LMFDB - Embedded newform 448.6.a.bd.1.4

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:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 198x^{2} + 43x + 5999 $ x^4 - x^3 - 198x^2 + 43x + 5999
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$13.1634$ 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)

$f(q)$	$=$	$q+30.3268 q^{3} +100.401 q^{5} -49.0000 q^{7} +676.717 q^{9} +O(q^{10})$	$q+30.3268 q^{3} +100.401 q^{5} -49.0000 q^{7} +676.717 q^{9} +140.379 q^{11} -725.325 q^{13} +3044.84 q^{15} -365.763 q^{17} +202.222 q^{19} -1486.02 q^{21} +2736.01 q^{23} +6955.34 q^{25} +13153.3 q^{27} -2176.24 q^{29} -8959.49 q^{31} +4257.26 q^{33} -4919.65 q^{35} +2606.44 q^{37} -21996.8 q^{39} +14223.6 q^{41} +11233.4 q^{43} +67943.0 q^{45} +17779.3 q^{47} +2401.00 q^{49} -11092.4 q^{51} -25931.4 q^{53} +14094.2 q^{55} +6132.75 q^{57} +17664.0 q^{59} -3934.67 q^{61} -33159.2 q^{63} -72823.3 q^{65} +854.278 q^{67} +82974.4 q^{69} -1911.99 q^{71} -48863.1 q^{73} +210934. q^{75} -6878.59 q^{77} -55577.8 q^{79} +234455. q^{81} -97672.1 q^{83} -36723.0 q^{85} -65998.6 q^{87} -31443.5 q^{89} +35540.9 q^{91} -271713. q^{93} +20303.3 q^{95} -29651.4 q^{97} +94997.1 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9}+O(q^{10}) $ 4 * q + 18 * q^3 + 30 * q^5 - 196 * q^7 + 696 * q^9	$ 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9} + 484 q^{11} - 686 q^{13} + 2184 q^{15} - 1700 q^{17} + 654 q^{19} - 882 q^{21} - 136 q^{23} + 3304 q^{25} + 14388 q^{27} - 3812 q^{29} - 12748 q^{31} - 1536 q^{33} - 1470 q^{35} - 820 q^{37} - 34704 q^{39} + 22340 q^{41} + 32924 q^{43} + 59070 q^{45} + 2620 q^{47} + 9604 q^{49} + 55788 q^{51} - 22984 q^{53} - 24360 q^{55} + 15132 q^{57} + 108158 q^{59} - 4258 q^{61} - 34104 q^{63} - 95028 q^{65} + 109496 q^{67} + 82392 q^{69} - 54600 q^{71} - 12384 q^{73} + 315198 q^{75} - 23716 q^{77} - 78184 q^{79} + 114384 q^{81} + 115582 q^{83} - 101652 q^{85} + 17772 q^{87} - 31560 q^{89} + 33614 q^{91} - 218040 q^{93} + 67032 q^{95} + 41068 q^{97} + 301284 q^{99}+O(q^{100}) $ 4 * q + 18 * q^3 + 30 * q^5 - 196 * q^7 + 696 * q^9 + 484 * q^11 - 686 * q^13 + 2184 * q^15 - 1700 * q^17 + 654 * q^19 - 882 * q^21 - 136 * q^23 + 3304 * q^25 + 14388 * q^27 - 3812 * q^29 - 12748 * q^31 - 1536 * q^33 - 1470 * q^35 - 820 * q^37 - 34704 * q^39 + 22340 * q^41 + 32924 * q^43 + 59070 * q^45 + 2620 * q^47 + 9604 * q^49 + 55788 * q^51 - 22984 * q^53 - 24360 * q^55 + 15132 * q^57 + 108158 * q^59 - 4258 * q^61 - 34104 * q^63 - 95028 * q^65 + 109496 * q^67 + 82392 * q^69 - 54600 * q^71 - 12384 * q^73 + 315198 * q^75 - 23716 * q^77 - 78184 * q^79 + 114384 * q^81 + 115582 * q^83 - 101652 * q^85 + 17772 * q^87 - 31560 * q^89 + 33614 * q^91 - 218040 * q^93 + 67032 * q^95 + 41068 * q^97 + 301284 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	30.3268	1.94547	0.972734	−	0.231924i	$-0.0745020\pi$
$3$	30.3268	1.94547	0.972734	+	0.231924i	$0.0745020\pi$
$4$	0	0
$5$	100.401	1.79603	0.898013	−	0.439969i	$-0.145010\pi$
$5$	100.401	1.79603	0.898013	+	0.439969i	$0.145010\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	676.717	2.78485
$10$	0	0
$11$	140.379	0.349801	0.174901	−	0.984586i	$-0.444040\pi$
$11$	140.379	0.349801	0.174901	+	0.984586i	$0.444040\pi$
$12$	0	0
$13$	−725.325	−1.19035	−0.595175	−	0.803596i	$-0.702917\pi$
$13$	−725.325	−1.19035	−0.595175	+	0.803596i	$0.702917\pi$
$14$	0	0
$15$	3044.84	3.49411
$16$	0	0
$17$	−365.763	−0.306957	−0.153479	−	0.988152i	$-0.549048\pi$
$17$	−365.763	−0.306957	−0.153479	+	0.988152i	$0.549048\pi$
$18$	0	0
$19$	202.222	0.128512	0.0642561	−	0.997933i	$-0.479533\pi$
$19$	202.222	0.128512	0.0642561	+	0.997933i	$0.479533\pi$
$20$	0	0
$21$	−1486.02	−0.735318
$22$	0	0
$23$	2736.01	1.07844	0.539222	−	0.842164i	$-0.318718\pi$
$23$	2736.01	1.07844	0.539222	+	0.842164i	$0.318718\pi$
$24$	0	0
$25$	6955.34	2.22571
$26$	0	0
$27$	13153.3	3.47236
$28$	0	0
$29$	−2176.24	−0.480521	−0.240261	−	0.970708i	$-0.577233\pi$
$29$	−2176.24	−0.480521	−0.240261	+	0.970708i	$0.577233\pi$
$30$	0	0
$31$	−8959.49	−1.67448	−0.837238	−	0.546838i	$-0.815831\pi$
$31$	−8959.49	−1.67448	−0.837238	+	0.546838i	$0.815831\pi$
$32$	0	0
$33$	4257.26	0.680527
$34$	0	0
$35$	−4919.65	−0.678834
$36$	0	0
$37$	2606.44	0.312999	0.156500	−	0.987678i	$-0.449979\pi$
$37$	2606.44	0.312999	0.156500	+	0.987678i	$0.449979\pi$
$38$	0	0
$39$	−21996.8	−2.31579
$40$	0	0
$41$	14223.6	1.32145	0.660725	−	0.750628i	$-0.270249\pi$
$41$	14223.6	1.32145	0.660725	+	0.750628i	$0.270249\pi$
$42$	0	0
$43$	11233.4	0.926485	0.463242	−	0.886232i	$-0.346686\pi$
$43$	11233.4	0.926485	0.463242	+	0.886232i	$0.346686\pi$
$44$	0	0
$45$	67943.0	5.00166
$46$	0	0
$47$	17779.3	1.17400	0.587002	−	0.809585i	$-0.300308\pi$
$47$	17779.3	1.17400	0.587002	+	0.809585i	$0.300308\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−11092.4	−0.597175
$52$	0	0
$53$	−25931.4	−1.26805	−0.634025	−	0.773313i	$-0.718598\pi$
$53$	−25931.4	−1.26805	−0.634025	+	0.773313i	$0.718598\pi$
$54$	0	0
$55$	14094.2	0.628252
$56$	0	0
$57$	6132.75	0.250016
$58$	0	0
$59$	17664.0	0.660632	0.330316	−	0.943870i	$-0.392845\pi$
$59$	17664.0	0.660632	0.330316	+	0.943870i	$0.392845\pi$
$60$	0	0
$61$	−3934.67	−0.135389	−0.0676945	−	0.997706i	$-0.521564\pi$
$61$	−3934.67	−0.135389	−0.0676945	+	0.997706i	$0.521564\pi$
$62$	0	0
$63$	−33159.2	−1.05257
$64$	0	0
$65$	−72823.3	−2.13790
$66$	0	0
$67$	854.278	0.0232494	0.0116247	−	0.999932i	$-0.496300\pi$
$67$	854.278	0.0232494	0.0116247	+	0.999932i	$0.496300\pi$
$68$	0	0
$69$	82974.4	2.09808
$70$	0	0
$71$	−1911.99	−0.0450131	−0.0225066	−	0.999747i	$-0.507165\pi$
$71$	−1911.99	−0.0450131	−0.0225066	+	0.999747i	$0.507165\pi$
$72$	0	0
$73$	−48863.1	−1.07318	−0.536592	−	0.843842i	$-0.680289\pi$
$73$	−48863.1	−1.07318	−0.536592	+	0.843842i	$0.680289\pi$
$74$	0	0
$75$	210934.	4.33005
$76$	0	0
$77$	−6878.59	−0.132212
$78$	0	0
$79$	−55577.8	−1.00192	−0.500961	−	0.865470i	$-0.667020\pi$
$79$	−55577.8	−1.00192	−0.500961	+	0.865470i	$0.667020\pi$
$80$	0	0
$81$	234455.	3.97052
$82$	0	0
$83$	−97672.1	−1.55624	−0.778118	−	0.628118i	$-0.783825\pi$
$83$	−97672.1	−1.55624	−0.778118	+	0.628118i	$0.783825\pi$
$84$	0	0
$85$	−36723.0	−0.551303
$86$	0	0
$87$	−65998.6	−0.934838
$88$	0	0
$89$	−31443.5	−0.420781	−0.210391	−	0.977617i	$-0.567474\pi$
$89$	−31443.5	−0.420781	−0.210391	+	0.977617i	$0.567474\pi$
$90$	0	0
$91$	35540.9	0.449910
$92$	0	0
$93$	−271713.	−3.25764
$94$	0	0
$95$	20303.3	0.230811
$96$	0	0
$97$	−29651.4	−0.319975	−0.159987	−	0.987119i	$-0.551145\pi$
$97$	−29651.4	−0.319975	−0.159987	+	0.987119i	$0.551145\pi$
$98$	0	0
$99$	94997.1	0.974143
$100$	0	0
$101$	−35681.7	−0.348051	−0.174025	−	0.984741i	$-0.555677\pi$
$101$	−35681.7	−0.348051	−0.174025	+	0.984741i	$0.555677\pi$
$102$	0	0
$103$	43614.4	0.405076	0.202538	−	0.979274i	$-0.435081\pi$
$103$	43614.4	0.405076	0.202538	+	0.979274i	$0.435081\pi$
$104$	0	0
$105$	−149197.	−1.32065
$106$	0	0
$107$	−188645.	−1.59289	−0.796444	−	0.604712i	$-0.793288\pi$
$107$	−188645.	−1.59289	−0.796444	+	0.604712i	$0.793288\pi$
$108$	0	0
$109$	21116.1	0.170235	0.0851173	−	0.996371i	$-0.472873\pi$
$109$	21116.1	0.170235	0.0851173	+	0.996371i	$0.472873\pi$
$110$	0	0
$111$	79045.1	0.608930
$112$	0	0
$113$	−134514.	−0.990998	−0.495499	−	0.868608i	$-0.665015\pi$
$113$	−134514.	−0.990998	−0.495499	+	0.868608i	$0.665015\pi$
$114$	0	0
$115$	274698.	1.93691
$116$	0	0
$117$	−490840.	−3.31494
$118$	0	0
$119$	17922.4	0.116019
$120$	0	0
$121$	−141345.	−0.877639
$122$	0	0
$123$	431358.	2.57084
$124$	0	0
$125$	384570.	2.20141
$126$	0	0
$127$	214981.	1.18274	0.591371	−	0.806400i	$-0.298587\pi$
$127$	214981.	1.18274	0.591371	+	0.806400i	$0.298587\pi$
$128$	0	0
$129$	340672.	1.80245
$130$	0	0
$131$	−68642.0	−0.349471	−0.174736	−	0.984615i	$-0.555907\pi$
$131$	−68642.0	−0.349471	−0.174736	+	0.984615i	$0.555907\pi$
$132$	0	0
$133$	−9908.87	−0.0485730
$134$	0	0
$135$	1.32060e6	6.23645
$136$	0	0
$137$	−203279.	−0.925317	−0.462658	−	0.886537i	$-0.653104\pi$
$137$	−203279.	−0.925317	−0.462658	+	0.886537i	$0.653104\pi$
$138$	0	0
$139$	165261.	0.725495	0.362747	−	0.931887i	$-0.381839\pi$
$139$	165261.	0.725495	0.362747	+	0.931887i	$0.381839\pi$
$140$	0	0
$141$	539190.	2.28399
$142$	0	0
$143$	−101821.	−0.416386
$144$	0	0
$145$	−218497.	−0.863028
$146$	0	0
$147$	72814.7	0.277924
$148$	0	0
$149$	−388936.	−1.43520	−0.717601	−	0.696455i	$-0.754760\pi$
$149$	−388936.	−1.43520	−0.717601	+	0.696455i	$0.754760\pi$
$150$	0	0
$151$	217670.	0.776884	0.388442	−	0.921473i	$-0.373013\pi$
$151$	217670.	0.776884	0.388442	+	0.921473i	$0.373013\pi$
$152$	0	0
$153$	−247518.	−0.854828
$154$	0	0
$155$	−899541.	−3.00740
$156$	0	0
$157$	365868.	1.18461	0.592305	−	0.805714i	$-0.298218\pi$
$157$	365868.	1.18461	0.592305	+	0.805714i	$0.298218\pi$
$158$	0	0
$159$	−786418.	−2.46695
$160$	0	0
$161$	−134064.	−0.407613
$162$	0	0
$163$	11755.0	0.0346539	0.0173269	−	0.999850i	$-0.494484\pi$
$163$	11755.0	0.0346539	0.0173269	+	0.999850i	$0.494484\pi$
$164$	0	0
$165$	427433.	1.22224
$166$	0	0
$167$	165760.	0.459925	0.229963	−	0.973199i	$-0.426140\pi$
$167$	165760.	0.459925	0.229963	+	0.973199i	$0.426140\pi$
$168$	0	0
$169$	154804.	0.416931
$170$	0	0
$171$	136847.	0.357886
$172$	0	0
$173$	84234.2	0.213980	0.106990	−	0.994260i	$-0.465879\pi$
$173$	84234.2	0.213980	0.106990	+	0.994260i	$0.465879\pi$
$174$	0	0
$175$	−340812.	−0.841239
$176$	0	0
$177$	535694.	1.28524
$178$	0	0
$179$	755801.	1.76309	0.881546	−	0.472098i	$-0.156503\pi$
$179$	755801.	1.76309	0.881546	+	0.472098i	$0.156503\pi$
$180$	0	0
$181$	−472599.	−1.07225	−0.536125	−	0.844139i	$-0.680112\pi$
$181$	−472599.	−1.07225	−0.536125	+	0.844139i	$0.680112\pi$
$182$	0	0
$183$	−119326.	−0.263395
$184$	0	0
$185$	261689.	0.562155
$186$	0	0
$187$	−51345.6	−0.107374
$188$	0	0
$189$	−644511.	−1.31243
$190$	0	0
$191$	143491.	0.284605	0.142302	−	0.989823i	$-0.454549\pi$
$191$	143491.	0.284605	0.142302	+	0.989823i	$0.454549\pi$
$192$	0	0
$193$	393410.	0.760242	0.380121	−	0.924937i	$-0.375882\pi$
$193$	393410.	0.760242	0.380121	+	0.924937i	$0.375882\pi$
$194$	0	0
$195$	−2.20850e6	−4.15921
$196$	0	0
$197$	39881.9	0.0732167	0.0366084	−	0.999330i	$-0.488345\pi$
$197$	39881.9	0.0732167	0.0366084	+	0.999330i	$0.488345\pi$
$198$	0	0
$199$	−782215.	−1.40021	−0.700106	−	0.714039i	$-0.746864\pi$
$199$	−782215.	−1.40021	−0.700106	+	0.714039i	$0.746864\pi$
$200$	0	0
$201$	25907.6	0.0452310
$202$	0	0
$203$	106636.	0.181620
$204$	0	0
$205$	1.42806e6	2.37336
$206$	0	0
$207$	1.85150e6	3.00330
$208$	0	0
$209$	28387.8	0.0449537
$210$	0	0
$211$	722272.	1.11685	0.558425	−	0.829555i	$-0.311406\pi$
$211$	722272.	1.11685	0.558425	+	0.829555i	$0.311406\pi$
$212$	0	0
$213$	−57984.5	−0.0875716
$214$	0	0
$215$	1.12784e6	1.66399
$216$	0	0
$217$	439015.	0.632893
$218$	0	0
$219$	−1.48186e6	−2.08785
$220$	0	0
$221$	265297.	0.365386
$222$	0	0
$223$	1.09497e6	1.47449	0.737244	−	0.675626i	$-0.236127\pi$
$223$	1.09497e6	1.47449	0.737244	+	0.675626i	$0.236127\pi$
$224$	0	0
$225$	4.70680e6	6.19826
$226$	0	0
$227$	−393901.	−0.507368	−0.253684	−	0.967287i	$-0.581642\pi$
$227$	−393901.	−0.507368	−0.253684	+	0.967287i	$0.581642\pi$
$228$	0	0
$229$	231353.	0.291532	0.145766	−	0.989319i	$-0.453435\pi$
$229$	231353.	0.291532	0.145766	+	0.989319i	$0.453435\pi$
$230$	0	0
$231$	−208606.	−0.257215
$232$	0	0
$233$	−453071.	−0.546735	−0.273367	−	0.961910i	$-0.588137\pi$
$233$	−453071.	−0.546735	−0.273367	+	0.961910i	$0.588137\pi$
$234$	0	0
$235$	1.78506e6	2.10854
$236$	0	0
$237$	−1.68550e6	−1.94921
$238$	0	0
$239$	−615308.	−0.696783	−0.348392	−	0.937349i	$-0.613272\pi$
$239$	−615308.	−0.696783	−0.348392	+	0.937349i	$0.613272\pi$
$240$	0	0
$241$	−1.56823e6	−1.73927	−0.869637	−	0.493692i	$-0.835647\pi$
$241$	−1.56823e6	−1.73927	−0.869637	+	0.493692i	$0.835647\pi$
$242$	0	0
$243$	3.91404e6	4.25215
$244$	0	0
$245$	241063.	0.256575
$246$	0	0
$247$	−146677.	−0.152974
$248$	0	0
$249$	−2.96209e6	−3.02761
$250$	0	0
$251$	1.02371e6	1.02563	0.512816	−	0.858499i	$-0.328602\pi$
$251$	1.02371e6	1.02563	0.512816	+	0.858499i	$0.328602\pi$
$252$	0	0
$253$	384079.	0.377241
$254$	0	0
$255$	−1.11369e6	−1.07254
$256$	0	0
$257$	−1.63004e6	−1.53945	−0.769724	−	0.638377i	$-0.779606\pi$
$257$	−1.63004e6	−1.53945	−0.769724	+	0.638377i	$0.779606\pi$
$258$	0	0
$259$	−127716.	−0.118303
$260$	0	0
$261$	−1.47270e6	−1.33818
$262$	0	0
$263$	−1.21819e6	−1.08599	−0.542995	−	0.839736i	$-0.682710\pi$
$263$	−1.21819e6	−1.08599	−0.542995	+	0.839736i	$0.682710\pi$
$264$	0	0
$265$	−2.60354e6	−2.27745
$266$	0	0
$267$	−953583.	−0.818617
$268$	0	0
$269$	1.17744e6	0.992103	0.496052	−	0.868293i	$-0.334783\pi$
$269$	1.17744e6	0.992103	0.496052	+	0.868293i	$0.334783\pi$
$270$	0	0
$271$	1.63480e6	1.35220	0.676100	−	0.736810i	$-0.263669\pi$
$271$	1.63480e6	1.35220	0.676100	+	0.736810i	$0.263669\pi$
$272$	0	0
$273$	1.07784e6	0.875285
$274$	0	0
$275$	976387.	0.778557
$276$	0	0
$277$	179217.	0.140339	0.0701695	−	0.997535i	$-0.477646\pi$
$277$	179217.	0.140339	0.0701695	+	0.997535i	$0.477646\pi$
$278$	0	0
$279$	−6.06304e6	−4.66316
$280$	0	0
$281$	−576238.	−0.435347	−0.217674	−	0.976022i	$-0.569847\pi$
$281$	−576238.	−0.435347	−0.217674	+	0.976022i	$0.569847\pi$
$282$	0	0
$283$	−1.57802e6	−1.17124	−0.585620	−	0.810586i	$-0.699149\pi$
$283$	−1.57802e6	−1.17124	−0.585620	+	0.810586i	$0.699149\pi$
$284$	0	0
$285$	615734.	0.449036
$286$	0	0
$287$	−696958.	−0.499461
$288$	0	0
$289$	−1.28607e6	−0.905777
$290$	0	0
$291$	−899234.	−0.622501
$292$	0	0
$293$	272790.	0.185635	0.0928175	−	0.995683i	$-0.470413\pi$
$293$	272790.	0.185635	0.0928175	+	0.995683i	$0.470413\pi$
$294$	0	0
$295$	1.77349e6	1.18651
$296$	0	0
$297$	1.84645e6	1.21464
$298$	0	0
$299$	−1.98449e6	−1.28372
$300$	0	0
$301$	−550434.	−0.350178
$302$	0	0
$303$	−1.08211e6	−0.677121
$304$	0	0
$305$	−395044.	−0.243162
$306$	0	0
$307$	421848.	0.255453	0.127726	−	0.991809i	$-0.459232\pi$
$307$	421848.	0.255453	0.127726	+	0.991809i	$0.459232\pi$
$308$	0	0
$309$	1.32269e6	0.788062
$310$	0	0
$311$	−2.63556e6	−1.54516	−0.772578	−	0.634920i	$-0.781033\pi$
$311$	−2.63556e6	−1.54516	−0.772578	+	0.634920i	$0.781033\pi$
$312$	0	0
$313$	2.74918e6	1.58615	0.793073	−	0.609127i	$-0.208480\pi$
$313$	2.74918e6	1.58615	0.793073	+	0.609127i	$0.208480\pi$
$314$	0	0
$315$	−3.32921e6	−1.89045
$316$	0	0
$317$	−1.90802e6	−1.06644	−0.533219	−	0.845977i	$-0.679018\pi$
$317$	−1.90802e6	−1.06644	−0.533219	+	0.845977i	$0.679018\pi$
$318$	0	0
$319$	−305500.	−0.168087
$320$	0	0
$321$	−5.72100e6	−3.09891
$322$	0	0
$323$	−73965.3	−0.0394477
$324$	0	0
$325$	−5.04489e6	−2.64937
$326$	0	0
$327$	640385.	0.331186
$328$	0	0
$329$	−871185.	−0.443732
$330$	0	0
$331$	2.83954e6	1.42455	0.712276	−	0.701899i	$-0.247664\pi$
$331$	2.83954e6	1.42455	0.712276	+	0.701899i	$0.247664\pi$
$332$	0	0
$333$	1.76382e6	0.871654
$334$	0	0
$335$	85770.3	0.0417566
$336$	0	0
$337$	−2.83223e6	−1.35848	−0.679241	−	0.733915i	$-0.737691\pi$
$337$	−2.83223e6	−1.35848	−0.679241	+	0.733915i	$0.737691\pi$
$338$	0	0
$339$	−4.07940e6	−1.92796
$340$	0	0
$341$	−1.25773e6	−0.585734
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	8.33071e6	3.76820
$346$	0	0
$347$	1.91706e6	0.854695	0.427347	−	0.904087i	$-0.359448\pi$
$347$	1.91706e6	0.854695	0.427347	+	0.904087i	$0.359448\pi$
$348$	0	0
$349$	−583594.	−0.256476	−0.128238	−	0.991743i	$-0.540932\pi$
$349$	−583594.	−0.256476	−0.128238	+	0.991743i	$0.540932\pi$
$350$	0	0
$351$	−9.54040e6	−4.13332
$352$	0	0
$353$	−879102.	−0.375494	−0.187747	−	0.982217i	$-0.560118\pi$
$353$	−879102.	−0.375494	−0.187747	+	0.982217i	$0.560118\pi$
$354$	0	0
$355$	−191965.	−0.0808448
$356$	0	0
$357$	543530.	0.225711
$358$	0	0
$359$	−1.63054e6	−0.667721	−0.333861	−	0.942622i	$-0.608351\pi$
$359$	−1.63054e6	−0.667721	−0.333861	+	0.942622i	$0.608351\pi$
$360$	0	0
$361$	−2.43521e6	−0.983485
$362$	0	0
$363$	−4.28654e6	−1.70742
$364$	0	0
$365$	−4.90590e6	−1.92747
$366$	0	0
$367$	−944409.	−0.366012	−0.183006	−	0.983112i	$-0.558583\pi$
$367$	−944409.	−0.366012	−0.183006	+	0.983112i	$0.558583\pi$
$368$	0	0
$369$	9.62537e6	3.68003
$370$	0	0
$371$	1.27064e6	0.479278
$372$	0	0
$373$	−3.38633e6	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$373$	−3.38633e6	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$374$	0	0
$375$	1.16628e7	4.28277
$376$	0	0
$377$	1.57848e6	0.571988
$378$	0	0
$379$	−1.34513e6	−0.481023	−0.240512	−	0.970646i	$-0.577315\pi$
$379$	−1.34513e6	−0.481023	−0.240512	+	0.970646i	$0.577315\pi$
$380$	0	0
$381$	6.51968e6	2.30099
$382$	0	0
$383$	857263.	0.298619	0.149309	−	0.988791i	$-0.452295\pi$
$383$	857263.	0.298619	0.149309	+	0.988791i	$0.452295\pi$
$384$	0	0
$385$	−690617.	−0.237457
$386$	0	0
$387$	7.60180e6	2.58012
$388$	0	0
$389$	−2.49667e6	−0.836540	−0.418270	−	0.908323i	$-0.637363\pi$
$389$	−2.49667e6	−0.836540	−0.418270	+	0.908323i	$0.637363\pi$
$390$	0	0
$391$	−1.00073e6	−0.331036
$392$	0	0
$393$	−2.08170e6	−0.679885
$394$	0	0
$395$	−5.58006e6	−1.79948
$396$	0	0
$397$	−1.96172e6	−0.624683	−0.312342	−	0.949970i	$-0.601113\pi$
$397$	−1.96172e6	−0.624683	−0.312342	+	0.949970i	$0.601113\pi$
$398$	0	0
$399$	−300505.	−0.0944972
$400$	0	0
$401$	−4.02823e6	−1.25099	−0.625493	−	0.780229i	$-0.715102\pi$
$401$	−4.02823e6	−1.25099	−0.625493	+	0.780229i	$0.715102\pi$
$402$	0	0
$403$	6.49854e6	1.99321
$404$	0	0
$405$	2.35395e7	7.13115
$406$	0	0
$407$	365890.	0.109488
$408$	0	0
$409$	3.45736e6	1.02197	0.510983	−	0.859591i	$-0.329282\pi$
$409$	3.45736e6	1.02197	0.510983	+	0.859591i	$0.329282\pi$
$410$	0	0
$411$	−6.16480e6	−1.80017
$412$	0	0
$413$	−865538.	−0.249696
$414$	0	0
$415$	−9.80637e6	−2.79504
$416$	0	0
$417$	5.01186e6	1.41143
$418$	0	0
$419$	−3.15192e6	−0.877082	−0.438541	−	0.898711i	$-0.644505\pi$
$419$	−3.15192e6	−0.877082	−0.438541	+	0.898711i	$0.644505\pi$
$420$	0	0
$421$	1.97118e6	0.542028	0.271014	−	0.962575i	$-0.412641\pi$
$421$	1.97118e6	0.542028	0.271014	+	0.962575i	$0.412641\pi$
$422$	0	0
$423$	1.20316e7	3.26942
$424$	0	0
$425$	−2.54401e6	−0.683198
$426$	0	0
$427$	192799.	0.0511723
$428$	0	0
$429$	−3.08790e6	−0.810065
$430$	0	0
$431$	7.37370e6	1.91202	0.956009	−	0.293336i	$-0.0947654\pi$
$431$	7.37370e6	1.91202	0.956009	+	0.293336i	$0.0947654\pi$
$432$	0	0
$433$	3.43414e6	0.880233	0.440117	−	0.897941i	$-0.354937\pi$
$433$	3.43414e6	0.880233	0.440117	+	0.897941i	$0.354937\pi$
$434$	0	0
$435$	−6.62632e6	−1.67899
$436$	0	0
$437$	553280.	0.138593
$438$	0	0
$439$	554385.	0.137294	0.0686468	−	0.997641i	$-0.478132\pi$
$439$	554385.	0.137294	0.0686468	+	0.997641i	$0.478132\pi$
$440$	0	0
$441$	1.62480e6	0.397835
$442$	0	0
$443$	2.47746e6	0.599788	0.299894	−	0.953973i	$-0.403049\pi$
$443$	2.47746e6	0.599788	0.299894	+	0.953973i	$0.403049\pi$
$444$	0	0
$445$	−3.15696e6	−0.755734
$446$	0	0
$447$	−1.17952e7	−2.79214
$448$	0	0
$449$	5.75792e6	1.34788	0.673938	−	0.738788i	$-0.264602\pi$
$449$	5.75792e6	1.34788	0.673938	+	0.738788i	$0.264602\pi$
$450$	0	0
$451$	1.99670e6	0.462245
$452$	0	0
$453$	6.60124e6	1.51140
$454$	0	0
$455$	3.56834e6	0.808050
$456$	0	0
$457$	1.27523e6	0.285626	0.142813	−	0.989750i	$-0.454385\pi$
$457$	1.27523e6	0.285626	0.142813	+	0.989750i	$0.454385\pi$
$458$	0	0
$459$	−4.81098e6	−1.06587
$460$	0	0
$461$	7.28110e6	1.59568	0.797838	−	0.602872i	$-0.205977\pi$
$461$	7.28110e6	1.59568	0.797838	+	0.602872i	$0.205977\pi$
$462$	0	0
$463$	578254.	0.125362	0.0626810	−	0.998034i	$-0.480035\pi$
$463$	578254.	0.125362	0.0626810	+	0.998034i	$0.480035\pi$
$464$	0	0
$465$	−2.72802e7	−5.85081
$466$	0	0
$467$	−3.96270e6	−0.840812	−0.420406	−	0.907336i	$-0.638112\pi$
$467$	−3.96270e6	−0.840812	−0.420406	+	0.907336i	$0.638112\pi$
$468$	0	0
$469$	−41859.6	−0.00878746
$470$	0	0
$471$	1.10956e7	2.30462
$472$	0	0
$473$	1.57693e6	0.324086
$474$	0	0
$475$	1.40652e6	0.286031
$476$	0	0
$477$	−1.75482e7	−3.53132
$478$	0	0
$479$	1.35618e6	0.270071	0.135036	−	0.990841i	$-0.456885\pi$
$479$	1.35618e6	0.270071	0.135036	+	0.990841i	$0.456885\pi$
$480$	0	0
$481$	−1.89052e6	−0.372578
$482$	0	0
$483$	−4.06575e6	−0.792999
$484$	0	0
$485$	−2.97703e6	−0.574683
$486$	0	0
$487$	−3.14278e6	−0.600470	−0.300235	−	0.953865i	$-0.597065\pi$
$487$	−3.14278e6	−0.600470	−0.300235	+	0.953865i	$0.597065\pi$
$488$	0	0
$489$	356491.	0.0674180
$490$	0	0
$491$	2.26023e6	0.423106	0.211553	−	0.977367i	$-0.432148\pi$
$491$	2.26023e6	0.423106	0.211553	+	0.977367i	$0.432148\pi$
$492$	0	0
$493$	795989.	0.147499
$494$	0	0
$495$	9.53780e6	1.74959
$496$	0	0
$497$	93687.4	0.0170134
$498$	0	0
$499$	−7.67916e6	−1.38058	−0.690291	−	0.723532i	$-0.742518\pi$
$499$	−7.67916e6	−1.38058	−0.690291	+	0.723532i	$0.742518\pi$
$500$	0	0
$501$	5.02696e6	0.894770
$502$	0	0
$503$	9.03752e6	1.59268	0.796342	−	0.604847i	$-0.206766\pi$
$503$	9.03752e6	1.59268	0.796342	+	0.604847i	$0.206766\pi$
$504$	0	0
$505$	−3.58248e6	−0.625108
$506$	0	0
$507$	4.69470e6	0.811126
$508$	0	0
$509$	863632.	0.147752	0.0738762	−	0.997267i	$-0.476463\pi$
$509$	863632.	0.147752	0.0738762	+	0.997267i	$0.476463\pi$
$510$	0	0
$511$	2.39429e6	0.405625
$512$	0	0
$513$	2.65988e6	0.446240
$514$	0	0
$515$	4.37892e6	0.727527
$516$	0	0
$517$	2.49585e6	0.410669
$518$	0	0
$519$	2.55456e6	0.416291
$520$	0	0
$521$	7.05642e6	1.13891	0.569456	−	0.822022i	$-0.307154\pi$
$521$	7.05642e6	1.13891	0.569456	+	0.822022i	$0.307154\pi$
$522$	0	0
$523$	7.53970e6	1.20531	0.602656	−	0.798001i	$-0.294109\pi$
$523$	7.53970e6	1.20531	0.602656	+	0.798001i	$0.294109\pi$
$524$	0	0
$525$	−1.03357e7	−1.63660
$526$	0	0
$527$	3.27705e6	0.513992
$528$	0	0
$529$	1.04939e6	0.163041
$530$	0	0
$531$	1.19536e7	1.83976
$532$	0	0
$533$	−1.03168e7	−1.57299
$534$	0	0
$535$	−1.89401e7	−2.86087
$536$	0	0
$537$	2.29211e7	3.43004
$538$	0	0
$539$	337051.	0.0499716
$540$	0	0
$541$	4.36838e6	0.641692	0.320846	−	0.947131i	$-0.396033\pi$
$541$	4.36838e6	0.641692	0.320846	+	0.947131i	$0.396033\pi$
$542$	0	0
$543$	−1.43324e7	−2.08603
$544$	0	0
$545$	2.12008e6	0.305746
$546$	0	0
$547$	1.24065e7	1.77289	0.886446	−	0.462832i	$-0.153167\pi$
$547$	1.24065e7	1.77289	0.886446	+	0.462832i	$0.153167\pi$
$548$	0	0
$549$	−2.66266e6	−0.377038
$550$	0	0
$551$	−440084.	−0.0617528
$552$	0	0
$553$	2.72331e6	0.378691
$554$	0	0
$555$	7.93620e6	1.09365
$556$	0	0
$557$	−5.81149e6	−0.793687	−0.396843	−	0.917886i	$-0.629894\pi$
$557$	−5.81149e6	−0.793687	−0.396843	+	0.917886i	$0.629894\pi$
$558$	0	0
$559$	−8.14783e6	−1.10284
$560$	0	0
$561$	−1.55715e6	−0.208893
$562$	0	0
$563$	−9.52315e6	−1.26622	−0.633111	−	0.774061i	$-0.718222\pi$
$563$	−9.52315e6	−1.26622	−0.633111	+	0.774061i	$0.718222\pi$
$564$	0	0
$565$	−1.35054e7	−1.77986
$566$	0	0
$567$	−1.14883e7	−1.50071
$568$	0	0
$569$	−4.08221e6	−0.528585	−0.264292	−	0.964443i	$-0.585138\pi$
$569$	−4.08221e6	−0.528585	−0.264292	+	0.964443i	$0.585138\pi$
$570$	0	0
$571$	−8.62191e6	−1.10666	−0.553329	−	0.832963i	$-0.686643\pi$
$571$	−8.62191e6	−1.10666	−0.553329	+	0.832963i	$0.686643\pi$
$572$	0	0
$573$	4.35164e6	0.553689
$574$	0	0
$575$	1.90299e7	2.40030
$576$	0	0
$577$	−4.83415e6	−0.604478	−0.302239	−	0.953232i	$-0.597734\pi$
$577$	−4.83415e6	−0.604478	−0.302239	+	0.953232i	$0.597734\pi$
$578$	0	0
$579$	1.19309e7	1.47903
$580$	0	0
$581$	4.78593e6	0.588202
$582$	0	0
$583$	−3.64023e6	−0.443566
$584$	0	0
$585$	−4.92808e7	−5.95372
$586$	0	0
$587$	−4.07962e6	−0.488680	−0.244340	−	0.969690i	$-0.578571\pi$
$587$	−4.07962e6	−0.488680	−0.244340	+	0.969690i	$0.578571\pi$
$588$	0	0
$589$	−1.81180e6	−0.215191
$590$	0	0
$591$	1.20949e6	0.142441
$592$	0	0
$593$	−6.90677e6	−0.806563	−0.403281	−	0.915076i	$-0.632130\pi$
$593$	−6.90677e6	−0.806563	−0.403281	+	0.915076i	$0.632130\pi$
$594$	0	0
$595$	1.79942e6	0.208373
$596$	0	0
$597$	−2.37221e7	−2.72407
$598$	0	0
$599$	4.82372e6	0.549307	0.274653	−	0.961543i	$-0.411437\pi$
$599$	4.82372e6	0.549307	0.274653	+	0.961543i	$0.411437\pi$
$600$	0	0
$601$	−2.61361e6	−0.295158	−0.147579	−	0.989050i	$-0.547148\pi$
$601$	−2.61361e6	−0.295158	−0.147579	+	0.989050i	$0.547148\pi$
$602$	0	0
$603$	578105.	0.0647460
$604$	0	0
$605$	−1.41911e7	−1.57626
$606$	0	0
$607$	−1.37755e7	−1.51752	−0.758762	−	0.651367i	$-0.774196\pi$
$607$	−1.37755e7	−1.51752	−0.758762	+	0.651367i	$0.774196\pi$
$608$	0	0
$609$	3.23393e6	0.353336
$610$	0	0
$611$	−1.28958e7	−1.39748
$612$	0	0
$613$	1.43107e6	0.153819	0.0769093	−	0.997038i	$-0.475495\pi$
$613$	1.43107e6	0.153819	0.0769093	+	0.997038i	$0.475495\pi$
$614$	0	0
$615$	4.33087e7	4.61729
$616$	0	0
$617$	1.04509e7	1.10520	0.552599	−	0.833447i	$-0.313636\pi$
$617$	1.04509e7	1.10520	0.552599	+	0.833447i	$0.313636\pi$
$618$	0	0
$619$	−1.29568e7	−1.35916	−0.679579	−	0.733602i	$-0.737838\pi$
$619$	−1.29568e7	−1.35916	−0.679579	+	0.733602i	$0.737838\pi$
$620$	0	0
$621$	3.59875e7	3.74474
$622$	0	0
$623$	1.54073e6	0.159040
$624$	0	0
$625$	1.68757e7	1.72808
$626$	0	0
$627$	860911.	0.0874560
$628$	0	0
$629$	−953339.	−0.0960773
$630$	0	0
$631$	−1.55481e7	−1.55455	−0.777275	−	0.629161i	$-0.783399\pi$
$631$	−1.55481e7	−1.55455	−0.777275	+	0.629161i	$0.783399\pi$
$632$	0	0
$633$	2.19042e7	2.17280
$634$	0	0
$635$	2.15843e7	2.12424
$636$	0	0
$637$	−1.74151e6	−0.170050
$638$	0	0
$639$	−1.29388e6	−0.125355
$640$	0	0
$641$	−8.29375e6	−0.797271	−0.398635	−	0.917109i	$-0.630516\pi$
$641$	−8.29375e6	−0.797271	−0.398635	+	0.917109i	$0.630516\pi$
$642$	0	0
$643$	5.60475e6	0.534600	0.267300	−	0.963613i	$-0.413869\pi$
$643$	5.60475e6	0.534600	0.267300	+	0.963613i	$0.413869\pi$
$644$	0	0
$645$	3.42038e7	3.23724
$646$	0	0
$647$	6.92863e6	0.650708	0.325354	−	0.945592i	$-0.394516\pi$
$647$	6.92863e6	0.650708	0.325354	+	0.945592i	$0.394516\pi$
$648$	0	0
$649$	2.47967e6	0.231090
$650$	0	0
$651$	1.33139e7	1.23127
$652$	0	0
$653$	4.78354e6	0.439002	0.219501	−	0.975612i	$-0.429557\pi$
$653$	4.78354e6	0.439002	0.219501	+	0.975612i	$0.429557\pi$
$654$	0	0
$655$	−6.89172e6	−0.627660
$656$	0	0
$657$	−3.30665e7	−2.98865
$658$	0	0
$659$	1.69493e7	1.52033	0.760166	−	0.649729i	$-0.225118\pi$
$659$	1.69493e7	1.52033	0.760166	+	0.649729i	$0.225118\pi$
$660$	0	0
$661$	−1.86673e6	−0.166180	−0.0830898	−	0.996542i	$-0.526479\pi$
$661$	−1.86673e6	−0.166180	−0.0830898	+	0.996542i	$0.526479\pi$
$662$	0	0
$663$	8.04563e6	0.710847
$664$	0	0
$665$	−994860.	−0.0872384
$666$	0	0
$667$	−5.95421e6	−0.518215
$668$	0	0
$669$	3.32071e7	2.86857
$670$	0	0
$671$	−552346.	−0.0473593
$672$	0	0
$673$	−640306.	−0.0544941	−0.0272471	−	0.999629i	$-0.508674\pi$
$673$	−640306.	−0.0544941	−0.0272471	+	0.999629i	$0.508674\pi$
$674$	0	0
$675$	9.14856e7	7.72846
$676$	0	0
$677$	9.94105e6	0.833606	0.416803	−	0.908997i	$-0.363151\pi$
$677$	9.94105e6	0.833606	0.416803	+	0.908997i	$0.363151\pi$
$678$	0	0
$679$	1.45292e6	0.120939
$680$	0	0
$681$	−1.19458e7	−0.987067
$682$	0	0
$683$	3.87368e6	0.317740	0.158870	−	0.987300i	$-0.449215\pi$
$683$	3.87368e6	0.317740	0.158870	+	0.987300i	$0.449215\pi$
$684$	0	0
$685$	−2.04094e7	−1.66189
$686$	0	0
$687$	7.01621e6	0.567166
$688$	0	0
$689$	1.88087e7	1.50942
$690$	0	0
$691$	6.87004e6	0.547349	0.273674	−	0.961822i	$-0.411761\pi$
$691$	6.87004e6	0.547349	0.273674	+	0.961822i	$0.411761\pi$
$692$	0	0
$693$	−4.65486e6	−0.368191
$694$	0	0
$695$	1.65924e7	1.30301
$696$	0	0
$697$	−5.20248e6	−0.405628
$698$	0	0
$699$	−1.37402e7	−1.06365
$700$	0	0
$701$	−1.31864e6	−0.101351	−0.0506757	−	0.998715i	$-0.516137\pi$
$701$	−1.31864e6	−0.101351	−0.0506757	+	0.998715i	$0.516137\pi$
$702$	0	0
$703$	527079.	0.0402242
$704$	0	0
$705$	5.41352e7	4.10210
$706$	0	0
$707$	1.74840e6	0.131551
$708$	0	0
$709$	4.06660e6	0.303820	0.151910	−	0.988394i	$-0.451458\pi$
$709$	4.06660e6	0.303820	0.151910	+	0.988394i	$0.451458\pi$
$710$	0	0
$711$	−3.76105e7	−2.79020
$712$	0	0
$713$	−2.45132e7	−1.80583
$714$	0	0
$715$	−1.02229e7	−0.747840
$716$	0	0
$717$	−1.86603e7	−1.35557
$718$	0	0
$719$	−1.62746e7	−1.17406	−0.587029	−	0.809566i	$-0.699702\pi$
$719$	−1.62746e7	−1.17406	−0.587029	+	0.809566i	$0.699702\pi$
$720$	0	0
$721$	−2.13710e6	−0.153104
$722$	0	0
$723$	−4.75595e7	−3.38370
$724$	0	0
$725$	−1.51365e7	−1.06950
$726$	0	0
$727$	3.53970e6	0.248388	0.124194	−	0.992258i	$-0.460366\pi$
$727$	3.53970e6	0.248388	0.124194	+	0.992258i	$0.460366\pi$
$728$	0	0
$729$	6.17278e7	4.30191
$730$	0	0
$731$	−4.10875e6	−0.284391
$732$	0	0
$733$	1.88003e7	1.29243	0.646213	−	0.763157i	$-0.276352\pi$
$733$	1.88003e7	1.29243	0.646213	+	0.763157i	$0.276352\pi$
$734$	0	0
$735$	7.31067e6	0.499159
$736$	0	0
$737$	119923.	0.00813268
$738$	0	0
$739$	1.50394e7	1.01302	0.506511	−	0.862233i	$-0.330935\pi$
$739$	1.50394e7	1.01302	0.506511	+	0.862233i	$0.330935\pi$
$740$	0	0
$741$	−4.44824e6	−0.297607
$742$	0	0
$743$	2.19765e7	1.46045	0.730225	−	0.683206i	$-0.239415\pi$
$743$	2.19765e7	1.46045	0.730225	+	0.683206i	$0.239415\pi$
$744$	0	0
$745$	−3.90496e7	−2.57766
$746$	0	0
$747$	−6.60964e7	−4.33387
$748$	0	0
$749$	9.24360e6	0.602055
$750$	0	0
$751$	1.13426e7	0.733861	0.366931	−	0.930248i	$-0.380409\pi$
$751$	1.13426e7	0.733861	0.366931	+	0.930248i	$0.380409\pi$
$752$	0	0
$753$	3.10458e7	1.99533
$754$	0	0
$755$	2.18543e7	1.39530
$756$	0	0
$757$	9.25255e6	0.586843	0.293421	−	0.955983i	$-0.405206\pi$
$757$	9.25255e6	0.586843	0.293421	+	0.955983i	$0.405206\pi$
$758$	0	0
$759$	1.16479e7	0.733910
$760$	0	0
$761$	−2.25165e7	−1.40941	−0.704707	−	0.709499i	$-0.748921\pi$
$761$	−2.25165e7	−1.40941	−0.704707	+	0.709499i	$0.748921\pi$
$762$	0	0
$763$	−1.03469e6	−0.0643427
$764$	0	0
$765$	−2.48511e7	−1.53529
$766$	0	0
$767$	−1.28122e7	−0.786383
$768$	0	0
$769$	1.76148e7	1.07414	0.537071	−	0.843537i	$-0.319531\pi$
$769$	1.76148e7	1.07414	0.537071	+	0.843537i	$0.319531\pi$
$770$	0	0
$771$	−4.94339e7	−2.99494
$772$	0	0
$773$	1.37818e7	0.829575	0.414788	−	0.909918i	$-0.363856\pi$
$773$	1.37818e7	0.829575	0.414788	+	0.909918i	$0.363856\pi$
$774$	0	0
$775$	−6.23163e7	−3.72690
$776$	0	0
$777$	−3.87321e6	−0.230154
$778$	0	0
$779$	2.87633e6	0.169822
$780$	0	0
$781$	−268404.	−0.0157457
$782$	0	0
$783$	−2.86247e7	−1.66854
$784$	0	0
$785$	3.67335e7	2.12759
$786$	0	0
$787$	−2.63416e7	−1.51602	−0.758011	−	0.652242i	$-0.773828\pi$
$787$	−2.63416e7	−1.51602	−0.758011	+	0.652242i	$0.773828\pi$
$788$	0	0
$789$	−3.69439e7	−2.11276
$790$	0	0
$791$	6.59121e6	0.374562
$792$	0	0
$793$	2.85391e6	0.161160
$794$	0	0
$795$	−7.89570e7	−4.43071
$796$	0	0
$797$	6.45081e6	0.359723	0.179862	−	0.983692i	$-0.442435\pi$
$797$	6.45081e6	0.359723	0.179862	+	0.983692i	$0.442435\pi$
$798$	0	0
$799$	−6.50301e6	−0.360369
$800$	0	0
$801$	−2.12784e7	−1.17181
$802$	0	0
$803$	−6.85938e6	−0.375401
$804$	0	0
$805$	−1.34602e7	−0.732084
$806$	0	0
$807$	3.57079e7	1.93011
$808$	0	0
$809$	−1.26311e7	−0.678534	−0.339267	−	0.940690i	$-0.610179\pi$
$809$	−1.26311e7	−0.678534	−0.339267	+	0.940690i	$0.610179\pi$
$810$	0	0
$811$	1.35397e7	0.722864	0.361432	−	0.932398i	$-0.382288\pi$
$811$	1.35397e7	0.722864	0.361432	+	0.932398i	$0.382288\pi$
$812$	0	0
$813$	4.95783e7	2.63066
$814$	0	0
$815$	1.18021e6	0.0622393
$816$	0	0
$817$	2.27163e6	0.119064
$818$	0	0
$819$	2.40512e7	1.25293
$820$	0	0
$821$	1.52801e7	0.791170	0.395585	−	0.918429i	$-0.370542\pi$
$821$	1.52801e7	0.791170	0.395585	+	0.918429i	$0.370542\pi$
$822$	0	0
$823$	−9.48109e6	−0.487931	−0.243966	−	0.969784i	$-0.578448\pi$
$823$	−9.48109e6	−0.487931	−0.243966	+	0.969784i	$0.578448\pi$
$824$	0	0
$825$	2.96107e7	1.51466
$826$	0	0
$827$	2.31016e7	1.17457	0.587285	−	0.809381i	$-0.300197\pi$
$827$	2.31016e7	1.17457	0.587285	+	0.809381i	$0.300197\pi$
$828$	0	0
$829$	3.25886e7	1.64695	0.823473	−	0.567356i	$-0.192034\pi$
$829$	3.25886e7	1.64695	0.823473	+	0.567356i	$0.192034\pi$
$830$	0	0
$831$	5.43507e6	0.273025
$832$	0	0
$833$	−878197.	−0.0438510
$834$	0	0
$835$	1.66424e7	0.826038
$836$	0	0
$837$	−1.17847e8	−5.81438
$838$	0	0
$839$	2.53876e7	1.24514	0.622568	−	0.782565i	$-0.286089\pi$
$839$	2.53876e7	1.24514	0.622568	+	0.782565i	$0.286089\pi$
$840$	0	0
$841$	−1.57751e7	−0.769100
$842$	0	0
$843$	−1.74755e7	−0.846954
$844$	0	0
$845$	1.55424e7	0.748819
$846$	0	0
$847$	6.92589e6	0.331716
$848$	0	0
$849$	−4.78563e7	−2.27861
$850$	0	0
$851$	7.13123e6	0.337552
$852$	0	0
$853$	1.37036e7	0.644857	0.322429	−	0.946594i	$-0.395501\pi$
$853$	1.37036e7	0.644857	0.322429	+	0.946594i	$0.395501\pi$
$854$	0	0
$855$	1.37396e7	0.642773
$856$	0	0
$857$	−1.06461e7	−0.495151	−0.247575	−	0.968869i	$-0.579634\pi$
$857$	−1.06461e7	−0.495151	−0.247575	+	0.968869i	$0.579634\pi$
$858$	0	0
$859$	−1.57882e7	−0.730046	−0.365023	−	0.930998i	$-0.618939\pi$
$859$	−1.57882e7	−0.730046	−0.365023	+	0.930998i	$0.618939\pi$
$860$	0	0
$861$	−2.11365e7	−0.971685
$862$	0	0
$863$	−1.30657e7	−0.597183	−0.298591	−	0.954381i	$-0.596517\pi$
$863$	−1.30657e7	−0.597183	−0.298591	+	0.954381i	$0.596517\pi$
$864$	0	0
$865$	8.45719e6	0.384314
$866$	0	0
$867$	−3.90026e7	−1.76216
$868$	0	0
$869$	−7.80198e6	−0.350474
$870$	0	0
$871$	−619629.	−0.0276749
$872$	0	0
$873$	−2.00656e7	−0.891081
$874$	0	0
$875$	−1.88439e7	−0.832054
$876$	0	0
$877$	−6.63045e6	−0.291101	−0.145551	−	0.989351i	$-0.546495\pi$
$877$	−6.63045e6	−0.291101	−0.145551	+	0.989351i	$0.546495\pi$
$878$	0	0
$879$	8.27287e6	0.361147
$880$	0	0
$881$	−4.30515e7	−1.86874	−0.934369	−	0.356307i	$-0.884036\pi$
$881$	−4.30515e7	−1.86874	−0.934369	+	0.356307i	$0.884036\pi$
$882$	0	0
$883$	1.47337e7	0.635931	0.317966	−	0.948102i	$-0.397000\pi$
$883$	1.47337e7	0.635931	0.317966	+	0.948102i	$0.397000\pi$
$884$	0	0
$885$	5.37842e7	2.30832
$886$	0	0
$887$	−4.96746e6	−0.211995	−0.105998	−	0.994366i	$-0.533804\pi$
$887$	−4.96746e6	−0.211995	−0.105998	+	0.994366i	$0.533804\pi$
$888$	0	0
$889$	−1.05341e7	−0.447034
$890$	0	0
$891$	3.29127e7	1.38889
$892$	0	0
$893$	3.59536e6	0.150874
$894$	0	0
$895$	7.58832e7	3.16656
$896$	0	0
$897$	−6.01834e7	−2.49744
$898$	0	0
$899$	1.94980e7	0.804621
$900$	0	0
$901$	9.48475e6	0.389237
$902$	0	0
$903$	−1.66929e7	−0.681260
$904$	0	0
$905$	−4.74493e7	−1.92579
$906$	0	0
$907$	1.11164e7	0.448691	0.224346	−	0.974510i	$-0.427976\pi$
$907$	1.11164e7	0.448691	0.224346	+	0.974510i	$0.427976\pi$
$908$	0	0
$909$	−2.41464e7	−0.969267
$910$	0	0
$911$	−4.01160e7	−1.60148	−0.800740	−	0.599012i	$-0.795560\pi$
$911$	−4.01160e7	−1.60148	−0.800740	+	0.599012i	$0.795560\pi$
$912$	0	0
$913$	−1.37111e7	−0.544373
$914$	0	0
$915$	−1.19805e7	−0.473065
$916$	0	0
$917$	3.36346e6	0.132088
$918$	0	0
$919$	−2.83366e7	−1.10677	−0.553387	−	0.832924i	$-0.686665\pi$
$919$	−2.83366e7	−1.10677	−0.553387	+	0.832924i	$0.686665\pi$
$920$	0	0
$921$	1.27933e7	0.496975
$922$	0	0
$923$	1.38681e6	0.0535813
$924$	0	0
$925$	1.81287e7	0.696646
$926$	0	0
$927$	2.95146e7	1.12807
$928$	0	0
$929$	3.64899e7	1.38718	0.693591	−	0.720369i	$-0.256027\pi$
$929$	3.64899e7	1.38718	0.693591	+	0.720369i	$0.256027\pi$
$930$	0	0
$931$	485535.	0.0183589
$932$	0	0
$933$	−7.99283e7	−3.00605
$934$	0	0
$935$	−5.15514e6	−0.192847
$936$	0	0
$937$	3.43802e7	1.27926	0.639631	−	0.768682i	$-0.279087\pi$
$937$	3.43802e7	1.27926	0.639631	+	0.768682i	$0.279087\pi$
$938$	0	0
$939$	8.33741e7	3.08580
$940$	0	0
$941$	−4.05799e7	−1.49395	−0.746977	−	0.664850i	$-0.768495\pi$
$941$	−4.05799e7	−1.49395	−0.746977	+	0.664850i	$0.768495\pi$
$942$	0	0
$943$	3.89159e7	1.42511
$944$	0	0
$945$	−6.47095e7	−2.35716
$946$	0	0
$947$	−1.77826e7	−0.644347	−0.322174	−	0.946681i	$-0.604414\pi$
$947$	−1.77826e7	−0.644347	−0.322174	+	0.946681i	$0.604414\pi$
$948$	0	0
$949$	3.54417e7	1.27746
$950$	0	0
$951$	−5.78644e7	−2.07472
$952$	0	0
$953$	3.28916e7	1.17315	0.586575	−	0.809895i	$-0.300476\pi$
$953$	3.28916e7	1.17315	0.586575	+	0.809895i	$0.300476\pi$
$954$	0	0
$955$	1.44067e7	0.511157
$956$	0	0
$957$	−9.26484e6	−0.327008
$958$	0	0
$959$	9.96066e6	0.349737
$960$	0	0
$961$	5.16433e7	1.80387
$962$	0	0
$963$	−1.27659e8	−4.43595
$964$	0	0
$965$	3.94987e7	1.36542
$966$	0	0
$967$	2.45654e7	0.844808	0.422404	−	0.906408i	$-0.361186\pi$
$967$	2.45654e7	0.844808	0.422404	+	0.906408i	$0.361186\pi$
$968$	0	0
$969$	−2.24313e6	−0.0767442
$970$	0	0
$971$	3.39341e7	1.15502	0.577509	−	0.816385i	$-0.304025\pi$
$971$	3.39341e7	1.15502	0.577509	+	0.816385i	$0.304025\pi$
$972$	0	0
$973$	−8.09781e6	−0.274211
$974$	0	0
$975$	−1.52995e8	−5.15427
$976$	0	0
$977$	2.11237e7	0.708001	0.354001	−	0.935245i	$-0.384821\pi$
$977$	2.11237e7	0.708001	0.354001	+	0.935245i	$0.384821\pi$
$978$	0	0
$979$	−4.41402e6	−0.147190
$980$	0	0
$981$	1.42896e7	0.474077
$982$	0	0
$983$	1.40112e7	0.462479	0.231240	−	0.972897i	$-0.425722\pi$
$983$	1.40112e7	0.462479	0.231240	+	0.972897i	$0.425722\pi$
$984$	0	0
$985$	4.00418e6	0.131499
$986$	0	0
$987$	−2.64203e7	−0.863267
$988$	0	0
$989$	3.07345e7	0.999161
$990$	0	0
$991$	−3.93443e7	−1.27262	−0.636308	−	0.771435i	$-0.719539\pi$
$991$	−3.93443e7	−1.27262	−0.636308	+	0.771435i	$0.719539\pi$
$992$	0	0
$993$	8.61144e7	2.77142
$994$	0	0
$995$	−7.85351e7	−2.51482
$996$	0	0
$997$	−1.99737e7	−0.636387	−0.318194	−	0.948026i	$-0.603076\pi$
$997$	−1.99737e7	−0.636387	−0.318194	+	0.948026i	$0.603076\pi$
$998$	0	0
$999$	3.42832e7	1.08685

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.bd.1.4		4
4.3	odd	2		448.6.a.bc.1.1		4
8.3	odd	2		224.6.a.h.1.4	yes	4
8.5	even	2		224.6.a.g.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.g.1.1	✓	4	8.5	even	2
224.6.a.h.1.4	yes	4	8.3	odd	2
448.6.a.bc.1.1		4	4.3	odd	2
448.6.a.bd.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+30.3268 q^{3} +100.401 q^{5} -49.0000 q^{7} +676.717 q^{9} +O(q^{10})\)	\(q+30.3268 q^{3} +100.401 q^{5} -49.0000 q^{7} +676.717 q^{9} +140.379 q^{11} -725.325 q^{13} +3044.84 q^{15} -365.763 q^{17} +202.222 q^{19} -1486.02 q^{21} +2736.01 q^{23} +6955.34 q^{25} +13153.3 q^{27} -2176.24 q^{29} -8959.49 q^{31} +4257.26 q^{33} -4919.65 q^{35} +2606.44 q^{37} -21996.8 q^{39} +14223.6 q^{41} +11233.4 q^{43} +67943.0 q^{45} +17779.3 q^{47} +2401.00 q^{49} -11092.4 q^{51} -25931.4 q^{53} +14094.2 q^{55} +6132.75 q^{57} +17664.0 q^{59} -3934.67 q^{61} -33159.2 q^{63} -72823.3 q^{65} +854.278 q^{67} +82974.4 q^{69} -1911.99 q^{71} -48863.1 q^{73} +210934. q^{75} -6878.59 q^{77} -55577.8 q^{79} +234455. q^{81} -97672.1 q^{83} -36723.0 q^{85} -65998.6 q^{87} -31443.5 q^{89} +35540.9 q^{91} -271713. q^{93} +20303.3 q^{95} -29651.4 q^{97} +94997.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9}+O(q^{10}) \) 4 * q + 18 * q^3 + 30 * q^5 - 196 * q^7 + 696 * q^9	\( 4 q + 18 q^{3} + 30 q^{5} - 196 q^{7} + 696 q^{9} + 484 q^{11} - 686 q^{13} + 2184 q^{15} - 1700 q^{17} + 654 q^{19} - 882 q^{21} - 136 q^{23} + 3304 q^{25} + 14388 q^{27} - 3812 q^{29} - 12748 q^{31} - 1536 q^{33} - 1470 q^{35} - 820 q^{37} - 34704 q^{39} + 22340 q^{41} + 32924 q^{43} + 59070 q^{45} + 2620 q^{47} + 9604 q^{49} + 55788 q^{51} - 22984 q^{53} - 24360 q^{55} + 15132 q^{57} + 108158 q^{59} - 4258 q^{61} - 34104 q^{63} - 95028 q^{65} + 109496 q^{67} + 82392 q^{69} - 54600 q^{71} - 12384 q^{73} + 315198 q^{75} - 23716 q^{77} - 78184 q^{79} + 114384 q^{81} + 115582 q^{83} - 101652 q^{85} + 17772 q^{87} - 31560 q^{89} + 33614 q^{91} - 218040 q^{93} + 67032 q^{95} + 41068 q^{97} + 301284 q^{99}+O(q^{100}) \) 4 * q + 18 * q^3 + 30 * q^5 - 196 * q^7 + 696 * q^9 + 484 * q^11 - 686 * q^13 + 2184 * q^15 - 1700 * q^17 + 654 * q^19 - 882 * q^21 - 136 * q^23 + 3304 * q^25 + 14388 * q^27 - 3812 * q^29 - 12748 * q^31 - 1536 * q^33 - 1470 * q^35 - 820 * q^37 - 34704 * q^39 + 22340 * q^41 + 32924 * q^43 + 59070 * q^45 + 2620 * q^47 + 9604 * q^49 + 55788 * q^51 - 22984 * q^53 - 24360 * q^55 + 15132 * q^57 + 108158 * q^59 - 4258 * q^61 - 34104 * q^63 - 95028 * q^65 + 109496 * q^67 + 82392 * q^69 - 54600 * q^71 - 12384 * q^73 + 315198 * q^75 - 23716 * q^77 - 78184 * q^79 + 114384 * q^81 + 115582 * q^83 - 101652 * q^85 + 17772 * q^87 - 31560 * q^89 + 33614 * q^91 - 218040 * q^93 + 67032 * q^95 + 41068 * q^97 + 301284 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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