LMFDB - Embedded newform 56.6.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [56,6,Mod(1,56)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("56.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(56, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 56 = 2^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	56.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.98149390953$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$7.44622$ of defining polynomial
Character	$\chi$	$=$	56.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-20.8924 q^{3} +90.4622 q^{5} -49.0000 q^{7} +193.494 q^{9} -552.494 q^{11} -593.172 q^{13} -1889.98 q^{15} -1422.19 q^{17} +318.997 q^{19} +1023.73 q^{21} +659.954 q^{23} +5058.41 q^{25} +1034.30 q^{27} -8185.27 q^{29} -9598.03 q^{31} +11543.0 q^{33} -4432.65 q^{35} +5180.56 q^{37} +12392.8 q^{39} -2192.46 q^{41} +7458.29 q^{43} +17503.9 q^{45} -19561.8 q^{47} +2401.00 q^{49} +29713.1 q^{51} +36569.6 q^{53} -49979.9 q^{55} -6664.63 q^{57} +16361.7 q^{59} -10893.1 q^{61} -9481.22 q^{63} -53659.6 q^{65} +8035.91 q^{67} -13788.0 q^{69} +55983.3 q^{71} -77752.5 q^{73} -105683. q^{75} +27072.2 q^{77} -3208.43 q^{79} -68628.1 q^{81} +76626.9 q^{83} -128655. q^{85} +171010. q^{87} -84288.6 q^{89} +29065.4 q^{91} +200526. q^{93} +28857.2 q^{95} +101273. q^{97} -106904. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{3} + 42 q^{5} - 98 q^{7} - 2 q^{9} - 716 q^{11} - 714 q^{13} - 2224 q^{15} - 1344 q^{17} - 1946 q^{19} + 686 q^{21} - 1792 q^{23} + 4282 q^{25} - 1988 q^{27} - 1200 q^{29} - 6804 q^{31} + 10416 q^{33}+ \cdots - 74940 q^{99}+O(q^{100}) $ 2 * q - 14 * q^3 + 42 * q^5 - 98 * q^7 - 2 * q^9 - 716 * q^11 - 714 * q^13 - 2224 * q^15 - 1344 * q^17 - 1946 * q^19 + 686 * q^21 - 1792 * q^23 + 4282 * q^25 - 1988 * q^27 - 1200 * q^29 - 6804 * q^31 + 10416 * q^33 - 2058 * q^35 + 14640 * q^37 + 11560 * q^39 + 7896 * q^41 + 524 * q^43 + 26978 * q^45 - 18396 * q^47 + 4802 * q^49 + 30252 * q^51 + 45132 * q^53 - 42056 * q^55 - 22276 * q^57 + 22582 * q^59 - 52822 * q^61 + 98 * q^63 - 47804 * q^65 + 9848 * q^67 - 30688 * q^69 - 840 * q^71 - 122052 * q^73 - 111034 * q^75 + 35084 * q^77 + 31704 * q^79 - 41954 * q^81 + 36974 * q^83 - 132444 * q^85 + 219156 * q^87 - 210588 * q^89 + 34986 * q^91 + 219784 * q^93 + 138624 * q^95 - 44240 * q^97 - 74940 * q^99

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
$3$	−20.8924	−1.34025	−0.670125	−	0.742248i	$-0.733760\pi$
$3$	−20.8924	−1.34025	−0.670125	+	0.742248i	$0.733760\pi$
$4$	0	0
$5$	90.4622	1.61824	0.809119	−	0.587645i	$-0.199945\pi$
$5$	90.4622	1.61824	0.809119	+	0.587645i	$0.199945\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	193.494	0.796272
$10$	0	0
$11$	−552.494	−1.37672	−0.688361	−	0.725369i	$-0.741669\pi$
$11$	−552.494	−1.37672	−0.688361	+	0.725369i	$0.741669\pi$
$12$	0	0
$13$	−593.172	−0.973469	−0.486734	−	0.873550i	$-0.661812\pi$
$13$	−593.172	−0.973469	−0.486734	+	0.873550i	$0.661812\pi$
$14$	0	0
$15$	−1889.98	−2.16884
$16$	0	0
$17$	−1422.19	−1.19354	−0.596769	−	0.802413i	$-0.703549\pi$
$17$	−1422.19	−1.19354	−0.596769	+	0.802413i	$0.703549\pi$
$18$	0	0
$19$	318.997	0.202723	0.101361	−	0.994850i	$-0.467680\pi$
$19$	318.997	0.202723	0.101361	+	0.994850i	$0.467680\pi$
$20$	0	0
$21$	1023.73	0.506567
$22$	0	0
$23$	659.954	0.260132	0.130066	−	0.991505i	$-0.458481\pi$
$23$	659.954	0.260132	0.130066	+	0.991505i	$0.458481\pi$
$24$	0	0
$25$	5058.41	1.61869
$26$	0	0
$27$	1034.30	0.273046
$28$	0	0
$29$	−8185.27	−1.80733	−0.903667	−	0.428237i	$-0.859135\pi$
$29$	−8185.27	−1.80733	−0.903667	+	0.428237i	$0.859135\pi$
$30$	0	0
$31$	−9598.03	−1.79382	−0.896908	−	0.442217i	$-0.854192\pi$
$31$	−9598.03	−1.79382	−0.896908	+	0.442217i	$0.854192\pi$
$32$	0	0
$33$	11543.0	1.84515
$34$	0	0
$35$	−4432.65	−0.611636
$36$	0	0
$37$	5180.56	0.622118	0.311059	−	0.950391i	$-0.399316\pi$
$37$	5180.56	0.622118	0.311059	+	0.950391i	$0.399316\pi$
$38$	0	0
$39$	12392.8	1.30469
$40$	0	0
$41$	−2192.46	−0.203691	−0.101846	−	0.994800i	$-0.532475\pi$
$41$	−2192.46	−0.203691	−0.101846	+	0.994800i	$0.532475\pi$
$42$	0	0
$43$	7458.29	0.615131	0.307566	−	0.951527i	$-0.400486\pi$
$43$	7458.29	0.615131	0.307566	+	0.951527i	$0.400486\pi$
$44$	0	0
$45$	17503.9	1.28856
$46$	0	0
$47$	−19561.8	−1.29171	−0.645853	−	0.763462i	$-0.723498\pi$
$47$	−19561.8	−1.29171	−0.645853	+	0.763462i	$0.723498\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	29713.1	1.59964
$52$	0	0
$53$	36569.6	1.78826	0.894129	−	0.447809i	$-0.147795\pi$
$53$	36569.6	1.78826	0.894129	+	0.447809i	$0.147795\pi$
$54$	0	0
$55$	−49979.9	−2.22786
$56$	0	0
$57$	−6664.63	−0.271700
$58$	0	0
$59$	16361.7	0.611927	0.305963	−	0.952043i	$-0.401021\pi$
$59$	16361.7	0.611927	0.305963	+	0.952043i	$0.401021\pi$
$60$	0	0
$61$	−10893.1	−0.374825	−0.187412	−	0.982281i	$-0.560010\pi$
$61$	−10893.1	−0.374825	−0.187412	+	0.982281i	$0.560010\pi$
$62$	0	0
$63$	−9481.22	−0.300963
$64$	0	0
$65$	−53659.6	−1.57530
$66$	0	0
$67$	8035.91	0.218700	0.109350	−	0.994003i	$-0.465123\pi$
$67$	8035.91	0.218700	0.109350	+	0.994003i	$0.465123\pi$
$68$	0	0
$69$	−13788.0	−0.348642
$70$	0	0
$71$	55983.3	1.31799	0.658996	−	0.752146i	$-0.270981\pi$
$71$	55983.3	1.31799	0.658996	+	0.752146i	$0.270981\pi$
$72$	0	0
$73$	−77752.5	−1.70768	−0.853841	−	0.520533i	$-0.825733\pi$
$73$	−77752.5	−1.70768	−0.853841	+	0.520533i	$0.825733\pi$
$74$	0	0
$75$	−105683.	−2.16945
$76$	0	0
$77$	27072.2	0.520352
$78$	0	0
$79$	−3208.43	−0.0578396	−0.0289198	−	0.999582i	$-0.509207\pi$
$79$	−3208.43	−0.0578396	−0.0289198	+	0.999582i	$0.509207\pi$
$80$	0	0
$81$	−68628.1	−1.16222
$82$	0	0
$83$	76626.9	1.22092	0.610458	−	0.792049i	$-0.290985\pi$
$83$	76626.9	1.22092	0.610458	+	0.792049i	$0.290985\pi$
$84$	0	0
$85$	−128655.	−1.93143
$86$	0	0
$87$	171010.	2.42228
$88$	0	0
$89$	−84288.6	−1.12796	−0.563980	−	0.825788i	$-0.690731\pi$
$89$	−84288.6	−1.12796	−0.563980	+	0.825788i	$0.690731\pi$
$90$	0	0
$91$	29065.4	0.367937
$92$	0	0
$93$	200526.	2.40416
$94$	0	0
$95$	28857.2	0.328054
$96$	0	0
$97$	101273.	1.09286	0.546428	−	0.837506i	$-0.315987\pi$
$97$	101273.	1.09286	0.546428	+	0.837506i	$0.315987\pi$
$98$	0	0
$99$	−106904.	−1.09625
$100$	0	0
$101$	77456.5	0.755535	0.377767	−	0.925901i	$-0.376692\pi$
$101$	77456.5	0.755535	0.377767	+	0.925901i	$0.376692\pi$
$102$	0	0
$103$	−56716.9	−0.526768	−0.263384	−	0.964691i	$-0.584839\pi$
$103$	−56716.9	−0.526768	−0.263384	+	0.964691i	$0.584839\pi$
$104$	0	0
$105$	92608.9	0.819746
$106$	0	0
$107$	−38699.2	−0.326770	−0.163385	−	0.986562i	$-0.552241\pi$
$107$	−38699.2	−0.326770	−0.163385	+	0.986562i	$0.552241\pi$
$108$	0	0
$109$	−16595.3	−0.133788	−0.0668941	−	0.997760i	$-0.521309\pi$
$109$	−16595.3	−0.133788	−0.0668941	+	0.997760i	$0.521309\pi$
$110$	0	0
$111$	−108235.	−0.833794
$112$	0	0
$113$	−76723.9	−0.565242	−0.282621	−	0.959232i	$-0.591204\pi$
$113$	−76723.9	−0.565242	−0.282621	+	0.959232i	$0.591204\pi$
$114$	0	0
$115$	59700.9	0.420955
$116$	0	0
$117$	−114775.	−0.775146
$118$	0	0
$119$	69687.4	0.451115
$120$	0	0
$121$	144199.	0.895361
$122$	0	0
$123$	45805.9	0.272997
$124$	0	0
$125$	174901.	1.00119
$126$	0	0
$127$	68539.3	0.377077	0.188539	−	0.982066i	$-0.439625\pi$
$127$	68539.3	0.377077	0.188539	+	0.982066i	$0.439625\pi$
$128$	0	0
$129$	−155822.	−0.824430
$130$	0	0
$131$	95936.2	0.488432	0.244216	−	0.969721i	$-0.421469\pi$
$131$	95936.2	0.488432	0.244216	+	0.969721i	$0.421469\pi$
$132$	0	0
$133$	−15630.9	−0.0766221
$134$	0	0
$135$	93564.8	0.441853
$136$	0	0
$137$	291295.	1.32597	0.662983	−	0.748635i	$-0.269290\pi$
$137$	291295.	1.32597	0.662983	+	0.748635i	$0.269290\pi$
$138$	0	0
$139$	−281554.	−1.23602	−0.618009	−	0.786171i	$-0.712061\pi$
$139$	−281554.	−1.23602	−0.618009	+	0.786171i	$0.712061\pi$
$140$	0	0
$141$	408693.	1.73121
$142$	0	0
$143$	327724.	1.34019
$144$	0	0
$145$	−740458.	−2.92469
$146$	0	0
$147$	−50162.8	−0.191464
$148$	0	0
$149$	−128283.	−0.473372	−0.236686	−	0.971586i	$-0.576061\pi$
$149$	−128283.	−0.473372	−0.236686	+	0.971586i	$0.576061\pi$
$150$	0	0
$151$	19017.7	0.0678760	0.0339380	−	0.999424i	$-0.489195\pi$
$151$	19017.7	0.0678760	0.0339380	+	0.999424i	$0.489195\pi$
$152$	0	0
$153$	−275186.	−0.950381
$154$	0	0
$155$	−868259.	−2.90282
$156$	0	0
$157$	−272988.	−0.883883	−0.441941	−	0.897044i	$-0.645710\pi$
$157$	−272988.	−0.883883	−0.441941	+	0.897044i	$0.645710\pi$
$158$	0	0
$159$	−764028.	−2.39672
$160$	0	0
$161$	−32337.7	−0.0983207
$162$	0	0
$163$	315849.	0.931131	0.465565	−	0.885013i	$-0.345851\pi$
$163$	315849.	0.931131	0.465565	+	0.885013i	$0.345851\pi$
$164$	0	0
$165$	1.04420e6	2.98589
$166$	0	0
$167$	−588655.	−1.63331	−0.816657	−	0.577123i	$-0.804175\pi$
$167$	−588655.	−1.63331	−0.816657	+	0.577123i	$0.804175\pi$
$168$	0	0
$169$	−19440.5	−0.0523590
$170$	0	0
$171$	61724.1	0.161423
$172$	0	0
$173$	718256.	1.82458	0.912292	−	0.409540i	$-0.134311\pi$
$173$	718256.	1.82458	0.912292	+	0.409540i	$0.134311\pi$
$174$	0	0
$175$	−247862.	−0.611808
$176$	0	0
$177$	−341837.	−0.820136
$178$	0	0
$179$	53637.2	0.125122	0.0625610	−	0.998041i	$-0.480073\pi$
$179$	53637.2	0.125122	0.0625610	+	0.998041i	$0.480073\pi$
$180$	0	0
$181$	−392213.	−0.889867	−0.444934	−	0.895564i	$-0.646773\pi$
$181$	−392213.	−0.889867	−0.444934	+	0.895564i	$0.646773\pi$
$182$	0	0
$183$	227584.	0.502359
$184$	0	0
$185$	468645.	1.00673
$186$	0	0
$187$	785753.	1.64317
$188$	0	0
$189$	−50680.5	−0.103202
$190$	0	0
$191$	395509.	0.784463	0.392232	−	0.919866i	$-0.371703\pi$
$191$	395509.	0.784463	0.392232	+	0.919866i	$0.371703\pi$
$192$	0	0
$193$	−562892.	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$193$	−562892.	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$194$	0	0
$195$	1.12108e6	2.11130
$196$	0	0
$197$	402202.	0.738378	0.369189	−	0.929354i	$-0.379635\pi$
$197$	402202.	0.738378	0.369189	+	0.929354i	$0.379635\pi$
$198$	0	0
$199$	−455975.	−0.816222	−0.408111	−	0.912932i	$-0.633812\pi$
$199$	−455975.	−0.816222	−0.408111	+	0.912932i	$0.633812\pi$
$200$	0	0
$201$	−167890.	−0.293112
$202$	0	0
$203$	401078.	0.683108
$204$	0	0
$205$	−198335.	−0.329621
$206$	0	0
$207$	127697.	0.207136
$208$	0	0
$209$	−176244.	−0.279093
$210$	0	0
$211$	−1.18264e6	−1.82871	−0.914356	−	0.404911i	$-0.867302\pi$
$211$	−1.18264e6	−1.82871	−0.914356	+	0.404911i	$0.867302\pi$
$212$	0	0
$213$	−1.16963e6	−1.76644
$214$	0	0
$215$	674693.	0.995429
$216$	0	0
$217$	470303.	0.677999
$218$	0	0
$219$	1.62444e6	2.28872
$220$	0	0
$221$	843604.	1.16187
$222$	0	0
$223$	931112.	1.25383	0.626917	−	0.779086i	$-0.284317\pi$
$223$	931112.	1.25383	0.626917	+	0.779086i	$0.284317\pi$
$224$	0	0
$225$	978774.	1.28892
$226$	0	0
$227$	192653.	0.248149	0.124074	−	0.992273i	$-0.460404\pi$
$227$	192653.	0.248149	0.124074	+	0.992273i	$0.460404\pi$
$228$	0	0
$229$	−783934.	−0.987850	−0.493925	−	0.869505i	$-0.664438\pi$
$229$	−783934.	−0.987850	−0.493925	+	0.869505i	$0.664438\pi$
$230$	0	0
$231$	−565605.	−0.697402
$232$	0	0
$233$	−1.39976e6	−1.68914	−0.844568	−	0.535448i	$-0.820143\pi$
$233$	−1.39976e6	−1.68914	−0.844568	+	0.535448i	$0.820143\pi$
$234$	0	0
$235$	−1.76960e6	−2.09029
$236$	0	0
$237$	67032.0	0.0775196
$238$	0	0
$239$	−643631.	−0.728857	−0.364429	−	0.931231i	$-0.618736\pi$
$239$	−643631.	−0.728857	−0.364429	+	0.931231i	$0.618736\pi$
$240$	0	0
$241$	−58756.4	−0.0651647	−0.0325824	−	0.999469i	$-0.510373\pi$
$241$	−58756.4	−0.0651647	−0.0325824	+	0.999469i	$0.510373\pi$
$242$	0	0
$243$	1.18247e6	1.28462
$244$	0	0
$245$	217200.	0.231177
$246$	0	0
$247$	−189220.	−0.197344
$248$	0	0
$249$	−1.60092e6	−1.63633
$250$	0	0
$251$	−641680.	−0.642886	−0.321443	−	0.946929i	$-0.604168\pi$
$251$	−641680.	−0.642886	−0.321443	+	0.946929i	$0.604168\pi$
$252$	0	0
$253$	−364621.	−0.358129
$254$	0	0
$255$	2.68791e6	2.58860
$256$	0	0
$257$	810671.	0.765617	0.382809	−	0.923828i	$-0.374957\pi$
$257$	810671.	0.765617	0.382809	+	0.923828i	$0.374957\pi$
$258$	0	0
$259$	−253848.	−0.235138
$260$	0	0
$261$	−1.58380e6	−1.43913
$262$	0	0
$263$	−2.04061e6	−1.81916	−0.909581	−	0.415526i	$-0.863597\pi$
$263$	−2.04061e6	−1.81916	−0.909581	+	0.415526i	$0.863597\pi$
$264$	0	0
$265$	3.30817e6	2.89383
$266$	0	0
$267$	1.76100e6	1.51175
$268$	0	0
$269$	−1.28300e6	−1.08105	−0.540524	−	0.841328i	$-0.681774\pi$
$269$	−1.28300e6	−1.08105	−0.540524	+	0.841328i	$0.681774\pi$
$270$	0	0
$271$	−22511.3	−0.0186199	−0.00930994	−	0.999957i	$-0.502963\pi$
$271$	−22511.3	−0.0186199	−0.00930994	+	0.999957i	$0.502963\pi$
$272$	0	0
$273$	−607247.	−0.493127
$274$	0	0
$275$	−2.79474e6	−2.22849
$276$	0	0
$277$	368210.	0.288334	0.144167	−	0.989553i	$-0.453950\pi$
$277$	368210.	0.288334	0.144167	+	0.989553i	$0.453950\pi$
$278$	0	0
$279$	−1.85716e6	−1.42837
$280$	0	0
$281$	2.03709e6	1.53902	0.769511	−	0.638634i	$-0.220500\pi$
$281$	2.03709e6	1.53902	0.769511	+	0.638634i	$0.220500\pi$
$282$	0	0
$283$	−656410.	−0.487202	−0.243601	−	0.969876i	$-0.578329\pi$
$283$	−656410.	−0.487202	−0.243601	+	0.969876i	$0.578329\pi$
$284$	0	0
$285$	−602897.	−0.439675
$286$	0	0
$287$	107431.	0.0769880
$288$	0	0
$289$	602773.	0.424531
$290$	0	0
$291$	−2.11583e6	−1.46470
$292$	0	0
$293$	−962295.	−0.654846	−0.327423	−	0.944878i	$-0.606180\pi$
$293$	−962295.	−0.654846	−0.327423	+	0.944878i	$0.606180\pi$
$294$	0	0
$295$	1.48012e6	0.990243
$296$	0	0
$297$	−571443.	−0.375908
$298$	0	0
$299$	−391466.	−0.253230
$300$	0	0
$301$	−365456.	−0.232498
$302$	0	0
$303$	−1.61826e6	−1.01261
$304$	0	0
$305$	−985418.	−0.606556
$306$	0	0
$307$	−296468.	−0.179528	−0.0897638	−	0.995963i	$-0.528611\pi$
$307$	−296468.	−0.179528	−0.0897638	+	0.995963i	$0.528611\pi$
$308$	0	0
$309$	1.18495e6	0.706001
$310$	0	0
$311$	1.12063e6	0.656996	0.328498	−	0.944505i	$-0.393458\pi$
$311$	1.12063e6	0.656996	0.328498	+	0.944505i	$0.393458\pi$
$312$	0	0
$313$	1.74908e6	1.00913	0.504567	−	0.863373i	$-0.331652\pi$
$313$	1.74908e6	1.00913	0.504567	+	0.863373i	$0.331652\pi$
$314$	0	0
$315$	−857692.	−0.487029
$316$	0	0
$317$	3.17498e6	1.77457	0.887284	−	0.461223i	$-0.152589\pi$
$317$	3.17498e6	1.77457	0.887284	+	0.461223i	$0.152589\pi$
$318$	0	0
$319$	4.52232e6	2.48819
$320$	0	0
$321$	808520.	0.437954
$322$	0	0
$323$	−453675.	−0.241957
$324$	0	0
$325$	−3.00051e6	−1.57575
$326$	0	0
$327$	346715.	0.179310
$328$	0	0
$329$	958526.	0.488219
$330$	0	0
$331$	878405.	0.440681	0.220341	−	0.975423i	$-0.429283\pi$
$331$	878405.	0.440681	0.220341	+	0.975423i	$0.429283\pi$
$332$	0	0
$333$	1.00241e6	0.495375
$334$	0	0
$335$	726946.	0.353908
$336$	0	0
$337$	2.05261e6	0.984536	0.492268	−	0.870444i	$-0.336168\pi$
$337$	2.05261e6	0.984536	0.492268	+	0.870444i	$0.336168\pi$
$338$	0	0
$339$	1.60295e6	0.757566
$340$	0	0
$341$	5.30286e6	2.46958
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−1.24730e6	−0.564186
$346$	0	0
$347$	476182.	0.212300	0.106150	−	0.994350i	$-0.466148\pi$
$347$	476182.	0.212300	0.106150	+	0.994350i	$0.466148\pi$
$348$	0	0
$349$	351681.	0.154556	0.0772780	−	0.997010i	$-0.475377\pi$
$349$	351681.	0.154556	0.0772780	+	0.997010i	$0.475377\pi$
$350$	0	0
$351$	−613515.	−0.265802
$352$	0	0
$353$	−1.73722e6	−0.742023	−0.371011	−	0.928628i	$-0.620989\pi$
$353$	−1.73722e6	−0.742023	−0.371011	+	0.928628i	$0.620989\pi$
$354$	0	0
$355$	5.06438e6	2.13282
$356$	0	0
$357$	−1.45594e6	−0.604607
$358$	0	0
$359$	−1.69731e6	−0.695063	−0.347531	−	0.937668i	$-0.612980\pi$
$359$	−1.69731e6	−0.695063	−0.347531	+	0.937668i	$0.612980\pi$
$360$	0	0
$361$	−2.37434e6	−0.958903
$362$	0	0
$363$	−3.01267e6	−1.20001
$364$	0	0
$365$	−7.03366e6	−2.76344
$366$	0	0
$367$	−1.11920e6	−0.433752	−0.216876	−	0.976199i	$-0.569587\pi$
$367$	−1.11920e6	−0.433752	−0.216876	+	0.976199i	$0.569587\pi$
$368$	0	0
$369$	−424228.	−0.162194
$370$	0	0
$371$	−1.79191e6	−0.675898
$372$	0	0
$373$	−837822.	−0.311803	−0.155901	−	0.987773i	$-0.549828\pi$
$373$	−837822.	−0.311803	−0.155901	+	0.987773i	$0.549828\pi$
$374$	0	0
$375$	−3.65411e6	−1.34185
$376$	0	0
$377$	4.85527e6	1.75938
$378$	0	0
$379$	−1.94713e6	−0.696300	−0.348150	−	0.937439i	$-0.613190\pi$
$379$	−1.94713e6	−0.696300	−0.348150	+	0.937439i	$0.613190\pi$
$380$	0	0
$381$	−1.43195e6	−0.505378
$382$	0	0
$383$	−202063.	−0.0703866	−0.0351933	−	0.999381i	$-0.511205\pi$
$383$	−202063.	−0.0703866	−0.0351933	+	0.999381i	$0.511205\pi$
$384$	0	0
$385$	2.44901e6	0.842053
$386$	0	0
$387$	1.44314e6	0.489812
$388$	0	0
$389$	3.23741e6	1.08473	0.542367	−	0.840141i	$-0.317528\pi$
$389$	3.23741e6	1.08473	0.542367	+	0.840141i	$0.317528\pi$
$390$	0	0
$391$	−938581.	−0.310477
$392$	0	0
$393$	−2.00434e6	−0.654622
$394$	0	0
$395$	−290242.	−0.0935982
$396$	0	0
$397$	823921.	0.262367	0.131183	−	0.991358i	$-0.458122\pi$
$397$	823921.	0.262367	0.131183	+	0.991358i	$0.458122\pi$
$398$	0	0
$399$	326567.	0.102693
$400$	0	0
$401$	2.39337e6	0.743273	0.371637	−	0.928378i	$-0.378797\pi$
$401$	2.39337e6	0.743273	0.371637	+	0.928378i	$0.378797\pi$
$402$	0	0
$403$	5.69328e6	1.74622
$404$	0	0
$405$	−6.20825e6	−1.88075
$406$	0	0
$407$	−2.86223e6	−0.856483
$408$	0	0
$409$	−581801.	−0.171975	−0.0859876	−	0.996296i	$-0.527405\pi$
$409$	−581801.	−0.171975	−0.0859876	+	0.996296i	$0.527405\pi$
$410$	0	0
$411$	−6.08587e6	−1.77713
$412$	0	0
$413$	−801725.	−0.231287
$414$	0	0
$415$	6.93184e6	1.97573
$416$	0	0
$417$	5.88236e6	1.65658
$418$	0	0
$419$	3.11898e6	0.867915	0.433957	−	0.900933i	$-0.357117\pi$
$419$	3.11898e6	0.867915	0.433957	+	0.900933i	$0.357117\pi$
$420$	0	0
$421$	−1.14481e6	−0.314796	−0.157398	−	0.987535i	$-0.550311\pi$
$421$	−1.14481e6	−0.314796	−0.157398	+	0.987535i	$0.550311\pi$
$422$	0	0
$423$	−3.78509e6	−1.02855
$424$	0	0
$425$	−7.19403e6	−1.93197
$426$	0	0
$427$	533764.	0.141671
$428$	0	0
$429$	−6.84695e6	−1.79620
$430$	0	0
$431$	−4.31012e6	−1.11762	−0.558812	−	0.829294i	$-0.688743\pi$
$431$	−4.31012e6	−1.11762	−0.558812	+	0.829294i	$0.688743\pi$
$432$	0	0
$433$	−2.79301e6	−0.715901	−0.357950	−	0.933741i	$-0.616524\pi$
$433$	−2.79301e6	−0.715901	−0.357950	+	0.933741i	$0.616524\pi$
$434$	0	0
$435$	1.54700e7	3.91982
$436$	0	0
$437$	210523.	0.0527347
$438$	0	0
$439$	252467.	0.0625236	0.0312618	−	0.999511i	$-0.490047\pi$
$439$	252467.	0.0625236	0.0312618	+	0.999511i	$0.490047\pi$
$440$	0	0
$441$	464580.	0.113753
$442$	0	0
$443$	−3.43282e6	−0.831077	−0.415539	−	0.909576i	$-0.636407\pi$
$443$	−3.43282e6	−0.831077	−0.415539	+	0.909576i	$0.636407\pi$
$444$	0	0
$445$	−7.62494e6	−1.82531
$446$	0	0
$447$	2.68014e6	0.634437
$448$	0	0
$449$	−1.68031e6	−0.393345	−0.196673	−	0.980469i	$-0.563014\pi$
$449$	−1.68031e6	−0.393345	−0.196673	+	0.980469i	$0.563014\pi$
$450$	0	0
$451$	1.21132e6	0.280426
$452$	0	0
$453$	−397327.	−0.0909709
$454$	0	0
$455$	2.62932e6	0.595409
$456$	0	0
$457$	−2.94585e6	−0.659812	−0.329906	−	0.944014i	$-0.607017\pi$
$457$	−2.94585e6	−0.659812	−0.329906	+	0.944014i	$0.607017\pi$
$458$	0	0
$459$	−1.47097e6	−0.325890
$460$	0	0
$461$	−1.91583e6	−0.419860	−0.209930	−	0.977716i	$-0.567324\pi$
$461$	−1.91583e6	−0.419860	−0.209930	+	0.977716i	$0.567324\pi$
$462$	0	0
$463$	−3.75401e6	−0.813847	−0.406924	−	0.913462i	$-0.633398\pi$
$463$	−3.75401e6	−0.813847	−0.406924	+	0.913462i	$0.633398\pi$
$464$	0	0
$465$	1.81401e7	3.89051
$466$	0	0
$467$	−2.10594e6	−0.446842	−0.223421	−	0.974722i	$-0.571722\pi$
$467$	−2.10594e6	−0.446842	−0.223421	+	0.974722i	$0.571722\pi$
$468$	0	0
$469$	−393759.	−0.0826607
$470$	0	0
$471$	5.70339e6	1.18462
$472$	0	0
$473$	−4.12066e6	−0.846864
$474$	0	0
$475$	1.61362e6	0.328146
$476$	0	0
$477$	7.07600e6	1.42394
$478$	0	0
$479$	−274926.	−0.0547491	−0.0273745	−	0.999625i	$-0.508715\pi$
$479$	−274926.	−0.0547491	−0.0273745	+	0.999625i	$0.508715\pi$
$480$	0	0
$481$	−3.07296e6	−0.605612
$482$	0	0
$483$	675614.	0.131774
$484$	0	0
$485$	9.16135e6	1.76850
$486$	0	0
$487$	7.67128e6	1.46570	0.732851	−	0.680389i	$-0.238189\pi$
$487$	7.67128e6	1.46570	0.732851	+	0.680389i	$0.238189\pi$
$488$	0	0
$489$	−6.59886e6	−1.24795
$490$	0	0
$491$	−2.42352e6	−0.453673	−0.226837	−	0.973933i	$-0.572838\pi$
$491$	−2.42352e6	−0.453673	−0.226837	+	0.973933i	$0.572838\pi$
$492$	0	0
$493$	1.16410e7	2.15712
$494$	0	0
$495$	−9.67081e6	−1.77399
$496$	0	0
$497$	−2.74318e6	−0.498154
$498$	0	0
$499$	−8.22317e6	−1.47839	−0.739193	−	0.673494i	$-0.764793\pi$
$499$	−8.22317e6	−1.47839	−0.739193	+	0.673494i	$0.764793\pi$
$500$	0	0
$501$	1.22984e7	2.18905
$502$	0	0
$503$	2.98186e6	0.525493	0.262746	−	0.964865i	$-0.415372\pi$
$503$	2.98186e6	0.525493	0.262746	+	0.964865i	$0.415372\pi$
$504$	0	0
$505$	7.00689e6	1.22263
$506$	0	0
$507$	406160.	0.0701741
$508$	0	0
$509$	−6.26867e6	−1.07246	−0.536230	−	0.844072i	$-0.680152\pi$
$509$	−6.26867e6	−1.07246	−0.536230	+	0.844072i	$0.680152\pi$
$510$	0	0
$511$	3.80987e6	0.645443
$512$	0	0
$513$	329938.	0.0553527
$514$	0	0
$515$	−5.13073e6	−0.852435
$516$	0	0
$517$	1.08078e7	1.77832
$518$	0	0
$519$	−1.50061e7	−2.44540
$520$	0	0
$521$	−4.12134e6	−0.665188	−0.332594	−	0.943070i	$-0.607924\pi$
$521$	−4.12134e6	−0.665188	−0.332594	+	0.943070i	$0.607924\pi$
$522$	0	0
$523$	−4.70469e6	−0.752102	−0.376051	−	0.926599i	$-0.622718\pi$
$523$	−4.70469e6	−0.752102	−0.376051	+	0.926599i	$0.622718\pi$
$524$	0	0
$525$	5.17845e6	0.819976
$526$	0	0
$527$	1.36502e7	2.14099
$528$	0	0
$529$	−6.00080e6	−0.932331
$530$	0	0
$531$	3.16590e6	0.487261
$532$	0	0
$533$	1.30051e6	0.198287
$534$	0	0
$535$	−3.50081e6	−0.528791
$536$	0	0
$537$	−1.12061e6	−0.167695
$538$	0	0
$539$	−1.32654e6	−0.196674
$540$	0	0
$541$	4.47840e6	0.657855	0.328927	−	0.944355i	$-0.393313\pi$
$541$	4.47840e6	0.657855	0.328927	+	0.944355i	$0.393313\pi$
$542$	0	0
$543$	8.19428e6	1.19265
$544$	0	0
$545$	−1.50124e6	−0.216501
$546$	0	0
$547$	−6.49187e6	−0.927687	−0.463843	−	0.885917i	$-0.653530\pi$
$547$	−6.49187e6	−0.927687	−0.463843	+	0.885917i	$0.653530\pi$
$548$	0	0
$549$	−2.10776e6	−0.298463
$550$	0	0
$551$	−2.61108e6	−0.366388
$552$	0	0
$553$	157213.	0.0218613
$554$	0	0
$555$	−9.79115e6	−1.34928
$556$	0	0
$557$	−2.50858e6	−0.342602	−0.171301	−	0.985219i	$-0.554797\pi$
$557$	−2.50858e6	−0.342602	−0.171301	+	0.985219i	$0.554797\pi$
$558$	0	0
$559$	−4.42404e6	−0.598811
$560$	0	0
$561$	−1.64163e7	−2.20226
$562$	0	0
$563$	−3.52702e6	−0.468961	−0.234480	−	0.972121i	$-0.575339\pi$
$563$	−3.52702e6	−0.468961	−0.234480	+	0.972121i	$0.575339\pi$
$564$	0	0
$565$	−6.94061e6	−0.914696
$566$	0	0
$567$	3.36278e6	0.439279
$568$	0	0
$569$	9.75283e6	1.26284	0.631422	−	0.775439i	$-0.282471\pi$
$569$	9.75283e6	1.26284	0.631422	+	0.775439i	$0.282471\pi$
$570$	0	0
$571$	1.70085e6	0.218312	0.109156	−	0.994025i	$-0.465185\pi$
$571$	1.70085e6	0.218312	0.109156	+	0.994025i	$0.465185\pi$
$572$	0	0
$573$	−8.26314e6	−1.05138
$574$	0	0
$575$	3.33832e6	0.421074
$576$	0	0
$577$	−1.17066e7	−1.46383	−0.731914	−	0.681397i	$-0.761373\pi$
$577$	−1.17066e7	−1.46383	−0.731914	+	0.681397i	$0.761373\pi$
$578$	0	0
$579$	1.17602e7	1.45787
$580$	0	0
$581$	−3.75472e6	−0.461463
$582$	0	0
$583$	−2.02045e7	−2.46193
$584$	0	0
$585$	−1.03828e7	−1.25437
$586$	0	0
$587$	−3.75683e6	−0.450015	−0.225007	−	0.974357i	$-0.572241\pi$
$587$	−3.75683e6	−0.450015	−0.225007	+	0.974357i	$0.572241\pi$
$588$	0	0
$589$	−3.06175e6	−0.363648
$590$	0	0
$591$	−8.40299e6	−0.989612
$592$	0	0
$593$	8.64338e6	1.00936	0.504681	−	0.863306i	$-0.331610\pi$
$593$	8.64338e6	1.00936	0.504681	+	0.863306i	$0.331610\pi$
$594$	0	0
$595$	6.30408e6	0.730011
$596$	0	0
$597$	9.52643e6	1.09394
$598$	0	0
$599$	9.31930e6	1.06125	0.530623	−	0.847608i	$-0.321958\pi$
$599$	9.31930e6	1.06125	0.530623	+	0.847608i	$0.321958\pi$
$600$	0	0
$601$	910438.	0.102817	0.0514084	−	0.998678i	$-0.483629\pi$
$601$	910438.	0.102817	0.0514084	+	0.998678i	$0.483629\pi$
$602$	0	0
$603$	1.55490e6	0.174144
$604$	0	0
$605$	1.30445e7	1.44891
$606$	0	0
$607$	5.70173e6	0.628109	0.314054	−	0.949405i	$-0.398313\pi$
$607$	5.70173e6	0.628109	0.314054	+	0.949405i	$0.398313\pi$
$608$	0	0
$609$	−8.37951e6	−0.915536
$610$	0	0
$611$	1.16035e7	1.25743
$612$	0	0
$613$	1.61327e7	1.73403	0.867016	−	0.498280i	$-0.166035\pi$
$613$	1.61327e7	1.73403	0.867016	+	0.498280i	$0.166035\pi$
$614$	0	0
$615$	4.14370e6	0.441774
$616$	0	0
$617$	−5.53575e6	−0.585415	−0.292708	−	0.956202i	$-0.594556\pi$
$617$	−5.53575e6	−0.585415	−0.292708	+	0.956202i	$0.594556\pi$
$618$	0	0
$619$	−1.88448e7	−1.97681	−0.988407	−	0.151828i	$-0.951484\pi$
$619$	−1.88448e7	−1.97681	−0.988407	+	0.151828i	$0.951484\pi$
$620$	0	0
$621$	682588.	0.0710280
$622$	0	0
$623$	4.13014e6	0.426329
$624$	0	0
$625$	14378.1	0.00147232
$626$	0	0
$627$	3.68217e6	0.374055
$628$	0	0
$629$	−7.36776e6	−0.742521
$630$	0	0
$631$	2.63269e6	0.263224	0.131612	−	0.991301i	$-0.457985\pi$
$631$	2.63269e6	0.263224	0.131612	+	0.991301i	$0.457985\pi$
$632$	0	0
$633$	2.47082e7	2.45093
$634$	0	0
$635$	6.20021e6	0.610200
$636$	0	0
$637$	−1.42420e6	−0.139067
$638$	0	0
$639$	1.08324e7	1.04948
$640$	0	0
$641$	1.42939e7	1.37406	0.687028	−	0.726631i	$-0.258915\pi$
$641$	1.42939e7	1.37406	0.687028	+	0.726631i	$0.258915\pi$
$642$	0	0
$643$	−1.63874e7	−1.56308	−0.781542	−	0.623853i	$-0.785566\pi$
$643$	−1.63874e7	−1.56308	−0.781542	+	0.623853i	$0.785566\pi$
$644$	0	0
$645$	−1.40960e7	−1.33412
$646$	0	0
$647$	1.19303e7	1.12045	0.560224	−	0.828341i	$-0.310715\pi$
$647$	1.19303e7	1.12045	0.560224	+	0.828341i	$0.310715\pi$
$648$	0	0
$649$	−9.03977e6	−0.842453
$650$	0	0
$651$	−9.82579e6	−0.908688
$652$	0	0
$653$	3.47862e6	0.319245	0.159622	−	0.987178i	$-0.448972\pi$
$653$	3.47862e6	0.319245	0.159622	+	0.987178i	$0.448972\pi$
$654$	0	0
$655$	8.67860e6	0.790399
$656$	0	0
$657$	−1.50447e7	−1.35978
$658$	0	0
$659$	2.09397e7	1.87826	0.939131	−	0.343560i	$-0.111633\pi$
$659$	2.09397e7	1.87826	0.939131	+	0.343560i	$0.111633\pi$
$660$	0	0
$661$	−1.02566e7	−0.913059	−0.456530	−	0.889708i	$-0.650908\pi$
$661$	−1.02566e7	−0.913059	−0.456530	+	0.889708i	$0.650908\pi$
$662$	0	0
$663$	−1.76249e7	−1.55720
$664$	0	0
$665$	−1.41400e6	−0.123993
$666$	0	0
$667$	−5.40190e6	−0.470145
$668$	0	0
$669$	−1.94532e7	−1.68045
$670$	0	0
$671$	6.01840e6	0.516029
$672$	0	0
$673$	716398.	0.0609701	0.0304851	−	0.999535i	$-0.490295\pi$
$673$	716398.	0.0609701	0.0304851	+	0.999535i	$0.490295\pi$
$674$	0	0
$675$	5.23190e6	0.441977
$676$	0	0
$677$	945625.	0.0792953	0.0396476	−	0.999214i	$-0.487376\pi$
$677$	945625.	0.0792953	0.0396476	+	0.999214i	$0.487376\pi$
$678$	0	0
$679$	−4.96236e6	−0.413061
$680$	0	0
$681$	−4.02500e6	−0.332582
$682$	0	0
$683$	3.56992e6	0.292824	0.146412	−	0.989224i	$-0.453227\pi$
$683$	3.56992e6	0.292824	0.146412	+	0.989224i	$0.453227\pi$
$684$	0	0
$685$	2.63512e7	2.14573
$686$	0	0
$687$	1.63783e7	1.32397
$688$	0	0
$689$	−2.16920e7	−1.74081
$690$	0	0
$691$	9.67811e6	0.771073	0.385536	−	0.922693i	$-0.374016\pi$
$691$	9.67811e6	0.771073	0.385536	+	0.922693i	$0.374016\pi$
$692$	0	0
$693$	5.23832e6	0.414342
$694$	0	0
$695$	−2.54700e7	−2.00017
$696$	0	0
$697$	3.11810e6	0.243113
$698$	0	0
$699$	2.92445e7	2.26387
$700$	0	0
$701$	2.52404e7	1.94000	0.970000	−	0.243105i	$-0.0781659\pi$
$701$	2.52404e7	1.94000	0.970000	+	0.243105i	$0.0781659\pi$
$702$	0	0
$703$	1.65259e6	0.126118
$704$	0	0
$705$	3.69713e7	2.80151
$706$	0	0
$707$	−3.79537e6	−0.285565
$708$	0	0
$709$	−1.74744e7	−1.30553	−0.652765	−	0.757560i	$-0.726391\pi$
$709$	−1.74744e7	−1.30553	−0.652765	+	0.757560i	$0.726391\pi$
$710$	0	0
$711$	−620813.	−0.0460561
$712$	0	0
$713$	−6.33426e6	−0.466629
$714$	0	0
$715$	2.96466e7	2.16875
$716$	0	0
$717$	1.34470e7	0.976852
$718$	0	0
$719$	−9.73361e6	−0.702185	−0.351093	−	0.936341i	$-0.614190\pi$
$719$	−9.73361e6	−0.702185	−0.351093	+	0.936341i	$0.614190\pi$
$720$	0	0
$721$	2.77913e6	0.199100
$722$	0	0
$723$	1.22756e6	0.0873371
$724$	0	0
$725$	−4.14045e7	−2.92552
$726$	0	0
$727$	−1.60454e7	−1.12594	−0.562971	−	0.826477i	$-0.690342\pi$
$727$	−1.60454e7	−1.12594	−0.562971	+	0.826477i	$0.690342\pi$
$728$	0	0
$729$	−8.02815e6	−0.559496
$730$	0	0
$731$	−1.06071e7	−0.734182
$732$	0	0
$733$	1.04534e7	0.718615	0.359307	−	0.933219i	$-0.383013\pi$
$733$	1.04534e7	0.718615	0.359307	+	0.933219i	$0.383013\pi$
$734$	0	0
$735$	−4.53783e6	−0.309835
$736$	0	0
$737$	−4.43979e6	−0.301088
$738$	0	0
$739$	−7.47638e6	−0.503594	−0.251797	−	0.967780i	$-0.581022\pi$
$739$	−7.47638e6	−0.503594	−0.251797	+	0.967780i	$0.581022\pi$
$740$	0	0
$741$	3.95327e6	0.264491
$742$	0	0
$743$	−1.33998e7	−0.890483	−0.445241	−	0.895411i	$-0.646882\pi$
$743$	−1.33998e7	−0.890483	−0.445241	+	0.895411i	$0.646882\pi$
$744$	0	0
$745$	−1.16047e7	−0.766028
$746$	0	0
$747$	1.48269e7	0.972182
$748$	0	0
$749$	1.89626e6	0.123507
$750$	0	0
$751$	9.13903e6	0.591290	0.295645	−	0.955298i	$-0.404465\pi$
$751$	9.13903e6	0.591290	0.295645	+	0.955298i	$0.404465\pi$
$752$	0	0
$753$	1.34063e7	0.861629
$754$	0	0
$755$	1.72039e6	0.109840
$756$	0	0
$757$	−1.16376e7	−0.738117	−0.369059	−	0.929406i	$-0.620320\pi$
$757$	−1.16376e7	−0.738117	−0.369059	+	0.929406i	$0.620320\pi$
$758$	0	0
$759$	7.61782e6	0.479983
$760$	0	0
$761$	−8.07072e6	−0.505186	−0.252593	−	0.967573i	$-0.581283\pi$
$761$	−8.07072e6	−0.505186	−0.252593	+	0.967573i	$0.581283\pi$
$762$	0	0
$763$	813167.	0.0505672
$764$	0	0
$765$	−2.48939e7	−1.53794
$766$	0	0
$767$	−9.70532e6	−0.595692
$768$	0	0
$769$	−1.30085e7	−0.793255	−0.396628	−	0.917980i	$-0.629820\pi$
$769$	−1.30085e7	−0.793255	−0.396628	+	0.917980i	$0.629820\pi$
$770$	0	0
$771$	−1.69369e7	−1.02612
$772$	0	0
$773$	2.37637e7	1.43043	0.715214	−	0.698906i	$-0.246329\pi$
$773$	2.37637e7	1.43043	0.715214	+	0.698906i	$0.246329\pi$
$774$	0	0
$775$	−4.85508e7	−2.90364
$776$	0	0
$777$	5.30350e6	0.315145
$778$	0	0
$779$	−699389.	−0.0412929
$780$	0	0
$781$	−3.09305e7	−1.81451
$782$	0	0
$783$	−8.46600e6	−0.493485
$784$	0	0
$785$	−2.46951e7	−1.43033
$786$	0	0
$787$	−2.84940e7	−1.63989	−0.819947	−	0.572439i	$-0.805997\pi$
$787$	−2.84940e7	−1.63989	−0.819947	+	0.572439i	$0.805997\pi$
$788$	0	0
$789$	4.26334e7	2.43813
$790$	0	0
$791$	3.75947e6	0.213641
$792$	0	0
$793$	6.46150e6	0.364880
$794$	0	0
$795$	−6.91157e7	−3.87845
$796$	0	0
$797$	1.43450e7	0.799938	0.399969	−	0.916529i	$-0.369021\pi$
$797$	1.43450e7	0.799938	0.399969	+	0.916529i	$0.369021\pi$
$798$	0	0
$799$	2.78206e7	1.54170
$800$	0	0
$801$	−1.63094e7	−0.898164
$802$	0	0
$803$	4.29578e7	2.35100
$804$	0	0
$805$	−2.92534e6	−0.159106
$806$	0	0
$807$	2.68050e7	1.44888
$808$	0	0
$809$	−3.59284e6	−0.193004	−0.0965020	−	0.995333i	$-0.530765\pi$
$809$	−3.59284e6	−0.193004	−0.0965020	+	0.995333i	$0.530765\pi$
$810$	0	0
$811$	3.52968e7	1.88445	0.942223	−	0.334988i	$-0.108732\pi$
$811$	3.52968e7	1.88445	0.942223	+	0.334988i	$0.108732\pi$
$812$	0	0
$813$	470316.	0.0249553
$814$	0	0
$815$	2.85724e7	1.50679
$816$	0	0
$817$	2.37917e6	0.124701
$818$	0	0
$819$	5.62399e6	0.292978
$820$	0	0
$821$	1.04558e7	0.541378	0.270689	−	0.962667i	$-0.412748\pi$
$821$	1.04558e7	0.541378	0.270689	+	0.962667i	$0.412748\pi$
$822$	0	0
$823$	4.99190e6	0.256901	0.128451	−	0.991716i	$-0.459000\pi$
$823$	4.99190e6	0.256901	0.128451	+	0.991716i	$0.459000\pi$
$824$	0	0
$825$	5.83890e7	2.98673
$826$	0	0
$827$	−2.06198e7	−1.04839	−0.524193	−	0.851599i	$-0.675633\pi$
$827$	−2.06198e7	−1.04839	−0.524193	+	0.851599i	$0.675633\pi$
$828$	0	0
$829$	−506843.	−0.0256146	−0.0128073	−	0.999918i	$-0.504077\pi$
$829$	−506843.	−0.0256146	−0.0128073	+	0.999918i	$0.504077\pi$
$830$	0	0
$831$	−7.69281e6	−0.386440
$832$	0	0
$833$	−3.41468e6	−0.170505
$834$	0	0
$835$	−5.32511e7	−2.64309
$836$	0	0
$837$	−9.92721e6	−0.489794
$838$	0	0
$839$	−4.41851e6	−0.216706	−0.108353	−	0.994112i	$-0.534558\pi$
$839$	−4.41851e6	−0.216706	−0.108353	+	0.994112i	$0.534558\pi$
$840$	0	0
$841$	4.64876e7	2.26645
$842$	0	0
$843$	−4.25598e7	−2.06267
$844$	0	0
$845$	−1.75863e6	−0.0847292
$846$	0	0
$847$	−7.06574e6	−0.338415
$848$	0	0
$849$	1.37140e7	0.652973
$850$	0	0
$851$	3.41893e6	0.161833
$852$	0	0
$853$	−1.11737e7	−0.525806	−0.262903	−	0.964822i	$-0.584680\pi$
$853$	−1.11737e7	−0.525806	−0.262903	+	0.964822i	$0.584680\pi$
$854$	0	0
$855$	5.58370e6	0.261220
$856$	0	0
$857$	2.12308e7	0.987448	0.493724	−	0.869619i	$-0.335635\pi$
$857$	2.12308e7	0.987448	0.493724	+	0.869619i	$0.335635\pi$
$858$	0	0
$859$	3.64229e7	1.68419	0.842096	−	0.539328i	$-0.181322\pi$
$859$	3.64229e7	1.68419	0.842096	+	0.539328i	$0.181322\pi$
$860$	0	0
$861$	−2.24449e6	−0.103183
$862$	0	0
$863$	−2.53298e7	−1.15772	−0.578861	−	0.815426i	$-0.696503\pi$
$863$	−2.53298e7	−1.15772	−0.578861	+	0.815426i	$0.696503\pi$
$864$	0	0
$865$	6.49750e7	2.95261
$866$	0	0
$867$	−1.25934e7	−0.568978
$868$	0	0
$869$	1.77264e6	0.0796290
$870$	0	0
$871$	−4.76667e6	−0.212897
$872$	0	0
$873$	1.95957e7	0.870211
$874$	0	0
$875$	−8.57014e6	−0.378415
$876$	0	0
$877$	−577138.	−0.0253385	−0.0126692	−	0.999920i	$-0.504033\pi$
$877$	−577138.	−0.0253385	−0.0126692	+	0.999920i	$0.504033\pi$
$878$	0	0
$879$	2.01047e7	0.877658
$880$	0	0
$881$	−2.15071e7	−0.933561	−0.466781	−	0.884373i	$-0.654586\pi$
$881$	−2.15071e7	−0.933561	−0.466781	+	0.884373i	$0.654586\pi$
$882$	0	0
$883$	−3.71948e6	−0.160539	−0.0802695	−	0.996773i	$-0.525578\pi$
$883$	−3.71948e6	−0.160539	−0.0802695	+	0.996773i	$0.525578\pi$
$884$	0	0
$885$	−3.09233e7	−1.32717
$886$	0	0
$887$	1.81760e7	0.775694	0.387847	−	0.921724i	$-0.373219\pi$
$887$	1.81760e7	0.775694	0.387847	+	0.921724i	$0.373219\pi$
$888$	0	0
$889$	−3.35842e6	−0.142522
$890$	0	0
$891$	3.79166e7	1.60006
$892$	0	0
$893$	−6.24015e6	−0.261858
$894$	0	0
$895$	4.85214e6	0.202477
$896$	0	0
$897$	8.17868e6	0.339392
$898$	0	0
$899$	7.85625e7	3.24202
$900$	0	0
$901$	−5.20090e7	−2.13435
$902$	0	0
$903$	7.63527e6	0.311605
$904$	0	0
$905$	−3.54804e7	−1.44002
$906$	0	0
$907$	−1.35430e7	−0.546635	−0.273317	−	0.961924i	$-0.588121\pi$
$907$	−1.35430e7	−0.546635	−0.273317	+	0.961924i	$0.588121\pi$
$908$	0	0
$909$	1.49874e7	0.601612
$910$	0	0
$911$	−412746.	−0.0164773	−0.00823866	−	0.999966i	$-0.502622\pi$
$911$	−412746.	−0.0164773	−0.00823866	+	0.999966i	$0.502622\pi$
$912$	0	0
$913$	−4.23359e7	−1.68086
$914$	0	0
$915$	2.05878e7	0.812937
$916$	0	0
$917$	−4.70087e6	−0.184610
$918$	0	0
$919$	2.79059e7	1.08995	0.544976	−	0.838452i	$-0.316539\pi$
$919$	2.79059e7	1.08995	0.544976	+	0.838452i	$0.316539\pi$
$920$	0	0
$921$	6.19394e6	0.240612
$922$	0	0
$923$	−3.32077e7	−1.28302
$924$	0	0
$925$	2.62054e7	1.00702
$926$	0	0
$927$	−1.09744e7	−0.419451
$928$	0	0
$929$	−1.94840e7	−0.740695	−0.370348	−	0.928893i	$-0.620761\pi$
$929$	−1.94840e7	−0.740695	−0.370348	+	0.928893i	$0.620761\pi$
$930$	0	0
$931$	765912.	0.0289604
$932$	0	0
$933$	−2.34128e7	−0.880539
$934$	0	0
$935$	7.10809e7	2.65904
$936$	0	0
$937$	−1.89631e7	−0.705602	−0.352801	−	0.935698i	$-0.614771\pi$
$937$	−1.89631e7	−0.705602	−0.352801	+	0.935698i	$0.614771\pi$
$938$	0	0
$939$	−3.65425e7	−1.35249
$940$	0	0
$941$	−2.30498e7	−0.848579	−0.424290	−	0.905526i	$-0.639476\pi$
$941$	−2.30498e7	−0.848579	−0.424290	+	0.905526i	$0.639476\pi$
$942$	0	0
$943$	−1.44692e6	−0.0529866
$944$	0	0
$945$	−4.58467e6	−0.167005
$946$	0	0
$947$	−3.49310e7	−1.26572	−0.632858	−	0.774268i	$-0.718118\pi$
$947$	−3.49310e7	−1.26572	−0.632858	+	0.774268i	$0.718118\pi$
$948$	0	0
$949$	4.61206e7	1.66238
$950$	0	0
$951$	−6.63331e7	−2.37837
$952$	0	0
$953$	2.74501e7	0.979066	0.489533	−	0.871985i	$-0.337167\pi$
$953$	2.74501e7	0.979066	0.489533	+	0.871985i	$0.337167\pi$
$954$	0	0
$955$	3.57786e7	1.26945
$956$	0	0
$957$	−9.44822e7	−3.33480
$958$	0	0
$959$	−1.42735e7	−0.501168
$960$	0	0
$961$	6.34930e7	2.21778
$962$	0	0
$963$	−7.48807e6	−0.260198
$964$	0	0
$965$	−5.09204e7	−1.76025
$966$	0	0
$967$	−9.37608e6	−0.322445	−0.161222	−	0.986918i	$-0.551544\pi$
$967$	−9.37608e6	−0.322445	−0.161222	+	0.986918i	$0.551544\pi$
$968$	0	0
$969$	9.47839e6	0.324284
$970$	0	0
$971$	7.23359e6	0.246210	0.123105	−	0.992394i	$-0.460715\pi$
$971$	7.23359e6	0.246210	0.123105	+	0.992394i	$0.460715\pi$
$972$	0	0
$973$	1.37962e7	0.467171
$974$	0	0
$975$	6.26879e7	2.11190
$976$	0	0
$977$	−2.88022e7	−0.965360	−0.482680	−	0.875797i	$-0.660337\pi$
$977$	−2.88022e7	−0.965360	−0.482680	+	0.875797i	$0.660337\pi$
$978$	0	0
$979$	4.65690e7	1.55289
$980$	0	0
$981$	−3.21109e6	−0.106532
$982$	0	0
$983$	5.21848e7	1.72250	0.861252	−	0.508178i	$-0.169681\pi$
$983$	5.21848e7	1.72250	0.861252	+	0.508178i	$0.169681\pi$
$984$	0	0
$985$	3.63841e7	1.19487
$986$	0	0
$987$	−2.00260e7	−0.654335
$988$	0	0
$989$	4.92212e6	0.160015
$990$	0	0
$991$	−3.89578e7	−1.26012	−0.630058	−	0.776548i	$-0.716969\pi$
$991$	−3.89578e7	−1.26012	−0.630058	+	0.776548i	$0.716969\pi$
$992$	0	0
$993$	−1.83520e7	−0.590624
$994$	0	0
$995$	−4.12485e7	−1.32084
$996$	0	0
$997$	−3.01716e7	−0.961305	−0.480652	−	0.876911i	$-0.659600\pi$
$997$	−3.01716e7	−0.961305	−0.480652	+	0.876911i	$0.659600\pi$
$998$	0	0
$999$	5.35824e6	0.169867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	56.6.a.d.1.1	✓	2
3.2	odd	2		504.6.a.m.1.1		2
4.3	odd	2		112.6.a.j.1.2		2
7.2	even	3		392.6.i.k.361.2		4
7.3	odd	6		392.6.i.h.177.1		4
7.4	even	3		392.6.i.k.177.2		4
7.5	odd	6		392.6.i.h.361.1		4
7.6	odd	2		392.6.a.e.1.2		2
8.3	odd	2		448.6.a.r.1.1		2
8.5	even	2		448.6.a.x.1.2		2
12.11	even	2		1008.6.a.bi.1.1		2
28.27	even	2		784.6.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.d.1.1	✓	2	1.1	even	1	trivial
112.6.a.j.1.2		2	4.3	odd	2
392.6.a.e.1.2		2	7.6	odd	2
392.6.i.h.177.1		4	7.3	odd	6
392.6.i.h.361.1		4	7.5	odd	6
392.6.i.k.177.2		4	7.4	even	3
392.6.i.k.361.2		4	7.2	even	3
448.6.a.r.1.1		2	8.3	odd	2
448.6.a.x.1.2		2	8.5	even	2
504.6.a.m.1.1		2	3.2	odd	2
784.6.a.q.1.1		2	28.27	even	2
1008.6.a.bi.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	56.a (trivial)

\(f(q)\)	\(=\)	\(q-20.8924 q^{3} +90.4622 q^{5} -49.0000 q^{7} +193.494 q^{9} -552.494 q^{11} -593.172 q^{13} -1889.98 q^{15} -1422.19 q^{17} +318.997 q^{19} +1023.73 q^{21} +659.954 q^{23} +5058.41 q^{25} +1034.30 q^{27} -8185.27 q^{29} -9598.03 q^{31} +11543.0 q^{33} -4432.65 q^{35} +5180.56 q^{37} +12392.8 q^{39} -2192.46 q^{41} +7458.29 q^{43} +17503.9 q^{45} -19561.8 q^{47} +2401.00 q^{49} +29713.1 q^{51} +36569.6 q^{53} -49979.9 q^{55} -6664.63 q^{57} +16361.7 q^{59} -10893.1 q^{61} -9481.22 q^{63} -53659.6 q^{65} +8035.91 q^{67} -13788.0 q^{69} +55983.3 q^{71} -77752.5 q^{73} -105683. q^{75} +27072.2 q^{77} -3208.43 q^{79} -68628.1 q^{81} +76626.9 q^{83} -128655. q^{85} +171010. q^{87} -84288.6 q^{89} +29065.4 q^{91} +200526. q^{93} +28857.2 q^{95} +101273. q^{97} -106904. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{3} + 42 q^{5} - 98 q^{7} - 2 q^{9} - 716 q^{11} - 714 q^{13} - 2224 q^{15} - 1344 q^{17} - 1946 q^{19} + 686 q^{21} - 1792 q^{23} + 4282 q^{25} - 1988 q^{27} - 1200 q^{29} - 6804 q^{31} + 10416 q^{33}+ \cdots - 74940 q^{99}+O(q^{100}) \) 2 * q - 14 * q^3 + 42 * q^5 - 98 * q^7 - 2 * q^9 - 716 * q^11 - 714 * q^13 - 2224 * q^15 - 1344 * q^17 - 1946 * q^19 + 686 * q^21 - 1792 * q^23 + 4282 * q^25 - 1988 * q^27 - 1200 * q^29 - 6804 * q^31 + 10416 * q^33 - 2058 * q^35 + 14640 * q^37 + 11560 * q^39 + 7896 * q^41 + 524 * q^43 + 26978 * q^45 - 18396 * q^47 + 4802 * q^49 + 30252 * q^51 + 45132 * q^53 - 42056 * q^55 - 22276 * q^57 + 22582 * q^59 - 52822 * q^61 + 98 * q^63 - 47804 * q^65 + 9848 * q^67 - 30688 * q^69 - 840 * q^71 - 122052 * q^73 - 111034 * q^75 + 35084 * q^77 + 31704 * q^79 - 41954 * q^81 + 36974 * q^83 - 132444 * q^85 + 219156 * q^87 - 210588 * q^89 + 34986 * q^91 + 219784 * q^93 + 138624 * q^95 - 44240 * q^97 - 74940 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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