LMFDB - Embedded newform 1104.6.a.i.1.2

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.5333.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 8 $ x^3 - x^2 - 11*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.714018$ 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)

$f(q)$	$=$	$q-9.00000 q^{3} -55.5168 q^{5} -2.50462 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -55.5168 q^{5} -2.50462 q^{7} +81.0000 q^{9} -228.550 q^{11} +658.703 q^{13} +499.651 q^{15} -1443.81 q^{17} +982.167 q^{19} +22.5416 q^{21} -529.000 q^{23} -42.8875 q^{25} -729.000 q^{27} -7157.56 q^{29} +9259.98 q^{31} +2056.95 q^{33} +139.049 q^{35} +2422.50 q^{37} -5928.33 q^{39} -4075.27 q^{41} +10417.3 q^{43} -4496.86 q^{45} -9358.08 q^{47} -16800.7 q^{49} +12994.3 q^{51} -34280.3 q^{53} +12688.4 q^{55} -8839.51 q^{57} +7268.79 q^{59} +26611.7 q^{61} -202.874 q^{63} -36569.1 q^{65} -53450.8 q^{67} +4761.00 q^{69} -21673.7 q^{71} -82856.3 q^{73} +385.987 q^{75} +572.432 q^{77} +23960.0 q^{79} +6561.00 q^{81} -81187.7 q^{83} +80155.6 q^{85} +64418.0 q^{87} +100115. q^{89} -1649.80 q^{91} -83339.8 q^{93} -54526.8 q^{95} +36122.1 q^{97} -18512.6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9}+O(q^{10}) $ 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9	$ 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9} + 376 q^{11} - 858 q^{13} + 504 q^{15} - 2548 q^{17} + 2846 q^{19} - 1026 q^{21} - 1587 q^{23} + 753 q^{25} - 2187 q^{27} - 16370 q^{29} + 14756 q^{31} - 3384 q^{33} + 18520 q^{35} + 15874 q^{37} + 7722 q^{39} + 12606 q^{41} - 3154 q^{43} - 4536 q^{45} - 29928 q^{47} + 4471 q^{49} + 22932 q^{51} - 44084 q^{53} - 38360 q^{55} - 25614 q^{57} + 29300 q^{59} + 54010 q^{61} + 9234 q^{63} - 51216 q^{65} - 43390 q^{67} + 14283 q^{69} - 23424 q^{71} - 91402 q^{73} - 6777 q^{75} - 97208 q^{77} + 49398 q^{79} + 19683 q^{81} + 103936 q^{83} + 5888 q^{85} + 147330 q^{87} + 96112 q^{89} - 129228 q^{91} - 132804 q^{93} + 55928 q^{95} - 135318 q^{97} + 30456 q^{99}+O(q^{100}) $ 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9 + 376 * q^11 - 858 * q^13 + 504 * q^15 - 2548 * q^17 + 2846 * q^19 - 1026 * q^21 - 1587 * q^23 + 753 * q^25 - 2187 * q^27 - 16370 * q^29 + 14756 * q^31 - 3384 * q^33 + 18520 * q^35 + 15874 * q^37 + 7722 * q^39 + 12606 * q^41 - 3154 * q^43 - 4536 * q^45 - 29928 * q^47 + 4471 * q^49 + 22932 * q^51 - 44084 * q^53 - 38360 * q^55 - 25614 * q^57 + 29300 * q^59 + 54010 * q^61 + 9234 * q^63 - 51216 * q^65 - 43390 * q^67 + 14283 * q^69 - 23424 * q^71 - 91402 * q^73 - 6777 * q^75 - 97208 * q^77 + 49398 * q^79 + 19683 * q^81 + 103936 * q^83 + 5888 * q^85 + 147330 * q^87 + 96112 * q^89 - 129228 * q^91 - 132804 * q^93 + 55928 * q^95 - 135318 * q^97 + 30456 * 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
$5$	−55.5168	−0.993114	−0.496557	−	0.868004i	$-0.665403\pi$
$5$	−55.5168	−0.993114	−0.496557	+	0.868004i	$0.665403\pi$
$6$	0	0
$7$	−2.50462	−0.0193196	−0.00965978	−	0.999953i	$-0.503075\pi$
$7$	−2.50462	−0.0193196	−0.00965978	+	0.999953i	$0.503075\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−228.550	−0.569508	−0.284754	−	0.958601i	$-0.591912\pi$
$11$	−228.550	−0.569508	−0.284754	+	0.958601i	$0.591912\pi$
$12$	0	0
$13$	658.703	1.08101	0.540507	−	0.841339i	$-0.318232\pi$
$13$	658.703	1.08101	0.540507	+	0.841339i	$0.318232\pi$
$14$	0	0
$15$	499.651	0.573375
$16$	0	0
$17$	−1443.81	−1.21168	−0.605839	−	0.795587i	$-0.707162\pi$
$17$	−1443.81	−1.21168	−0.605839	+	0.795587i	$0.707162\pi$
$18$	0	0
$19$	982.167	0.624168	0.312084	−	0.950055i	$-0.398973\pi$
$19$	982.167	0.624168	0.312084	+	0.950055i	$0.398973\pi$
$20$	0	0
$21$	22.5416	0.0111542
$22$	0	0
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	−42.8875	−0.0137240
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−7157.56	−1.58041	−0.790205	−	0.612842i	$-0.790026\pi$
$29$	−7157.56	−1.58041	−0.790205	+	0.612842i	$0.790026\pi$
$30$	0	0
$31$	9259.98	1.73064	0.865318	−	0.501223i	$-0.167116\pi$
$31$	9259.98	1.73064	0.865318	+	0.501223i	$0.167116\pi$
$32$	0	0
$33$	2056.95	0.328805
$34$	0	0
$35$	139.049	0.0191865
$36$	0	0
$37$	2422.50	0.290911	0.145455	−	0.989365i	$-0.453535\pi$
$37$	2422.50	0.290911	0.145455	+	0.989365i	$0.453535\pi$
$38$	0	0
$39$	−5928.33	−0.624124
$40$	0	0
$41$	−4075.27	−0.378614	−0.189307	−	0.981918i	$-0.560624\pi$
$41$	−4075.27	−0.378614	−0.189307	+	0.981918i	$0.560624\pi$
$42$	0	0
$43$	10417.3	0.859178	0.429589	−	0.903024i	$-0.358658\pi$
$43$	10417.3	0.859178	0.429589	+	0.903024i	$0.358658\pi$
$44$	0	0
$45$	−4496.86	−0.331038
$46$	0	0
$47$	−9358.08	−0.617934	−0.308967	−	0.951073i	$-0.599983\pi$
$47$	−9358.08	−0.617934	−0.308967	+	0.951073i	$0.599983\pi$
$48$	0	0
$49$	−16800.7	−0.999627
$50$	0	0
$51$	12994.3	0.699563
$52$	0	0
$53$	−34280.3	−1.67631	−0.838157	−	0.545428i	$-0.816367\pi$
$53$	−34280.3	−1.67631	−0.838157	+	0.545428i	$0.816367\pi$
$54$	0	0
$55$	12688.4	0.565586
$56$	0	0
$57$	−8839.51	−0.360364
$58$	0	0
$59$	7268.79	0.271852	0.135926	−	0.990719i	$-0.456599\pi$
$59$	7268.79	0.271852	0.135926	+	0.990719i	$0.456599\pi$
$60$	0	0
$61$	26611.7	0.915687	0.457844	−	0.889033i	$-0.348622\pi$
$61$	26611.7	0.915687	0.457844	+	0.889033i	$0.348622\pi$
$62$	0	0
$63$	−202.874	−0.00643985
$64$	0	0
$65$	−36569.1	−1.07357
$66$	0	0
$67$	−53450.8	−1.45468	−0.727339	−	0.686278i	$-0.759243\pi$
$67$	−53450.8	−1.45468	−0.727339	+	0.686278i	$0.759243\pi$
$68$	0	0
$69$	4761.00	0.120386
$70$	0	0
$71$	−21673.7	−0.510255	−0.255128	−	0.966907i	$-0.582118\pi$
$71$	−21673.7	−0.510255	−0.255128	+	0.966907i	$0.582118\pi$
$72$	0	0
$73$	−82856.3	−1.81978	−0.909889	−	0.414851i	$-0.863834\pi$
$73$	−82856.3	−1.81978	−0.909889	+	0.414851i	$0.863834\pi$
$74$	0	0
$75$	385.987	0.00792355
$76$	0	0
$77$	572.432	0.0110026
$78$	0	0
$79$	23960.0	0.431936	0.215968	−	0.976400i	$-0.430709\pi$
$79$	23960.0	0.431936	0.215968	+	0.976400i	$0.430709\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−81187.7	−1.29359	−0.646793	−	0.762666i	$-0.723890\pi$
$83$	−81187.7	−1.29359	−0.646793	+	0.762666i	$0.723890\pi$
$84$	0	0
$85$	80155.6	1.20334
$86$	0	0
$87$	64418.0	0.912451
$88$	0	0
$89$	100115.	1.33975	0.669876	−	0.742473i	$-0.266347\pi$
$89$	100115.	1.33975	0.669876	+	0.742473i	$0.266347\pi$
$90$	0	0
$91$	−1649.80	−0.0208847
$92$	0	0
$93$	−83339.8	−0.999183
$94$	0	0
$95$	−54526.8	−0.619870
$96$	0	0
$97$	36122.1	0.389802	0.194901	−	0.980823i	$-0.437562\pi$
$97$	36122.1	0.389802	0.194901	+	0.980823i	$0.437562\pi$
$98$	0	0
$99$	−18512.6	−0.189836
$100$	0	0
$101$	−41151.8	−0.401408	−0.200704	−	0.979652i	$-0.564323\pi$
$101$	−41151.8	−0.401408	−0.200704	+	0.979652i	$0.564323\pi$
$102$	0	0
$103$	172534.	1.60244	0.801221	−	0.598369i	$-0.204184\pi$
$103$	172534.	1.60244	0.801221	+	0.598369i	$0.204184\pi$
$104$	0	0
$105$	−1251.44	−0.0110774
$106$	0	0
$107$	−178228.	−1.50493	−0.752464	−	0.658633i	$-0.771135\pi$
$107$	−178228.	−1.50493	−0.752464	+	0.658633i	$0.771135\pi$
$108$	0	0
$109$	−138352.	−1.11537	−0.557686	−	0.830052i	$-0.688311\pi$
$109$	−138352.	−1.11537	−0.557686	+	0.830052i	$0.688311\pi$
$110$	0	0
$111$	−21802.5	−0.167958
$112$	0	0
$113$	−13523.5	−0.0996307	−0.0498153	−	0.998758i	$-0.515863\pi$
$113$	−13523.5	−0.0996307	−0.0498153	+	0.998758i	$0.515863\pi$
$114$	0	0
$115$	29368.4	0.207079
$116$	0	0
$117$	53355.0	0.360338
$118$	0	0
$119$	3616.20	0.0234091
$120$	0	0
$121$	−108816.	−0.675661
$122$	0	0
$123$	36677.4	0.218593
$124$	0	0
$125$	175871.	1.00674
$126$	0	0
$127$	−158567.	−0.872376	−0.436188	−	0.899856i	$-0.643672\pi$
$127$	−158567.	−0.872376	−0.436188	+	0.899856i	$0.643672\pi$
$128$	0	0
$129$	−93755.6	−0.496047
$130$	0	0
$131$	383514.	1.95255	0.976277	−	0.216526i	$-0.0694727\pi$
$131$	383514.	1.95255	0.976277	+	0.216526i	$0.0694727\pi$
$132$	0	0
$133$	−2459.96	−0.0120587
$134$	0	0
$135$	40471.7	0.191125
$136$	0	0
$137$	−33171.1	−0.150994	−0.0754968	−	0.997146i	$-0.524054\pi$
$137$	−33171.1	−0.150994	−0.0754968	+	0.997146i	$0.524054\pi$
$138$	0	0
$139$	128488.	0.564060	0.282030	−	0.959406i	$-0.408992\pi$
$139$	128488.	0.564060	0.282030	+	0.959406i	$0.408992\pi$
$140$	0	0
$141$	84222.7	0.356764
$142$	0	0
$143$	−150547.	−0.615646
$144$	0	0
$145$	397365.	1.56953
$146$	0	0
$147$	151207.	0.577135
$148$	0	0
$149$	−184228.	−0.679814	−0.339907	−	0.940459i	$-0.610396\pi$
$149$	−184228.	−0.679814	−0.339907	+	0.940459i	$0.610396\pi$
$150$	0	0
$151$	19798.1	0.0706610	0.0353305	−	0.999376i	$-0.488752\pi$
$151$	19798.1	0.0706610	0.0353305	+	0.999376i	$0.488752\pi$
$152$	0	0
$153$	−116948.	−0.403893
$154$	0	0
$155$	−514084.	−1.71872
$156$	0	0
$157$	193317.	0.625923	0.312962	−	0.949766i	$-0.398679\pi$
$157$	193317.	0.625923	0.312962	+	0.949766i	$0.398679\pi$
$158$	0	0
$159$	308523.	0.967821
$160$	0	0
$161$	1324.95	0.00402841
$162$	0	0
$163$	−106286.	−0.313334	−0.156667	−	0.987651i	$-0.550075\pi$
$163$	−106286.	−0.313334	−0.156667	+	0.987651i	$0.550075\pi$
$164$	0	0
$165$	−114195.	−0.326541
$166$	0	0
$167$	77715.5	0.215634	0.107817	−	0.994171i	$-0.465614\pi$
$167$	77715.5	0.215634	0.107817	+	0.994171i	$0.465614\pi$
$168$	0	0
$169$	62597.3	0.168593
$170$	0	0
$171$	79555.5	0.208056
$172$	0	0
$173$	218524.	0.555115	0.277558	−	0.960709i	$-0.410475\pi$
$173$	218524.	0.555115	0.277558	+	0.960709i	$0.410475\pi$
$174$	0	0
$175$	107.417	0.000265142 0
$176$	0	0
$177$	−65419.1	−0.156954
$178$	0	0
$179$	545493.	1.27250	0.636248	−	0.771485i	$-0.280486\pi$
$179$	545493.	1.27250	0.636248	+	0.771485i	$0.280486\pi$
$180$	0	0
$181$	510878.	1.15910	0.579549	−	0.814937i	$-0.303229\pi$
$181$	510878.	1.15910	0.579549	+	0.814937i	$0.303229\pi$
$182$	0	0
$183$	−239505.	−0.528672
$184$	0	0
$185$	−134490.	−0.288908
$186$	0	0
$187$	329982.	0.690060
$188$	0	0
$189$	1825.87	0.00371805
$190$	0	0
$191$	57685.2	0.114415	0.0572073	−	0.998362i	$-0.481780\pi$
$191$	57685.2	0.114415	0.0572073	+	0.998362i	$0.481780\pi$
$192$	0	0
$193$	−822091.	−1.58864	−0.794322	−	0.607497i	$-0.792174\pi$
$193$	−822091.	−1.58864	−0.794322	+	0.607497i	$0.792174\pi$
$194$	0	0
$195$	329122.	0.619827
$196$	0	0
$197$	971571.	1.78365	0.891824	−	0.452383i	$-0.149426\pi$
$197$	971571.	1.78365	0.891824	+	0.452383i	$0.149426\pi$
$198$	0	0
$199$	−806110.	−1.44298	−0.721492	−	0.692423i	$-0.756543\pi$
$199$	−806110.	−1.44298	−0.721492	+	0.692423i	$0.756543\pi$
$200$	0	0
$201$	481057.	0.839859
$202$	0	0
$203$	17927.0	0.0305329
$204$	0	0
$205$	226246.	0.376007
$206$	0	0
$207$	−42849.0	−0.0695048
$208$	0	0
$209$	−224474.	−0.355468
$210$	0	0
$211$	821913.	1.27092	0.635462	−	0.772132i	$-0.280810\pi$
$211$	821913.	1.27092	0.635462	+	0.772132i	$0.280810\pi$
$212$	0	0
$213$	195063.	0.294596
$214$	0	0
$215$	−578334.	−0.853262
$216$	0	0
$217$	−23192.8	−0.0334351
$218$	0	0
$219$	745707.	1.05065
$220$	0	0
$221$	−951042.	−1.30984
$222$	0	0
$223$	−511948.	−0.689388	−0.344694	−	0.938715i	$-0.612017\pi$
$223$	−511948.	−0.689388	−0.344694	+	0.938715i	$0.612017\pi$
$224$	0	0
$225$	−3473.89	−0.00457467
$226$	0	0
$227$	1.45074e6	1.86864	0.934321	−	0.356433i	$-0.116007\pi$
$227$	1.45074e6	1.86864	0.934321	+	0.356433i	$0.116007\pi$
$228$	0	0
$229$	621169.	0.782747	0.391373	−	0.920232i	$-0.372000\pi$
$229$	621169.	0.782747	0.391373	+	0.920232i	$0.372000\pi$
$230$	0	0
$231$	−5151.89	−0.00635238
$232$	0	0
$233$	490334.	0.591701	0.295851	−	0.955234i	$-0.404397\pi$
$233$	490334.	0.591701	0.295851	+	0.955234i	$0.404397\pi$
$234$	0	0
$235$	519530.	0.613679
$236$	0	0
$237$	−215640.	−0.249378
$238$	0	0
$239$	−109503.	−0.124003	−0.0620014	−	0.998076i	$-0.519748\pi$
$239$	−109503.	−0.124003	−0.0620014	+	0.998076i	$0.519748\pi$
$240$	0	0
$241$	1.36171e6	1.51022	0.755111	−	0.655597i	$-0.227583\pi$
$241$	1.36171e6	1.51022	0.755111	+	0.655597i	$0.227583\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	932722.	0.992744
$246$	0	0
$247$	646957.	0.674735
$248$	0	0
$249$	730690.	0.746852
$250$	0	0
$251$	−8042.27	−0.00805739	−0.00402869	−	0.999992i	$-0.501282\pi$
$251$	−8042.27	−0.00805739	−0.00402869	+	0.999992i	$0.501282\pi$
$252$	0	0
$253$	120903.	0.118751
$254$	0	0
$255$	−721400.	−0.694746
$256$	0	0
$257$	1.16730e6	1.10243	0.551214	−	0.834364i	$-0.314165\pi$
$257$	1.16730e6	1.10243	0.551214	+	0.834364i	$0.314165\pi$
$258$	0	0
$259$	−6067.46	−0.00562027
$260$	0	0
$261$	−579762.	−0.526804
$262$	0	0
$263$	558051.	0.497490	0.248745	−	0.968569i	$-0.419982\pi$
$263$	558051.	0.497490	0.248745	+	0.968569i	$0.419982\pi$
$264$	0	0
$265$	1.90313e6	1.66477
$266$	0	0
$267$	−901036.	−0.773506
$268$	0	0
$269$	−1.21947e6	−1.02752	−0.513759	−	0.857935i	$-0.671747\pi$
$269$	−1.21947e6	−1.02752	−0.513759	+	0.857935i	$0.671747\pi$
$270$	0	0
$271$	−1.40581e6	−1.16280	−0.581399	−	0.813618i	$-0.697495\pi$
$271$	−1.40581e6	−1.16280	−0.581399	+	0.813618i	$0.697495\pi$
$272$	0	0
$273$	14848.2	0.0120578
$274$	0	0
$275$	9801.94	0.00781592
$276$	0	0
$277$	1.87799e6	1.47060	0.735300	−	0.677742i	$-0.237041\pi$
$277$	1.87799e6	1.47060	0.735300	+	0.677742i	$0.237041\pi$
$278$	0	0
$279$	750058.	0.576879
$280$	0	0
$281$	−2.11759e6	−1.59984	−0.799920	−	0.600106i	$-0.795125\pi$
$281$	−2.11759e6	−1.59984	−0.799920	+	0.600106i	$0.795125\pi$
$282$	0	0
$283$	−1.16042e6	−0.861288	−0.430644	−	0.902522i	$-0.641714\pi$
$283$	−1.16042e6	−0.861288	−0.430644	+	0.902522i	$0.641714\pi$
$284$	0	0
$285$	490741.	0.357882
$286$	0	0
$287$	10207.0	0.00731466
$288$	0	0
$289$	664726.	0.468164
$290$	0	0
$291$	−325099.	−0.225052
$292$	0	0
$293$	−1.27325e6	−0.866449	−0.433225	−	0.901286i	$-0.642624\pi$
$293$	−1.27325e6	−0.866449	−0.433225	+	0.901286i	$0.642624\pi$
$294$	0	0
$295$	−403540.	−0.269980
$296$	0	0
$297$	166613.	0.109602
$298$	0	0
$299$	−348454.	−0.225407
$300$	0	0
$301$	−26091.4	−0.0165990
$302$	0	0
$303$	370367.	0.231753
$304$	0	0
$305$	−1.47739e6	−0.909382
$306$	0	0
$307$	2.26175e6	1.36961	0.684807	−	0.728724i	$-0.259886\pi$
$307$	2.26175e6	1.36961	0.684807	+	0.728724i	$0.259886\pi$
$308$	0	0
$309$	−1.55281e6	−0.925170
$310$	0	0
$311$	−1.20263e6	−0.705066	−0.352533	−	0.935799i	$-0.614680\pi$
$311$	−1.20263e6	−0.705066	−0.352533	+	0.935799i	$0.614680\pi$
$312$	0	0
$313$	−484311.	−0.279424	−0.139712	−	0.990192i	$-0.544618\pi$
$313$	−484311.	−0.279424	−0.139712	+	0.990192i	$0.544618\pi$
$314$	0	0
$315$	11262.9	0.00639551
$316$	0	0
$317$	796624.	0.445251	0.222626	−	0.974904i	$-0.428537\pi$
$317$	796624.	0.445251	0.222626	+	0.974904i	$0.428537\pi$
$318$	0	0
$319$	1.63586e6	0.900056
$320$	0	0
$321$	1.60405e6	0.868871
$322$	0	0
$323$	−1.41806e6	−0.756291
$324$	0	0
$325$	−28250.1	−0.0148358
$326$	0	0
$327$	1.24517e6	0.643961
$328$	0	0
$329$	23438.5	0.0119382
$330$	0	0
$331$	262856.	0.131871	0.0659353	−	0.997824i	$-0.478997\pi$
$331$	262856.	0.131871	0.0659353	+	0.997824i	$0.478997\pi$
$332$	0	0
$333$	196223.	0.0969703
$334$	0	0
$335$	2.96741e6	1.44466
$336$	0	0
$337$	1.23123e6	0.590558	0.295279	−	0.955411i	$-0.404587\pi$
$337$	1.23123e6	0.590558	0.295279	+	0.955411i	$0.404587\pi$
$338$	0	0
$339$	121712.	0.0575218
$340$	0	0
$341$	−2.11637e6	−0.985611
$342$	0	0
$343$	84174.7	0.0386319
$344$	0	0
$345$	−264315.	−0.119557
$346$	0	0
$347$	3.70344e6	1.65113	0.825567	−	0.564305i	$-0.190856\pi$
$347$	3.70344e6	1.65113	0.825567	+	0.564305i	$0.190856\pi$
$348$	0	0
$349$	1.62884e6	0.715837	0.357918	−	0.933753i	$-0.383487\pi$
$349$	1.62884e6	0.715837	0.357918	+	0.933753i	$0.383487\pi$
$350$	0	0
$351$	−480195.	−0.208041
$352$	0	0
$353$	2.56589e6	1.09598	0.547989	−	0.836486i	$-0.315394\pi$
$353$	2.56589e6	1.09598	0.547989	+	0.836486i	$0.315394\pi$
$354$	0	0
$355$	1.20326e6	0.506742
$356$	0	0
$357$	−32545.8	−0.0135152
$358$	0	0
$359$	3.46895e6	1.42057	0.710284	−	0.703916i	$-0.248567\pi$
$359$	3.46895e6	1.42057	0.710284	+	0.703916i	$0.248567\pi$
$360$	0	0
$361$	−1.51145e6	−0.610414
$362$	0	0
$363$	979343.	0.390093
$364$	0	0
$365$	4.59992e6	1.80725
$366$	0	0
$367$	−1.64135e6	−0.636115	−0.318058	−	0.948071i	$-0.603031\pi$
$367$	−1.64135e6	−0.636115	−0.318058	+	0.948071i	$0.603031\pi$
$368$	0	0
$369$	−330097.	−0.126205
$370$	0	0
$371$	85859.4	0.0323857
$372$	0	0
$373$	1.58607e6	0.590268	0.295134	−	0.955456i	$-0.404636\pi$
$373$	1.58607e6	0.590268	0.295134	+	0.955456i	$0.404636\pi$
$374$	0	0
$375$	−1.58284e6	−0.581244
$376$	0	0
$377$	−4.71471e6	−1.70845
$378$	0	0
$379$	966030.	0.345456	0.172728	−	0.984970i	$-0.444742\pi$
$379$	966030.	0.345456	0.172728	+	0.984970i	$0.444742\pi$
$380$	0	0
$381$	1.42710e6	0.503667
$382$	0	0
$383$	3.10322e6	1.08097	0.540487	−	0.841352i	$-0.318240\pi$
$383$	3.10322e6	1.08097	0.540487	+	0.841352i	$0.318240\pi$
$384$	0	0
$385$	−31779.6	−0.0109269
$386$	0	0
$387$	843800.	0.286393
$388$	0	0
$389$	624568.	0.209269	0.104635	−	0.994511i	$-0.466633\pi$
$389$	624568.	0.209269	0.104635	+	0.994511i	$0.466633\pi$
$390$	0	0
$391$	763775.	0.252652
$392$	0	0
$393$	−3.45163e6	−1.12731
$394$	0	0
$395$	−1.33018e6	−0.428962
$396$	0	0
$397$	2.80814e6	0.894216	0.447108	−	0.894480i	$-0.352454\pi$
$397$	2.80814e6	0.894216	0.447108	+	0.894480i	$0.352454\pi$
$398$	0	0
$399$	22139.6	0.00696207
$400$	0	0
$401$	3.81287e6	1.18411	0.592053	−	0.805899i	$-0.298318\pi$
$401$	3.81287e6	1.18411	0.592053	+	0.805899i	$0.298318\pi$
$402$	0	0
$403$	6.09958e6	1.87084
$404$	0	0
$405$	−364246.	−0.110346
$406$	0	0
$407$	−553663.	−0.165676
$408$	0	0
$409$	−3.86268e6	−1.14178	−0.570888	−	0.821028i	$-0.693401\pi$
$409$	−3.86268e6	−1.14178	−0.570888	+	0.821028i	$0.693401\pi$
$410$	0	0
$411$	298540.	0.0871762
$412$	0	0
$413$	−18205.6	−0.00525206
$414$	0	0
$415$	4.50728e6	1.28468
$416$	0	0
$417$	−1.15639e6	−0.325660
$418$	0	0
$419$	4.00521e6	1.11453	0.557263	−	0.830336i	$-0.311851\pi$
$419$	4.00521e6	1.11453	0.557263	+	0.830336i	$0.311851\pi$
$420$	0	0
$421$	4.89697e6	1.34655	0.673275	−	0.739392i	$-0.264887\pi$
$421$	4.89697e6	1.34655	0.673275	+	0.739392i	$0.264887\pi$
$422$	0	0
$423$	−758005.	−0.205978
$424$	0	0
$425$	61921.3	0.0166291
$426$	0	0
$427$	−66652.2	−0.0176907
$428$	0	0
$429$	1.35492e6	0.355443
$430$	0	0
$431$	−1.81874e6	−0.471605	−0.235803	−	0.971801i	$-0.575772\pi$
$431$	−1.81874e6	−0.471605	−0.235803	+	0.971801i	$0.575772\pi$
$432$	0	0
$433$	545672.	0.139866	0.0699330	−	0.997552i	$-0.477721\pi$
$433$	545672.	0.139866	0.0699330	+	0.997552i	$0.477721\pi$
$434$	0	0
$435$	−3.57628e6	−0.906168
$436$	0	0
$437$	−519566.	−0.130148
$438$	0	0
$439$	−2.81114e6	−0.696179	−0.348090	−	0.937461i	$-0.613170\pi$
$439$	−2.81114e6	−0.696179	−0.348090	+	0.937461i	$0.613170\pi$
$440$	0	0
$441$	−1.36086e6	−0.333209
$442$	0	0
$443$	2.81428e6	0.681332	0.340666	−	0.940184i	$-0.389348\pi$
$443$	2.81428e6	0.681332	0.340666	+	0.940184i	$0.389348\pi$
$444$	0	0
$445$	−5.55807e6	−1.33053
$446$	0	0
$447$	1.65805e6	0.392491
$448$	0	0
$449$	4.00644e6	0.937871	0.468936	−	0.883232i	$-0.344638\pi$
$449$	4.00644e6	0.937871	0.468936	+	0.883232i	$0.344638\pi$
$450$	0	0
$451$	931403.	0.215624
$452$	0	0
$453$	−178182.	−0.0407962
$454$	0	0
$455$	91591.8	0.0207409
$456$	0	0
$457$	−5.43801e6	−1.21801	−0.609003	−	0.793168i	$-0.708430\pi$
$457$	−5.43801e6	−1.21801	−0.609003	+	0.793168i	$0.708430\pi$
$458$	0	0
$459$	1.05254e6	0.233188
$460$	0	0
$461$	2.50966e6	0.550001	0.275000	−	0.961444i	$-0.411322\pi$
$461$	2.50966e6	0.550001	0.275000	+	0.961444i	$0.411322\pi$
$462$	0	0
$463$	−108662.	−0.0235573	−0.0117787	−	0.999931i	$-0.503749\pi$
$463$	−108662.	−0.0235573	−0.0117787	+	0.999931i	$0.503749\pi$
$464$	0	0
$465$	4.62676e6	0.992303
$466$	0	0
$467$	−3.09125e6	−0.655906	−0.327953	−	0.944694i	$-0.606359\pi$
$467$	−3.09125e6	−0.655906	−0.327953	+	0.944694i	$0.606359\pi$
$468$	0	0
$469$	133874.	0.0281037
$470$	0	0
$471$	−1.73985e6	−0.361377
$472$	0	0
$473$	−2.38087e6	−0.489309
$474$	0	0
$475$	−42122.7	−0.00856608
$476$	0	0
$477$	−2.77671e6	−0.558772
$478$	0	0
$479$	7.74743e6	1.54283	0.771417	−	0.636330i	$-0.219548\pi$
$479$	7.74743e6	1.54283	0.771417	+	0.636330i	$0.219548\pi$
$480$	0	0
$481$	1.59571e6	0.314479
$482$	0	0
$483$	−11924.5	−0.00232580
$484$	0	0
$485$	−2.00538e6	−0.387118
$486$	0	0
$487$	−6.60929e6	−1.26279	−0.631397	−	0.775460i	$-0.717518\pi$
$487$	−6.60929e6	−1.26279	−0.631397	+	0.775460i	$0.717518\pi$
$488$	0	0
$489$	956575.	0.180903
$490$	0	0
$491$	−1.68453e6	−0.315337	−0.157669	−	0.987492i	$-0.550398\pi$
$491$	−1.68453e6	−0.315337	−0.157669	+	0.987492i	$0.550398\pi$
$492$	0	0
$493$	1.03341e7	1.91495
$494$	0	0
$495$	1.02776e6	0.188529
$496$	0	0
$497$	54284.5	0.00985791
$498$	0	0
$499$	−6.29003e6	−1.13084	−0.565420	−	0.824803i	$-0.691286\pi$
$499$	−6.29003e6	−1.13084	−0.565420	+	0.824803i	$0.691286\pi$
$500$	0	0
$501$	−699440.	−0.124496
$502$	0	0
$503$	−3.68350e6	−0.649143	−0.324572	−	0.945861i	$-0.605220\pi$
$503$	−3.68350e6	−0.649143	−0.324572	+	0.945861i	$0.605220\pi$
$504$	0	0
$505$	2.28462e6	0.398644
$506$	0	0
$507$	−563376.	−0.0973371
$508$	0	0
$509$	3.33612e6	0.570752	0.285376	−	0.958416i	$-0.407882\pi$
$509$	3.33612e6	0.570752	0.285376	+	0.958416i	$0.407882\pi$
$510$	0	0
$511$	207524.	0.0351573
$512$	0	0
$513$	−716000.	−0.120121
$514$	0	0
$515$	−9.57854e6	−1.59141
$516$	0	0
$517$	2.13879e6	0.351918
$518$	0	0
$519$	−1.96671e6	−0.320496
$520$	0	0
$521$	−987399.	−0.159367	−0.0796835	−	0.996820i	$-0.525391\pi$
$521$	−987399.	−0.159367	−0.0796835	+	0.996820i	$0.525391\pi$
$522$	0	0
$523$	−6.57834e6	−1.05163	−0.525814	−	0.850600i	$-0.676239\pi$
$523$	−6.57834e6	−1.05163	−0.525814	+	0.850600i	$0.676239\pi$
$524$	0	0
$525$	−966.753	−0.000153080 0
$526$	0	0
$527$	−1.33696e7	−2.09697
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	588772.	0.0906172
$532$	0	0
$533$	−2.68439e6	−0.409287
$534$	0	0
$535$	9.89463e6	1.49457
$536$	0	0
$537$	−4.90943e6	−0.734676
$538$	0	0
$539$	3.83981e6	0.569295
$540$	0	0
$541$	−6.47470e6	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$541$	−6.47470e6	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$542$	0	0
$543$	−4.59790e6	−0.669206
$544$	0	0
$545$	7.68087e6	1.10769
$546$	0	0
$547$	495799.	0.0708496	0.0354248	−	0.999372i	$-0.488722\pi$
$547$	495799.	0.0708496	0.0354248	+	0.999372i	$0.488722\pi$
$548$	0	0
$549$	2.15554e6	0.305229
$550$	0	0
$551$	−7.02992e6	−0.986442
$552$	0	0
$553$	−60010.8	−0.00834481
$554$	0	0
$555$	1.21041e6	0.166801
$556$	0	0
$557$	3.94887e6	0.539306	0.269653	−	0.962958i	$-0.413091\pi$
$557$	3.94887e6	0.539306	0.269653	+	0.962958i	$0.413091\pi$
$558$	0	0
$559$	6.86190e6	0.928784
$560$	0	0
$561$	−2.96984e6	−0.398406
$562$	0	0
$563$	2.70237e6	0.359314	0.179657	−	0.983729i	$-0.442501\pi$
$563$	2.70237e6	0.359314	0.179657	+	0.983729i	$0.442501\pi$
$564$	0	0
$565$	750781.	0.0989447
$566$	0	0
$567$	−16432.8	−0.00214662
$568$	0	0
$569$	1.33108e7	1.72355	0.861773	−	0.507294i	$-0.169354\pi$
$569$	1.33108e7	1.72355	0.861773	+	0.507294i	$0.169354\pi$
$570$	0	0
$571$	1.10053e7	1.41257	0.706286	−	0.707926i	$-0.250369\pi$
$571$	1.10053e7	1.41257	0.706286	+	0.707926i	$0.250369\pi$
$572$	0	0
$573$	−519167.	−0.0660572
$574$	0	0
$575$	22687.5	0.00286165
$576$	0	0
$577$	−480544.	−0.0600888	−0.0300444	−	0.999549i	$-0.509565\pi$
$577$	−480544.	−0.0600888	−0.0300444	+	0.999549i	$0.509565\pi$
$578$	0	0
$579$	7.39882e6	0.917204
$580$	0	0
$581$	203345.	0.0249915
$582$	0	0
$583$	7.83477e6	0.954674
$584$	0	0
$585$	−2.96210e6	−0.357857
$586$	0	0
$587$	−9.12805e6	−1.09341	−0.546705	−	0.837325i	$-0.684118\pi$
$587$	−9.12805e6	−1.09341	−0.546705	+	0.837325i	$0.684118\pi$
$588$	0	0
$589$	9.09485e6	1.08021
$590$	0	0
$591$	−8.74414e6	−1.02979
$592$	0	0
$593$	−2.66114e6	−0.310765	−0.155382	−	0.987854i	$-0.549661\pi$
$593$	−2.66114e6	−0.310765	−0.155382	+	0.987854i	$0.549661\pi$
$594$	0	0
$595$	−200760.	−0.0232479
$596$	0	0
$597$	7.25499e6	0.833107
$598$	0	0
$599$	−6.44915e6	−0.734405	−0.367202	−	0.930141i	$-0.619684\pi$
$599$	−6.44915e6	−0.734405	−0.367202	+	0.930141i	$0.619684\pi$
$600$	0	0
$601$	−1.48491e6	−0.167693	−0.0838463	−	0.996479i	$-0.526720\pi$
$601$	−1.48491e6	−0.167693	−0.0838463	+	0.996479i	$0.526720\pi$
$602$	0	0
$603$	−4.32951e6	−0.484893
$604$	0	0
$605$	6.04111e6	0.671009
$606$	0	0
$607$	6.56656e6	0.723379	0.361690	−	0.932299i	$-0.382200\pi$
$607$	6.56656e6	0.723379	0.361690	+	0.932299i	$0.382200\pi$
$608$	0	0
$609$	−161343.	−0.0176281
$610$	0	0
$611$	−6.16420e6	−0.667996
$612$	0	0
$613$	−9.33817e6	−1.00372	−0.501858	−	0.864950i	$-0.667350\pi$
$613$	−9.33817e6	−1.00372	−0.501858	+	0.864950i	$0.667350\pi$
$614$	0	0
$615$	−2.03621e6	−0.217088
$616$	0	0
$617$	7.56128e6	0.799617	0.399809	−	0.916599i	$-0.369077\pi$
$617$	7.56128e6	0.799617	0.399809	+	0.916599i	$0.369077\pi$
$618$	0	0
$619$	−1.16180e7	−1.21872	−0.609359	−	0.792895i	$-0.708573\pi$
$619$	−1.16180e7	−1.21872	−0.609359	+	0.792895i	$0.708573\pi$
$620$	0	0
$621$	385641.	0.0401286
$622$	0	0
$623$	−250751.	−0.0258834
$624$	0	0
$625$	−9.62976e6	−0.986088
$626$	0	0
$627$	2.02027e6	0.205230
$628$	0	0
$629$	−3.49763e6	−0.352491
$630$	0	0
$631$	8.43872e6	0.843730	0.421865	−	0.906659i	$-0.361376\pi$
$631$	8.43872e6	0.843730	0.421865	+	0.906659i	$0.361376\pi$
$632$	0	0
$633$	−7.39722e6	−0.733768
$634$	0	0
$635$	8.80313e6	0.866369
$636$	0	0
$637$	−1.10667e7	−1.08061
$638$	0	0
$639$	−1.75557e6	−0.170085
$640$	0	0
$641$	1.32010e7	1.26900	0.634500	−	0.772923i	$-0.281206\pi$
$641$	1.32010e7	1.26900	0.634500	+	0.772923i	$0.281206\pi$
$642$	0	0
$643$	1.57321e7	1.50058	0.750290	−	0.661109i	$-0.229914\pi$
$643$	1.57321e7	1.50058	0.750290	+	0.661109i	$0.229914\pi$
$644$	0	0
$645$	5.20501e6	0.492631
$646$	0	0
$647$	6.22066e6	0.584219	0.292109	−	0.956385i	$-0.405643\pi$
$647$	6.22066e6	0.584219	0.292109	+	0.956385i	$0.405643\pi$
$648$	0	0
$649$	−1.66128e6	−0.154822
$650$	0	0
$651$	208735.	0.0193038
$652$	0	0
$653$	−1.57628e7	−1.44660	−0.723302	−	0.690532i	$-0.757376\pi$
$653$	−1.57628e7	−1.44660	−0.723302	+	0.690532i	$0.757376\pi$
$654$	0	0
$655$	−2.12915e7	−1.93911
$656$	0	0
$657$	−6.71136e6	−0.606593
$658$	0	0
$659$	5.77008e6	0.517569	0.258784	−	0.965935i	$-0.416678\pi$
$659$	5.77008e6	0.517569	0.258784	+	0.965935i	$0.416678\pi$
$660$	0	0
$661$	−1.48192e7	−1.31924	−0.659618	−	0.751601i	$-0.729282\pi$
$661$	−1.48192e7	−1.31924	−0.659618	+	0.751601i	$0.729282\pi$
$662$	0	0
$663$	8.55938e6	0.756238
$664$	0	0
$665$	136569.	0.0119756
$666$	0	0
$667$	3.78635e6	0.329538
$668$	0	0
$669$	4.60753e6	0.398018
$670$	0	0
$671$	−6.08209e6	−0.521491
$672$	0	0
$673$	−1.74615e7	−1.48609	−0.743045	−	0.669242i	$-0.766619\pi$
$673$	−1.74615e7	−1.48609	−0.743045	+	0.669242i	$0.766619\pi$
$674$	0	0
$675$	31265.0	0.00264118
$676$	0	0
$677$	−2.10975e7	−1.76913	−0.884564	−	0.466418i	$-0.845544\pi$
$677$	−2.10975e7	−1.76913	−0.884564	+	0.466418i	$0.845544\pi$
$678$	0	0
$679$	−90472.2	−0.00753080
$680$	0	0
$681$	−1.30567e7	−1.07886
$682$	0	0
$683$	1.43654e7	1.17833	0.589163	−	0.808014i	$-0.299458\pi$
$683$	1.43654e7	1.17833	0.589163	+	0.808014i	$0.299458\pi$
$684$	0	0
$685$	1.84155e6	0.149954
$686$	0	0
$687$	−5.59052e6	−0.451919
$688$	0	0
$689$	−2.25806e7	−1.81212
$690$	0	0
$691$	−1.64154e7	−1.30785	−0.653923	−	0.756561i	$-0.726878\pi$
$691$	−1.64154e7	−1.30785	−0.653923	+	0.756561i	$0.726878\pi$
$692$	0	0
$693$	46367.0	0.00366755
$694$	0	0
$695$	−7.13324e6	−0.560176
$696$	0	0
$697$	5.88391e6	0.458758
$698$	0	0
$699$	−4.41301e6	−0.341619
$700$	0	0
$701$	−1.07940e7	−0.829635	−0.414818	−	0.909905i	$-0.636155\pi$
$701$	−1.07940e7	−0.829635	−0.414818	+	0.909905i	$0.636155\pi$
$702$	0	0
$703$	2.37930e6	0.181577
$704$	0	0
$705$	−4.67577e6	−0.354308
$706$	0	0
$707$	103070.	0.00775502
$708$	0	0
$709$	2.44390e7	1.82586	0.912930	−	0.408115i	$-0.133814\pi$
$709$	2.44390e7	1.82586	0.912930	+	0.408115i	$0.133814\pi$
$710$	0	0
$711$	1.94076e6	0.143979
$712$	0	0
$713$	−4.89853e6	−0.360863
$714$	0	0
$715$	8.35787e6	0.611407
$716$	0	0
$717$	985527.	0.0715930
$718$	0	0
$719$	6.41280e6	0.462621	0.231310	−	0.972880i	$-0.425699\pi$
$719$	6.41280e6	0.462621	0.231310	+	0.972880i	$0.425699\pi$
$720$	0	0
$721$	−432133.	−0.0309585
$722$	0	0
$723$	−1.22554e7	−0.871928
$724$	0	0
$725$	306970.	0.0216896
$726$	0	0
$727$	1.58788e7	1.11424	0.557122	−	0.830431i	$-0.311905\pi$
$727$	1.58788e7	1.11424	0.557122	+	0.830431i	$0.311905\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.50406e7	−1.04105
$732$	0	0
$733$	2.52517e7	1.73592	0.867962	−	0.496631i	$-0.165430\pi$
$733$	2.52517e7	1.73592	0.867962	+	0.496631i	$0.165430\pi$
$734$	0	0
$735$	−8.39450e6	−0.573161
$736$	0	0
$737$	1.22162e7	0.828450
$738$	0	0
$739$	1.56260e7	1.05253	0.526267	−	0.850319i	$-0.323591\pi$
$739$	1.56260e7	1.05253	0.526267	+	0.850319i	$0.323591\pi$
$740$	0	0
$741$	−5.82261e6	−0.389558
$742$	0	0
$743$	−1.42696e7	−0.948284	−0.474142	−	0.880448i	$-0.657242\pi$
$743$	−1.42696e7	−0.948284	−0.474142	+	0.880448i	$0.657242\pi$
$744$	0	0
$745$	1.02278e7	0.675133
$746$	0	0
$747$	−6.57621e6	−0.431195
$748$	0	0
$749$	446393.	0.0290746
$750$	0	0
$751$	−2.22359e7	−1.43865	−0.719323	−	0.694676i	$-0.755548\pi$
$751$	−2.22359e7	−1.43865	−0.719323	+	0.694676i	$0.755548\pi$
$752$	0	0
$753$	72380.4	0.00465193
$754$	0	0
$755$	−1.09912e6	−0.0701745
$756$	0	0
$757$	−1.09113e7	−0.692048	−0.346024	−	0.938226i	$-0.612468\pi$
$757$	−1.09113e7	−0.692048	−0.346024	+	0.938226i	$0.612468\pi$
$758$	0	0
$759$	−1.08813e6	−0.0685607
$760$	0	0
$761$	2.42444e6	0.151758	0.0758788	−	0.997117i	$-0.475824\pi$
$761$	2.42444e6	0.151758	0.0758788	+	0.997117i	$0.475824\pi$
$762$	0	0
$763$	346520.	0.0215485
$764$	0	0
$765$	6.49260e6	0.401112
$766$	0	0
$767$	4.78798e6	0.293876
$768$	0	0
$769$	2.00363e6	0.122181	0.0610903	−	0.998132i	$-0.480542\pi$
$769$	2.00363e6	0.122181	0.0610903	+	0.998132i	$0.480542\pi$
$770$	0	0
$771$	−1.05057e7	−0.636487
$772$	0	0
$773$	3.91181e6	0.235466	0.117733	−	0.993045i	$-0.462437\pi$
$773$	3.91181e6	0.235466	0.117733	+	0.993045i	$0.462437\pi$
$774$	0	0
$775$	−397137.	−0.0237512
$776$	0	0
$777$	54607.1	0.00324487
$778$	0	0
$779$	−4.00260e6	−0.236319
$780$	0	0
$781$	4.95353e6	0.290594
$782$	0	0
$783$	5.21786e6	0.304150
$784$	0	0
$785$	−1.07323e7	−0.621614
$786$	0	0
$787$	2.37516e7	1.36696	0.683479	−	0.729970i	$-0.260466\pi$
$787$	2.37516e7	1.36696	0.683479	+	0.729970i	$0.260466\pi$
$788$	0	0
$789$	−5.02246e6	−0.287226
$790$	0	0
$791$	33871.3	0.00192482
$792$	0	0
$793$	1.75292e7	0.989872
$794$	0	0
$795$	−1.71282e7	−0.961157
$796$	0	0
$797$	−2.50486e7	−1.39681	−0.698405	−	0.715703i	$-0.746106\pi$
$797$	−2.50486e7	−1.39681	−0.698405	+	0.715703i	$0.746106\pi$
$798$	0	0
$799$	1.35113e7	0.748737
$800$	0	0
$801$	8.10932e6	0.446584
$802$	0	0
$803$	1.89368e7	1.03638
$804$	0	0
$805$	−73556.7	−0.00400067
$806$	0	0
$807$	1.09752e7	0.593237
$808$	0	0
$809$	−2.26709e7	−1.21786	−0.608931	−	0.793223i	$-0.708402\pi$
$809$	−2.26709e7	−1.21786	−0.608931	+	0.793223i	$0.708402\pi$
$810$	0	0
$811$	1.66885e7	0.890974	0.445487	−	0.895288i	$-0.353031\pi$
$811$	1.66885e7	0.890974	0.445487	+	0.895288i	$0.353031\pi$
$812$	0	0
$813$	1.26523e7	0.671342
$814$	0	0
$815$	5.90066e6	0.311176
$816$	0	0
$817$	1.02315e7	0.536272
$818$	0	0
$819$	−133634.	−0.00696158
$820$	0	0
$821$	−2.41094e7	−1.24833	−0.624164	−	0.781293i	$-0.714560\pi$
$821$	−2.41094e7	−1.24833	−0.624164	+	0.781293i	$0.714560\pi$
$822$	0	0
$823$	9.03070e6	0.464753	0.232376	−	0.972626i	$-0.425350\pi$
$823$	9.03070e6	0.464753	0.232376	+	0.972626i	$0.425350\pi$
$824$	0	0
$825$	−88217.4	−0.00451252
$826$	0	0
$827$	−1.22307e7	−0.621852	−0.310926	−	0.950434i	$-0.600639\pi$
$827$	−1.22307e7	−0.621852	−0.310926	+	0.950434i	$0.600639\pi$
$828$	0	0
$829$	2.01683e7	1.01925	0.509627	−	0.860395i	$-0.329783\pi$
$829$	2.01683e7	1.01925	0.509627	+	0.860395i	$0.329783\pi$
$830$	0	0
$831$	−1.69019e7	−0.849051
$832$	0	0
$833$	2.42570e7	1.21123
$834$	0	0
$835$	−4.31452e6	−0.214149
$836$	0	0
$837$	−6.75053e6	−0.333061
$838$	0	0
$839$	−1.80995e7	−0.887690	−0.443845	−	0.896103i	$-0.646386\pi$
$839$	−1.80995e7	−0.887690	−0.443845	+	0.896103i	$0.646386\pi$
$840$	0	0
$841$	3.07195e7	1.49770
$842$	0	0
$843$	1.90583e7	0.923668
$844$	0	0
$845$	−3.47520e6	−0.167432
$846$	0	0
$847$	272543.	0.0130535
$848$	0	0
$849$	1.04438e7	0.497265
$850$	0	0
$851$	−1.28150e6	−0.0606591
$852$	0	0
$853$	−1.82781e7	−0.860121	−0.430060	−	0.902800i	$-0.641508\pi$
$853$	−1.82781e7	−0.860121	−0.430060	+	0.902800i	$0.641508\pi$
$854$	0	0
$855$	−4.41667e6	−0.206623
$856$	0	0
$857$	−1.47267e7	−0.684940	−0.342470	−	0.939529i	$-0.611264\pi$
$857$	−1.47267e7	−0.684940	−0.342470	+	0.939529i	$0.611264\pi$
$858$	0	0
$859$	1.82990e7	0.846142	0.423071	−	0.906096i	$-0.360952\pi$
$859$	1.82990e7	0.846142	0.423071	+	0.906096i	$0.360952\pi$
$860$	0	0
$861$	−91863.1	−0.00422312
$862$	0	0
$863$	3.92569e6	0.179427	0.0897137	−	0.995968i	$-0.471405\pi$
$863$	3.92569e6	0.179427	0.0897137	+	0.995968i	$0.471405\pi$
$864$	0	0
$865$	−1.21317e7	−0.551293
$866$	0	0
$867$	−5.98254e6	−0.270295
$868$	0	0
$869$	−5.47606e6	−0.245991
$870$	0	0
$871$	−3.52082e7	−1.57253
$872$	0	0
$873$	2.92589e6	0.129934
$874$	0	0
$875$	−440490.	−0.0194499
$876$	0	0
$877$	−3.49463e7	−1.53427	−0.767135	−	0.641486i	$-0.778318\pi$
$877$	−3.49463e7	−1.53427	−0.767135	+	0.641486i	$0.778318\pi$
$878$	0	0
$879$	1.14592e7	0.500245
$880$	0	0
$881$	−3.45541e7	−1.49989	−0.749945	−	0.661501i	$-0.769920\pi$
$881$	−3.45541e7	−1.49989	−0.749945	+	0.661501i	$0.769920\pi$
$882$	0	0
$883$	−2.29538e6	−0.0990722	−0.0495361	−	0.998772i	$-0.515774\pi$
$883$	−2.29538e6	−0.0990722	−0.0495361	+	0.998772i	$0.515774\pi$
$884$	0	0
$885$	3.63186e6	0.155873
$886$	0	0
$887$	−2.48902e7	−1.06223	−0.531116	−	0.847299i	$-0.678227\pi$
$887$	−2.48902e7	−1.06223	−0.531116	+	0.847299i	$0.678227\pi$
$888$	0	0
$889$	397151.	0.0168539
$890$	0	0
$891$	−1.49952e6	−0.0632786
$892$	0	0
$893$	−9.19120e6	−0.385695
$894$	0	0
$895$	−3.02840e7	−1.26373
$896$	0	0
$897$	3.13609e6	0.130139
$898$	0	0
$899$	−6.62789e7	−2.73512
$900$	0	0
$901$	4.94943e7	2.03115
$902$	0	0
$903$	234822.	0.00958341
$904$	0	0
$905$	−2.83623e7	−1.15112
$906$	0	0
$907$	1.12581e7	0.454411	0.227205	−	0.973847i	$-0.427041\pi$
$907$	1.12581e7	0.454411	0.227205	+	0.973847i	$0.427041\pi$
$908$	0	0
$909$	−3.33330e6	−0.133803
$910$	0	0
$911$	2.07869e7	0.829837	0.414919	−	0.909859i	$-0.363810\pi$
$911$	2.07869e7	0.829837	0.414919	+	0.909859i	$0.363810\pi$
$912$	0	0
$913$	1.85555e7	0.736707
$914$	0	0
$915$	1.32965e7	0.525032
$916$	0	0
$917$	−960558.	−0.0377225
$918$	0	0
$919$	1.17836e7	0.460244	0.230122	−	0.973162i	$-0.426087\pi$
$919$	1.17836e7	0.460244	0.230122	+	0.973162i	$0.426087\pi$
$920$	0	0
$921$	−2.03557e7	−0.790747
$922$	0	0
$923$	−1.42766e7	−0.551594
$924$	0	0
$925$	−103895.	−0.00399246
$926$	0	0
$927$	1.39753e7	0.534147
$928$	0	0
$929$	−2.82219e7	−1.07287	−0.536435	−	0.843942i	$-0.680229\pi$
$929$	−2.82219e7	−1.07287	−0.536435	+	0.843942i	$0.680229\pi$
$930$	0	0
$931$	−1.65011e7	−0.623935
$932$	0	0
$933$	1.08236e7	0.407070
$934$	0	0
$935$	−1.83196e7	−0.685308
$936$	0	0
$937$	1.34325e7	0.499814	0.249907	−	0.968270i	$-0.419600\pi$
$937$	1.34325e7	0.499814	0.249907	+	0.968270i	$0.419600\pi$
$938$	0	0
$939$	4.35880e6	0.161325
$940$	0	0
$941$	4.06264e7	1.49566	0.747832	−	0.663888i	$-0.231095\pi$
$941$	4.06264e7	1.49566	0.747832	+	0.663888i	$0.231095\pi$
$942$	0	0
$943$	2.15582e6	0.0789465
$944$	0	0
$945$	−101366.	−0.00369245
$946$	0	0
$947$	3.34341e7	1.21148	0.605738	−	0.795664i	$-0.292878\pi$
$947$	3.34341e7	1.21148	0.605738	+	0.795664i	$0.292878\pi$
$948$	0	0
$949$	−5.45778e7	−1.96721
$950$	0	0
$951$	−7.16962e6	−0.257066
$952$	0	0
$953$	−8.95158e6	−0.319277	−0.159638	−	0.987176i	$-0.551033\pi$
$953$	−8.95158e6	−0.319277	−0.159638	+	0.987176i	$0.551033\pi$
$954$	0	0
$955$	−3.20250e6	−0.113627
$956$	0	0
$957$	−1.47227e7	−0.519648
$958$	0	0
$959$	83081.1	0.00291713
$960$	0	0
$961$	5.71181e7	1.99510
$962$	0	0
$963$	−1.44365e7	−0.501643
$964$	0	0
$965$	4.56399e7	1.57771
$966$	0	0
$967$	2.94663e7	1.01335	0.506675	−	0.862137i	$-0.330874\pi$
$967$	2.94663e7	1.01335	0.506675	+	0.862137i	$0.330874\pi$
$968$	0	0
$969$	1.27626e7	0.436645
$970$	0	0
$971$	3.03617e7	1.03342	0.516712	−	0.856159i	$-0.327156\pi$
$971$	3.03617e7	1.03342	0.516712	+	0.856159i	$0.327156\pi$
$972$	0	0
$973$	−321814.	−0.0108974
$974$	0	0
$975$	254251.	0.00856548
$976$	0	0
$977$	3.77587e6	0.126555	0.0632777	−	0.997996i	$-0.479845\pi$
$977$	3.77587e6	0.126555	0.0632777	+	0.997996i	$0.479845\pi$
$978$	0	0
$979$	−2.28813e7	−0.762999
$980$	0	0
$981$	−1.12065e7	−0.371791
$982$	0	0
$983$	−1.74788e7	−0.576938	−0.288469	−	0.957489i	$-0.593146\pi$
$983$	−1.74788e7	−0.576938	−0.288469	+	0.957489i	$0.593146\pi$
$984$	0	0
$985$	−5.39385e7	−1.77137
$986$	0	0
$987$	−210946.	−0.00689253
$988$	0	0
$989$	−5.51074e6	−0.179151
$990$	0	0
$991$	−4.83380e7	−1.56352	−0.781762	−	0.623576i	$-0.785679\pi$
$991$	−4.83380e7	−1.56352	−0.781762	+	0.623576i	$0.785679\pi$
$992$	0	0
$993$	−2.36570e6	−0.0761355
$994$	0	0
$995$	4.47526e7	1.43305
$996$	0	0
$997$	−1.74335e7	−0.555452	−0.277726	−	0.960660i	$-0.589581\pi$
$997$	−1.74335e7	−0.555452	−0.277726	+	0.960660i	$0.589581\pi$
$998$	0	0
$999$	−1.76601e6	−0.0559858

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.i.1.2		3
4.3	odd	2		69.6.a.b.1.1	✓	3
12.11	even	2		207.6.a.c.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.b.1.1	✓	3	4.3	odd	2
207.6.a.c.1.3		3	12.11	even	2
1104.6.a.i.1.2		3	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)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -55.5168 q^{5} -2.50462 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -55.5168 q^{5} -2.50462 q^{7} +81.0000 q^{9} -228.550 q^{11} +658.703 q^{13} +499.651 q^{15} -1443.81 q^{17} +982.167 q^{19} +22.5416 q^{21} -529.000 q^{23} -42.8875 q^{25} -729.000 q^{27} -7157.56 q^{29} +9259.98 q^{31} +2056.95 q^{33} +139.049 q^{35} +2422.50 q^{37} -5928.33 q^{39} -4075.27 q^{41} +10417.3 q^{43} -4496.86 q^{45} -9358.08 q^{47} -16800.7 q^{49} +12994.3 q^{51} -34280.3 q^{53} +12688.4 q^{55} -8839.51 q^{57} +7268.79 q^{59} +26611.7 q^{61} -202.874 q^{63} -36569.1 q^{65} -53450.8 q^{67} +4761.00 q^{69} -21673.7 q^{71} -82856.3 q^{73} +385.987 q^{75} +572.432 q^{77} +23960.0 q^{79} +6561.00 q^{81} -81187.7 q^{83} +80155.6 q^{85} +64418.0 q^{87} +100115. q^{89} -1649.80 q^{91} -83339.8 q^{93} -54526.8 q^{95} +36122.1 q^{97} -18512.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9}+O(q^{10}) \) 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9	\( 3 q - 27 q^{3} - 56 q^{5} + 114 q^{7} + 243 q^{9} + 376 q^{11} - 858 q^{13} + 504 q^{15} - 2548 q^{17} + 2846 q^{19} - 1026 q^{21} - 1587 q^{23} + 753 q^{25} - 2187 q^{27} - 16370 q^{29} + 14756 q^{31} - 3384 q^{33} + 18520 q^{35} + 15874 q^{37} + 7722 q^{39} + 12606 q^{41} - 3154 q^{43} - 4536 q^{45} - 29928 q^{47} + 4471 q^{49} + 22932 q^{51} - 44084 q^{53} - 38360 q^{55} - 25614 q^{57} + 29300 q^{59} + 54010 q^{61} + 9234 q^{63} - 51216 q^{65} - 43390 q^{67} + 14283 q^{69} - 23424 q^{71} - 91402 q^{73} - 6777 q^{75} - 97208 q^{77} + 49398 q^{79} + 19683 q^{81} + 103936 q^{83} + 5888 q^{85} + 147330 q^{87} + 96112 q^{89} - 129228 q^{91} - 132804 q^{93} + 55928 q^{95} - 135318 q^{97} + 30456 q^{99}+O(q^{100}) \) 3 * q - 27 * q^3 - 56 * q^5 + 114 * q^7 + 243 * q^9 + 376 * q^11 - 858 * q^13 + 504 * q^15 - 2548 * q^17 + 2846 * q^19 - 1026 * q^21 - 1587 * q^23 + 753 * q^25 - 2187 * q^27 - 16370 * q^29 + 14756 * q^31 - 3384 * q^33 + 18520 * q^35 + 15874 * q^37 + 7722 * q^39 + 12606 * q^41 - 3154 * q^43 - 4536 * q^45 - 29928 * q^47 + 4471 * q^49 + 22932 * q^51 - 44084 * q^53 - 38360 * q^55 - 25614 * q^57 + 29300 * q^59 + 54010 * q^61 + 9234 * q^63 - 51216 * q^65 - 43390 * q^67 + 14283 * q^69 - 23424 * q^71 - 91402 * q^73 - 6777 * q^75 - 97208 * q^77 + 49398 * q^79 + 19683 * q^81 + 103936 * q^83 + 5888 * q^85 + 147330 * q^87 + 96112 * q^89 - 129228 * q^91 - 132804 * q^93 + 55928 * q^95 - 135318 * q^97 + 30456 * q^99

Introduction

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