LMFDB - Embedded newform 448.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,6,Mod(1,448)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.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:	$71.8519512762$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 28)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q-26.0000 q^{3} -16.0000 q^{5} -49.0000 q^{7} +433.000 q^{9} +O(q^{10})$

$q-26.0000 q^{3} -16.0000 q^{5} -49.0000 q^{7} +433.000 q^{9} -8.00000 q^{11} -684.000 q^{13} +416.000 q^{15} -2218.00 q^{17} +2698.00 q^{19} +1274.00 q^{21} +3344.00 q^{23} -2869.00 q^{25} -4940.00 q^{27} +3254.00 q^{29} +4788.00 q^{31} +208.000 q^{33} +784.000 q^{35} +11470.0 q^{37} +17784.0 q^{39} +13350.0 q^{41} +928.000 q^{43} -6928.00 q^{45} +1212.00 q^{47} +2401.00 q^{49} +57668.0 q^{51} -13110.0 q^{53} +128.000 q^{55} -70148.0 q^{57} -34702.0 q^{59} +1032.00 q^{61} -21217.0 q^{63} +10944.0 q^{65} -10108.0 q^{67} -86944.0 q^{69} +62720.0 q^{71} -18926.0 q^{73} +74594.0 q^{75} +392.000 q^{77} +11400.0 q^{79} +23221.0 q^{81} -88958.0 q^{83} +35488.0 q^{85} -84604.0 q^{87} +19722.0 q^{89} +33516.0 q^{91} -124488. q^{93} -43168.0 q^{95} +17062.0 q^{97} -3464.00 q^{99} +O(q^{100})$

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
$3$	−26.0000	−1.66790	−0.833950	−	0.551839i	$-0.813926\pi$
$3$	−26.0000	−1.66790	−0.833950	+	0.551839i	$0.813926\pi$
$4$	0	0
$5$	−16.0000	−0.286217	−0.143108	−	0.989707i	$-0.545710\pi$
$5$	−16.0000	−0.286217	−0.143108	+	0.989707i	$0.545710\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	433.000	1.78189
$10$	0	0
$11$	−8.00000	−0.0199346	−0.00996732	−	0.999950i	$-0.503173\pi$
$11$	−8.00000	−0.0199346	−0.00996732	+	0.999950i	$0.503173\pi$
$12$	0	0
$13$	−684.000	−1.12253	−0.561265	−	0.827636i	$-0.689685\pi$
$13$	−684.000	−1.12253	−0.561265	+	0.827636i	$0.689685\pi$
$14$	0	0
$15$	416.000	0.477381
$16$	0	0
$17$	−2218.00	−1.86140	−0.930699	−	0.365786i	$-0.880800\pi$
$17$	−2218.00	−1.86140	−0.930699	+	0.365786i	$0.880800\pi$
$18$	0	0
$19$	2698.00	1.71458	0.857290	−	0.514833i	$-0.172146\pi$
$19$	2698.00	1.71458	0.857290	+	0.514833i	$0.172146\pi$
$20$	0	0
$21$	1274.00	0.630407
$22$	0	0
$23$	3344.00	1.31809	0.659047	−	0.752101i	$-0.270960\pi$
$23$	3344.00	1.31809	0.659047	+	0.752101i	$0.270960\pi$
$24$	0	0
$25$	−2869.00	−0.918080
$26$	0	0
$27$	−4940.00	−1.30412
$28$	0	0
$29$	3254.00	0.718493	0.359247	−	0.933243i	$-0.383034\pi$
$29$	3254.00	0.718493	0.359247	+	0.933243i	$0.383034\pi$
$30$	0	0
$31$	4788.00	0.894849	0.447425	−	0.894322i	$-0.352341\pi$
$31$	4788.00	0.894849	0.447425	+	0.894322i	$0.352341\pi$
$32$	0	0
$33$	208.000	0.0332490
$34$	0	0
$35$	784.000	0.108180
$36$	0	0
$37$	11470.0	1.37740	0.688698	−	0.725048i	$-0.258182\pi$
$37$	11470.0	1.37740	0.688698	+	0.725048i	$0.258182\pi$
$38$	0	0
$39$	17784.0	1.87227
$40$	0	0
$41$	13350.0	1.24029	0.620143	−	0.784489i	$-0.287075\pi$
$41$	13350.0	1.24029	0.620143	+	0.784489i	$0.287075\pi$
$42$	0	0
$43$	928.000	0.0765380	0.0382690	−	0.999267i	$-0.487816\pi$
$43$	928.000	0.0765380	0.0382690	+	0.999267i	$0.487816\pi$
$44$	0	0
$45$	−6928.00	−0.510008
$46$	0	0
$47$	1212.00	0.0800310	0.0400155	−	0.999199i	$-0.487259\pi$
$47$	1212.00	0.0800310	0.0400155	+	0.999199i	$0.487259\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	57668.0	3.10463
$52$	0	0
$53$	−13110.0	−0.641081	−0.320541	−	0.947235i	$-0.603865\pi$
$53$	−13110.0	−0.641081	−0.320541	+	0.947235i	$0.603865\pi$
$54$	0	0
$55$	128.000	0.00570563
$56$	0	0
$57$	−70148.0	−2.85975
$58$	0	0
$59$	−34702.0	−1.29785	−0.648925	−	0.760852i	$-0.724781\pi$
$59$	−34702.0	−1.29785	−0.648925	+	0.760852i	$0.724781\pi$
$60$	0	0
$61$	1032.00	0.0355104	0.0177552	−	0.999842i	$-0.494348\pi$
$61$	1032.00	0.0355104	0.0177552	+	0.999842i	$0.494348\pi$
$62$	0	0
$63$	−21217.0	−0.673492
$64$	0	0
$65$	10944.0	0.321287
$66$	0	0
$67$	−10108.0	−0.275092	−0.137546	−	0.990495i	$-0.543922\pi$
$67$	−10108.0	−0.275092	−0.137546	+	0.990495i	$0.543922\pi$
$68$	0	0
$69$	−86944.0	−2.19845
$70$	0	0
$71$	62720.0	1.47659	0.738295	−	0.674477i	$-0.235631\pi$
$71$	62720.0	1.47659	0.738295	+	0.674477i	$0.235631\pi$
$72$	0	0
$73$	−18926.0	−0.415673	−0.207836	−	0.978164i	$-0.566642\pi$
$73$	−18926.0	−0.415673	−0.207836	+	0.978164i	$0.566642\pi$
$74$	0	0
$75$	74594.0	1.53127
$76$	0	0
$77$	392.000	0.00753458
$78$	0	0
$79$	11400.0	0.205512	0.102756	−	0.994707i	$-0.467234\pi$
$79$	11400.0	0.205512	0.102756	+	0.994707i	$0.467234\pi$
$80$	0	0
$81$	23221.0	0.393250
$82$	0	0
$83$	−88958.0	−1.41739	−0.708696	−	0.705514i	$-0.750716\pi$
$83$	−88958.0	−1.41739	−0.708696	+	0.705514i	$0.750716\pi$
$84$	0	0
$85$	35488.0	0.532763
$86$	0	0
$87$	−84604.0	−1.19838
$88$	0	0
$89$	19722.0	0.263922	0.131961	−	0.991255i	$-0.457873\pi$
$89$	19722.0	0.263922	0.131961	+	0.991255i	$0.457873\pi$
$90$	0	0
$91$	33516.0	0.424276
$92$	0	0
$93$	−124488.	−1.49252
$94$	0	0
$95$	−43168.0	−0.490742
$96$	0	0
$97$	17062.0	0.184120	0.0920599	−	0.995753i	$-0.470655\pi$
$97$	17062.0	0.184120	0.0920599	+	0.995753i	$0.470655\pi$
$98$	0	0
$99$	−3464.00	−0.0355214
$100$	0	0
$101$	−45904.0	−0.447762	−0.223881	−	0.974617i	$-0.571873\pi$
$101$	−45904.0	−0.447762	−0.223881	+	0.974617i	$0.571873\pi$
$102$	0	0
$103$	−136012.	−1.26324	−0.631618	−	0.775280i	$-0.717609\pi$
$103$	−136012.	−1.26324	−0.631618	+	0.775280i	$0.717609\pi$
$104$	0	0
$105$	−20384.0	−0.180433
$106$	0	0
$107$	69156.0	0.583943	0.291971	−	0.956427i	$-0.405689\pi$
$107$	69156.0	0.583943	0.291971	+	0.956427i	$0.405689\pi$
$108$	0	0
$109$	146414.	1.18037	0.590183	−	0.807270i	$-0.299056\pi$
$109$	146414.	1.18037	0.590183	+	0.807270i	$0.299056\pi$
$110$	0	0
$111$	−298220.	−2.29736
$112$	0	0
$113$	−80186.0	−0.590748	−0.295374	−	0.955382i	$-0.595444\pi$
$113$	−80186.0	−0.590748	−0.295374	+	0.955382i	$0.595444\pi$
$114$	0	0
$115$	−53504.0	−0.377261
$116$	0	0
$117$	−296172.	−2.00023
$118$	0	0
$119$	108682.	0.703542
$120$	0	0
$121$	−160987.	−0.999603
$122$	0	0
$123$	−347100.	−2.06867
$124$	0	0
$125$	95904.0	0.548987
$126$	0	0
$127$	274800.	1.51185	0.755923	−	0.654661i	$-0.227189\pi$
$127$	274800.	1.51185	0.755923	+	0.654661i	$0.227189\pi$
$128$	0	0
$129$	−24128.0	−0.127658
$130$	0	0
$131$	−180742.	−0.920197	−0.460099	−	0.887868i	$-0.652186\pi$
$131$	−180742.	−0.920197	−0.460099	+	0.887868i	$0.652186\pi$
$132$	0	0
$133$	−132202.	−0.648051
$134$	0	0
$135$	79040.0	0.373261
$136$	0	0
$137$	−209678.	−0.954446	−0.477223	−	0.878782i	$-0.658357\pi$
$137$	−209678.	−0.954446	−0.477223	+	0.878782i	$0.658357\pi$
$138$	0	0
$139$	−17242.0	−0.0756921	−0.0378461	−	0.999284i	$-0.512050\pi$
$139$	−17242.0	−0.0756921	−0.0378461	+	0.999284i	$0.512050\pi$
$140$	0	0
$141$	−31512.0	−0.133484
$142$	0	0
$143$	5472.00	0.0223772
$144$	0	0
$145$	−52064.0	−0.205645
$146$	0	0
$147$	−62426.0	−0.238272
$148$	0	0
$149$	−59358.0	−0.219035	−0.109518	−	0.993985i	$-0.534931\pi$
$149$	−59358.0	−0.219035	−0.109518	+	0.993985i	$0.534931\pi$
$150$	0	0
$151$	−336344.	−1.20044	−0.600221	−	0.799834i	$-0.704921\pi$
$151$	−336344.	−1.20044	−0.600221	+	0.799834i	$0.704921\pi$
$152$	0	0
$153$	−960394.	−3.31681
$154$	0	0
$155$	−76608.0	−0.256121
$156$	0	0
$157$	−464588.	−1.50425	−0.752123	−	0.659023i	$-0.770970\pi$
$157$	−464588.	−1.50425	−0.752123	+	0.659023i	$0.770970\pi$
$158$	0	0
$159$	340860.	1.06926
$160$	0	0
$161$	−163856.	−0.498193
$162$	0	0
$163$	−314792.	−0.928014	−0.464007	−	0.885831i	$-0.653589\pi$
$163$	−314792.	−0.928014	−0.464007	+	0.885831i	$0.653589\pi$
$164$	0	0
$165$	−3328.00	−0.00951642
$166$	0	0
$167$	285724.	0.792785	0.396393	−	0.918081i	$-0.370262\pi$
$167$	285724.	0.792785	0.396393	+	0.918081i	$0.370262\pi$
$168$	0	0
$169$	96563.0	0.260072
$170$	0	0
$171$	1.16823e6	3.05520
$172$	0	0
$173$	709148.	1.80145	0.900724	−	0.434392i	$-0.143037\pi$
$173$	709148.	1.80145	0.900724	+	0.434392i	$0.143037\pi$
$174$	0	0
$175$	140581.	0.347002
$176$	0	0
$177$	902252.	2.16468
$178$	0	0
$179$	617148.	1.43965	0.719825	−	0.694156i	$-0.244222\pi$
$179$	617148.	1.43965	0.719825	+	0.694156i	$0.244222\pi$
$180$	0	0
$181$	−237828.	−0.539593	−0.269797	−	0.962917i	$-0.586956\pi$
$181$	−237828.	−0.539593	−0.269797	+	0.962917i	$0.586956\pi$
$182$	0	0
$183$	−26832.0	−0.0592278
$184$	0	0
$185$	−183520.	−0.394234
$186$	0	0
$187$	17744.0	0.0371063
$188$	0	0
$189$	242060.	0.492911
$190$	0	0
$191$	−133512.	−0.264812	−0.132406	−	0.991196i	$-0.542270\pi$
$191$	−133512.	−0.264812	−0.132406	+	0.991196i	$0.542270\pi$
$192$	0	0
$193$	270446.	0.522622	0.261311	−	0.965255i	$-0.415845\pi$
$193$	270446.	0.522622	0.261311	+	0.965255i	$0.415845\pi$
$194$	0	0
$195$	−284544.	−0.535874
$196$	0	0
$197$	−875102.	−1.60655	−0.803273	−	0.595611i	$-0.796910\pi$
$197$	−875102.	−1.60655	−0.803273	+	0.595611i	$0.796910\pi$
$198$	0	0
$199$	−347620.	−0.622260	−0.311130	−	0.950367i	$-0.600708\pi$
$199$	−347620.	−0.622260	−0.311130	+	0.950367i	$0.600708\pi$
$200$	0	0
$201$	262808.	0.458826
$202$	0	0
$203$	−159446.	−0.271565
$204$	0	0
$205$	−213600.	−0.354990
$206$	0	0
$207$	1.44795e6	2.34870
$208$	0	0
$209$	−21584.0	−0.0341795
$210$	0	0
$211$	425380.	0.657765	0.328883	−	0.944371i	$-0.393328\pi$
$211$	425380.	0.657765	0.328883	+	0.944371i	$0.393328\pi$
$212$	0	0
$213$	−1.63072e6	−2.46281
$214$	0	0
$215$	−14848.0	−0.0219064
$216$	0	0
$217$	−234612.	−0.338221
$218$	0	0
$219$	492076.	0.693301
$220$	0	0
$221$	1.51711e6	2.08947
$222$	0	0
$223$	481592.	0.648511	0.324255	−	0.945970i	$-0.394886\pi$
$223$	481592.	0.648511	0.324255	+	0.945970i	$0.394886\pi$
$224$	0	0
$225$	−1.24228e6	−1.63592
$226$	0	0
$227$	6042.00	0.00778245	0.00389122	−	0.999992i	$-0.498761\pi$
$227$	6042.00	0.00778245	0.00389122	+	0.999992i	$0.498761\pi$
$228$	0	0
$229$	−1804.00	−0.00227325	−0.00113663	−	0.999999i	$-0.500362\pi$
$229$	−1804.00	−0.00227325	−0.00113663	+	0.999999i	$0.500362\pi$
$230$	0	0
$231$	−10192.0	−0.0125669
$232$	0	0
$233$	−1.61153e6	−1.94468	−0.972339	−	0.233576i	$-0.924957\pi$
$233$	−1.61153e6	−1.94468	−0.972339	+	0.233576i	$0.924957\pi$
$234$	0	0
$235$	−19392.0	−0.0229062
$236$	0	0
$237$	−296400.	−0.342774
$238$	0	0
$239$	−987096.	−1.11780	−0.558901	−	0.829235i	$-0.688777\pi$
$239$	−987096.	−1.11780	−0.558901	+	0.829235i	$0.688777\pi$
$240$	0	0
$241$	893510.	0.990962	0.495481	−	0.868619i	$-0.334992\pi$
$241$	893510.	0.990962	0.495481	+	0.868619i	$0.334992\pi$
$242$	0	0
$243$	596674.	0.648219
$244$	0	0
$245$	−38416.0	−0.0408881
$246$	0	0
$247$	−1.84543e6	−1.92467
$248$	0	0
$249$	2.31291e6	2.36407
$250$	0	0
$251$	−365946.	−0.366634	−0.183317	−	0.983054i	$-0.558683\pi$
$251$	−365946.	−0.366634	−0.183317	+	0.983054i	$0.558683\pi$
$252$	0	0
$253$	−26752.0	−0.0262757
$254$	0	0
$255$	−922688.	−0.888596
$256$	0	0
$257$	1.40459e6	1.32653	0.663266	−	0.748383i	$-0.269170\pi$
$257$	1.40459e6	1.32653	0.663266	+	0.748383i	$0.269170\pi$
$258$	0	0
$259$	−562030.	−0.520607
$260$	0	0
$261$	1.40898e6	1.28028
$262$	0	0
$263$	1.09968e6	0.980341	0.490170	−	0.871627i	$-0.336935\pi$
$263$	1.09968e6	0.980341	0.490170	+	0.871627i	$0.336935\pi$
$264$	0	0
$265$	209760.	0.183488
$266$	0	0
$267$	−512772.	−0.440196
$268$	0	0
$269$	−814948.	−0.686672	−0.343336	−	0.939213i	$-0.611557\pi$
$269$	−814948.	−0.686672	−0.343336	+	0.939213i	$0.611557\pi$
$270$	0	0
$271$	−1.69906e6	−1.40535	−0.702675	−	0.711511i	$-0.748011\pi$
$271$	−1.69906e6	−1.40535	−0.702675	+	0.711511i	$0.748011\pi$
$272$	0	0
$273$	−871416.	−0.707651
$274$	0	0
$275$	22952.0	0.0183016
$276$	0	0
$277$	1.36508e6	1.06895	0.534477	−	0.845183i	$-0.320508\pi$
$277$	1.36508e6	1.06895	0.534477	+	0.845183i	$0.320508\pi$
$278$	0	0
$279$	2.07320e6	1.59453
$280$	0	0
$281$	−715846.	−0.540821	−0.270411	−	0.962745i	$-0.587159\pi$
$281$	−715846.	−0.540821	−0.270411	+	0.962745i	$0.587159\pi$
$282$	0	0
$283$	−217726.	−0.161601	−0.0808005	−	0.996730i	$-0.525748\pi$
$283$	−217726.	−0.161601	−0.0808005	+	0.996730i	$0.525748\pi$
$284$	0	0
$285$	1.12237e6	0.818508
$286$	0	0
$287$	−654150.	−0.468784
$288$	0	0
$289$	3.49967e6	2.46480
$290$	0	0
$291$	−443612.	−0.307094
$292$	0	0
$293$	−1.50708e6	−1.02557	−0.512787	−	0.858516i	$-0.671387\pi$
$293$	−1.50708e6	−1.02557	−0.512787	+	0.858516i	$0.671387\pi$
$294$	0	0
$295$	555232.	0.371466
$296$	0	0
$297$	39520.0	0.0259972
$298$	0	0
$299$	−2.28730e6	−1.47960
$300$	0	0
$301$	−45472.0	−0.0289286
$302$	0	0
$303$	1.19350e6	0.746822
$304$	0	0
$305$	−16512.0	−0.0101637
$306$	0	0
$307$	−12502.0	−0.00757066	−0.00378533	−	0.999993i	$-0.501205\pi$
$307$	−12502.0	−0.00757066	−0.00378533	+	0.999993i	$0.501205\pi$
$308$	0	0
$309$	3.53631e6	2.10695
$310$	0	0
$311$	−647432.	−0.379571	−0.189786	−	0.981826i	$-0.560779\pi$
$311$	−647432.	−0.379571	−0.189786	+	0.981826i	$0.560779\pi$
$312$	0	0
$313$	−935978.	−0.540014	−0.270007	−	0.962858i	$-0.587026\pi$
$313$	−935978.	−0.540014	−0.270007	+	0.962858i	$0.587026\pi$
$314$	0	0
$315$	339472.	0.192765
$316$	0	0
$317$	−705942.	−0.394567	−0.197284	−	0.980346i	$-0.563212\pi$
$317$	−705942.	−0.394567	−0.197284	+	0.980346i	$0.563212\pi$
$318$	0	0
$319$	−26032.0	−0.0143229
$320$	0	0
$321$	−1.79806e6	−0.973959
$322$	0	0
$323$	−5.98416e6	−3.19152
$324$	0	0
$325$	1.96240e6	1.03057
$326$	0	0
$327$	−3.80676e6	−1.96873
$328$	0	0
$329$	−59388.0	−0.0302489
$330$	0	0
$331$	1.14304e6	0.573445	0.286722	−	0.958014i	$-0.407434\pi$
$331$	1.14304e6	0.573445	0.286722	+	0.958014i	$0.407434\pi$
$332$	0	0
$333$	4.96651e6	2.45437
$334$	0	0
$335$	161728.	0.0787360
$336$	0	0
$337$	−2.36402e6	−1.13390	−0.566952	−	0.823751i	$-0.691877\pi$
$337$	−2.36402e6	−1.13390	−0.566952	+	0.823751i	$0.691877\pi$
$338$	0	0
$339$	2.08484e6	0.985309
$340$	0	0
$341$	−38304.0	−0.0178385
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	1.39110e6	0.629234
$346$	0	0
$347$	−726240.	−0.323785	−0.161892	−	0.986808i	$-0.551760\pi$
$347$	−726240.	−0.323785	−0.161892	+	0.986808i	$0.551760\pi$
$348$	0	0
$349$	−136180.	−0.0598480	−0.0299240	−	0.999552i	$-0.509527\pi$
$349$	−136180.	−0.0598480	−0.0299240	+	0.999552i	$0.509527\pi$
$350$	0	0
$351$	3.37896e6	1.46391
$352$	0	0
$353$	−1.16907e6	−0.499349	−0.249674	−	0.968330i	$-0.580324\pi$
$353$	−1.16907e6	−0.499349	−0.249674	+	0.968330i	$0.580324\pi$
$354$	0	0
$355$	−1.00352e6	−0.422625
$356$	0	0
$357$	−2.82573e6	−1.17344
$358$	0	0
$359$	−4280.00	−0.00175270	−0.000876350	−	1.00000i	$-0.500279\pi$
$359$	−4280.00	−0.00175270	−0.000876350		1.00000i	$0.500279\pi$
$360$	0	0
$361$	4.80310e6	1.93979
$362$	0	0
$363$	4.18566e6	1.66724
$364$	0	0
$365$	302816.	0.118973
$366$	0	0
$367$	2.44796e6	0.948722	0.474361	−	0.880330i	$-0.342679\pi$
$367$	2.44796e6	0.948722	0.474361	+	0.880330i	$0.342679\pi$
$368$	0	0
$369$	5.78055e6	2.21006
$370$	0	0
$371$	642390.	0.242306
$372$	0	0
$373$	904514.	0.336623	0.168311	−	0.985734i	$-0.446169\pi$
$373$	904514.	0.336623	0.168311	+	0.985734i	$0.446169\pi$
$374$	0	0
$375$	−2.49350e6	−0.915655
$376$	0	0
$377$	−2.22574e6	−0.806530
$378$	0	0
$379$	4.23034e6	1.51279	0.756393	−	0.654117i	$-0.226960\pi$
$379$	4.23034e6	1.51279	0.756393	+	0.654117i	$0.226960\pi$
$380$	0	0
$381$	−7.14480e6	−2.52161
$382$	0	0
$383$	4.55400e6	1.58634	0.793169	−	0.609002i	$-0.208430\pi$
$383$	4.55400e6	1.58634	0.793169	+	0.609002i	$0.208430\pi$
$384$	0	0
$385$	−6272.00	−0.00215652
$386$	0	0
$387$	401824.	0.136382
$388$	0	0
$389$	3.98541e6	1.33536	0.667680	−	0.744448i	$-0.267287\pi$
$389$	3.98541e6	1.33536	0.667680	+	0.744448i	$0.267287\pi$
$390$	0	0
$391$	−7.41699e6	−2.45350
$392$	0	0
$393$	4.69929e6	1.53480
$394$	0	0
$395$	−182400.	−0.0588210
$396$	0	0
$397$	−552420.	−0.175911	−0.0879555	−	0.996124i	$-0.528033\pi$
$397$	−552420.	−0.175911	−0.0879555	+	0.996124i	$0.528033\pi$
$398$	0	0
$399$	3.43725e6	1.08088
$400$	0	0
$401$	38190.0	0.0118601	0.00593006	−	0.999982i	$-0.498112\pi$
$401$	38190.0	0.0118601	0.00593006	+	0.999982i	$0.498112\pi$
$402$	0	0
$403$	−3.27499e6	−1.00449
$404$	0	0
$405$	−371536.	−0.112555
$406$	0	0
$407$	−91760.0	−0.0274579
$408$	0	0
$409$	−3.92475e6	−1.16012	−0.580062	−	0.814573i	$-0.696972\pi$
$409$	−3.92475e6	−1.16012	−0.580062	+	0.814573i	$0.696972\pi$
$410$	0	0
$411$	5.45163e6	1.59192
$412$	0	0
$413$	1.70040e6	0.490541
$414$	0	0
$415$	1.42333e6	0.405681
$416$	0	0
$417$	448292.	0.126247
$418$	0	0
$419$	−598386.	−0.166512	−0.0832562	−	0.996528i	$-0.526532\pi$
$419$	−598386.	−0.166512	−0.0832562	+	0.996528i	$0.526532\pi$
$420$	0	0
$421$	−4.61597e6	−1.26928	−0.634641	−	0.772807i	$-0.718852\pi$
$421$	−4.61597e6	−1.26928	−0.634641	+	0.772807i	$0.718852\pi$
$422$	0	0
$423$	524796.	0.142607
$424$	0	0
$425$	6.36344e6	1.70891
$426$	0	0
$427$	−50568.0	−0.0134217
$428$	0	0
$429$	−142272.	−0.0373230
$430$	0	0
$431$	−61560.0	−0.0159627	−0.00798133	−	0.999968i	$-0.502541\pi$
$431$	−61560.0	−0.0159627	−0.00798133	+	0.999968i	$0.502541\pi$
$432$	0	0
$433$	−3.79727e6	−0.973310	−0.486655	−	0.873594i	$-0.661783\pi$
$433$	−3.79727e6	−0.973310	−0.486655	+	0.873594i	$0.661783\pi$
$434$	0	0
$435$	1.35366e6	0.342995
$436$	0	0
$437$	9.02211e6	2.25998
$438$	0	0
$439$	−2.28852e6	−0.566752	−0.283376	−	0.959009i	$-0.591455\pi$
$439$	−2.28852e6	−0.566752	−0.283376	+	0.959009i	$0.591455\pi$
$440$	0	0
$441$	1.03963e6	0.254556
$442$	0	0
$443$	4.75976e6	1.15233	0.576163	−	0.817335i	$-0.304549\pi$
$443$	4.75976e6	1.15233	0.576163	+	0.817335i	$0.304549\pi$
$444$	0	0
$445$	−315552.	−0.0755389
$446$	0	0
$447$	1.54331e6	0.365329
$448$	0	0
$449$	−4.36715e6	−1.02231	−0.511155	−	0.859489i	$-0.670782\pi$
$449$	−4.36715e6	−1.02231	−0.511155	+	0.859489i	$0.670782\pi$
$450$	0	0
$451$	−106800.	−0.0247246
$452$	0	0
$453$	8.74494e6	2.00222
$454$	0	0
$455$	−536256.	−0.121435
$456$	0	0
$457$	5.44994e6	1.22068	0.610339	−	0.792140i	$-0.291033\pi$
$457$	5.44994e6	1.22068	0.610339	+	0.792140i	$0.291033\pi$
$458$	0	0
$459$	1.09569e7	2.42749
$460$	0	0
$461$	−1.66966e6	−0.365911	−0.182956	−	0.983121i	$-0.558566\pi$
$461$	−1.66966e6	−0.365911	−0.182956	+	0.983121i	$0.558566\pi$
$462$	0	0
$463$	70768.0	0.0153421	0.00767104	−	0.999971i	$-0.497558\pi$
$463$	70768.0	0.0153421	0.00767104	+	0.999971i	$0.497558\pi$
$464$	0	0
$465$	1.99181e6	0.427184
$466$	0	0
$467$	−5.66083e6	−1.20112	−0.600562	−	0.799578i	$-0.705056\pi$
$467$	−5.66083e6	−1.20112	−0.600562	+	0.799578i	$0.705056\pi$
$468$	0	0
$469$	495292.	0.103975
$470$	0	0
$471$	1.20793e7	2.50893
$472$	0	0
$473$	−7424.00	−0.00152576
$474$	0	0
$475$	−7.74056e6	−1.57412
$476$	0	0
$477$	−5.67663e6	−1.14234
$478$	0	0
$479$	−1.44948e6	−0.288652	−0.144326	−	0.989530i	$-0.546101\pi$
$479$	−1.44948e6	−0.288652	−0.144326	+	0.989530i	$0.546101\pi$
$480$	0	0
$481$	−7.84548e6	−1.54617
$482$	0	0
$483$	4.26026e6	0.830937
$484$	0	0
$485$	−272992.	−0.0526982
$486$	0	0
$487$	4.07504e6	0.778591	0.389296	−	0.921113i	$-0.372718\pi$
$487$	4.07504e6	0.778591	0.389296	+	0.921113i	$0.372718\pi$
$488$	0	0
$489$	8.18459e6	1.54784
$490$	0	0
$491$	−986100.	−0.184594	−0.0922969	−	0.995732i	$-0.529421\pi$
$491$	−986100.	−0.184594	−0.0922969	+	0.995732i	$0.529421\pi$
$492$	0	0
$493$	−7.21737e6	−1.33740
$494$	0	0
$495$	55424.0	0.0101668
$496$	0	0
$497$	−3.07328e6	−0.558099
$498$	0	0
$499$	−5.98342e6	−1.07572	−0.537859	−	0.843035i	$-0.680767\pi$
$499$	−5.98342e6	−1.07572	−0.537859	+	0.843035i	$0.680767\pi$
$500$	0	0
$501$	−7.42882e6	−1.32229
$502$	0	0
$503$	−3.49373e6	−0.615700	−0.307850	−	0.951435i	$-0.599609\pi$
$503$	−3.49373e6	−0.615700	−0.307850	+	0.951435i	$0.599609\pi$
$504$	0	0
$505$	734464.	0.128157
$506$	0	0
$507$	−2.51064e6	−0.433775
$508$	0	0
$509$	−2.15711e6	−0.369043	−0.184522	−	0.982828i	$-0.559074\pi$
$509$	−2.15711e6	−0.369043	−0.184522	+	0.982828i	$0.559074\pi$
$510$	0	0
$511$	927374.	0.157110
$512$	0	0
$513$	−1.33281e7	−2.23602
$514$	0	0
$515$	2.17619e6	0.361559
$516$	0	0
$517$	−9696.00	−0.00159539
$518$	0	0
$519$	−1.84378e7	−3.00464
$520$	0	0
$521$	−6.65817e6	−1.07463	−0.537317	−	0.843380i	$-0.680562\pi$
$521$	−6.65817e6	−1.07463	−0.537317	+	0.843380i	$0.680562\pi$
$522$	0	0
$523$	−5.95223e6	−0.951537	−0.475768	−	0.879571i	$-0.657830\pi$
$523$	−5.95223e6	−0.951537	−0.475768	+	0.879571i	$0.657830\pi$
$524$	0	0
$525$	−3.65511e6	−0.578764
$526$	0	0
$527$	−1.06198e7	−1.66567
$528$	0	0
$529$	4.74599e6	0.737374
$530$	0	0
$531$	−1.50260e7	−2.31263
$532$	0	0
$533$	−9.13140e6	−1.39226
$534$	0	0
$535$	−1.10650e6	−0.167134
$536$	0	0
$537$	−1.60458e7	−2.40119
$538$	0	0
$539$	−19208.0	−0.00284780
$540$	0	0
$541$	6.39681e6	0.939659	0.469830	−	0.882757i	$-0.344315\pi$
$541$	6.39681e6	0.939659	0.469830	+	0.882757i	$0.344315\pi$
$542$	0	0
$543$	6.18353e6	0.899988
$544$	0	0
$545$	−2.34262e6	−0.337840
$546$	0	0
$547$	5.51851e6	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$547$	5.51851e6	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$548$	0	0
$549$	446856.	0.0632757
$550$	0	0
$551$	8.77929e6	1.23191
$552$	0	0
$553$	−558600.	−0.0776762
$554$	0	0
$555$	4.77152e6	0.657543
$556$	0	0
$557$	−2.02159e6	−0.276093	−0.138046	−	0.990426i	$-0.544082\pi$
$557$	−2.02159e6	−0.276093	−0.138046	+	0.990426i	$0.544082\pi$
$558$	0	0
$559$	−634752.	−0.0859161
$560$	0	0
$561$	−461344.	−0.0618896
$562$	0	0
$563$	−8.14678e6	−1.08322	−0.541608	−	0.840631i	$-0.682184\pi$
$563$	−8.14678e6	−1.08322	−0.541608	+	0.840631i	$0.682184\pi$
$564$	0	0
$565$	1.28298e6	0.169082
$566$	0	0
$567$	−1.13783e6	−0.148634
$568$	0	0
$569$	−1.19824e7	−1.55154	−0.775772	−	0.631013i	$-0.782639\pi$
$569$	−1.19824e7	−1.55154	−0.775772	+	0.631013i	$0.782639\pi$
$570$	0	0
$571$	−1.39582e6	−0.179159	−0.0895793	−	0.995980i	$-0.528552\pi$
$571$	−1.39582e6	−0.179159	−0.0895793	+	0.995980i	$0.528552\pi$
$572$	0	0
$573$	3.47131e6	0.441679
$574$	0	0
$575$	−9.59394e6	−1.21012
$576$	0	0
$577$	1.96784e6	0.246065	0.123033	−	0.992403i	$-0.460738\pi$
$577$	1.96784e6	0.246065	0.123033	+	0.992403i	$0.460738\pi$
$578$	0	0
$579$	−7.03160e6	−0.871681
$580$	0	0
$581$	4.35894e6	0.535724
$582$	0	0
$583$	104880.	0.0127797
$584$	0	0
$585$	4.73875e6	0.572498
$586$	0	0
$587$	3.18897e6	0.381993	0.190997	−	0.981591i	$-0.438828\pi$
$587$	3.18897e6	0.381993	0.190997	+	0.981591i	$0.438828\pi$
$588$	0	0
$589$	1.29180e7	1.53429
$590$	0	0
$591$	2.27527e7	2.67956
$592$	0	0
$593$	1.67500e6	0.195604	0.0978022	−	0.995206i	$-0.468819\pi$
$593$	1.67500e6	0.195604	0.0978022	+	0.995206i	$0.468819\pi$
$594$	0	0
$595$	−1.73891e6	−0.201366
$596$	0	0
$597$	9.03812e6	1.03787
$598$	0	0
$599$	−1.00635e7	−1.14599	−0.572994	−	0.819559i	$-0.694218\pi$
$599$	−1.00635e7	−1.14599	−0.572994	+	0.819559i	$0.694218\pi$
$600$	0	0
$601$	1.72798e6	0.195143	0.0975713	−	0.995229i	$-0.468893\pi$
$601$	1.72798e6	0.195143	0.0975713	+	0.995229i	$0.468893\pi$
$602$	0	0
$603$	−4.37676e6	−0.490185
$604$	0	0
$605$	2.57579e6	0.286103
$606$	0	0
$607$	−1.69523e7	−1.86748	−0.933740	−	0.357953i	$-0.883475\pi$
$607$	−1.69523e7	−1.86748	−0.933740	+	0.357953i	$0.883475\pi$
$608$	0	0
$609$	4.14560e6	0.452943
$610$	0	0
$611$	−829008.	−0.0898371
$612$	0	0
$613$	−1.01942e7	−1.09572	−0.547861	−	0.836569i	$-0.684558\pi$
$613$	−1.01942e7	−1.09572	−0.547861	+	0.836569i	$0.684558\pi$
$614$	0	0
$615$	5.55360e6	0.592089
$616$	0	0
$617$	1.57452e7	1.66508	0.832540	−	0.553965i	$-0.186886\pi$
$617$	1.57452e7	1.66508	0.832540	+	0.553965i	$0.186886\pi$
$618$	0	0
$619$	332690.	0.0348990	0.0174495	−	0.999848i	$-0.494445\pi$
$619$	332690.	0.0348990	0.0174495	+	0.999848i	$0.494445\pi$
$620$	0	0
$621$	−1.65194e7	−1.71895
$622$	0	0
$623$	−966378.	−0.0997532
$624$	0	0
$625$	7.43116e6	0.760951
$626$	0	0
$627$	561184.	0.0570081
$628$	0	0
$629$	−2.54405e7	−2.56388
$630$	0	0
$631$	3.59720e6	0.359659	0.179830	−	0.983698i	$-0.442445\pi$
$631$	3.59720e6	0.359659	0.179830	+	0.983698i	$0.442445\pi$
$632$	0	0
$633$	−1.10599e7	−1.09709
$634$	0	0
$635$	−4.39680e6	−0.432715
$636$	0	0
$637$	−1.64228e6	−0.160361
$638$	0	0
$639$	2.71578e7	2.63113
$640$	0	0
$641$	−1.46389e7	−1.40723	−0.703614	−	0.710583i	$-0.748431\pi$
$641$	−1.46389e7	−1.40723	−0.703614	+	0.710583i	$0.748431\pi$
$642$	0	0
$643$	1.38386e7	1.31997	0.659987	−	0.751277i	$-0.270562\pi$
$643$	1.38386e7	1.31997	0.659987	+	0.751277i	$0.270562\pi$
$644$	0	0
$645$	386048.	0.0365378
$646$	0	0
$647$	−1.40358e7	−1.31819	−0.659093	−	0.752061i	$-0.729060\pi$
$647$	−1.40358e7	−1.31819	−0.659093	+	0.752061i	$0.729060\pi$
$648$	0	0
$649$	277616.	0.0258722
$650$	0	0
$651$	6.09991e6	0.564119
$652$	0	0
$653$	−1.61063e7	−1.47813	−0.739064	−	0.673635i	$-0.764732\pi$
$653$	−1.61063e7	−1.47813	−0.739064	+	0.673635i	$0.764732\pi$
$654$	0	0
$655$	2.89187e6	0.263376
$656$	0	0
$657$	−8.19496e6	−0.740685
$658$	0	0
$659$	−4.80075e6	−0.430622	−0.215311	−	0.976546i	$-0.569076\pi$
$659$	−4.80075e6	−0.430622	−0.215311	+	0.976546i	$0.569076\pi$
$660$	0	0
$661$	1.76565e7	1.57181	0.785905	−	0.618347i	$-0.212197\pi$
$661$	1.76565e7	1.57181	0.785905	+	0.618347i	$0.212197\pi$
$662$	0	0
$663$	−3.94449e7	−3.48504
$664$	0	0
$665$	2.11523e6	0.185483
$666$	0	0
$667$	1.08814e7	0.947042
$668$	0	0
$669$	−1.25214e7	−1.08165
$670$	0	0
$671$	−8256.00	−0.000707886 0
$672$	0	0
$673$	6.59225e6	0.561043	0.280521	−	0.959848i	$-0.409493\pi$
$673$	6.59225e6	0.561043	0.280521	+	0.959848i	$0.409493\pi$
$674$	0	0
$675$	1.41729e7	1.19729
$676$	0	0
$677$	−9.77178e6	−0.819411	−0.409706	−	0.912218i	$-0.634369\pi$
$677$	−9.77178e6	−0.819411	−0.409706	+	0.912218i	$0.634369\pi$
$678$	0	0
$679$	−836038.	−0.0695908
$680$	0	0
$681$	−157092.	−0.0129803
$682$	0	0
$683$	1.88663e7	1.54752	0.773758	−	0.633481i	$-0.218374\pi$
$683$	1.88663e7	1.54752	0.773758	+	0.633481i	$0.218374\pi$
$684$	0	0
$685$	3.35485e6	0.273178
$686$	0	0
$687$	46904.0	0.00379156
$688$	0	0
$689$	8.96724e6	0.719632
$690$	0	0
$691$	−8.67018e6	−0.690769	−0.345385	−	0.938461i	$-0.612252\pi$
$691$	−8.67018e6	−0.690769	−0.345385	+	0.938461i	$0.612252\pi$
$692$	0	0
$693$	169736.	0.0134258
$694$	0	0
$695$	275872.	0.0216643
$696$	0	0
$697$	−2.96103e7	−2.30866
$698$	0	0
$699$	4.18997e7	3.24353
$700$	0	0
$701$	−7.93482e6	−0.609877	−0.304938	−	0.952372i	$-0.598636\pi$
$701$	−7.93482e6	−0.609877	−0.304938	+	0.952372i	$0.598636\pi$
$702$	0	0
$703$	3.09461e7	2.36166
$704$	0	0
$705$	504192.	0.0382053
$706$	0	0
$707$	2.24930e6	0.169238
$708$	0	0
$709$	−2.62600e7	−1.96191	−0.980956	−	0.194228i	$-0.937780\pi$
$709$	−2.62600e7	−1.96191	−0.980956	+	0.194228i	$0.937780\pi$
$710$	0	0
$711$	4.93620e6	0.366200
$712$	0	0
$713$	1.60111e7	1.17950
$714$	0	0
$715$	−87552.0	−0.00640473
$716$	0	0
$717$	2.56645e7	1.86438
$718$	0	0
$719$	2.20763e7	1.59259	0.796295	−	0.604909i	$-0.206790\pi$
$719$	2.20763e7	1.59259	0.796295	+	0.604909i	$0.206790\pi$
$720$	0	0
$721$	6.66459e6	0.477458
$722$	0	0
$723$	−2.32313e7	−1.65283
$724$	0	0
$725$	−9.33573e6	−0.659634
$726$	0	0
$727$	8.49245e6	0.595933	0.297966	−	0.954576i	$-0.403692\pi$
$727$	8.49245e6	0.595933	0.297966	+	0.954576i	$0.403692\pi$
$728$	0	0
$729$	−2.11562e7	−1.47441
$730$	0	0
$731$	−2.05830e6	−0.142468
$732$	0	0
$733$	1.90713e7	1.31105	0.655526	−	0.755172i	$-0.272447\pi$
$733$	1.90713e7	1.31105	0.655526	+	0.755172i	$0.272447\pi$
$734$	0	0
$735$	998816.	0.0681973
$736$	0	0
$737$	80864.0	0.00548386
$738$	0	0
$739$	1.46832e7	0.989032	0.494516	−	0.869169i	$-0.335345\pi$
$739$	1.46832e7	0.989032	0.494516	+	0.869169i	$0.335345\pi$
$740$	0	0
$741$	4.79812e7	3.21015
$742$	0	0
$743$	1.64265e7	1.09162	0.545812	−	0.837908i	$-0.316221\pi$
$743$	1.64265e7	1.09162	0.545812	+	0.837908i	$0.316221\pi$
$744$	0	0
$745$	949728.	0.0626915
$746$	0	0
$747$	−3.85188e7	−2.52564
$748$	0	0
$749$	−3.38864e6	−0.220710
$750$	0	0
$751$	−2.44357e7	−1.58097	−0.790486	−	0.612479i	$-0.790172\pi$
$751$	−2.44357e7	−1.58097	−0.790486	+	0.612479i	$0.790172\pi$
$752$	0	0
$753$	9.51460e6	0.611509
$754$	0	0
$755$	5.38150e6	0.343587
$756$	0	0
$757$	295566.	0.0187463	0.00937313	−	0.999956i	$-0.497016\pi$
$757$	295566.	0.0187463	0.00937313	+	0.999956i	$0.497016\pi$
$758$	0	0
$759$	695552.	0.0438253
$760$	0	0
$761$	−473842.	−0.0296601	−0.0148300	−	0.999890i	$-0.504721\pi$
$761$	−473842.	−0.0296601	−0.0148300	+	0.999890i	$0.504721\pi$
$762$	0	0
$763$	−7.17429e6	−0.446136
$764$	0	0
$765$	1.53663e7	0.949327
$766$	0	0
$767$	2.37362e7	1.45687
$768$	0	0
$769$	2.33241e7	1.42229	0.711145	−	0.703045i	$-0.248177\pi$
$769$	2.33241e7	1.42229	0.711145	+	0.703045i	$0.248177\pi$
$770$	0	0
$771$	−3.65194e7	−2.21253
$772$	0	0
$773$	−1.55583e7	−0.936511	−0.468255	−	0.883593i	$-0.655117\pi$
$773$	−1.55583e7	−0.936511	−0.468255	+	0.883593i	$0.655117\pi$
$774$	0	0
$775$	−1.37368e7	−0.821543
$776$	0	0
$777$	1.46128e7	0.868321
$778$	0	0
$779$	3.60183e7	2.12657
$780$	0	0
$781$	−501760.	−0.0294353
$782$	0	0
$783$	−1.60748e7	−0.937001
$784$	0	0
$785$	7.43341e6	0.430540
$786$	0	0
$787$	−6.66843e6	−0.383784	−0.191892	−	0.981416i	$-0.561462\pi$
$787$	−6.66843e6	−0.383784	−0.191892	+	0.981416i	$0.561462\pi$
$788$	0	0
$789$	−2.85917e7	−1.63511
$790$	0	0
$791$	3.92911e6	0.223282
$792$	0	0
$793$	−705888.	−0.0398614
$794$	0	0
$795$	−5.45376e6	−0.306040
$796$	0	0
$797$	1.22461e7	0.682892	0.341446	−	0.939901i	$-0.389083\pi$
$797$	1.22461e7	0.682892	0.341446	+	0.939901i	$0.389083\pi$
$798$	0	0
$799$	−2.68822e6	−0.148969
$800$	0	0
$801$	8.53963e6	0.470281
$802$	0	0
$803$	151408.	0.00828629
$804$	0	0
$805$	2.62170e6	0.142591
$806$	0	0
$807$	2.11886e7	1.14530
$808$	0	0
$809$	−2.91495e7	−1.56588	−0.782941	−	0.622095i	$-0.786282\pi$
$809$	−2.91495e7	−1.56588	−0.782941	+	0.622095i	$0.786282\pi$
$810$	0	0
$811$	−7.58849e6	−0.405138	−0.202569	−	0.979268i	$-0.564929\pi$
$811$	−7.58849e6	−0.405138	−0.202569	+	0.979268i	$0.564929\pi$
$812$	0	0
$813$	4.41755e7	2.34398
$814$	0	0
$815$	5.03667e6	0.265613
$816$	0	0
$817$	2.50374e6	0.131230
$818$	0	0
$819$	1.45124e7	0.756015
$820$	0	0
$821$	5.98849e6	0.310070	0.155035	−	0.987909i	$-0.450451\pi$
$821$	5.98849e6	0.310070	0.155035	+	0.987909i	$0.450451\pi$
$822$	0	0
$823$	817960.	0.0420952	0.0210476	−	0.999778i	$-0.493300\pi$
$823$	817960.	0.0420952	0.0210476	+	0.999778i	$0.493300\pi$
$824$	0	0
$825$	−596752.	−0.0305252
$826$	0	0
$827$	−2.51963e6	−0.128107	−0.0640535	−	0.997946i	$-0.520403\pi$
$827$	−2.51963e6	−0.128107	−0.0640535	+	0.997946i	$0.520403\pi$
$828$	0	0
$829$	1.61006e7	0.813684	0.406842	−	0.913499i	$-0.366630\pi$
$829$	1.61006e7	0.813684	0.406842	+	0.913499i	$0.366630\pi$
$830$	0	0
$831$	−3.54921e7	−1.78291
$832$	0	0
$833$	−5.32542e6	−0.265914
$834$	0	0
$835$	−4.57158e6	−0.226908
$836$	0	0
$837$	−2.36527e7	−1.16699
$838$	0	0
$839$	−2.58167e7	−1.26618	−0.633091	−	0.774077i	$-0.718214\pi$
$839$	−2.58167e7	−1.26618	−0.633091	+	0.774077i	$0.718214\pi$
$840$	0	0
$841$	−9.92263e6	−0.483768
$842$	0	0
$843$	1.86120e7	0.902036
$844$	0	0
$845$	−1.54501e6	−0.0744370
$846$	0	0
$847$	7.88836e6	0.377814
$848$	0	0
$849$	5.66088e6	0.269535
$850$	0	0
$851$	3.83557e7	1.81554
$852$	0	0
$853$	1.54270e7	0.725954	0.362977	−	0.931798i	$-0.381760\pi$
$853$	1.54270e7	0.725954	0.362977	+	0.931798i	$0.381760\pi$
$854$	0	0
$855$	−1.86917e7	−0.874449
$856$	0	0
$857$	3.60517e6	0.167677	0.0838384	−	0.996479i	$-0.473282\pi$
$857$	3.60517e6	0.167677	0.0838384	+	0.996479i	$0.473282\pi$
$858$	0	0
$859$	−4.06995e6	−0.188194	−0.0940970	−	0.995563i	$-0.529996\pi$
$859$	−4.06995e6	−0.188194	−0.0940970	+	0.995563i	$0.529996\pi$
$860$	0	0
$861$	1.70079e7	0.781885
$862$	0	0
$863$	7.25111e6	0.331419	0.165710	−	0.986175i	$-0.447009\pi$
$863$	7.25111e6	0.331419	0.165710	+	0.986175i	$0.447009\pi$
$864$	0	0
$865$	−1.13464e7	−0.515604
$866$	0	0
$867$	−9.09913e7	−4.11105
$868$	0	0
$869$	−91200.0	−0.00409681
$870$	0	0
$871$	6.91387e6	0.308799
$872$	0	0
$873$	7.38785e6	0.328082
$874$	0	0
$875$	−4.69930e6	−0.207497
$876$	0	0
$877$	−2.37414e6	−0.104233	−0.0521167	−	0.998641i	$-0.516597\pi$
$877$	−2.37414e6	−0.104233	−0.0521167	+	0.998641i	$0.516597\pi$
$878$	0	0
$879$	3.91841e7	1.71056
$880$	0	0
$881$	3.03558e7	1.31766	0.658828	−	0.752293i	$-0.271052\pi$
$881$	3.03558e7	1.31766	0.658828	+	0.752293i	$0.271052\pi$
$882$	0	0
$883$	−1.53338e6	−0.0661832	−0.0330916	−	0.999452i	$-0.510535\pi$
$883$	−1.53338e6	−0.0661832	−0.0330916	+	0.999452i	$0.510535\pi$
$884$	0	0
$885$	−1.44360e7	−0.619569
$886$	0	0
$887$	−2.92379e7	−1.24778	−0.623888	−	0.781514i	$-0.714448\pi$
$887$	−2.92379e7	−1.24778	−0.623888	+	0.781514i	$0.714448\pi$
$888$	0	0
$889$	−1.34652e7	−0.571424
$890$	0	0
$891$	−185768.	−0.00783929
$892$	0	0
$893$	3.26998e6	0.137220
$894$	0	0
$895$	−9.87437e6	−0.412052
$896$	0	0
$897$	5.94697e7	2.46783
$898$	0	0
$899$	1.55802e7	0.642943
$900$	0	0
$901$	2.90780e7	1.19331
$902$	0	0
$903$	1.18227e6	0.0482501
$904$	0	0
$905$	3.80525e6	0.154441
$906$	0	0
$907$	−4.48227e7	−1.80917	−0.904587	−	0.426289i	$-0.859821\pi$
$907$	−4.48227e7	−1.80917	−0.904587	+	0.426289i	$0.859821\pi$
$908$	0	0
$909$	−1.98764e7	−0.797864
$910$	0	0
$911$	−3.62906e7	−1.44877	−0.724384	−	0.689397i	$-0.757876\pi$
$911$	−3.62906e7	−1.44877	−0.724384	+	0.689397i	$0.757876\pi$
$912$	0	0
$913$	711664.	0.0282552
$914$	0	0
$915$	429312.	0.0169520
$916$	0	0
$917$	8.85636e6	0.347802
$918$	0	0
$919$	3.25350e7	1.27076	0.635378	−	0.772201i	$-0.280844\pi$
$919$	3.25350e7	1.27076	0.635378	+	0.772201i	$0.280844\pi$
$920$	0	0
$921$	325052.	0.0126271
$922$	0	0
$923$	−4.29005e7	−1.65752
$924$	0	0
$925$	−3.29074e7	−1.26456
$926$	0	0
$927$	−5.88932e7	−2.25095
$928$	0	0
$929$	−4.46676e7	−1.69806	−0.849030	−	0.528344i	$-0.822813\pi$
$929$	−4.46676e7	−1.69806	−0.849030	+	0.528344i	$0.822813\pi$
$930$	0	0
$931$	6.47790e6	0.244940
$932$	0	0
$933$	1.68332e7	0.633087
$934$	0	0
$935$	−283904.	−0.0106204
$936$	0	0
$937$	1.56680e7	0.582995	0.291498	−	0.956572i	$-0.405846\pi$
$937$	1.56680e7	0.582995	0.291498	+	0.956572i	$0.405846\pi$
$938$	0	0
$939$	2.43354e7	0.900689
$940$	0	0
$941$	2.01175e7	0.740627	0.370313	−	0.928907i	$-0.379250\pi$
$941$	2.01175e7	0.740627	0.370313	+	0.928907i	$0.379250\pi$
$942$	0	0
$943$	4.46424e7	1.63481
$944$	0	0
$945$	−3.87296e6	−0.141079
$946$	0	0
$947$	8.84518e6	0.320503	0.160251	−	0.987076i	$-0.448769\pi$
$947$	8.84518e6	0.320503	0.160251	+	0.987076i	$0.448769\pi$
$948$	0	0
$949$	1.29454e7	0.466605
$950$	0	0
$951$	1.83545e7	0.658099
$952$	0	0
$953$	−3.14364e7	−1.12124	−0.560622	−	0.828072i	$-0.689438\pi$
$953$	−3.14364e7	−1.12124	−0.560622	+	0.828072i	$0.689438\pi$
$954$	0	0
$955$	2.13619e6	0.0757935
$956$	0	0
$957$	676832.	0.0238892
$958$	0	0
$959$	1.02742e7	0.360747
$960$	0	0
$961$	−5.70421e6	−0.199245
$962$	0	0
$963$	2.99445e7	1.04052
$964$	0	0
$965$	−4.32714e6	−0.149583
$966$	0	0
$967$	−9.52158e6	−0.327449	−0.163724	−	0.986506i	$-0.552351\pi$
$967$	−9.52158e6	−0.327449	−0.163724	+	0.986506i	$0.552351\pi$
$968$	0	0
$969$	1.55588e8	5.32313
$970$	0	0
$971$	−1.06520e7	−0.362564	−0.181282	−	0.983431i	$-0.558025\pi$
$971$	−1.06520e7	−0.362564	−0.181282	+	0.983431i	$0.558025\pi$
$972$	0	0
$973$	844858.	0.0286089
$974$	0	0
$975$	−5.10223e7	−1.71889
$976$	0	0
$977$	2.72931e7	0.914779	0.457389	−	0.889266i	$-0.348785\pi$
$977$	2.72931e7	0.914779	0.457389	+	0.889266i	$0.348785\pi$
$978$	0	0
$979$	−157776.	−0.00526119
$980$	0	0
$981$	6.33973e7	2.10328
$982$	0	0
$983$	1.04764e7	0.345802	0.172901	−	0.984939i	$-0.444686\pi$
$983$	1.04764e7	0.345802	0.172901	+	0.984939i	$0.444686\pi$
$984$	0	0
$985$	1.40016e7	0.459820
$986$	0	0
$987$	1.54409e6	0.0504521
$988$	0	0
$989$	3.10323e6	0.100884
$990$	0	0
$991$	1.88230e6	0.0608843	0.0304422	−	0.999537i	$-0.490308\pi$
$991$	1.88230e6	0.0608843	0.0304422	+	0.999537i	$0.490308\pi$
$992$	0	0
$993$	−2.97190e7	−0.956449
$994$	0	0
$995$	5.56192e6	0.178101
$996$	0	0
$997$	−2.71518e7	−0.865090	−0.432545	−	0.901612i	$-0.642384\pi$
$997$	−2.71518e7	−0.865090	−0.432545	+	0.901612i	$0.642384\pi$
$998$	0	0
$999$	−5.66618e7	−1.79629

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.b.1.1		1
4.3	odd	2		448.6.a.o.1.1		1
8.3	odd	2		112.6.a.b.1.1		1
8.5	even	2		28.6.a.b.1.1	✓	1
24.5	odd	2		252.6.a.a.1.1		1
24.11	even	2		1008.6.a.l.1.1		1
40.13	odd	4		700.6.e.b.449.2		2
40.29	even	2		700.6.a.b.1.1		1
40.37	odd	4		700.6.e.b.449.1		2
56.5	odd	6		196.6.e.i.165.1		2
56.13	odd	2		196.6.a.a.1.1		1
56.27	even	2		784.6.a.m.1.1		1
56.37	even	6		196.6.e.a.165.1		2
56.45	odd	6		196.6.e.i.177.1		2
56.53	even	6		196.6.e.a.177.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.6.a.b.1.1	✓	1	8.5	even	2
112.6.a.b.1.1		1	8.3	odd	2
196.6.a.a.1.1		1	56.13	odd	2
196.6.e.a.165.1		2	56.37	even	6
196.6.e.a.177.1		2	56.53	even	6
196.6.e.i.165.1		2	56.5	odd	6
196.6.e.i.177.1		2	56.45	odd	6
252.6.a.a.1.1		1	24.5	odd	2
448.6.a.b.1.1		1	1.1	even	1	trivial
448.6.a.o.1.1		1	4.3	odd	2
700.6.a.b.1.1		1	40.29	even	2
700.6.e.b.449.1		2	40.37	odd	4
700.6.e.b.449.2		2	40.13	odd	4
784.6.a.m.1.1		1	56.27	even	2
1008.6.a.l.1.1		1	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 $\F_{q}$

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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