	0.869517i	$-0.335570\pi$
$751$	1.52676e7	0.987806	0.493903	+	0.869517i	$0.335570\pi$
$752$	0	0
$753$	−1.68625e7	−1.08376
$754$	0	0
$755$	7.09843e6	0.453205
$756$	0	0
$757$	4.38752e6	0.278278	0.139139	−	0.990273i	$-0.455566\pi$
$757$	4.38752e6	0.278278	0.139139	+	0.990273i	$0.455566\pi$
$758$	0	0
$759$	224897.	0.0141703
$760$	0	0
$761$	1.89521e7	1.18630	0.593151	−	0.805092i	$-0.297884\pi$
$761$	1.89521e7	1.18630	0.593151	+	0.805092i	$0.297884\pi$
$762$	0	0
$763$	1.78018e7	1.10701
$764$	0	0
$765$	−9.23180e6	−0.570338
$766$	0	0
$767$	3.92175e6	0.240709
$768$	0	0
$769$	−2.04531e7	−1.24722	−0.623610	−	0.781736i	$-0.714334\pi$
$769$	−2.04531e7	−1.24722	−0.623610	+	0.781736i	$0.714334\pi$
$770$	0	0
$771$	2.47413e6	0.149895
$772$	0	0
$773$	−2.19141e7	−1.31909	−0.659546	−	0.751664i	$-0.729251\pi$
$773$	−2.19141e7	−1.31909	−0.659546	+	0.751664i	$0.729251\pi$
$774$	0	0
$775$	−842688.	−0.0503979
$776$	0	0
$777$	−8.41218e6	−0.499869
$778$	0	0
$779$	−1.10030e7	−0.649630
$780$	0	0
$781$	192684.	0.0113037
$782$	0	0
$783$	−2.25724e6	−0.131575
$784$	0	0
$785$	−1.92185e7	−1.11313
$786$	0	0
$787$	4.31130e6	0.248125	0.124063	−	0.992274i	$-0.460408\pi$
$787$	4.31130e6	0.248125	0.124063	+	0.992274i	$0.460408\pi$
$788$	0	0
$789$	1.22826e7	0.702423
$790$	0	0
$791$	1.94351e7	1.10445
$792$	0	0
$793$	−1.35578e6	−0.0765608
$794$	0	0
$795$	−2.05251e6	−0.115177
$796$	0	0
$797$	−6.52495e6	−0.363857	−0.181929	−	0.983312i	$-0.558234\pi$
$797$	−6.52495e6	−0.363857	−0.181929	+	0.983312i	$0.558234\pi$
$798$	0	0
$799$	1.33361e7	0.739030
$800$	0	0
$801$	4.64226e6	0.255652
$802$	0	0
$803$	−89852.6	−0.00491747
$804$	0	0
$805$	−2.93349e6	−0.159549
$806$	0	0
$807$	−6.79287e6	−0.367172
$808$	0	0
$809$	3.02775e7	1.62648	0.813241	−	0.581927i	$-0.197701\pi$
$809$	3.02775e7	1.62648	0.813241	+	0.581927i	$0.197701\pi$
$810$	0	0
$811$	8.94106e6	0.477350	0.238675	−	0.971099i	$-0.423287\pi$
$811$	8.94106e6	0.477350	0.238675	+	0.971099i	$0.423287\pi$
$812$	0	0
$813$	4.87563e6	0.258704
$814$	0	0
$815$	−8.02468e6	−0.423188
$816$	0	0
$817$	−9.90448e6	−0.519130
$818$	0	0
$819$	−2.85598e6	−0.148780
$820$	0	0
$821$	−6.39276e6	−0.331002	−0.165501	−	0.986210i	$-0.552924\pi$
$821$	−6.39276e6	−0.331002	−0.165501	+	0.986210i	$0.552924\pi$
$822$	0	0
$823$	3.40740e7	1.75357	0.876786	−	0.480881i	$-0.159683\pi$
$823$	3.40740e7	1.75357	0.876786	+	0.480881i	$0.159683\pi$
$824$	0	0
$825$	−62826.1	−0.00321370
$826$	0	0
$827$	−4.08299e6	−0.207594	−0.103797	−	0.994599i	$-0.533099\pi$
$827$	−4.08299e6	−0.207594	−0.103797	+	0.994599i	$0.533099\pi$
$828$	0	0
$829$	7.26894e6	0.367354	0.183677	−	0.982987i	$-0.441200\pi$
$829$	7.26894e6	0.367354	0.183677	+	0.982987i	$0.441200\pi$
$830$	0	0
$831$	−6.57025e6	−0.330050
$832$	0	0
$833$	1.35318e7	0.675685
$834$	0	0
$835$	1.57017e7	0.779346
$836$	0	0
$837$	4.15702e6	0.205101
$838$	0	0
$839$	2.77230e7	1.35968	0.679838	−	0.733363i	$-0.262050\pi$
$839$	2.77230e7	1.35968	0.679838	+	0.733363i	$0.262050\pi$
$840$	0	0
$841$	−1.09238e7	−0.532578
$842$	0	0
$843$	−1.93782e7	−0.939171
$844$	0	0
$845$	−1.36917e7	−0.659653
$846$	0	0
$847$	1.61409e7	0.773069
$848$	0	0
$849$	−1.55196e7	−0.738942
$850$	0	0
$851$	4.86518e6	0.230290
$852$	0	0
$853$	4.17385e7	1.96410	0.982052	−	0.188609i	$-0.0603979\pi$
$853$	4.17385e7	1.96410	0.982052	+	0.188609i	$0.0603979\pi$
$854$	0	0
$855$	2.89601e6	0.135483
$856$	0	0
$857$	1.87575e7	0.872413	0.436207	−	0.899846i	$-0.356322\pi$
$857$	1.87575e7	0.872413	0.436207	+	0.899846i	$0.356322\pi$
$858$	0	0
$859$	9.25974e6	0.428170	0.214085	−	0.976815i	$-0.431323\pi$
$859$	9.25974e6	0.428170	0.214085	+	0.976815i	$0.431323\pi$
$860$	0	0
$861$	1.53591e7	0.706086
$862$	0	0
$863$	−3.44200e7	−1.57320	−0.786599	−	0.617464i	$-0.788160\pi$
$863$	−3.44200e7	−1.57320	−0.786599	+	0.617464i	$0.788160\pi$
$864$	0	0
$865$	1.09360e7	0.496956
$866$	0	0
$867$	2.64888e7	1.19678
$868$	0	0
$869$	−824117.	−0.0370203
$870$	0	0
$871$	1.12798e7	0.503796
$872$	0	0
$873$	64556.9	0.00286686
$874$	0	0
$875$	1.81487e7	0.801355
$876$	0	0
$877$	2.84532e7	1.24920	0.624601	−	0.780944i	$-0.285262\pi$
$877$	2.84532e7	1.24920	0.624601	+	0.780944i	$0.285262\pi$
$878$	0	0
$879$	−2.00082e7	−0.873445
$880$	0	0
$881$	4.18914e7	1.81838	0.909192	−	0.416377i	$-0.136700\pi$
$881$	4.18914e7	1.81838	0.909192	+	0.416377i	$0.136700\pi$
$882$	0	0
$883$	6.58336e6	0.284149	0.142074	−	0.989856i	$-0.454623\pi$
$883$	6.58336e6	0.284149	0.142074	+	0.989856i	$0.454623\pi$
$884$	0	0
$885$	5.55113e6	0.238245
$886$	0	0
$887$	−2.52718e7	−1.07852	−0.539258	−	0.842141i	$-0.681295\pi$
$887$	−2.52718e7	−1.07852	−0.539258	+	0.842141i	$0.681295\pi$
$888$	0	0
$889$	−5.09691e6	−0.216298
$890$	0	0
$891$	309924.	0.0130786
$892$	0	0
$893$	−4.18353e6	−0.175555
$894$	0	0
$895$	3.21675e7	1.34233
$896$	0	0
$897$	1.65175e6	0.0685432
$898$	0	0
$899$	−1.76565e7	−0.728626
$900$	0	0
$901$	8.73038e6	0.358279
$902$	0	0
$903$	1.38257e7	0.564245
$904$	0	0
$905$	3.44556e6	0.139842
$906$	0	0
$907$	2.22752e7	0.899090	0.449545	−	0.893258i	$-0.351586\pi$
$907$	2.22752e7	0.899090	0.449545	+	0.893258i	$0.351586\pi$
$908$	0	0
$909$	−8.63501e6	−0.346620
$910$	0	0
$911$	−4.85042e7	−1.93635	−0.968173	−	0.250281i	$-0.919477\pi$
$911$	−4.85042e7	−1.93635	−0.968173	+	0.250281i	$0.919477\pi$
$912$	0	0
$913$	−3.90782e6	−0.155152
$914$	0	0
$915$	−1.91907e6	−0.0757770
$916$	0	0
$917$	4.63698e6	0.182101
$918$	0	0
$919$	−3.20478e7	−1.25173	−0.625863	−	0.779933i	$-0.715253\pi$
$919$	−3.20478e7	−1.25173	−0.625863	+	0.779933i	$0.715253\pi$
$920$	0	0
$921$	−2.17236e7	−0.843884
$922$	0	0
$923$	1.41517e6	0.0546769
$924$	0	0
$925$	−1.35911e6	−0.0522277
$926$	0	0
$927$	−1.36190e6	−0.0520529
$928$	0	0
$929$	1.97018e7	0.748975	0.374487	−	0.927232i	$-0.377819\pi$
$929$	1.97018e7	0.748975	0.374487	+	0.927232i	$0.377819\pi$
$930$	0	0
$931$	−4.24494e6	−0.160508
$932$	0	0
$933$	644685.	0.0242462
$934$	0	0
$935$	−5.38377e6	−0.201399
$936$	0	0
$937$	2.60647e7	0.969846	0.484923	−	0.874557i	$-0.338848\pi$
$937$	2.60647e7	0.969846	0.484923	+	0.874557i	$0.338848\pi$
$938$	0	0
$939$	−1.22631e7	−0.453876
$940$	0	0
$941$	254464.	0.00936813	0.00468406	−	0.999989i	$-0.498509\pi$
$941$	254464.	0.00936813	0.00468406	+	0.999989i	$0.498509\pi$
$942$	0	0
$943$	−8.88292e6	−0.325294
$944$	0	0
$945$	−4.04255e6	−0.147257
$946$	0	0
$947$	4.16819e6	0.151033	0.0755167	−	0.997145i	$-0.475939\pi$
$947$	4.16819e6	0.151033	0.0755167	+	0.997145i	$0.475939\pi$
$948$	0	0
$949$	−659921.	−0.0237863
$950$	0	0
$951$	−2.84971e7	−1.02176
$952$	0	0
$953$	−3.88840e6	−0.138688	−0.0693440	−	0.997593i	$-0.522091\pi$
$953$	−3.88840e6	−0.138688	−0.0693440	+	0.997593i	$0.522091\pi$
$954$	0	0
$955$	2.38665e7	0.846799
$956$	0	0
$957$	−1.31637e6	−0.0464620
$958$	0	0
$959$	−2.01349e7	−0.706972
$960$	0	0
$961$	3.88772e6	0.135796
$962$	0	0
$963$	−184535.	−0.00641227
$964$	0	0
$965$	6.31836e6	0.218417
$966$	0	0
$967$	−4.40569e7	−1.51512	−0.757561	−	0.652764i	$-0.773609\pi$
$967$	−4.40569e7	−1.51512	−0.757561	+	0.652764i	$0.773609\pi$
$968$	0	0
$969$	−1.23182e7	−0.421443
$970$	0	0
$971$	−2.81305e7	−0.957479	−0.478740	−	0.877957i	$-0.658906\pi$
$971$	−2.81305e7	−0.957479	−0.478740	+	0.877957i	$0.658906\pi$
$972$	0	0
$973$	−1.13563e6	−0.0384552
$974$	0	0
$975$	−461426.	−0.0155450
$976$	0	0
$977$	−6.86915e6	−0.230232	−0.115116	−	0.993352i	$-0.536724\pi$
$977$	−6.86915e6	−0.230232	−0.115116	+	0.993352i	$0.536724\pi$
$978$	0	0
$979$	2.70726e6	0.0902763
$980$	0	0
$981$	−1.41882e7	−0.470711
$982$	0	0
$983$	8.74183e6	0.288548	0.144274	−	0.989538i	$-0.453915\pi$
$983$	8.74183e6	0.288548	0.144274	+	0.989538i	$0.453915\pi$
$984$	0	0
$985$	−3.68277e7	−1.20944
$986$	0	0
$987$	5.83981e6	0.190812
$988$	0	0
$989$	−7.99609e6	−0.259948
$990$	0	0
$991$	−1.13447e7	−0.366953	−0.183477	−	0.983024i	$-0.558735\pi$
$991$	−1.13447e7	−0.366953	−0.183477	+	0.983024i	$0.558735\pi$
$992$	0	0
$993$	1.50710e6	0.0485030
$994$	0	0
$995$	−6.77425e6	−0.216922
$996$	0	0
$997$	660298.	0.0210379	0.0105189	−	0.999945i	$-0.496652\pi$
$997$	660298.	0.0210379	0.0105189	+	0.999945i	$0.496652\pi$
$998$	0	0
$999$	6.70457e6	0.212548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.u.1.5		6
4.3	odd	2		552.6.a.d.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.6.a.d.1.5	✓	6	4.3	odd	2
1104.6.a.u.1.5		6	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q+9.00000 q^{3} +54.5639 q^{5} -101.630 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q+9.00000 q^{3} +54.5639 q^{5} -101.630 q^{7} +81.0000 q^{9} +47.2374 q^{11} +346.934 q^{13} +491.075 q^{15} -2088.79 q^{17} +655.254 q^{19} -914.672 q^{21} +529.000 q^{23} -147.779 q^{25} +729.000 q^{27} -3096.35 q^{29} +5702.36 q^{31} +425.136 q^{33} -5545.34 q^{35} +9196.94 q^{37} +3122.41 q^{39} -16791.9 q^{41} -15115.5 q^{43} +4419.68 q^{45} -6384.59 q^{47} -6478.31 q^{49} -18799.1 q^{51} -4179.63 q^{53} +2577.46 q^{55} +5897.29 q^{57} +11304.0 q^{59} -3907.89 q^{61} -8232.04 q^{63} +18930.1 q^{65} +32512.7 q^{67} +4761.00 q^{69} +4079.07 q^{71} -1902.15 q^{73} -1330.01 q^{75} -4800.74 q^{77} -17446.3 q^{79} +6561.00 q^{81} -82727.3 q^{83} -113973. q^{85} -27867.1 q^{87} +57311.9 q^{89} -35259.0 q^{91} +51321.2 q^{93} +35753.2 q^{95} +796.998 q^{97} +3826.23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 54 q^{3} - 70 q^{5} - 144 q^{7} + 486 q^{9}+O(q^{10}) \) 6 * q + 54 * q^3 - 70 * q^5 - 144 * q^7 + 486 * q^9	\( 6 q + 54 q^{3} - 70 q^{5} - 144 q^{7} + 486 q^{9} + 118 q^{11} - 692 q^{13} - 630 q^{15} - 1200 q^{17} - 470 q^{19} - 1296 q^{21} + 3174 q^{23} + 6586 q^{25} + 4374 q^{27} + 2728 q^{29} - 6540 q^{31} + 1062 q^{33} - 6208 q^{35} + 7246 q^{37} - 6228 q^{39} + 13396 q^{41} - 13394 q^{43} - 5670 q^{45} - 35996 q^{47} - 2334 q^{49} - 10800 q^{51} + 6274 q^{53} - 44164 q^{55} - 4230 q^{57} - 3756 q^{59} - 4526 q^{61} - 11664 q^{63} - 33828 q^{65} - 9646 q^{67} + 28566 q^{69} + 4912 q^{71} - 29352 q^{73} + 59274 q^{75} - 146060 q^{77} - 17608 q^{79} + 39366 q^{81} + 175218 q^{83} - 99652 q^{85} + 24552 q^{87} - 157740 q^{89} + 62512 q^{91} - 58860 q^{93} + 215644 q^{95} - 150008 q^{97} + 9558 q^{99}+O(q^{100}) \) 6 * q + 54 * q^3 - 70 * q^5 - 144 * q^7 + 486 * q^9 + 118 * q^11 - 692 * q^13 - 630 * q^15 - 1200 * q^17 - 470 * q^19 - 1296 * q^21 + 3174 * q^23 + 6586 * q^25 + 4374 * q^27 + 2728 * q^29 - 6540 * q^31 + 1062 * q^33 - 6208 * q^35 + 7246 * q^37 - 6228 * q^39 + 13396 * q^41 - 13394 * q^43 - 5670 * q^45 - 35996 * q^47 - 2334 * q^49 - 10800 * q^51 + 6274 * q^53 - 44164 * q^55 - 4230 * q^57 - 3756 * q^59 - 4526 * q^61 - 11664 * q^63 - 33828 * q^65 - 9646 * q^67 + 28566 * q^69 + 4912 * q^71 - 29352 * q^73 + 59274 * q^75 - 146060 * q^77 - 17608 * q^79 + 39366 * q^81 + 175218 * q^83 - 99652 * q^85 + 24552 * q^87 - 157740 * q^89 + 62512 * q^91 - 58860 * q^93 + 215644 * q^95 - 150008 * q^97 + 9558 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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