LMFDB - Embedded newform 448.6.a.x.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,6,Mod(1,448)] 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("448.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,14,0,-42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$71.8519512762$
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:	no (minimal twist has level 56)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.44622$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-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	$-0.266240\pi$
$3$	20.8924	1.34025	0.670125	+	0.742248i	$0.266240\pi$
$4$	0	0
$5$	−90.4622	−1.61824	−0.809119	−	0.587645i	$-0.800055\pi$
$5$	−90.4622	−1.61824	−0.809119	+	0.587645i	$0.800055\pi$
$6$	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	$-0.258331\pi$
$11$	552.494	1.37672	0.688361	+	0.725369i	$0.258331\pi$
$12$	0	0
$13$	593.172	0.973469	0.486734	−	0.873550i	$-0.338188\pi$
$13$	593.172	0.973469	0.486734	+	0.873550i	$0.338188\pi$
$14$	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	$-0.532320\pi$
$19$	−318.997	−0.202723	−0.101361	+	0.994850i	$0.532320\pi$
$20$	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	$-0.140865\pi$
$29$	8185.27	1.80733	0.903667	+	0.428237i	$0.140865\pi$
$30$	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	$-0.600684\pi$
$37$	−5180.56	−0.622118	−0.311059	+	0.950391i	$0.600684\pi$
$38$	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	$-0.599514\pi$
$43$	−7458.29	−0.615131	−0.307566	+	0.951527i	$0.599514\pi$
$44$	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	$-0.852205\pi$
$53$	−36569.6	−1.78826	−0.894129	+	0.447809i	$0.852205\pi$
$54$	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	$-0.598979\pi$
$59$	−16361.7	−0.611927	−0.305963	+	0.952043i	$0.598979\pi$
$60$	0	0
$61$	10893.1	0.374825	0.187412	−	0.982281i	$-0.439990\pi$
$61$	10893.1	0.374825	0.187412	+	0.982281i	$0.439990\pi$
$62$	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	$-0.534877\pi$
$67$	−8035.91	−0.218700	−0.109350	+	0.994003i	$0.534877\pi$
$68$	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	$-0.709015\pi$
$83$	−76626.9	−1.22092	−0.610458	+	0.792049i	$0.709015\pi$
$84$	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	$-0.623308\pi$
$101$	−77456.5	−0.755535	−0.377767	+	0.925901i	$0.623308\pi$
$102$	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	$-0.447759\pi$
$107$	38699.2	0.326770	0.163385	+	0.986562i	$0.447759\pi$
$108$	0	0
$109$	16595.3	0.133788	0.0668941	−	0.997760i	$-0.478691\pi$
$109$	16595.3	0.133788	0.0668941	+	0.997760i	$0.478691\pi$
$110$	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	$-0.578531\pi$
$131$	−95936.2	−0.488432	−0.244216	+	0.969721i	$0.578531\pi$
$132$	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	$-0.287939\pi$
$139$	281554.	1.23602	0.618009	+	0.786171i	$0.287939\pi$
$140$	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	$-0.423939\pi$
$149$	128283.	0.473372	0.236686	+	0.971586i	$0.423939\pi$
$150$	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	$-0.354290\pi$
$157$	272988.	0.883883	0.441941	+	0.897044i	$0.354290\pi$
$158$	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	$-0.654149\pi$
$163$	−315849.	−0.931131	−0.465565	+	0.885013i	$0.654149\pi$
$164$	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	$-0.865689\pi$
$173$	−718256.	−1.82458	−0.912292	+	0.409540i	$0.865689\pi$
$174$	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	$-0.519927\pi$
$179$	−53637.2	−0.125122	−0.0625610	+	0.998041i	$0.519927\pi$
$180$	0	0
$181$	392213.	0.889867	0.444934	−	0.895564i	$-0.353227\pi$
$181$	392213.	0.889867	0.444934	+	0.895564i	$0.353227\pi$
$182$	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	$-0.620365\pi$
$197$	−402202.	−0.738378	−0.369189	+	0.929354i	$0.620365\pi$
$198$	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	$-0.132698\pi$
$211$	1.18264e6	1.82871	0.914356	+	0.404911i	$0.132698\pi$
$212$	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	$-0.539596\pi$
$227$	−192653.	−0.248149	−0.124074	+	0.992273i	$0.539596\pi$
$228$	0	0
$229$	783934.	0.987850	0.493925	−	0.869505i	$-0.335562\pi$
$229$	783934.	0.987850	0.493925	+	0.869505i	$0.335562\pi$
$230$	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	$-0.395832\pi$
$251$	641680.	0.642886	0.321443	+	0.946929i	$0.395832\pi$
$252$	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	$-0.318226\pi$
$269$	1.28300e6	1.08105	0.540524	+	0.841328i	$0.318226\pi$
$270$	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	$-0.546050\pi$
$277$	−368210.	−0.288334	−0.144167	+	0.989553i	$0.546050\pi$
$278$	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	$-0.421671\pi$
$283$	656410.	0.487202	0.243601	+	0.969876i	$0.421671\pi$
$284$	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	$-0.393820\pi$
$293$	962295.	0.654846	0.327423	+	0.944878i	$0.393820\pi$
$294$	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	$-0.471389\pi$
$307$	296468.	0.179528	0.0897638	+	0.995963i	$0.471389\pi$
$308$	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	$-0.847411\pi$
$317$	−3.17498e6	−1.77457	−0.887284	+	0.461223i	$0.847411\pi$
$318$	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	$-0.570717\pi$
$331$	−878405.	−0.440681	−0.220341	+	0.975423i	$0.570717\pi$
$332$	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	$-0.533852\pi$
$347$	−476182.	−0.212300	−0.106150	+	0.994350i	$0.533852\pi$
$348$	0	0
$349$	−351681.	−0.154556	−0.0772780	−	0.997010i	$-0.524623\pi$
$349$	−351681.	−0.154556	−0.0772780	+	0.997010i	$0.524623\pi$
$350$	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	$-0.450172\pi$
$373$	837822.	0.311803	0.155901	+	0.987773i	$0.450172\pi$
$374$	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	$-0.386810\pi$
$379$	1.94713e6	0.696300	0.348150	+	0.937439i	$0.386810\pi$
$380$	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	$-0.682472\pi$
$389$	−3.23741e6	−1.08473	−0.542367	+	0.840141i	$0.682472\pi$
$390$	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	$-0.541878\pi$
$397$	−823921.	−0.262367	−0.131183	+	0.991358i	$0.541878\pi$
$398$	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	$-0.642883\pi$
$419$	−3.11898e6	−0.867915	−0.433957	+	0.900933i	$0.642883\pi$
$420$	0	0
$421$	1.14481e6	0.314796	0.157398	−	0.987535i	$-0.449689\pi$
$421$	1.14481e6	0.314796	0.157398	+	0.987535i	$0.449689\pi$
$422$	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	$-0.363593\pi$
$443$	3.43282e6	0.831077	0.415539	+	0.909576i	$0.363593\pi$
$444$	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	$-0.432676\pi$
$461$	1.91583e6	0.419860	0.209930	+	0.977716i	$0.432676\pi$
$462$	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	$-0.428278\pi$
$467$	2.10594e6	0.446842	0.223421	+	0.974722i	$0.428278\pi$
$468$	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	$-0.427162\pi$
$491$	2.42352e6	0.453673	0.226837	+	0.973933i	$0.427162\pi$
$492$	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	$-0.235207\pi$
$499$	8.22317e6	1.47839	0.739193	+	0.673494i	$0.235207\pi$
$500$	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	$-0.319848\pi$
$509$	6.26867e6	1.07246	0.536230	+	0.844072i	$0.319848\pi$
$510$	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	$-0.377282\pi$
$523$	4.70469e6	0.752102	0.376051	+	0.926599i	$0.377282\pi$
$524$	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	$-0.606687\pi$
$541$	−4.47840e6	−0.657855	−0.328927	+	0.944355i	$0.606687\pi$
$542$	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	$-0.346470\pi$
$547$	6.49187e6	0.927687	0.463843	+	0.885917i	$0.346470\pi$
$548$	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	$-0.445203\pi$
$557$	2.50858e6	0.342602	0.171301	+	0.985219i	$0.445203\pi$
$558$	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	$-0.424661\pi$
$563$	3.52702e6	0.468961	0.234480	+	0.972121i	$0.424661\pi$
$564$	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	$-0.534815\pi$
$571$	−1.70085e6	−0.218312	−0.109156	+	0.994025i	$0.534815\pi$
$572$	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	$-0.427759\pi$
$587$	3.75683e6	0.450015	0.225007	+	0.974357i	$0.427759\pi$
$588$	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	$-0.833965\pi$
$613$	−1.61327e7	−1.73403	−0.867016	+	0.498280i	$0.833965\pi$
$614$	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	$-0.0485161\pi$
$619$	1.88448e7	1.97681	0.988407	+	0.151828i	$0.0485161\pi$
$620$	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	$-0.214434\pi$
$643$	1.63874e7	1.56308	0.781542	+	0.623853i	$0.214434\pi$
$644$	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	$-0.551028\pi$
$653$	−3.47862e6	−0.319245	−0.159622	+	0.987178i	$0.551028\pi$
$654$	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	$-0.888367\pi$
$659$	−2.09397e7	−1.87826	−0.939131	+	0.343560i	$0.888367\pi$
$660$	0	0
$661$	1.02566e7	0.913059	0.456530	−	0.889708i	$-0.349092\pi$
$661$	1.02566e7	0.913059	0.456530	+	0.889708i	$0.349092\pi$
$662$	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	$-0.512624\pi$
$677$	−945625.	−0.0792953	−0.0396476	+	0.999214i	$0.512624\pi$
$678$	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	$-0.546773\pi$
$683$	−3.56992e6	−0.292824	−0.146412	+	0.989224i	$0.546773\pi$
$684$	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	$-0.625984\pi$
$691$	−9.67811e6	−0.771073	−0.385536	+	0.922693i	$0.625984\pi$
$692$	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	$-0.921834\pi$
$701$	−2.52404e7	−1.94000	−0.970000	+	0.243105i	$0.921834\pi$
$702$	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	$-0.273609\pi$
$709$	1.74744e7	1.30553	0.652765	+	0.757560i	$0.273609\pi$
$710$	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	$-0.616987\pi$
$733$	−1.04534e7	−0.718615	−0.359307	+	0.933219i	$0.616987\pi$
$734$	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	$-0.418978\pi$
$739$	7.47638e6	0.503594	0.251797	+	0.967780i	$0.418978\pi$
$740$	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	$-0.379680\pi$
$757$	1.16376e7	0.738117	0.369059	+	0.929406i	$0.379680\pi$
$758$	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	$-0.753671\pi$
$773$	−2.37637e7	−1.43043	−0.715214	+	0.698906i	$0.753671\pi$
$774$	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	$-0.194003\pi$
$787$	2.84940e7	1.63989	0.819947	+	0.572439i	$0.194003\pi$
$788$	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	$-0.630979\pi$
$797$	−1.43450e7	−0.799938	−0.399969	+	0.916529i	$0.630979\pi$
$798$	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	$-0.891268\pi$
$811$	−3.52968e7	−1.88445	−0.942223	+	0.334988i	$0.891268\pi$
$812$	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	$-0.587252\pi$
$821$	−1.04558e7	−0.541378	−0.270689	+	0.962667i	$0.587252\pi$
$822$	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	$-0.324367\pi$
$827$	2.06198e7	1.04839	0.524193	+	0.851599i	$0.324367\pi$
$828$	0	0
$829$	506843.	0.0256146	0.0128073	−	0.999918i	$-0.495923\pi$
$829$	506843.	0.0256146	0.0128073	+	0.999918i	$0.495923\pi$
$830$	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	$-0.415320\pi$
$853$	1.11737e7	0.525806	0.262903	+	0.964822i	$0.415320\pi$
$854$	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	$-0.818678\pi$
$859$	−3.64229e7	−1.68419	−0.842096	+	0.539328i	$0.818678\pi$
$860$	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	$-0.495967\pi$
$877$	577138.	0.0253385	0.0126692	+	0.999920i	$0.495967\pi$
$878$	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	$-0.474422\pi$
$883$	3.71948e6	0.160539	0.0802695	+	0.996773i	$0.474422\pi$
$884$	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	$-0.411879\pi$
$907$	1.35430e7	0.546635	0.273317	+	0.961924i	$0.411879\pi$
$908$	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	$-0.360524\pi$
$941$	2.30498e7	0.848579	0.424290	+	0.905526i	$0.360524\pi$
$942$	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	$-0.281882\pi$
$947$	3.49310e7	1.26572	0.632858	+	0.774268i	$0.281882\pi$
$948$	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	$-0.539285\pi$
$971$	−7.23359e6	−0.246210	−0.123105	+	0.992394i	$0.539285\pi$
$972$	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	$-0.340400\pi$
$997$	3.01716e7	0.961305	0.480652	+	0.876911i	$0.340400\pi$
$998$	0	0
$999$	5.35824e6	0.169867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.x.1.2		2
4.3	odd	2		448.6.a.r.1.1		2
8.3	odd	2		112.6.a.j.1.2		2
8.5	even	2		56.6.a.d.1.1	✓	2
24.5	odd	2		504.6.a.m.1.1		2
24.11	even	2		1008.6.a.bi.1.1		2
56.5	odd	6		392.6.i.h.361.1		4
56.13	odd	2		392.6.a.e.1.2		2
56.27	even	2		784.6.a.q.1.1		2
56.37	even	6		392.6.i.k.361.2		4
56.45	odd	6		392.6.i.h.177.1		4
56.53	even	6		392.6.i.k.177.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.d.1.1	✓	2	8.5	even	2
112.6.a.j.1.2		2	8.3	odd	2
392.6.a.e.1.2		2	56.13	odd	2
392.6.i.h.177.1		4	56.45	odd	6
392.6.i.h.361.1		4	56.5	odd	6
392.6.i.k.177.2		4	56.53	even	6
392.6.i.k.361.2		4	56.37	even	6
448.6.a.r.1.1		2	4.3	odd	2
448.6.a.x.1.2		2	1.1	even	1	trivial
504.6.a.m.1.1		2	24.5	odd	2
784.6.a.q.1.1		2	56.27	even	2
1008.6.a.bi.1.1		2	24.11	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:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.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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