LMFDB - Embedded newform 448.6.a.bf.1.5

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:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 229x^{3} - 272x^{2} + 7973x - 13998 $ x^5 - 229x^3 - 272x^2 + 7973*x - 13998
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-11.8926$ 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)

$f(q)$	$=$	$q+25.7852 q^{3} +34.4729 q^{5} +49.0000 q^{7} +421.876 q^{9} +O(q^{10})$	$q+25.7852 q^{3} +34.4729 q^{5} +49.0000 q^{7} +421.876 q^{9} +783.692 q^{11} +894.228 q^{13} +888.891 q^{15} +237.449 q^{17} +1774.00 q^{19} +1263.47 q^{21} -4263.54 q^{23} -1936.62 q^{25} +4612.35 q^{27} -5262.44 q^{29} -7517.81 q^{31} +20207.6 q^{33} +1689.17 q^{35} -676.241 q^{37} +23057.8 q^{39} +5947.49 q^{41} -22338.9 q^{43} +14543.3 q^{45} -11667.4 q^{47} +2401.00 q^{49} +6122.66 q^{51} +10119.6 q^{53} +27016.2 q^{55} +45742.9 q^{57} +22057.3 q^{59} +4521.65 q^{61} +20671.9 q^{63} +30826.7 q^{65} -10550.2 q^{67} -109936. q^{69} +32626.7 q^{71} +34637.5 q^{73} -49936.0 q^{75} +38400.9 q^{77} +19453.7 q^{79} +16414.5 q^{81} +66507.7 q^{83} +8185.56 q^{85} -135693. q^{87} -43269.5 q^{89} +43817.2 q^{91} -193848. q^{93} +61155.0 q^{95} -86454.2 q^{97} +330621. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 10 q^{3} - 36 q^{5} + 245 q^{7} + 637 q^{9}+O(q^{10}) $ 5 * q + 10 * q^3 - 36 * q^5 + 245 * q^7 + 637 * q^9	$ 5 q + 10 q^{3} - 36 q^{5} + 245 q^{7} + 637 q^{9} - 116 q^{11} - 40 q^{13} + 16 q^{15} - 402 q^{17} + 3582 q^{19} + 490 q^{21} + 472 q^{23} + 9615 q^{25} - 356 q^{27} - 4754 q^{29} - 10500 q^{31} + 15864 q^{33} - 1764 q^{35} - 19642 q^{37} + 10872 q^{39} + 23398 q^{41} - 22044 q^{43} - 49476 q^{45} + 16004 q^{47} + 12005 q^{49} + 45676 q^{51} - 54246 q^{53} + 53456 q^{55} + 109556 q^{57} + 74366 q^{59} - 68316 q^{61} + 31213 q^{63} + 152568 q^{65} + 26560 q^{67} - 214720 q^{69} + 93072 q^{71} + 136098 q^{73} + 124510 q^{75} - 5684 q^{77} + 96080 q^{79} + 104801 q^{81} + 145894 q^{83} - 117352 q^{85} + 168876 q^{87} + 188554 q^{89} - 1960 q^{91} - 86296 q^{93} + 74736 q^{95} - 88146 q^{97} + 260236 q^{99}+O(q^{100}) $ 5 * q + 10 * q^3 - 36 * q^5 + 245 * q^7 + 637 * q^9 - 116 * q^11 - 40 * q^13 + 16 * q^15 - 402 * q^17 + 3582 * q^19 + 490 * q^21 + 472 * q^23 + 9615 * q^25 - 356 * q^27 - 4754 * q^29 - 10500 * q^31 + 15864 * q^33 - 1764 * q^35 - 19642 * q^37 + 10872 * q^39 + 23398 * q^41 - 22044 * q^43 - 49476 * q^45 + 16004 * q^47 + 12005 * q^49 + 45676 * q^51 - 54246 * q^53 + 53456 * q^55 + 109556 * q^57 + 74366 * q^59 - 68316 * q^61 + 31213 * q^63 + 152568 * q^65 + 26560 * q^67 - 214720 * q^69 + 93072 * q^71 + 136098 * q^73 + 124510 * q^75 - 5684 * q^77 + 96080 * q^79 + 104801 * q^81 + 145894 * q^83 - 117352 * q^85 + 168876 * q^87 + 188554 * q^89 - 1960 * q^91 - 86296 * q^93 + 74736 * q^95 - 88146 * q^97 + 260236 * 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
$3$	25.7852	1.65412	0.827060	−	0.562113i	$-0.190011\pi$
$3$	25.7852	1.65412	0.827060	+	0.562113i	$0.190011\pi$
$4$	0	0
$5$	34.4729	0.616671	0.308335	−	0.951278i	$-0.400228\pi$
$5$	34.4729	0.616671	0.308335	+	0.951278i	$0.400228\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	421.876	1.73611
$10$	0	0
$11$	783.692	1.95283	0.976413	−	0.215912i	$-0.0692725\pi$
$11$	783.692	1.95283	0.976413	+	0.215912i	$0.0692725\pi$
$12$	0	0
$13$	894.228	1.46754	0.733770	−	0.679398i	$-0.237759\pi$
$13$	894.228	1.46754	0.733770	+	0.679398i	$0.237759\pi$
$14$	0	0
$15$	888.891	1.02005
$16$	0	0
$17$	237.449	0.199273	0.0996363	−	0.995024i	$-0.468232\pi$
$17$	237.449	0.199273	0.0996363	+	0.995024i	$0.468232\pi$
$18$	0	0
$19$	1774.00	1.12738	0.563689	−	0.825987i	$-0.309382\pi$
$19$	1774.00	1.12738	0.563689	+	0.825987i	$0.309382\pi$
$20$	0	0
$21$	1263.47	0.625199
$22$	0	0
$23$	−4263.54	−1.68055	−0.840274	−	0.542161i	$-0.817606\pi$
$23$	−4263.54	−1.68055	−0.840274	+	0.542161i	$0.817606\pi$
$24$	0	0
$25$	−1936.62	−0.619717
$26$	0	0
$27$	4612.35	1.21762
$28$	0	0
$29$	−5262.44	−1.16196	−0.580981	−	0.813917i	$-0.697331\pi$
$29$	−5262.44	−1.16196	−0.580981	+	0.813917i	$0.697331\pi$
$30$	0	0
$31$	−7517.81	−1.40503	−0.702517	−	0.711667i	$-0.747941\pi$
$31$	−7517.81	−1.40503	−0.702517	+	0.711667i	$0.747941\pi$
$32$	0	0
$33$	20207.6	3.23021
$34$	0	0
$35$	1689.17	0.233080
$36$	0	0
$37$	−676.241	−0.0812077	−0.0406038	−	0.999175i	$-0.512928\pi$
$37$	−676.241	−0.0812077	−0.0406038	+	0.999175i	$0.512928\pi$
$38$	0	0
$39$	23057.8	2.42749
$40$	0	0
$41$	5947.49	0.552553	0.276276	−	0.961078i	$-0.410899\pi$
$41$	5947.49	0.552553	0.276276	+	0.961078i	$0.410899\pi$
$42$	0	0
$43$	−22338.9	−1.84243	−0.921213	−	0.389058i	$-0.872801\pi$
$43$	−22338.9	−1.84243	−0.921213	+	0.389058i	$0.872801\pi$
$44$	0	0
$45$	14543.3	1.07061
$46$	0	0
$47$	−11667.4	−0.770423	−0.385211	−	0.922828i	$-0.625871\pi$
$47$	−11667.4	−0.770423	−0.385211	+	0.922828i	$0.625871\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	6122.66	0.329621
$52$	0	0
$53$	10119.6	0.494851	0.247425	−	0.968907i	$-0.420416\pi$
$53$	10119.6	0.494851	0.247425	+	0.968907i	$0.420416\pi$
$54$	0	0
$55$	27016.2	1.20425
$56$	0	0
$57$	45742.9	1.86482
$58$	0	0
$59$	22057.3	0.824941	0.412471	−	0.910971i	$-0.364666\pi$
$59$	22057.3	0.824941	0.412471	+	0.910971i	$0.364666\pi$
$60$	0	0
$61$	4521.65	0.155587	0.0777934	−	0.996970i	$-0.475213\pi$
$61$	4521.65	0.155587	0.0777934	+	0.996970i	$0.475213\pi$
$62$	0	0
$63$	20671.9	0.656190
$64$	0	0
$65$	30826.7	0.904989
$66$	0	0
$67$	−10550.2	−0.287126	−0.143563	−	0.989641i	$-0.545856\pi$
$67$	−10550.2	−0.287126	−0.143563	+	0.989641i	$0.545856\pi$
$68$	0	0
$69$	−109936.	−2.77983
$70$	0	0
$71$	32626.7	0.768117	0.384059	−	0.923309i	$-0.374526\pi$
$71$	32626.7	0.768117	0.384059	+	0.923309i	$0.374526\pi$
$72$	0	0
$73$	34637.5	0.760746	0.380373	−	0.924833i	$-0.375796\pi$
$73$	34637.5	0.760746	0.380373	+	0.924833i	$0.375796\pi$
$74$	0	0
$75$	−49936.0	−1.02509
$76$	0	0
$77$	38400.9	0.738099
$78$	0	0
$79$	19453.7	0.350699	0.175349	−	0.984506i	$-0.443895\pi$
$79$	19453.7	0.350699	0.175349	+	0.984506i	$0.443895\pi$
$80$	0	0
$81$	16414.5	0.277980
$82$	0	0
$83$	66507.7	1.05968	0.529842	−	0.848096i	$-0.322251\pi$
$83$	66507.7	1.05968	0.529842	+	0.848096i	$0.322251\pi$
$84$	0	0
$85$	8185.56	0.122886
$86$	0	0
$87$	−135693.	−1.92203
$88$	0	0
$89$	−43269.5	−0.579038	−0.289519	−	0.957172i	$-0.593495\pi$
$89$	−43269.5	−0.579038	−0.289519	+	0.957172i	$0.593495\pi$
$90$	0	0
$91$	43817.2	0.554678
$92$	0	0
$93$	−193848.	−2.32410
$94$	0	0
$95$	61155.0	0.695221
$96$	0	0
$97$	−86454.2	−0.932947	−0.466474	−	0.884535i	$-0.654476\pi$
$97$	−86454.2	−0.932947	−0.466474	+	0.884535i	$0.654476\pi$
$98$	0	0
$99$	330621.	3.39033
$100$	0	0
$101$	−38338.3	−0.373964	−0.186982	−	0.982363i	$-0.559871\pi$
$101$	−38338.3	−0.373964	−0.186982	+	0.982363i	$0.559871\pi$
$102$	0	0
$103$	54228.7	0.503658	0.251829	−	0.967772i	$-0.418968\pi$
$103$	54228.7	0.503658	0.251829	+	0.967772i	$0.418968\pi$
$104$	0	0
$105$	43555.7	0.385542
$106$	0	0
$107$	−36211.6	−0.305765	−0.152883	−	0.988244i	$-0.548856\pi$
$107$	−36211.6	−0.305765	−0.152883	+	0.988244i	$0.548856\pi$
$108$	0	0
$109$	−168039.	−1.35470	−0.677351	−	0.735660i	$-0.736872\pi$
$109$	−168039.	−1.35470	−0.677351	+	0.735660i	$0.736872\pi$
$110$	0	0
$111$	−17437.0	−0.134327
$112$	0	0
$113$	−122969.	−0.905941	−0.452971	−	0.891525i	$-0.649636\pi$
$113$	−122969.	−0.905941	−0.452971	+	0.891525i	$0.649636\pi$
$114$	0	0
$115$	−146977.	−1.03635
$116$	0	0
$117$	377253.	2.54782
$118$	0	0
$119$	11635.0	0.0753180
$120$	0	0
$121$	453121.	2.81353
$122$	0	0
$123$	153357.	0.913989
$124$	0	0
$125$	−174489.	−0.998832
$126$	0	0
$127$	223397.	1.22905	0.614523	−	0.788899i	$-0.289349\pi$
$127$	223397.	1.22905	0.614523	+	0.788899i	$0.289349\pi$
$128$	0	0
$129$	−576012.	−3.04760
$130$	0	0
$131$	27341.0	0.139199	0.0695996	−	0.997575i	$-0.477828\pi$
$131$	27341.0	0.139199	0.0695996	+	0.997575i	$0.477828\pi$
$132$	0	0
$133$	86926.0	0.426109
$134$	0	0
$135$	159001.	0.750873
$136$	0	0
$137$	−432656.	−1.96943	−0.984716	−	0.174169i	$-0.944276\pi$
$137$	−432656.	−1.96943	−0.984716	+	0.174169i	$0.944276\pi$
$138$	0	0
$139$	325455.	1.42874	0.714371	−	0.699767i	$-0.246713\pi$
$139$	325455.	1.42874	0.714371	+	0.699767i	$0.246713\pi$
$140$	0	0
$141$	−300846.	−1.27437
$142$	0	0
$143$	700799.	2.86585
$144$	0	0
$145$	−181412.	−0.716548
$146$	0	0
$147$	61910.2	0.236303
$148$	0	0
$149$	4308.39	0.0158982	0.00794912	−	0.999968i	$-0.497470\pi$
$149$	4308.39	0.0158982	0.00794912	+	0.999968i	$0.497470\pi$
$150$	0	0
$151$	−108842.	−0.388467	−0.194233	−	0.980955i	$-0.562222\pi$
$151$	−108842.	−0.388467	−0.194233	+	0.980955i	$0.562222\pi$
$152$	0	0
$153$	100174.	0.345960
$154$	0	0
$155$	−259161.	−0.866444
$156$	0	0
$157$	364241.	1.17934	0.589671	−	0.807643i	$-0.299257\pi$
$157$	364241.	1.17934	0.589671	+	0.807643i	$0.299257\pi$
$158$	0	0
$159$	260936.	0.818543
$160$	0	0
$161$	−208914.	−0.635188
$162$	0	0
$163$	−308830.	−0.910439	−0.455220	−	0.890379i	$-0.650439\pi$
$163$	−308830.	−0.910439	−0.455220	+	0.890379i	$0.650439\pi$
$164$	0	0
$165$	696617.	1.99198
$166$	0	0
$167$	32535.9	0.0902757	0.0451379	−	0.998981i	$-0.485627\pi$
$167$	32535.9	0.0902757	0.0451379	+	0.998981i	$0.485627\pi$
$168$	0	0
$169$	428350.	1.15367
$170$	0	0
$171$	748408.	1.95726
$172$	0	0
$173$	72288.2	0.183634	0.0918169	−	0.995776i	$-0.470733\pi$
$173$	72288.2	0.183634	0.0918169	+	0.995776i	$0.470733\pi$
$174$	0	0
$175$	−94894.2	−0.234231
$176$	0	0
$177$	568753.	1.36455
$178$	0	0
$179$	−715750.	−1.66966	−0.834832	−	0.550505i	$-0.814435\pi$
$179$	−715750.	−1.66966	−0.834832	+	0.550505i	$0.814435\pi$
$180$	0	0
$181$	108339.	0.245804	0.122902	−	0.992419i	$-0.460780\pi$
$181$	108339.	0.245804	0.122902	+	0.992419i	$0.460780\pi$
$182$	0	0
$183$	116592.	0.257359
$184$	0	0
$185$	−23312.0	−0.0500784
$186$	0	0
$187$	186087.	0.389145
$188$	0	0
$189$	226005.	0.460218
$190$	0	0
$191$	−358754.	−0.711562	−0.355781	−	0.934569i	$-0.615785\pi$
$191$	−358754.	−0.711562	−0.355781	+	0.934569i	$0.615785\pi$
$192$	0	0
$193$	−471503.	−0.911153	−0.455577	−	0.890197i	$-0.650567\pi$
$193$	−471503.	−0.911153	−0.455577	+	0.890197i	$0.650567\pi$
$194$	0	0
$195$	794871.	1.49696
$196$	0	0
$197$	679797.	1.24800	0.623998	−	0.781426i	$-0.285507\pi$
$197$	679797.	1.24800	0.623998	+	0.781426i	$0.285507\pi$
$198$	0	0
$199$	−42375.1	−0.0758540	−0.0379270	−	0.999281i	$-0.512075\pi$
$199$	−42375.1	−0.0758540	−0.0379270	+	0.999281i	$0.512075\pi$
$200$	0	0
$201$	−272038.	−0.474941
$202$	0	0
$203$	−257859.	−0.439180
$204$	0	0
$205$	205027.	0.340743
$206$	0	0
$207$	−1.79869e6	−2.91763
$208$	0	0
$209$	1.39027e6	2.20157
$210$	0	0
$211$	−142585.	−0.220479	−0.110240	−	0.993905i	$-0.535162\pi$
$211$	−142585.	−0.220479	−0.110240	+	0.993905i	$0.535162\pi$
$212$	0	0
$213$	841286.	1.27056
$214$	0	0
$215$	−770087.	−1.13617
$216$	0	0
$217$	−368373.	−0.531053
$218$	0	0
$219$	893135.	1.25837
$220$	0	0
$221$	212333.	0.292440
$222$	0	0
$223$	1.39633e6	1.88030	0.940148	−	0.340767i	$-0.110687\pi$
$223$	1.39633e6	1.88030	0.940148	+	0.340767i	$0.110687\pi$
$224$	0	0
$225$	−817012.	−1.07590
$226$	0	0
$227$	1.34493e6	1.73234	0.866172	−	0.499745i	$-0.166573\pi$
$227$	1.34493e6	1.73234	0.866172	+	0.499745i	$0.166573\pi$
$228$	0	0
$229$	−404550.	−0.509781	−0.254891	−	0.966970i	$-0.582039\pi$
$229$	−404550.	−0.509781	−0.254891	+	0.966970i	$0.582039\pi$
$230$	0	0
$231$	990174.	1.22090
$232$	0	0
$233$	565574.	0.682495	0.341247	−	0.939974i	$-0.389151\pi$
$233$	565574.	0.682495	0.341247	+	0.939974i	$0.389151\pi$
$234$	0	0
$235$	−402209.	−0.475097
$236$	0	0
$237$	501617.	0.580098
$238$	0	0
$239$	445640.	0.504649	0.252325	−	0.967643i	$-0.418805\pi$
$239$	445640.	0.504649	0.252325	+	0.967643i	$0.418805\pi$
$240$	0	0
$241$	−1.15458e6	−1.28051	−0.640254	−	0.768163i	$-0.721171\pi$
$241$	−1.15458e6	−1.28051	−0.640254	+	0.768163i	$0.721171\pi$
$242$	0	0
$243$	−697551.	−0.757810
$244$	0	0
$245$	82769.5	0.0880958
$246$	0	0
$247$	1.58636e6	1.65447
$248$	0	0
$249$	1.71491e6	1.75285
$250$	0	0
$251$	101592.	0.101783	0.0508914	−	0.998704i	$-0.483794\pi$
$251$	101592.	0.101783	0.0508914	+	0.998704i	$0.483794\pi$
$252$	0	0
$253$	−3.34130e6	−3.28182
$254$	0	0
$255$	211066.	0.203268
$256$	0	0
$257$	−624321.	−0.589624	−0.294812	−	0.955555i	$-0.595257\pi$
$257$	−624321.	−0.589624	−0.294812	+	0.955555i	$0.595257\pi$
$258$	0	0
$259$	−33135.8	−0.0306936
$260$	0	0
$261$	−2.22010e6	−2.01730
$262$	0	0
$263$	−1.01644e6	−0.906134	−0.453067	−	0.891476i	$-0.649670\pi$
$263$	−1.01644e6	−0.906134	−0.453067	+	0.891476i	$0.649670\pi$
$264$	0	0
$265$	348853.	0.305160
$266$	0	0
$267$	−1.11571e6	−0.957799
$268$	0	0
$269$	−1.55436e6	−1.30970	−0.654848	−	0.755760i	$-0.727267\pi$
$269$	−1.55436e6	−1.30970	−0.654848	+	0.755760i	$0.727267\pi$
$270$	0	0
$271$	543634.	0.449659	0.224830	−	0.974398i	$-0.427817\pi$
$271$	543634.	0.449659	0.224830	+	0.974398i	$0.427817\pi$
$272$	0	0
$273$	1.12983e6	0.917504
$274$	0	0
$275$	−1.51771e6	−1.21020
$276$	0	0
$277$	−945309.	−0.740243	−0.370122	−	0.928983i	$-0.620684\pi$
$277$	−945309.	−0.740243	−0.370122	+	0.928983i	$0.620684\pi$
$278$	0	0
$279$	−3.17158e6	−2.43930
$280$	0	0
$281$	−1.59362e6	−1.20398	−0.601990	−	0.798504i	$-0.705625\pi$
$281$	−1.59362e6	−1.20398	−0.601990	+	0.798504i	$0.705625\pi$
$282$	0	0
$283$	1.07141e6	0.795224	0.397612	−	0.917554i	$-0.369839\pi$
$283$	1.07141e6	0.795224	0.397612	+	0.917554i	$0.369839\pi$
$284$	0	0
$285$	1.57689e6	1.14998
$286$	0	0
$287$	291427.	0.208845
$288$	0	0
$289$	−1.36348e6	−0.960290
$290$	0	0
$291$	−2.22924e6	−1.54321
$292$	0	0
$293$	1.68935e6	1.14961	0.574806	−	0.818290i	$-0.305078\pi$
$293$	1.68935e6	1.14961	0.574806	+	0.818290i	$0.305078\pi$
$294$	0	0
$295$	760381.	0.508717
$296$	0	0
$297$	3.61466e6	2.37781
$298$	0	0
$299$	−3.81258e6	−2.46627
$300$	0	0
$301$	−1.09461e6	−0.696372
$302$	0	0
$303$	−988561.	−0.618581
$304$	0	0
$305$	155875.	0.0959458
$306$	0	0
$307$	394676.	0.238998	0.119499	−	0.992834i	$-0.461871\pi$
$307$	394676.	0.238998	0.119499	+	0.992834i	$0.461871\pi$
$308$	0	0
$309$	1.39830e6	0.833111
$310$	0	0
$311$	2.47454e6	1.45076	0.725378	−	0.688351i	$-0.241665\pi$
$311$	2.47454e6	1.45076	0.725378	+	0.688351i	$0.241665\pi$
$312$	0	0
$313$	434473.	0.250670	0.125335	−	0.992114i	$-0.459999\pi$
$313$	434473.	0.250670	0.125335	+	0.992114i	$0.459999\pi$
$314$	0	0
$315$	712622.	0.404653
$316$	0	0
$317$	−330594.	−0.184777	−0.0923884	−	0.995723i	$-0.529450\pi$
$317$	−330594.	−0.184777	−0.0923884	+	0.995723i	$0.529450\pi$
$318$	0	0
$319$	−4.12413e6	−2.26911
$320$	0	0
$321$	−933723.	−0.505773
$322$	0	0
$323$	421234.	0.224656
$324$	0	0
$325$	−1.73178e6	−0.909459
$326$	0	0
$327$	−4.33292e6	−2.24084
$328$	0	0
$329$	−571702.	−0.291192
$330$	0	0
$331$	−3.29539e6	−1.65325	−0.826623	−	0.562757i	$-0.809741\pi$
$331$	−3.29539e6	−1.65325	−0.826623	+	0.562757i	$0.809741\pi$
$332$	0	0
$333$	−285290.	−0.140986
$334$	0	0
$335$	−363695.	−0.177062
$336$	0	0
$337$	708165.	0.339672	0.169836	−	0.985472i	$-0.445676\pi$
$337$	708165.	0.339672	0.169836	+	0.985472i	$0.445676\pi$
$338$	0	0
$339$	−3.17078e6	−1.49854
$340$	0	0
$341$	−5.89164e6	−2.74379
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−3.78983e6	−1.71424
$346$	0	0
$347$	2.22853e6	0.993563	0.496782	−	0.867876i	$-0.334515\pi$
$347$	2.22853e6	0.993563	0.496782	+	0.867876i	$0.334515\pi$
$348$	0	0
$349$	−3.30754e6	−1.45359	−0.726794	−	0.686855i	$-0.758991\pi$
$349$	−3.30754e6	−1.45359	−0.726794	+	0.686855i	$0.758991\pi$
$350$	0	0
$351$	4.12449e6	1.78691
$352$	0	0
$353$	1.53241e6	0.654542	0.327271	−	0.944930i	$-0.393871\pi$
$353$	1.53241e6	0.654542	0.327271	+	0.944930i	$0.393871\pi$
$354$	0	0
$355$	1.12474e6	0.473676
$356$	0	0
$357$	300010.	0.124585
$358$	0	0
$359$	−2.62516e6	−1.07503	−0.537513	−	0.843255i	$-0.680636\pi$
$359$	−2.62516e6	−1.07503	−0.537513	+	0.843255i	$0.680636\pi$
$360$	0	0
$361$	670975.	0.270981
$362$	0	0
$363$	1.16838e7	4.65391
$364$	0	0
$365$	1.19406e6	0.469130
$366$	0	0
$367$	−2.88852e6	−1.11946	−0.559732	−	0.828674i	$-0.689096\pi$
$367$	−2.88852e6	−1.11946	−0.559732	+	0.828674i	$0.689096\pi$
$368$	0	0
$369$	2.50910e6	0.959295
$370$	0	0
$371$	495861.	0.187036
$372$	0	0
$373$	4.56674e6	1.69955	0.849776	−	0.527144i	$-0.176737\pi$
$373$	4.56674e6	1.69955	0.849776	+	0.527144i	$0.176737\pi$
$374$	0	0
$375$	−4.49923e6	−1.65219
$376$	0	0
$377$	−4.70582e6	−1.70523
$378$	0	0
$379$	3.54135e6	1.26640	0.633200	−	0.773988i	$-0.281741\pi$
$379$	3.54135e6	1.26640	0.633200	+	0.773988i	$0.281741\pi$
$380$	0	0
$381$	5.76033e6	2.03299
$382$	0	0
$383$	2.57609e6	0.897355	0.448678	−	0.893694i	$-0.351895\pi$
$383$	2.57609e6	0.897355	0.448678	+	0.893694i	$0.351895\pi$
$384$	0	0
$385$	1.32379e6	0.455164
$386$	0	0
$387$	−9.42424e6	−3.19866
$388$	0	0
$389$	2.59202e6	0.868489	0.434244	−	0.900795i	$-0.357015\pi$
$389$	2.59202e6	0.868489	0.434244	+	0.900795i	$0.357015\pi$
$390$	0	0
$391$	−1.01237e6	−0.334887
$392$	0	0
$393$	704994.	0.230252
$394$	0	0
$395$	670626.	0.216266
$396$	0	0
$397$	−1.30804e6	−0.416528	−0.208264	−	0.978073i	$-0.566781\pi$
$397$	−1.30804e6	−0.416528	−0.208264	+	0.978073i	$0.566781\pi$
$398$	0	0
$399$	2.24140e6	0.704835
$400$	0	0
$401$	−2.65761e6	−0.825335	−0.412667	−	0.910882i	$-0.635403\pi$
$401$	−2.65761e6	−0.825335	−0.412667	+	0.910882i	$0.635403\pi$
$402$	0	0
$403$	−6.72263e6	−2.06194
$404$	0	0
$405$	565855.	0.171422
$406$	0	0
$407$	−529964.	−0.158584
$408$	0	0
$409$	2.29384e6	0.678039	0.339019	−	0.940779i	$-0.389905\pi$
$409$	2.29384e6	0.678039	0.339019	+	0.940779i	$0.389905\pi$
$410$	0	0
$411$	−1.11561e7	−3.25768
$412$	0	0
$413$	1.08081e6	0.311799
$414$	0	0
$415$	2.29272e6	0.653477
$416$	0	0
$417$	8.39191e6	2.36331
$418$	0	0
$419$	4.15963e6	1.15750	0.578748	−	0.815506i	$-0.303541\pi$
$419$	4.15963e6	1.15750	0.578748	+	0.815506i	$0.303541\pi$
$420$	0	0
$421$	2.33306e6	0.641536	0.320768	−	0.947158i	$-0.396059\pi$
$421$	2.33306e6	0.641536	0.320768	+	0.947158i	$0.396059\pi$
$422$	0	0
$423$	−4.92219e6	−1.33754
$424$	0	0
$425$	−459847.	−0.123493
$426$	0	0
$427$	221561.	0.0588063
$428$	0	0
$429$	1.80702e7	4.74046
$430$	0	0
$431$	2.92094e6	0.757408	0.378704	−	0.925518i	$-0.376370\pi$
$431$	2.92094e6	0.757408	0.378704	+	0.925518i	$0.376370\pi$
$432$	0	0
$433$	−370204.	−0.0948902	−0.0474451	−	0.998874i	$-0.515108\pi$
$433$	−370204.	−0.0948902	−0.0474451	+	0.998874i	$0.515108\pi$
$434$	0	0
$435$	−4.67774e6	−1.18526
$436$	0	0
$437$	−7.56352e6	−1.89461
$438$	0	0
$439$	265105.	0.0656534	0.0328267	−	0.999461i	$-0.489549\pi$
$439$	265105.	0.0656534	0.0328267	+	0.999461i	$0.489549\pi$
$440$	0	0
$441$	1.01292e6	0.248016
$442$	0	0
$443$	3.90579e6	0.945583	0.472792	−	0.881174i	$-0.343246\pi$
$443$	3.90579e6	0.945583	0.472792	+	0.881174i	$0.343246\pi$
$444$	0	0
$445$	−1.49163e6	−0.357076
$446$	0	0
$447$	111093.	0.0262976
$448$	0	0
$449$	−74062.6	−0.0173374	−0.00866869	−	0.999962i	$-0.502759\pi$
$449$	−74062.6	−0.0173374	−0.00866869	+	0.999962i	$0.502759\pi$
$450$	0	0
$451$	4.66099e6	1.07904
$452$	0	0
$453$	−2.80651e6	−0.642571
$454$	0	0
$455$	1.51051e6	0.342054
$456$	0	0
$457$	3.01112e6	0.674430	0.337215	−	0.941428i	$-0.390515\pi$
$457$	3.01112e6	0.674430	0.337215	+	0.941428i	$0.390515\pi$
$458$	0	0
$459$	1.09520e6	0.242639
$460$	0	0
$461$	6.54908e6	1.43525	0.717625	−	0.696429i	$-0.245229\pi$
$461$	6.54908e6	1.43525	0.717625	+	0.696429i	$0.245229\pi$
$462$	0	0
$463$	4.90007e6	1.06231	0.531153	−	0.847276i	$-0.321759\pi$
$463$	4.90007e6	1.06231	0.531153	+	0.847276i	$0.321759\pi$
$464$	0	0
$465$	−6.68252e6	−1.43320
$466$	0	0
$467$	4.83001e6	1.02484	0.512420	−	0.858735i	$-0.328749\pi$
$467$	4.83001e6	1.02484	0.512420	+	0.858735i	$0.328749\pi$
$468$	0	0
$469$	−516958.	−0.108523
$470$	0	0
$471$	9.39203e6	1.95077
$472$	0	0
$473$	−1.75068e7	−3.59794
$474$	0	0
$475$	−3.43556e6	−0.698655
$476$	0	0
$477$	4.26922e6	0.859118
$478$	0	0
$479$	711695.	0.141728	0.0708639	−	0.997486i	$-0.477424\pi$
$479$	711695.	0.141728	0.0708639	+	0.997486i	$0.477424\pi$
$480$	0	0
$481$	−604713.	−0.119175
$482$	0	0
$483$	−5.38688e6	−1.05068
$484$	0	0
$485$	−2.98033e6	−0.575321
$486$	0	0
$487$	−1.01760e7	−1.94426	−0.972130	−	0.234442i	$-0.924674\pi$
$487$	−1.01760e7	−1.94426	−0.972130	+	0.234442i	$0.924674\pi$
$488$	0	0
$489$	−7.96325e6	−1.50598
$490$	0	0
$491$	−4.18310e6	−0.783060	−0.391530	−	0.920165i	$-0.628054\pi$
$491$	−4.18310e6	−0.783060	−0.391530	+	0.920165i	$0.628054\pi$
$492$	0	0
$493$	−1.24956e6	−0.231547
$494$	0	0
$495$	1.13975e7	2.09072
$496$	0	0
$497$	1.59871e6	0.290321
$498$	0	0
$499$	2.12165e6	0.381436	0.190718	−	0.981645i	$-0.438918\pi$
$499$	2.12165e6	0.381436	0.190718	+	0.981645i	$0.438918\pi$
$500$	0	0
$501$	838943.	0.149327
$502$	0	0
$503$	−6.38603e6	−1.12541	−0.562705	−	0.826658i	$-0.690239\pi$
$503$	−6.38603e6	−1.12541	−0.562705	+	0.826658i	$0.690239\pi$
$504$	0	0
$505$	−1.32163e6	−0.230613
$506$	0	0
$507$	1.10451e7	1.90831
$508$	0	0
$509$	702103.	0.120118	0.0600588	−	0.998195i	$-0.480871\pi$
$509$	702103.	0.120118	0.0600588	+	0.998195i	$0.480871\pi$
$510$	0	0
$511$	1.69724e6	0.287535
$512$	0	0
$513$	8.18231e6	1.37272
$514$	0	0
$515$	1.86942e6	0.310591
$516$	0	0
$517$	−9.14364e6	−1.50450
$518$	0	0
$519$	1.86397e6	0.303752
$520$	0	0
$521$	9.62033e6	1.55273	0.776365	−	0.630284i	$-0.217061\pi$
$521$	9.62033e6	1.55273	0.776365	+	0.630284i	$0.217061\pi$
$522$	0	0
$523$	9.71019e6	1.55229	0.776146	−	0.630553i	$-0.217172\pi$
$523$	9.71019e6	1.55229	0.776146	+	0.630553i	$0.217172\pi$
$524$	0	0
$525$	−2.44686e6	−0.387446
$526$	0	0
$527$	−1.78509e6	−0.279985
$528$	0	0
$529$	1.17415e7	1.82424
$530$	0	0
$531$	9.30546e6	1.43219
$532$	0	0
$533$	5.31841e6	0.810893
$534$	0	0
$535$	−1.24832e6	−0.188557
$536$	0	0
$537$	−1.84558e7	−2.76183
$538$	0	0
$539$	1.88164e6	0.278975
$540$	0	0
$541$	−1.14556e7	−1.68277	−0.841385	−	0.540436i	$-0.818259\pi$
$541$	−1.14556e7	−1.68277	−0.841385	+	0.540436i	$0.818259\pi$
$542$	0	0
$543$	2.79354e6	0.406589
$544$	0	0
$545$	−5.79280e6	−0.835405
$546$	0	0
$547$	−5.22923e6	−0.747256	−0.373628	−	0.927579i	$-0.621886\pi$
$547$	−5.22923e6	−0.747256	−0.373628	+	0.927579i	$0.621886\pi$
$548$	0	0
$549$	1.90758e6	0.270117
$550$	0	0
$551$	−9.33556e6	−1.30997
$552$	0	0
$553$	953231.	0.132552
$554$	0	0
$555$	−601105.	−0.0828357
$556$	0	0
$557$	−5.01059e6	−0.684306	−0.342153	−	0.939644i	$-0.611156\pi$
$557$	−5.01059e6	−0.684306	−0.342153	+	0.939644i	$0.611156\pi$
$558$	0	0
$559$	−1.99760e7	−2.70383
$560$	0	0
$561$	4.79828e6	0.643692
$562$	0	0
$563$	6.57927e6	0.874796	0.437398	−	0.899268i	$-0.355900\pi$
$563$	6.57927e6	0.874796	0.437398	+	0.899268i	$0.355900\pi$
$564$	0	0
$565$	−4.23911e6	−0.558668
$566$	0	0
$567$	804308.	0.105067
$568$	0	0
$569$	578247.	0.0748743	0.0374372	−	0.999299i	$-0.488081\pi$
$569$	578247.	0.0748743	0.0374372	+	0.999299i	$0.488081\pi$
$570$	0	0
$571$	1.95159e6	0.250495	0.125247	−	0.992126i	$-0.460028\pi$
$571$	1.95159e6	0.250495	0.125247	+	0.992126i	$0.460028\pi$
$572$	0	0
$573$	−9.25053e6	−1.17701
$574$	0	0
$575$	8.25685e6	1.04146
$576$	0	0
$577$	−1.64277e6	−0.205418	−0.102709	−	0.994711i	$-0.532751\pi$
$577$	−1.64277e6	−0.205418	−0.102709	+	0.994711i	$0.532751\pi$
$578$	0	0
$579$	−1.21578e7	−1.50716
$580$	0	0
$581$	3.25888e6	0.400523
$582$	0	0
$583$	7.93065e6	0.966357
$584$	0	0
$585$	1.30050e7	1.57116
$586$	0	0
$587$	1.22641e7	1.46907	0.734534	−	0.678572i	$-0.237401\pi$
$587$	1.22641e7	1.46907	0.734534	+	0.678572i	$0.237401\pi$
$588$	0	0
$589$	−1.33366e7	−1.58401
$590$	0	0
$591$	1.75287e7	2.06434
$592$	0	0
$593$	−1.42193e7	−1.66051	−0.830256	−	0.557383i	$-0.811806\pi$
$593$	−1.42193e7	−1.66051	−0.830256	+	0.557383i	$0.811806\pi$
$594$	0	0
$595$	401092.	0.0464464
$596$	0	0
$597$	−1.09265e6	−0.125472
$598$	0	0
$599$	827193.	0.0941976	0.0470988	−	0.998890i	$-0.485002\pi$
$599$	827193.	0.0941976	0.0470988	+	0.998890i	$0.485002\pi$
$600$	0	0
$601$	1.04128e7	1.17593	0.587965	−	0.808886i	$-0.299929\pi$
$601$	1.04128e7	1.17593	0.587965	+	0.808886i	$0.299929\pi$
$602$	0	0
$603$	−4.45086e6	−0.498484
$604$	0	0
$605$	1.56204e7	1.73502
$606$	0	0
$607$	6.16297e6	0.678920	0.339460	−	0.940620i	$-0.389756\pi$
$607$	6.16297e6	0.678920	0.339460	+	0.940620i	$0.389756\pi$
$608$	0	0
$609$	−6.64895e6	−0.726457
$610$	0	0
$611$	−1.04333e7	−1.13063
$612$	0	0
$613$	3.83024e6	0.411694	0.205847	−	0.978584i	$-0.434005\pi$
$613$	3.83024e6	0.411694	0.205847	+	0.978584i	$0.434005\pi$
$614$	0	0
$615$	5.28667e6	0.563630
$616$	0	0
$617$	−6.11879e6	−0.647072	−0.323536	−	0.946216i	$-0.604872\pi$
$617$	−6.11879e6	−0.647072	−0.323536	+	0.946216i	$0.604872\pi$
$618$	0	0
$619$	−9.04795e6	−0.949125	−0.474563	−	0.880222i	$-0.657394\pi$
$619$	−9.04795e6	−0.949125	−0.474563	+	0.880222i	$0.657394\pi$
$620$	0	0
$621$	−1.96650e7	−2.04627
$622$	0	0
$623$	−2.12021e6	−0.218856
$624$	0	0
$625$	36780.5	0.00376632
$626$	0	0
$627$	3.58483e7	3.64167
$628$	0	0
$629$	−160573.	−0.0161825
$630$	0	0
$631$	−752447.	−0.0752320	−0.0376160	−	0.999292i	$-0.511976\pi$
$631$	−752447.	−0.0752320	−0.0376160	+	0.999292i	$0.511976\pi$
$632$	0	0
$633$	−3.67658e6	−0.364699
$634$	0	0
$635$	7.70115e6	0.757917
$636$	0	0
$637$	2.14704e6	0.209648
$638$	0	0
$639$	1.37644e7	1.33354
$640$	0	0
$641$	−2.57777e6	−0.247799	−0.123899	−	0.992295i	$-0.539540\pi$
$641$	−2.57777e6	−0.247799	−0.123899	+	0.992295i	$0.539540\pi$
$642$	0	0
$643$	−1.42710e7	−1.36121	−0.680606	−	0.732650i	$-0.738283\pi$
$643$	−1.42710e7	−1.36121	−0.680606	+	0.732650i	$0.738283\pi$
$644$	0	0
$645$	−1.98568e7	−1.87936
$646$	0	0
$647$	1.19672e7	1.12391	0.561957	−	0.827166i	$-0.310048\pi$
$647$	1.19672e7	1.12391	0.561957	+	0.827166i	$0.310048\pi$
$648$	0	0
$649$	1.72861e7	1.61097
$650$	0	0
$651$	−9.49856e6	−0.878426
$652$	0	0
$653$	−9.80818e6	−0.900131	−0.450065	−	0.892996i	$-0.648599\pi$
$653$	−9.80818e6	−0.900131	−0.450065	+	0.892996i	$0.648599\pi$
$654$	0	0
$655$	942526.	0.0858401
$656$	0	0
$657$	1.46127e7	1.32074
$658$	0	0
$659$	−9.11613e6	−0.817705	−0.408853	−	0.912600i	$-0.634071\pi$
$659$	−9.11613e6	−0.817705	−0.408853	+	0.912600i	$0.634071\pi$
$660$	0	0
$661$	5.54645e6	0.493755	0.246877	−	0.969047i	$-0.420595\pi$
$661$	5.54645e6	0.493755	0.246877	+	0.969047i	$0.420595\pi$
$662$	0	0
$663$	5.47505e6	0.483732
$664$	0	0
$665$	2.99659e6	0.262769
$666$	0	0
$667$	2.24366e7	1.95273
$668$	0	0
$669$	3.60046e7	3.11024
$670$	0	0
$671$	3.54358e6	0.303834
$672$	0	0
$673$	−2.78690e6	−0.237183	−0.118592	−	0.992943i	$-0.537838\pi$
$673$	−2.78690e6	−0.237183	−0.118592	+	0.992943i	$0.537838\pi$
$674$	0	0
$675$	−8.93235e6	−0.754582
$676$	0	0
$677$	8.98428e6	0.753376	0.376688	−	0.926340i	$-0.377063\pi$
$677$	8.98428e6	0.753376	0.376688	+	0.926340i	$0.377063\pi$
$678$	0	0
$679$	−4.23626e6	−0.352621
$680$	0	0
$681$	3.46792e7	2.86551
$682$	0	0
$683$	7.51256e6	0.616221	0.308110	−	0.951351i	$-0.400303\pi$
$683$	7.51256e6	0.616221	0.308110	+	0.951351i	$0.400303\pi$
$684$	0	0
$685$	−1.49149e7	−1.21449
$686$	0	0
$687$	−1.04314e7	−0.843240
$688$	0	0
$689$	9.04924e6	0.726213
$690$	0	0
$691$	−1.55185e7	−1.23639	−0.618193	−	0.786027i	$-0.712135\pi$
$691$	−1.55185e7	−1.23639	−0.618193	+	0.786027i	$0.712135\pi$
$692$	0	0
$693$	1.62004e7	1.28142
$694$	0	0
$695$	1.12194e7	0.881063
$696$	0	0
$697$	1.41222e6	0.110109
$698$	0	0
$699$	1.45834e7	1.12893
$700$	0	0
$701$	−1.50186e7	−1.15434	−0.577169	−	0.816624i	$-0.695843\pi$
$701$	−1.50186e7	−1.15434	−0.577169	+	0.816624i	$0.695843\pi$
$702$	0	0
$703$	−1.19965e6	−0.0915517
$704$	0	0
$705$	−1.03710e7	−0.785868
$706$	0	0
$707$	−1.87858e6	−0.141345
$708$	0	0
$709$	−1.92459e7	−1.43788	−0.718939	−	0.695073i	$-0.755372\pi$
$709$	−1.92459e7	−1.43788	−0.718939	+	0.695073i	$0.755372\pi$
$710$	0	0
$711$	8.20704e6	0.608853
$712$	0	0
$713$	3.20525e7	2.36123
$714$	0	0
$715$	2.41586e7	1.76729
$716$	0	0
$717$	1.14909e7	0.834751
$718$	0	0
$719$	1.83615e7	1.32461	0.662303	−	0.749236i	$-0.269579\pi$
$719$	1.83615e7	1.32461	0.662303	+	0.749236i	$0.269579\pi$
$720$	0	0
$721$	2.65720e6	0.190365
$722$	0	0
$723$	−2.97711e7	−2.11812
$724$	0	0
$725$	1.01913e7	0.720088
$726$	0	0
$727$	−2.02012e7	−1.41756	−0.708779	−	0.705430i	$-0.750754\pi$
$727$	−2.02012e7	−1.41756	−0.708779	+	0.705430i	$0.750754\pi$
$728$	0	0
$729$	−2.19752e7	−1.53149
$730$	0	0
$731$	−5.30434e6	−0.367145
$732$	0	0
$733$	4.45403e6	0.306192	0.153096	−	0.988211i	$-0.451076\pi$
$733$	4.45403e6	0.306192	0.153096	+	0.988211i	$0.451076\pi$
$734$	0	0
$735$	2.13423e6	0.145721
$736$	0	0
$737$	−8.26808e6	−0.560707
$738$	0	0
$739$	5.62011e6	0.378559	0.189280	−	0.981923i	$-0.439385\pi$
$739$	5.62011e6	0.378559	0.189280	+	0.981923i	$0.439385\pi$
$740$	0	0
$741$	4.09046e7	2.73670
$742$	0	0
$743$	−4.93468e6	−0.327934	−0.163967	−	0.986466i	$-0.552429\pi$
$743$	−4.93468e6	−0.327934	−0.163967	+	0.986466i	$0.552429\pi$
$744$	0	0
$745$	148523.	0.00980398
$746$	0	0
$747$	2.80580e7	1.83973
$748$	0	0
$749$	−1.77437e6	−0.115568
$750$	0	0
$751$	−2.07492e6	−0.134246	−0.0671229	−	0.997745i	$-0.521382\pi$
$751$	−2.07492e6	−0.134246	−0.0671229	+	0.997745i	$0.521382\pi$
$752$	0	0
$753$	2.61956e6	0.168361
$754$	0	0
$755$	−3.75210e6	−0.239556
$756$	0	0
$757$	−1.35911e7	−0.862014	−0.431007	−	0.902349i	$-0.641842\pi$
$757$	−1.35911e7	−0.862014	−0.431007	+	0.902349i	$0.641842\pi$
$758$	0	0
$759$	−8.61561e7	−5.42852
$760$	0	0
$761$	−2.17441e7	−1.36107	−0.680533	−	0.732718i	$-0.738252\pi$
$761$	−2.17441e7	−1.36107	−0.680533	+	0.732718i	$0.738252\pi$
$762$	0	0
$763$	−8.23391e6	−0.512029
$764$	0	0
$765$	3.45329e6	0.213344
$766$	0	0
$767$	1.97243e7	1.21063
$768$	0	0
$769$	−1.62731e7	−0.992327	−0.496163	−	0.868229i	$-0.665258\pi$
$769$	−1.62731e7	−0.992327	−0.496163	+	0.868229i	$0.665258\pi$
$770$	0	0
$771$	−1.60982e7	−0.975309
$772$	0	0
$773$	3.31413e7	1.99490	0.997448	−	0.0713922i	$-0.0227442\pi$
$773$	3.31413e7	1.99490	0.997448	+	0.0713922i	$0.0227442\pi$
$774$	0	0
$775$	1.45591e7	0.870724
$776$	0	0
$777$	−854413.	−0.0507709
$778$	0	0
$779$	1.05508e7	0.622936
$780$	0	0
$781$	2.55693e7	1.50000
$782$	0	0
$783$	−2.42722e7	−1.41483
$784$	0	0
$785$	1.25565e7	0.727266
$786$	0	0
$787$	−2.03844e7	−1.17317	−0.586585	−	0.809888i	$-0.699528\pi$
$787$	−2.03844e7	−1.17317	−0.586585	+	0.809888i	$0.699528\pi$
$788$	0	0
$789$	−2.62091e7	−1.49886
$790$	0	0
$791$	−6.02549e6	−0.342414
$792$	0	0
$793$	4.04339e6	0.228330
$794$	0	0
$795$	8.99523e6	0.504771
$796$	0	0
$797$	−2.20439e7	−1.22925	−0.614627	−	0.788818i	$-0.710693\pi$
$797$	−2.20439e7	−1.22925	−0.614627	+	0.788818i	$0.710693\pi$
$798$	0	0
$799$	−2.77041e6	−0.153524
$800$	0	0
$801$	−1.82544e7	−1.00528
$802$	0	0
$803$	2.71451e7	1.48560
$804$	0	0
$805$	−7.20187e6	−0.391702
$806$	0	0
$807$	−4.00794e7	−2.16640
$808$	0	0
$809$	1.43352e7	0.770073	0.385036	−	0.922901i	$-0.374189\pi$
$809$	1.43352e7	0.770073	0.385036	+	0.922901i	$0.374189\pi$
$810$	0	0
$811$	−3.11761e7	−1.66445	−0.832223	−	0.554441i	$-0.812932\pi$
$811$	−3.11761e7	−1.66445	−0.832223	+	0.554441i	$0.812932\pi$
$812$	0	0
$813$	1.40177e7	0.743791
$814$	0	0
$815$	−1.06463e7	−0.561441
$816$	0	0
$817$	−3.96292e7	−2.07711
$818$	0	0
$819$	1.84854e7	0.962984
$820$	0	0
$821$	2.59310e7	1.34265	0.671323	−	0.741165i	$-0.265726\pi$
$821$	2.59310e7	1.34265	0.671323	+	0.741165i	$0.265726\pi$
$822$	0	0
$823$	1.79958e7	0.926131	0.463065	−	0.886324i	$-0.346749\pi$
$823$	1.79958e7	0.926131	0.463065	+	0.886324i	$0.346749\pi$
$824$	0	0
$825$	−3.91344e7	−2.00182
$826$	0	0
$827$	−1.92432e7	−0.978395	−0.489198	−	0.872173i	$-0.662710\pi$
$827$	−1.92432e7	−0.978395	−0.489198	+	0.872173i	$0.662710\pi$
$828$	0	0
$829$	−3.66154e7	−1.85045	−0.925226	−	0.379416i	$-0.876125\pi$
$829$	−3.66154e7	−1.85045	−0.925226	+	0.379416i	$0.876125\pi$
$830$	0	0
$831$	−2.43750e7	−1.22445
$832$	0	0
$833$	570115.	0.0284675
$834$	0	0
$835$	1.12161e6	0.0556704
$836$	0	0
$837$	−3.46748e7	−1.71080
$838$	0	0
$839$	2.94963e7	1.44665	0.723323	−	0.690510i	$-0.242614\pi$
$839$	2.94963e7	1.44665	0.723323	+	0.690510i	$0.242614\pi$
$840$	0	0
$841$	7.18210e6	0.350156
$842$	0	0
$843$	−4.10918e7	−1.99153
$844$	0	0
$845$	1.47665e7	0.711436
$846$	0	0
$847$	2.22030e7	1.06341
$848$	0	0
$849$	2.76265e7	1.31540
$850$	0	0
$851$	2.88318e6	0.136473
$852$	0	0
$853$	3.48820e6	0.164146	0.0820728	−	0.996626i	$-0.473846\pi$
$853$	3.48820e6	0.164146	0.0820728	+	0.996626i	$0.473846\pi$
$854$	0	0
$855$	2.57998e7	1.20698
$856$	0	0
$857$	−1.95738e7	−0.910383	−0.455191	−	0.890394i	$-0.650429\pi$
$857$	−1.95738e7	−0.910383	−0.455191	+	0.890394i	$0.650429\pi$
$858$	0	0
$859$	−2.89258e7	−1.33753	−0.668763	−	0.743476i	$-0.733176\pi$
$859$	−2.89258e7	−1.33753	−0.668763	+	0.743476i	$0.733176\pi$
$860$	0	0
$861$	7.51449e6	0.345455
$862$	0	0
$863$	−5.04644e6	−0.230652	−0.115326	−	0.993328i	$-0.536791\pi$
$863$	−5.04644e6	−0.230652	−0.115326	+	0.993328i	$0.536791\pi$
$864$	0	0
$865$	2.49199e6	0.113242
$866$	0	0
$867$	−3.51575e7	−1.58844
$868$	0	0
$869$	1.52457e7	0.684853
$870$	0	0
$871$	−9.43425e6	−0.421369
$872$	0	0
$873$	−3.64730e7	−1.61970
$874$	0	0
$875$	−8.54995e6	−0.377523
$876$	0	0
$877$	4.26312e6	0.187167	0.0935834	−	0.995611i	$-0.470168\pi$
$877$	4.26312e6	0.187167	0.0935834	+	0.995611i	$0.470168\pi$
$878$	0	0
$879$	4.35603e7	1.90160
$880$	0	0
$881$	−6.69047e6	−0.290414	−0.145207	−	0.989401i	$-0.546385\pi$
$881$	−6.69047e6	−0.290414	−0.145207	+	0.989401i	$0.546385\pi$
$882$	0	0
$883$	−1.04361e7	−0.450439	−0.225220	−	0.974308i	$-0.572310\pi$
$883$	−1.04361e7	−0.450439	−0.225220	+	0.974308i	$0.572310\pi$
$884$	0	0
$885$	1.96066e7	0.841480
$886$	0	0
$887$	1.00763e7	0.430022	0.215011	−	0.976612i	$-0.431021\pi$
$887$	1.00763e7	0.430022	0.215011	+	0.976612i	$0.431021\pi$
$888$	0	0
$889$	1.09465e7	0.464536
$890$	0	0
$891$	1.28639e7	0.542847
$892$	0	0
$893$	−2.06979e7	−0.868558
$894$	0	0
$895$	−2.46740e7	−1.02963
$896$	0	0
$897$	−9.83081e7	−4.07951
$898$	0	0
$899$	3.95620e7	1.63260
$900$	0	0
$901$	2.40289e6	0.0986102
$902$	0	0
$903$	−2.82246e7	−1.15188
$904$	0	0
$905$	3.73476e6	0.151580
$906$	0	0
$907$	−3.11368e7	−1.25677	−0.628384	−	0.777903i	$-0.716283\pi$
$907$	−3.11368e7	−1.25677	−0.628384	+	0.777903i	$0.716283\pi$
$908$	0	0
$909$	−1.61740e7	−0.649244
$910$	0	0
$911$	−3.70378e6	−0.147860	−0.0739298	−	0.997263i	$-0.523554\pi$
$911$	−3.70378e6	−0.147860	−0.0739298	+	0.997263i	$0.523554\pi$
$912$	0	0
$913$	5.21215e7	2.06938
$914$	0	0
$915$	4.01926e6	0.158706
$916$	0	0
$917$	1.33971e6	0.0526123
$918$	0	0
$919$	3.03563e7	1.18566	0.592830	−	0.805328i	$-0.298011\pi$
$919$	3.03563e7	1.18566	0.592830	+	0.805328i	$0.298011\pi$
$920$	0	0
$921$	1.01768e7	0.395332
$922$	0	0
$923$	2.91757e7	1.12724
$924$	0	0
$925$	1.30962e6	0.0503258
$926$	0	0
$927$	2.28778e7	0.874408
$928$	0	0
$929$	−1.79519e6	−0.0682449	−0.0341225	−	0.999418i	$-0.510864\pi$
$929$	−1.79519e6	−0.0682449	−0.0341225	+	0.999418i	$0.510864\pi$
$930$	0	0
$931$	4.25937e6	0.161054
$932$	0	0
$933$	6.38066e7	2.39972
$934$	0	0
$935$	6.41495e6	0.239974
$936$	0	0
$937$	2.48124e7	0.923250	0.461625	−	0.887075i	$-0.347267\pi$
$937$	2.48124e7	0.923250	0.461625	+	0.887075i	$0.347267\pi$
$938$	0	0
$939$	1.12030e7	0.414638
$940$	0	0
$941$	−3.95832e7	−1.45726	−0.728629	−	0.684909i	$-0.759842\pi$
$941$	−3.95832e7	−1.45726	−0.728629	+	0.684909i	$0.759842\pi$
$942$	0	0
$943$	−2.53574e7	−0.928592
$944$	0	0
$945$	7.79106e6	0.283803
$946$	0	0
$947$	−5.65582e6	−0.204937	−0.102469	−	0.994736i	$-0.532674\pi$
$947$	−5.65582e6	−0.204937	−0.102469	+	0.994736i	$0.532674\pi$
$948$	0	0
$949$	3.09738e7	1.11642
$950$	0	0
$951$	−8.52444e6	−0.305643
$952$	0	0
$953$	1.34828e7	0.480891	0.240445	−	0.970663i	$-0.422706\pi$
$953$	1.34828e7	0.480891	0.240445	+	0.970663i	$0.422706\pi$
$954$	0	0
$955$	−1.23673e7	−0.438800
$956$	0	0
$957$	−1.06341e8	−3.75338
$958$	0	0
$959$	−2.12001e7	−0.744375
$960$	0	0
$961$	2.78883e7	0.974123
$962$	0	0
$963$	−1.52768e7	−0.530844
$964$	0	0
$965$	−1.62541e7	−0.561882
$966$	0	0
$967$	−4.14135e7	−1.42422	−0.712108	−	0.702070i	$-0.752260\pi$
$967$	−4.14135e7	−1.42422	−0.712108	+	0.702070i	$0.752260\pi$
$968$	0	0
$969$	1.08616e7	0.371607
$970$	0	0
$971$	−2.05618e7	−0.699863	−0.349931	−	0.936775i	$-0.613795\pi$
$971$	−2.05618e7	−0.699863	−0.349931	+	0.936775i	$0.613795\pi$
$972$	0	0
$973$	1.59473e7	0.540014
$974$	0	0
$975$	−4.46542e7	−1.50436
$976$	0	0
$977$	−2.96209e7	−0.992800	−0.496400	−	0.868094i	$-0.665345\pi$
$977$	−2.96209e7	−0.992800	−0.496400	+	0.868094i	$0.665345\pi$
$978$	0	0
$979$	−3.39100e7	−1.13076
$980$	0	0
$981$	−7.08916e7	−2.35192
$982$	0	0
$983$	2.86906e7	0.947014	0.473507	−	0.880790i	$-0.342988\pi$
$983$	2.86906e7	0.947014	0.473507	+	0.880790i	$0.342988\pi$
$984$	0	0
$985$	2.34346e7	0.769603
$986$	0	0
$987$	−1.47414e7	−0.481667
$988$	0	0
$989$	9.52428e7	3.09629
$990$	0	0
$991$	1.09947e7	0.355630	0.177815	−	0.984064i	$-0.443097\pi$
$991$	1.09947e7	0.355630	0.177815	+	0.984064i	$0.443097\pi$
$992$	0	0
$993$	−8.49723e7	−2.73467
$994$	0	0
$995$	−1.46080e6	−0.0467769
$996$	0	0
$997$	−3.56162e7	−1.13477	−0.567387	−	0.823451i	$-0.692046\pi$
$997$	−3.56162e7	−1.13477	−0.567387	+	0.823451i	$0.692046\pi$
$998$	0	0
$999$	−3.11906e6	−0.0988803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.bf.1.5		5
4.3	odd	2		448.6.a.be.1.1		5
8.3	odd	2		224.6.a.j.1.5	yes	5
8.5	even	2		224.6.a.i.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.i.1.1	✓	5	8.5	even	2
224.6.a.j.1.5	yes	5	8.3	odd	2
448.6.a.be.1.1		5	4.3	odd	2
448.6.a.bf.1.5		5	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)

\(f(q)\)	\(=\)	\(q+25.7852 q^{3} +34.4729 q^{5} +49.0000 q^{7} +421.876 q^{9} +O(q^{10})\)	\(q+25.7852 q^{3} +34.4729 q^{5} +49.0000 q^{7} +421.876 q^{9} +783.692 q^{11} +894.228 q^{13} +888.891 q^{15} +237.449 q^{17} +1774.00 q^{19} +1263.47 q^{21} -4263.54 q^{23} -1936.62 q^{25} +4612.35 q^{27} -5262.44 q^{29} -7517.81 q^{31} +20207.6 q^{33} +1689.17 q^{35} -676.241 q^{37} +23057.8 q^{39} +5947.49 q^{41} -22338.9 q^{43} +14543.3 q^{45} -11667.4 q^{47} +2401.00 q^{49} +6122.66 q^{51} +10119.6 q^{53} +27016.2 q^{55} +45742.9 q^{57} +22057.3 q^{59} +4521.65 q^{61} +20671.9 q^{63} +30826.7 q^{65} -10550.2 q^{67} -109936. q^{69} +32626.7 q^{71} +34637.5 q^{73} -49936.0 q^{75} +38400.9 q^{77} +19453.7 q^{79} +16414.5 q^{81} +66507.7 q^{83} +8185.56 q^{85} -135693. q^{87} -43269.5 q^{89} +43817.2 q^{91} -193848. q^{93} +61155.0 q^{95} -86454.2 q^{97} +330621. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 10 q^{3} - 36 q^{5} + 245 q^{7} + 637 q^{9}+O(q^{10}) \) 5 * q + 10 * q^3 - 36 * q^5 + 245 * q^7 + 637 * q^9	\( 5 q + 10 q^{3} - 36 q^{5} + 245 q^{7} + 637 q^{9} - 116 q^{11} - 40 q^{13} + 16 q^{15} - 402 q^{17} + 3582 q^{19} + 490 q^{21} + 472 q^{23} + 9615 q^{25} - 356 q^{27} - 4754 q^{29} - 10500 q^{31} + 15864 q^{33} - 1764 q^{35} - 19642 q^{37} + 10872 q^{39} + 23398 q^{41} - 22044 q^{43} - 49476 q^{45} + 16004 q^{47} + 12005 q^{49} + 45676 q^{51} - 54246 q^{53} + 53456 q^{55} + 109556 q^{57} + 74366 q^{59} - 68316 q^{61} + 31213 q^{63} + 152568 q^{65} + 26560 q^{67} - 214720 q^{69} + 93072 q^{71} + 136098 q^{73} + 124510 q^{75} - 5684 q^{77} + 96080 q^{79} + 104801 q^{81} + 145894 q^{83} - 117352 q^{85} + 168876 q^{87} + 188554 q^{89} - 1960 q^{91} - 86296 q^{93} + 74736 q^{95} - 88146 q^{97} + 260236 q^{99}+O(q^{100}) \) 5 * q + 10 * q^3 - 36 * q^5 + 245 * q^7 + 637 * q^9 - 116 * q^11 - 40 * q^13 + 16 * q^15 - 402 * q^17 + 3582 * q^19 + 490 * q^21 + 472 * q^23 + 9615 * q^25 - 356 * q^27 - 4754 * q^29 - 10500 * q^31 + 15864 * q^33 - 1764 * q^35 - 19642 * q^37 + 10872 * q^39 + 23398 * q^41 - 22044 * q^43 - 49476 * q^45 + 16004 * q^47 + 12005 * q^49 + 45676 * q^51 - 54246 * q^53 + 53456 * q^55 + 109556 * q^57 + 74366 * q^59 - 68316 * q^61 + 31213 * q^63 + 152568 * q^65 + 26560 * q^67 - 214720 * q^69 + 93072 * q^71 + 136098 * q^73 + 124510 * q^75 - 5684 * q^77 + 96080 * q^79 + 104801 * q^81 + 145894 * q^83 - 117352 * q^85 + 168876 * q^87 + 188554 * q^89 - 1960 * q^91 - 86296 * q^93 + 74736 * q^95 - 88146 * q^97 + 260236 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more