LMFDB - Embedded newform 650.6.b.a.599.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [650,6,Mod(599,650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("650.599");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 650 = 2 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	650.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$104.249482878$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			599.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	650.599
Dual form			650.6.b.a.599.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000i q^{2} -16.0000 q^{4} -170.000i q^{7} +64.0000i q^{8} +243.000 q^{9} +O(q^{10})$	$q-4.00000i q^{2} -16.0000 q^{4} -170.000i q^{7} +64.0000i q^{8} +243.000 q^{9} -250.000 q^{11} +169.000i q^{13} -680.000 q^{14} +256.000 q^{16} +1062.00i q^{17} -972.000i q^{18} +78.0000 q^{19} +1000.00i q^{22} -1576.00i q^{23} +676.000 q^{26} +2720.00i q^{28} -2578.00 q^{29} -8654.00 q^{31} -1024.00i q^{32} +4248.00 q^{34} -3888.00 q^{36} +10986.0i q^{37} -312.000i q^{38} +1050.00 q^{41} +5900.00i q^{43} +4000.00 q^{44} -6304.00 q^{46} -5962.00i q^{47} -12093.0 q^{49} -2704.00i q^{52} -29046.0i q^{53} +10880.0 q^{56} +10312.0i q^{58} +13922.0 q^{59} -32882.0 q^{61} +34616.0i q^{62} -41310.0i q^{63} -4096.00 q^{64} -69566.0i q^{67} -16992.0i q^{68} -50542.0 q^{71} +15552.0i q^{72} +46750.0i q^{73} +43944.0 q^{74} -1248.00 q^{76} +42500.0i q^{77} +19348.0 q^{79} +59049.0 q^{81} -4200.00i q^{82} +87438.0i q^{83} +23600.0 q^{86} -16000.0i q^{88} -94170.0 q^{89} +28730.0 q^{91} +25216.0i q^{92} -23848.0 q^{94} +182786. i q^{97} +48372.0i q^{98} -60750.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{4} + 486 q^{9}+O(q^{10}) $ 2 * q - 32 * q^4 + 486 * q^9	$ 2 q - 32 q^{4} + 486 q^{9} - 500 q^{11} - 1360 q^{14} + 512 q^{16} + 156 q^{19} + 1352 q^{26} - 5156 q^{29} - 17308 q^{31} + 8496 q^{34} - 7776 q^{36} + 2100 q^{41} + 8000 q^{44} - 12608 q^{46} - 24186 q^{49} + 21760 q^{56} + 27844 q^{59} - 65764 q^{61} - 8192 q^{64} - 101084 q^{71} + 87888 q^{74} - 2496 q^{76} + 38696 q^{79} + 118098 q^{81} + 47200 q^{86} - 188340 q^{89} + 57460 q^{91} - 47696 q^{94} - 121500 q^{99}+O(q^{100}) $ 2 * q - 32 * q^4 + 486 * q^9 - 500 * q^11 - 1360 * q^14 + 512 * q^16 + 156 * q^19 + 1352 * q^26 - 5156 * q^29 - 17308 * q^31 + 8496 * q^34 - 7776 * q^36 + 2100 * q^41 + 8000 * q^44 - 12608 * q^46 - 24186 * q^49 + 21760 * q^56 + 27844 * q^59 - 65764 * q^61 - 8192 * q^64 - 101084 * q^71 + 87888 * q^74 - 2496 * q^76 + 38696 * q^79 + 118098 * q^81 + 47200 * q^86 - 188340 * q^89 + 57460 * q^91 - 47696 * q^94 - 121500 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/650\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 4.00000i	− 0.707107i
$2$	− 4.00000i	− 0.707107i
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	−16.0000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 170.000i	− 1.31131i	−0.755063	−	0.655653i	$-0.772394\pi$
$7$	− 170.000i	− 1.31131i	0.755063	−	0.655653i	$-0.227606\pi$
$8$	64.0000i	0.353553i
$9$	243.000	1.00000
$10$	0	0
$11$	−250.000	−0.622957	−0.311479	−	0.950253i	$-0.600824\pi$
$11$	−250.000	−0.622957	−0.311479	+	0.950253i	$0.600824\pi$
$12$	0	0
$13$	169.000i	0.277350i
$13$	169.000i	0.277350i
$14$	−680.000	−0.927233
$15$	0	0
$16$	256.000	0.250000
$17$	1062.00i	0.891255i	0.895218	+	0.445628i	$0.147019\pi$
$17$	1062.00i	0.891255i	−0.895218	+	0.445628i	$0.852981\pi$
$18$	− 972.000i	− 0.707107i
$19$	78.0000	0.0495691	0.0247845	−	0.999693i	$-0.492110\pi$
$19$	78.0000	0.0495691	0.0247845	+	0.999693i	$0.492110\pi$
$20$	0	0
$21$	0	0
$22$	1000.00i	0.440497i
$23$	− 1576.00i	− 0.621207i	−0.950539	−	0.310604i	$-0.899469\pi$
$23$	− 1576.00i	− 0.621207i	0.950539	−	0.310604i	$-0.100531\pi$
$24$	0	0
$25$	0	0
$26$	676.000	0.196116
$27$	0	0
$28$	2720.00i	0.655653i
$29$	−2578.00	−0.569230	−0.284615	−	0.958642i	$-0.591866\pi$
$29$	−2578.00	−0.569230	−0.284615	+	0.958642i	$0.591866\pi$
$30$	0	0
$31$	−8654.00	−1.61738	−0.808691	−	0.588234i	$-0.799824\pi$
$31$	−8654.00	−1.61738	−0.808691	+	0.588234i	$0.799824\pi$
$32$	− 1024.00i	− 0.176777i
$33$	0	0
$34$	4248.00	0.630213
$35$	0	0
$36$	−3888.00	−0.500000
$37$	10986.0i	1.31927i	0.751584	+	0.659637i	$0.229290\pi$
$37$	10986.0i	1.31927i	−0.751584	+	0.659637i	$0.770710\pi$
$38$	− 312.000i	− 0.0350506i
$39$	0	0
$40$	0	0
$41$	1050.00	0.0975505	0.0487753	−	0.998810i	$-0.484468\pi$
$41$	1050.00	0.0975505	0.0487753	+	0.998810i	$0.484468\pi$
$42$	0	0
$43$	5900.00i	0.486610i	0.969950	+	0.243305i	$0.0782316\pi$
$43$	5900.00i	0.486610i	−0.969950	+	0.243305i	$0.921768\pi$
$44$	4000.00	0.311479
$45$	0	0
$46$	−6304.00	−0.439260
$47$	− 5962.00i	− 0.393684i	−0.980435	−	0.196842i	$-0.936931\pi$
$47$	− 5962.00i	− 0.393684i	0.980435	−	0.196842i	$-0.0630685\pi$
$48$	0	0
$49$	−12093.0	−0.719522
$50$	0	0
$51$	0	0
$52$	− 2704.00i	− 0.138675i
$53$	− 29046.0i	− 1.42035i	−0.704023	−	0.710177i	$-0.748615\pi$
$53$	− 29046.0i	− 1.42035i	0.704023	−	0.710177i	$-0.251385\pi$
$54$	0	0
$55$	0	0
$56$	10880.0	0.463616
$57$	0	0
$58$	10312.0i	0.402507i
$59$	13922.0	0.520681	0.260340	−	0.965517i	$-0.416165\pi$
$59$	13922.0	0.520681	0.260340	+	0.965517i	$0.416165\pi$
$60$	0	0
$61$	−32882.0	−1.13145	−0.565723	−	0.824596i	$-0.691403\pi$
$61$	−32882.0	−1.13145	−0.565723	+	0.824596i	$0.691403\pi$
$62$	34616.0i	1.14366i
$63$	− 41310.0i	− 1.31131i
$64$	−4096.00	−0.125000
$65$	0	0
$66$	0	0
$67$	− 69566.0i	− 1.89326i	−0.322324	−	0.946629i	$-0.604464\pi$
$67$	− 69566.0i	− 1.89326i	0.322324	−	0.946629i	$-0.395536\pi$
$68$	− 16992.0i	− 0.445628i
$69$	0	0
$70$	0	0
$71$	−50542.0	−1.18989	−0.594945	−	0.803767i	$-0.702826\pi$
$71$	−50542.0	−1.18989	−0.594945	+	0.803767i	$0.702826\pi$
$72$	15552.0i	0.353553i
$73$	46750.0i	1.02677i	0.858157	+	0.513387i	$0.171609\pi$
$73$	46750.0i	1.02677i	−0.858157	+	0.513387i	$0.828391\pi$
$74$	43944.0	0.932868
$75$	0	0
$76$	−1248.00	−0.0247845
$77$	42500.0i	0.816887i
$78$	0	0
$79$	19348.0	0.348793	0.174397	−	0.984675i	$-0.444202\pi$
$79$	19348.0	0.348793	0.174397	+	0.984675i	$0.444202\pi$
$80$	0	0
$81$	59049.0	1.00000
$82$	− 4200.00i	− 0.0689786i
$83$	87438.0i	1.39317i	0.717473	+	0.696586i	$0.245299\pi$
$83$	87438.0i	1.39317i	−0.717473	+	0.696586i	$0.754701\pi$
$84$	0	0
$85$	0	0
$86$	23600.0	0.344085
$87$	0	0
$88$	− 16000.0i	− 0.220249i
$89$	−94170.0	−1.26019	−0.630097	−	0.776516i	$-0.716985\pi$
$89$	−94170.0	−1.26019	−0.630097	+	0.776516i	$0.716985\pi$
$90$	0	0
$91$	28730.0	0.363691
$92$	25216.0i	0.310604i
$93$	0	0
$94$	−23848.0	−0.278376
$95$	0	0
$96$	0	0
$97$	182786.i	1.97248i	0.165307	+	0.986242i	$0.447139\pi$
$97$	182786.i	1.97248i	−0.165307	+	0.986242i	$0.552861\pi$
$98$	48372.0i	0.508779i
$99$	−60750.0	−0.622957
$100$	0	0
$101$	−18514.0	−0.180591	−0.0902957	−	0.995915i	$-0.528781\pi$
$101$	−18514.0	−0.180591	−0.0902957	+	0.995915i	$0.528781\pi$
$102$	0	0
$103$	− 116056.i	− 1.07789i	−0.842341	−	0.538945i	$-0.818823\pi$
$103$	− 116056.i	− 1.07789i	0.842341	−	0.538945i	$-0.181177\pi$
$104$	−10816.0	−0.0980581
$105$	0	0
$106$	−116184.	−1.00434
$107$	153520.i	1.29630i	0.761513	+	0.648150i	$0.224457\pi$
$107$	153520.i	1.29630i	−0.761513	+	0.648150i	$0.775543\pi$
$108$	0	0
$109$	178622.	1.44002	0.720010	−	0.693963i	$-0.244137\pi$
$109$	178622.	1.44002	0.720010	+	0.693963i	$0.244137\pi$
$110$	0	0
$111$	0	0
$112$	− 43520.0i	− 0.327826i
$113$	244754.i	1.80316i	0.432615	+	0.901579i	$0.357591\pi$
$113$	244754.i	1.80316i	−0.432615	+	0.901579i	$0.642409\pi$
$114$	0	0
$115$	0	0
$116$	41248.0	0.284615
$117$	41067.0i	0.277350i
$118$	− 55688.0i	− 0.368177i
$119$	180540.	1.16871
$120$	0	0
$121$	−98551.0	−0.611924
$122$	131528.i	0.800053i
$123$	0	0
$124$	138464.	0.808691
$125$	0	0
$126$	−165240.	−0.927233
$127$	256600.i	1.41172i	0.708353	+	0.705858i	$0.249438\pi$
$127$	256600.i	1.41172i	−0.708353	+	0.705858i	$0.750562\pi$
$128$	16384.0i	0.0883883i
$129$	0	0
$130$	0	0
$131$	−262736.	−1.33765	−0.668823	−	0.743421i	$-0.733202\pi$
$131$	−262736.	−1.33765	−0.668823	+	0.743421i	$0.733202\pi$
$132$	0	0
$133$	− 13260.0i	− 0.0650002i
$134$	−278264.	−1.33874
$135$	0	0
$136$	−67968.0	−0.315106
$137$	− 38286.0i	− 0.174276i	−0.996196	−	0.0871382i	$-0.972228\pi$
$137$	− 38286.0i	− 0.174276i	0.996196	−	0.0871382i	$-0.0277722\pi$
$138$	0	0
$139$	57776.0	0.253636	0.126818	−	0.991926i	$-0.459524\pi$
$139$	57776.0	0.253636	0.126818	+	0.991926i	$0.459524\pi$
$140$	0	0
$141$	0	0
$142$	202168.i	0.841379i
$143$	− 42250.0i	− 0.172777i
$144$	62208.0	0.250000
$145$	0	0
$146$	187000.	0.726038
$147$	0	0
$148$	− 175776.i	− 0.659637i
$149$	−28866.0	−0.106517	−0.0532587	−	0.998581i	$-0.516961\pi$
$149$	−28866.0	−0.106517	−0.0532587	+	0.998581i	$0.516961\pi$
$150$	0	0
$151$	39870.0	0.142300	0.0711498	−	0.997466i	$-0.477333\pi$
$151$	39870.0	0.142300	0.0711498	+	0.997466i	$0.477333\pi$
$152$	4992.00i	0.0175253i
$153$	258066.i	0.891255i
$154$	170000.	0.577627
$155$	0	0
$156$	0	0
$157$	161042.i	0.521423i	0.965417	+	0.260711i	$0.0839571\pi$
$157$	161042.i	0.521423i	−0.965417	+	0.260711i	$0.916043\pi$
$158$	− 77392.0i	− 0.246634i
$159$	0	0
$160$	0	0
$161$	−267920.	−0.814593
$162$	− 236196.i	− 0.707107i
$163$	− 312830.i	− 0.922230i	−0.887340	−	0.461115i	$-0.847450\pi$
$163$	− 312830.i	− 0.922230i	0.887340	−	0.461115i	$-0.152550\pi$
$164$	−16800.0	−0.0487753
$165$	0	0
$166$	349752.	0.985122
$167$	532926.i	1.47869i	0.673329	+	0.739343i	$0.264864\pi$
$167$	532926.i	1.47869i	−0.673329	+	0.739343i	$0.735136\pi$
$168$	0	0
$169$	−28561.0	−0.0769231
$170$	0	0
$171$	18954.0	0.0495691
$172$	− 94400.0i	− 0.243305i
$173$	630458.i	1.60155i	0.598964	+	0.800776i	$0.295579\pi$
$173$	630458.i	1.60155i	−0.598964	+	0.800776i	$0.704421\pi$
$174$	0	0
$175$	0	0
$176$	−64000.0	−0.155739
$177$	0	0
$178$	376680.i	0.891092i
$179$	674916.	1.57441	0.787204	−	0.616693i	$-0.211528\pi$
$179$	674916.	1.57441	0.787204	+	0.616693i	$0.211528\pi$
$180$	0	0
$181$	186282.	0.422644	0.211322	−	0.977417i	$-0.432223\pi$
$181$	186282.	0.422644	0.211322	+	0.977417i	$0.432223\pi$
$182$	− 114920.i	− 0.257168i
$183$	0	0
$184$	100864.	0.219630
$185$	0	0
$186$	0	0
$187$	− 265500.i	− 0.555214i
$188$	95392.0i	0.196842i
$189$	0	0
$190$	0	0
$191$	812180.	1.61090	0.805451	−	0.592663i	$-0.201923\pi$
$191$	812180.	1.61090	0.805451	+	0.592663i	$0.201923\pi$
$192$	0	0
$193$	150142.i	0.290141i	0.989421	+	0.145070i	$0.0463409\pi$
$193$	150142.i	0.290141i	−0.989421	+	0.145070i	$0.953659\pi$
$194$	731144.	1.39476
$195$	0	0
$196$	193488.	0.359761
$197$	236394.i	0.433981i	0.976174	+	0.216991i	$0.0696241\pi$
$197$	236394.i	0.433981i	−0.976174	+	0.216991i	$0.930376\pi$
$198$	243000.i	0.440497i
$199$	39376.0	0.0704854	0.0352427	−	0.999379i	$-0.488780\pi$
$199$	39376.0	0.0704854	0.0352427	+	0.999379i	$0.488780\pi$
$200$	0	0
$201$	0	0
$202$	74056.0i	0.127697i
$203$	438260.i	0.746435i
$204$	0	0
$205$	0	0
$206$	−464224.	−0.762183
$207$	− 382968.i	− 0.621207i
$208$	43264.0i	0.0693375i
$209$	−19500.0	−0.0308794
$210$	0	0
$211$	−410776.	−0.635183	−0.317592	−	0.948228i	$-0.602874\pi$
$211$	−410776.	−0.635183	−0.317592	+	0.948228i	$0.602874\pi$
$212$	464736.i	0.710177i
$213$	0	0
$214$	614080.	0.916623
$215$	0	0
$216$	0	0
$217$	1.47118e6i	2.12088i
$218$	− 714488.i	− 1.01825i
$219$	0	0
$220$	0	0
$221$	−179478.	−0.247190
$222$	0	0
$223$	− 1.08688e6i	− 1.46359i	−0.681523	−	0.731796i	$-0.738682\pi$
$223$	− 1.08688e6i	− 1.46359i	0.681523	−	0.731796i	$-0.261318\pi$
$224$	−174080.	−0.231808
$225$	0	0
$226$	979016.	1.27502
$227$	− 256470.i	− 0.330348i	−0.986264	−	0.165174i	$-0.947181\pi$
$227$	− 256470.i	− 0.330348i	0.986264	−	0.165174i	$-0.0528186\pi$
$228$	0	0
$229$	298110.	0.375654	0.187827	−	0.982202i	$-0.439856\pi$
$229$	298110.	0.375654	0.187827	+	0.982202i	$0.439856\pi$
$230$	0	0
$231$	0	0
$232$	− 164992.i	− 0.201253i
$233$	611926.i	0.738430i	0.929344	+	0.369215i	$0.120373\pi$
$233$	611926.i	0.738430i	−0.929344	+	0.369215i	$0.879627\pi$
$234$	164268.	0.196116
$235$	0	0
$236$	−222752.	−0.260340
$237$	0	0
$238$	− 722160.i	− 0.826401i
$239$	−36570.0	−0.0414124	−0.0207062	−	0.999786i	$-0.506591\pi$
$239$	−36570.0	−0.0414124	−0.0207062	+	0.999786i	$0.506591\pi$
$240$	0	0
$241$	380922.	0.422468	0.211234	−	0.977436i	$-0.432252\pi$
$241$	380922.	0.422468	0.211234	+	0.977436i	$0.432252\pi$
$242$	394204.i	0.432696i
$243$	0	0
$244$	526112.	0.565723
$245$	0	0
$246$	0	0
$247$	13182.0i	0.0137480i
$248$	− 553856.i	− 0.571831i
$249$	0	0
$250$	0	0
$251$	−1.22807e6	−1.23038	−0.615188	−	0.788380i	$-0.710920\pi$
$251$	−1.22807e6	−1.23038	−0.615188	+	0.788380i	$0.710920\pi$
$252$	660960.i	0.655653i
$253$	394000.i	0.386986i
$254$	1.02640e6	0.998234
$255$	0	0
$256$	65536.0	0.0625000
$257$	− 439278.i	− 0.414865i	−0.978249	−	0.207432i	$-0.933489\pi$
$257$	− 439278.i	− 0.414865i	0.978249	−	0.207432i	$-0.0665107\pi$
$258$	0	0
$259$	1.86762e6	1.72997
$260$	0	0
$261$	−626454.	−0.569230
$262$	1.05094e6i	0.945859i
$263$	1.67987e6i	1.49757i	0.662816	+	0.748783i	$0.269361\pi$
$263$	1.67987e6i	1.49757i	−0.662816	+	0.748783i	$0.730639\pi$
$264$	0	0
$265$	0	0
$266$	−53040.0	−0.0459621
$267$	0	0
$268$	1.11306e6i	0.946629i
$269$	−1.93840e6	−1.63329	−0.816645	−	0.577141i	$-0.804168\pi$
$269$	−1.93840e6	−1.63329	−0.816645	+	0.577141i	$0.804168\pi$
$270$	0	0
$271$	−695498.	−0.575271	−0.287636	−	0.957740i	$-0.592869\pi$
$271$	−695498.	−0.575271	−0.287636	+	0.957740i	$0.592869\pi$
$272$	271872.i	0.222814i
$273$	0	0
$274$	−153144.	−0.123232
$275$	0	0
$276$	0	0
$277$	− 1.13138e6i	− 0.885948i	−0.896534	−	0.442974i	$-0.853923\pi$
$277$	− 1.13138e6i	− 0.885948i	0.896534	−	0.442974i	$-0.146077\pi$
$278$	− 231104.i	− 0.179348i
$279$	−2.10292e6	−1.61738
$280$	0	0
$281$	1.73122e6	1.30793	0.653967	−	0.756523i	$-0.273103\pi$
$281$	1.73122e6	1.30793	0.653967	+	0.756523i	$0.273103\pi$
$282$	0	0
$283$	1.47124e6i	1.09199i	0.837788	+	0.545995i	$0.183848\pi$
$283$	1.47124e6i	1.09199i	−0.837788	+	0.545995i	$0.816152\pi$
$284$	808672.	0.594945
$285$	0	0
$286$	−169000.	−0.122172
$287$	− 178500.i	− 0.127919i
$288$	− 248832.i	− 0.176777i
$289$	292013.	0.205664
$290$	0	0
$291$	0	0
$292$	− 748000.i	− 0.513387i
$293$	− 2.88855e6i	− 1.96567i	−0.184491	−	0.982834i	$-0.559064\pi$
$293$	− 2.88855e6i	− 1.96567i	0.184491	−	0.982834i	$-0.440936\pi$
$294$	0	0
$295$	0	0
$296$	−703104.	−0.466434
$297$	0	0
$298$	115464.i	0.0753192i
$299$	266344.	0.172292
$300$	0	0
$301$	1.00300e6	0.638094
$302$	− 159480.i	− 0.100621i
$303$	0	0
$304$	19968.0	0.0123923
$305$	0	0
$306$	1.03226e6	0.630213
$307$	874118.i	0.529327i	0.964341	+	0.264664i	$0.0852609\pi$
$307$	874118.i	0.529327i	−0.964341	+	0.264664i	$0.914739\pi$
$308$	− 680000.i	− 0.408444i
$309$	0	0
$310$	0	0
$311$	2.68224e6	1.57252	0.786261	−	0.617895i	$-0.212014\pi$
$311$	2.68224e6	1.57252	0.786261	+	0.617895i	$0.212014\pi$
$312$	0	0
$313$	1.34459e6i	0.775761i	0.921710	+	0.387880i	$0.126793\pi$
$313$	1.34459e6i	0.775761i	−0.921710	+	0.387880i	$0.873207\pi$
$314$	644168.	0.368702
$315$	0	0
$316$	−309568.	−0.174397
$317$	1.32074e6i	0.738191i	0.929392	+	0.369095i	$0.120332\pi$
$317$	1.32074e6i	0.738191i	−0.929392	+	0.369095i	$0.879668\pi$
$318$	0	0
$319$	644500.	0.354606
$320$	0	0
$321$	0	0
$322$	1.07168e6i	0.576004i
$323$	82836.0i	0.0441787i
$324$	−944784.	−0.500000
$325$	0	0
$326$	−1.25132e6	−0.652115
$327$	0	0
$328$	67200.0i	0.0344893i
$329$	−1.01354e6	−0.516239
$330$	0	0
$331$	−2.05728e6	−1.03210	−0.516051	−	0.856558i	$-0.672599\pi$
$331$	−2.05728e6	−1.03210	−0.516051	+	0.856558i	$0.672599\pi$
$332$	− 1.39901e6i	− 0.696586i
$333$	2.66960e6i	1.31927i
$334$	2.13170e6	1.04559
$335$	0	0
$336$	0	0
$337$	453398.i	0.217473i	0.994071	+	0.108736i	$0.0346804\pi$
$337$	453398.i	0.217473i	−0.994071	+	0.108736i	$0.965320\pi$
$338$	114244.i	0.0543928i
$339$	0	0
$340$	0	0
$341$	2.16350e6	1.00756
$342$	− 75816.0i	− 0.0350506i
$343$	− 801380.i	− 0.367793i
$344$	−377600.	−0.172043
$345$	0	0
$346$	2.52183e6	1.13247
$347$	− 1.23065e6i	− 0.548669i	−0.961634	−	0.274334i	$-0.911543\pi$
$347$	− 1.23065e6i	− 0.548669i	0.961634	−	0.274334i	$-0.0884575\pi$
$348$	0	0
$349$	2.43825e6	1.07155	0.535777	−	0.844360i	$-0.320019\pi$
$349$	2.43825e6	1.07155	0.535777	+	0.844360i	$0.320019\pi$
$350$	0	0
$351$	0	0
$352$	256000.i	0.110124i
$353$	2.68315e6i	1.14606i	0.819534	+	0.573031i	$0.194233\pi$
$353$	2.68315e6i	1.14606i	−0.819534	+	0.573031i	$0.805767\pi$
$354$	0	0
$355$	0	0
$356$	1.50672e6	0.630097
$357$	0	0
$358$	− 2.69966e6i	− 1.11327i
$359$	−1.58693e6	−0.649864	−0.324932	−	0.945737i	$-0.605341\pi$
$359$	−1.58693e6	−0.649864	−0.324932	+	0.945737i	$0.605341\pi$
$360$	0	0
$361$	−2.47002e6	−0.997543
$362$	− 745128.i	− 0.298854i
$363$	0	0
$364$	−459680.	−0.181845
$365$	0	0
$366$	0	0
$367$	− 60052.0i	− 0.0232735i	−0.999932	−	0.0116368i	$-0.996296\pi$
$367$	− 60052.0i	− 0.0232735i	0.999932	−	0.0116368i	$-0.00370418\pi$
$368$	− 403456.i	− 0.155302i
$369$	255150.	0.0975505
$370$	0	0
$371$	−4.93782e6	−1.86252
$372$	0	0
$373$	4.01853e6i	1.49553i	0.663963	+	0.747766i	$0.268873\pi$
$373$	4.01853e6i	1.49553i	−0.663963	+	0.747766i	$0.731127\pi$
$374$	−1.06200e6	−0.392596
$375$	0	0
$376$	381568.	0.139188
$377$	− 435682.i	− 0.157876i
$378$	0	0
$379$	−1.67581e6	−0.599276	−0.299638	−	0.954053i	$-0.596866\pi$
$379$	−1.67581e6	−0.599276	−0.299638	+	0.954053i	$0.596866\pi$
$380$	0	0
$381$	0	0
$382$	− 3.24872e6i	− 1.13908i
$383$	− 687258.i	− 0.239399i	−0.992810	−	0.119700i	$-0.961807\pi$
$383$	− 687258.i	− 0.239399i	0.992810	−	0.119700i	$-0.0381932\pi$
$384$	0	0
$385$	0	0
$386$	600568.	0.205161
$387$	1.43370e6i	0.486610i
$388$	− 2.92458e6i	− 0.986242i
$389$	−1.37611e6	−0.461082	−0.230541	−	0.973063i	$-0.574050\pi$
$389$	−1.37611e6	−0.461082	−0.230541	+	0.973063i	$0.574050\pi$
$390$	0	0
$391$	1.67371e6	0.553655
$392$	− 773952.i	− 0.254389i
$393$	0	0
$394$	945576.	0.306871
$395$	0	0
$396$	972000.	0.311479
$397$	− 721198.i	− 0.229656i	−0.993385	−	0.114828i	$-0.963368\pi$
$397$	− 721198.i	− 0.229656i	0.993385	−	0.114828i	$-0.0366317\pi$
$398$	− 157504.i	− 0.0498407i
$399$	0	0
$400$	0	0
$401$	2.22681e6	0.691548	0.345774	−	0.938318i	$-0.387616\pi$
$401$	2.22681e6	0.691548	0.345774	+	0.938318i	$0.387616\pi$
$402$	0	0
$403$	− 1.46253e6i	− 0.448581i
$404$	296224.	0.0902957
$405$	0	0
$406$	1.75304e6	0.527809
$407$	− 2.74650e6i	− 0.821852i
$408$	0	0
$409$	−2.00783e6	−0.593496	−0.296748	−	0.954956i	$-0.595902\pi$
$409$	−2.00783e6	−0.593496	−0.296748	+	0.954956i	$0.595902\pi$
$410$	0	0
$411$	0	0
$412$	1.85690e6i	0.538945i
$413$	− 2.36674e6i	− 0.682772i
$414$	−1.53187e6	−0.439260
$415$	0	0
$416$	173056.	0.0490290
$417$	0	0
$418$	78000.0i	0.0218350i
$419$	−5.99378e6	−1.66788	−0.833942	−	0.551852i	$-0.813921\pi$
$419$	−5.99378e6	−1.66788	−0.833942	+	0.551852i	$0.813921\pi$
$420$	0	0
$421$	−5.32737e6	−1.46490	−0.732449	−	0.680822i	$-0.761623\pi$
$421$	−5.32737e6	−1.46490	−0.732449	+	0.680822i	$0.761623\pi$
$422$	1.64310e6i	0.449142i
$423$	− 1.44877e6i	− 0.393684i
$424$	1.85894e6	0.502171
$425$	0	0
$426$	0	0
$427$	5.58994e6i	1.48367i
$428$	− 2.45632e6i	− 0.648150i
$429$	0	0
$430$	0	0
$431$	−5.42972e6	−1.40794	−0.703970	−	0.710230i	$-0.748591\pi$
$431$	−5.42972e6	−1.40794	−0.703970	+	0.710230i	$0.748591\pi$
$432$	0	0
$433$	− 7.43979e6i	− 1.90696i	−0.301459	−	0.953479i	$-0.597474\pi$
$433$	− 7.43979e6i	− 1.90696i	0.301459	−	0.953479i	$-0.402526\pi$
$434$	5.88472e6	1.49969
$435$	0	0
$436$	−2.85795e6	−0.720010
$437$	− 122928.i	− 0.0307927i
$438$	0	0
$439$	−6.86418e6	−1.69991	−0.849957	−	0.526852i	$-0.823372\pi$
$439$	−6.86418e6	−1.69991	−0.849957	+	0.526852i	$0.823372\pi$
$440$	0	0
$441$	−2.93860e6	−0.719522
$442$	717912.i	0.174790i
$443$	3.46630e6i	0.839182i	0.907713	+	0.419591i	$0.137827\pi$
$443$	3.46630e6i	0.839182i	−0.907713	+	0.419591i	$0.862173\pi$
$444$	0	0
$445$	0	0
$446$	−4.34753e6	−1.03492
$447$	0	0
$448$	696320.i	0.163913i
$449$	1.40426e6	0.328725	0.164362	−	0.986400i	$-0.447443\pi$
$449$	1.40426e6	0.328725	0.164362	+	0.986400i	$0.447443\pi$
$450$	0	0
$451$	−262500.	−0.0607698
$452$	− 3.91606e6i	− 0.901579i
$453$	0	0
$454$	−1.02588e6	−0.233591
$455$	0	0
$456$	0	0
$457$	− 5.95072e6i	− 1.33284i	−0.745575	−	0.666421i	$-0.767825\pi$
$457$	− 5.95072e6i	− 1.33284i	0.745575	−	0.666421i	$-0.232175\pi$
$458$	− 1.19244e6i	− 0.265627i
$459$	0	0
$460$	0	0
$461$	−6.25465e6	−1.37073	−0.685363	−	0.728202i	$-0.740356\pi$
$461$	−6.25465e6	−1.37073	−0.685363	+	0.728202i	$0.740356\pi$
$462$	0	0
$463$	1.55055e6i	0.336149i	0.985774	+	0.168075i	$0.0537550\pi$
$463$	1.55055e6i	0.336149i	−0.985774	+	0.168075i	$0.946245\pi$
$464$	−659968.	−0.142308
$465$	0	0
$466$	2.44770e6	0.522149
$467$	− 1.80480e6i	− 0.382945i	−0.981498	−	0.191472i	$-0.938674\pi$
$467$	− 1.80480e6i	− 0.382945i	0.981498	−	0.191472i	$-0.0613262\pi$
$468$	− 657072.i	− 0.138675i
$469$	−1.18262e7	−2.48264
$470$	0	0
$471$	0	0
$472$	891008.i	0.184088i
$473$	− 1.47500e6i	− 0.303137i
$474$	0	0
$475$	0	0
$476$	−2.88864e6	−0.584354
$477$	− 7.05818e6i	− 1.42035i
$478$	146280.i	0.0292830i
$479$	−2.21809e6	−0.441712	−0.220856	−	0.975306i	$-0.570885\pi$
$479$	−2.21809e6	−0.441712	−0.220856	+	0.975306i	$0.570885\pi$
$480$	0	0
$481$	−1.85663e6	−0.365901
$482$	− 1.52369e6i	− 0.298730i
$483$	0	0
$484$	1.57682e6	0.305962
$485$	0	0
$486$	0	0
$487$	− 6.14268e6i	− 1.17364i	−0.809717	−	0.586821i	$-0.800379\pi$
$487$	− 6.14268e6i	− 1.17364i	0.809717	−	0.586821i	$-0.199621\pi$
$488$	− 2.10445e6i	− 0.400026i
$489$	0	0
$490$	0	0
$491$	6.44486e6	1.20645	0.603226	−	0.797571i	$-0.293882\pi$
$491$	6.44486e6	1.20645	0.603226	+	0.797571i	$0.293882\pi$
$492$	0	0
$493$	− 2.73784e6i	− 0.507330i
$494$	52728.0	0.00972129
$495$	0	0
$496$	−2.21542e6	−0.404346
$497$	8.59214e6i	1.56031i
$498$	0	0
$499$	−4.25838e6	−0.765584	−0.382792	−	0.923835i	$-0.625037\pi$
$499$	−4.25838e6	−0.765584	−0.382792	+	0.923835i	$0.625037\pi$
$500$	0	0
$501$	0	0
$502$	4.91227e6i	0.870008i
$503$	3.56242e6i	0.627806i	0.949455	+	0.313903i	$0.101637\pi$
$503$	3.56242e6i	0.627806i	−0.949455	+	0.313903i	$0.898363\pi$
$504$	2.64384e6	0.463616
$505$	0	0
$506$	1.57600e6	0.273640
$507$	0	0
$508$	− 4.10560e6i	− 0.705858i
$509$	−4.23936e6	−0.725281	−0.362640	−	0.931929i	$-0.618125\pi$
$509$	−4.23936e6	−0.725281	−0.362640	+	0.931929i	$0.618125\pi$
$510$	0	0
$511$	7.94750e6	1.34641
$512$	− 262144.i	− 0.0441942i
$513$	0	0
$514$	−1.75711e6	−0.293354
$515$	0	0
$516$	0	0
$517$	1.49050e6i	0.245248i
$518$	− 7.47048e6i	− 1.22328i
$519$	0	0
$520$	0	0
$521$	2.38657e6	0.385194	0.192597	−	0.981278i	$-0.438309\pi$
$521$	2.38657e6	0.385194	0.192597	+	0.981278i	$0.438309\pi$
$522$	2.50582e6i	0.402507i
$523$	8.84129e6i	1.41339i	0.707519	+	0.706694i	$0.249814\pi$
$523$	8.84129e6i	1.41339i	−0.707519	+	0.706694i	$0.750186\pi$
$524$	4.20378e6	0.668823
$525$	0	0
$526$	6.71947e6	1.05894
$527$	− 9.19055e6i	− 1.44150i
$528$	0	0
$529$	3.95257e6	0.614101
$530$	0	0
$531$	3.38305e6	0.520681
$532$	212160.i	0.0325001i
$533$	177450.i	0.0270557i
$534$	0	0
$535$	0	0
$536$	4.45222e6	0.669368
$537$	0	0
$538$	7.75361e6i	1.15491i
$539$	3.02325e6	0.448231
$540$	0	0
$541$	70058.0	0.0102912	0.00514558	−	0.999987i	$-0.498362\pi$
$541$	70058.0	0.0102912	0.00514558	+	0.999987i	$0.498362\pi$
$542$	2.78199e6i	0.406778i
$543$	0	0
$544$	1.08749e6	0.157553
$545$	0	0
$546$	0	0
$547$	− 6.60752e6i	− 0.944213i	−0.881541	−	0.472107i	$-0.843494\pi$
$547$	− 6.60752e6i	− 0.944213i	0.881541	−	0.472107i	$-0.156506\pi$
$548$	612576.i	0.0871382i
$549$	−7.99033e6	−1.13145
$550$	0	0
$551$	−201084.	−0.0282162
$552$	0	0
$553$	− 3.28916e6i	− 0.457375i
$554$	−4.52551e6	−0.626460
$555$	0	0
$556$	−924416.	−0.126818
$557$	− 1.10726e7i	− 1.51221i	−0.654448	−	0.756107i	$-0.727099\pi$
$557$	− 1.10726e7i	− 1.51221i	0.654448	−	0.756107i	$-0.272901\pi$
$558$	8.41169e6i	1.14366i
$559$	−997100.	−0.134961
$560$	0	0
$561$	0	0
$562$	− 6.92487e6i	− 0.924849i
$563$	1.43532e6i	0.190843i	0.995437	+	0.0954216i	$0.0304199\pi$
$563$	1.43532e6i	0.190843i	−0.995437	+	0.0954216i	$0.969580\pi$
$564$	0	0
$565$	0	0
$566$	5.88498e6	0.772153
$567$	− 1.00383e7i	− 1.31131i
$568$	− 3.23469e6i	− 0.420689i
$569$	1.17051e7	1.51564	0.757818	−	0.652466i	$-0.226266\pi$
$569$	1.17051e7	1.51564	0.757818	+	0.652466i	$0.226266\pi$
$570$	0	0
$571$	4.81885e6	0.618519	0.309260	−	0.950978i	$-0.399919\pi$
$571$	4.81885e6	0.618519	0.309260	+	0.950978i	$0.399919\pi$
$572$	676000.i	0.0863886i
$573$	0	0
$574$	−714000.	−0.0904521
$575$	0	0
$576$	−995328.	−0.125000
$577$	− 1.35572e6i	− 0.169523i	−0.996401	−	0.0847617i	$-0.972987\pi$
$577$	− 1.35572e6i	− 0.169523i	0.996401	−	0.0847617i	$-0.0270129\pi$
$578$	− 1.16805e6i	− 0.145426i
$579$	0	0
$580$	0	0
$581$	1.48645e7	1.82687
$582$	0	0
$583$	7.26150e6i	0.884820i
$584$	−2.99200e6	−0.363019
$585$	0	0
$586$	−1.15542e7	−1.38994
$587$	5.03941e6i	0.603649i	0.953364	+	0.301824i	$0.0975957\pi$
$587$	5.03941e6i	0.603649i	−0.953364	+	0.301824i	$0.902404\pi$
$588$	0	0
$589$	−675012.	−0.0801721
$590$	0	0
$591$	0	0
$592$	2.81242e6i	0.329819i
$593$	− 9.16124e6i	− 1.06984i	−0.844904	−	0.534919i	$-0.820342\pi$
$593$	− 9.16124e6i	− 1.06984i	0.844904	−	0.534919i	$-0.179658\pi$
$594$	0	0
$595$	0	0
$596$	461856.	0.0532587
$597$	0	0
$598$	− 1.06538e6i	− 0.121829i
$599$	6.46635e6	0.736363	0.368182	−	0.929754i	$-0.379980\pi$
$599$	6.46635e6	0.736363	0.368182	+	0.929754i	$0.379980\pi$
$600$	0	0
$601$	−1.18021e7	−1.33282	−0.666411	−	0.745585i	$-0.732170\pi$
$601$	−1.18021e7	−1.33282	−0.666411	+	0.745585i	$0.732170\pi$
$602$	− 4.01200e6i	− 0.451201i
$603$	− 1.69045e7i	− 1.89326i
$604$	−637920.	−0.0711498
$605$	0	0
$606$	0	0
$607$	2.25748e6i	0.248686i	0.992239	+	0.124343i	$0.0396823\pi$
$607$	2.25748e6i	0.248686i	−0.992239	+	0.124343i	$0.960318\pi$
$608$	− 79872.0i	− 0.00876265i
$609$	0	0
$610$	0	0
$611$	1.00758e6	0.109188
$612$	− 4.12906e6i	− 0.445628i
$613$	2.75378e6i	0.295991i	0.988988	+	0.147995i	$0.0472821\pi$
$613$	2.75378e6i	0.295991i	−0.988988	+	0.147995i	$0.952718\pi$
$614$	3.49647e6	0.374291
$615$	0	0
$616$	−2.72000e6	−0.288813
$617$	3.41607e6i	0.361255i	0.983552	+	0.180627i	$0.0578128\pi$
$617$	3.41607e6i	0.361255i	−0.983552	+	0.180627i	$0.942187\pi$
$618$	0	0
$619$	−9.43169e6	−0.989379	−0.494690	−	0.869070i	$-0.664718\pi$
$619$	−9.43169e6	−0.989379	−0.494690	+	0.869070i	$0.664718\pi$
$620$	0	0
$621$	0	0
$622$	− 1.07290e7i	− 1.11194i
$623$	1.60089e7i	1.65250i
$624$	0	0
$625$	0	0
$626$	5.37834e6	0.548546
$627$	0	0
$628$	− 2.57667e6i	− 0.260711i
$629$	−1.16671e7	−1.17581
$630$	0	0
$631$	−4.87474e6	−0.487391	−0.243696	−	0.969852i	$-0.578360\pi$
$631$	−4.87474e6	−0.487391	−0.243696	+	0.969852i	$0.578360\pi$
$632$	1.23827e6i	0.123317i
$633$	0	0
$634$	5.28295e6	0.521980
$635$	0	0
$636$	0	0
$637$	− 2.04372e6i	− 0.199559i
$638$	− 2.57800e6i	− 0.250744i
$639$	−1.22817e7	−1.18989
$640$	0	0
$641$	9.74279e6	0.936566	0.468283	−	0.883579i	$-0.344873\pi$
$641$	9.74279e6	0.936566	0.468283	+	0.883579i	$0.344873\pi$
$642$	0	0
$643$	− 1.63894e6i	− 0.156327i	−0.996941	−	0.0781637i	$-0.975094\pi$
$643$	− 1.63894e6i	− 0.156327i	0.996941	−	0.0781637i	$-0.0249057\pi$
$644$	4.28672e6	0.407296
$645$	0	0
$646$	331344.	0.0312390
$647$	− 1.59069e6i	− 0.149391i	−0.997206	−	0.0746955i	$-0.976202\pi$
$647$	− 1.59069e6i	− 0.149391i	0.997206	−	0.0746955i	$-0.0237985\pi$
$648$	3.77914e6i	0.353553i
$649$	−3.48050e6	−0.324362
$650$	0	0
$651$	0	0
$652$	5.00528e6i	0.461115i
$653$	− 1.59778e7i	− 1.46634i	−0.680045	−	0.733170i	$-0.738040\pi$
$653$	− 1.59778e7i	− 1.46634i	0.680045	−	0.733170i	$-0.261960\pi$
$654$	0	0
$655$	0	0
$656$	268800.	0.0243876
$657$	1.13602e7i	1.02677i
$658$	4.05416e6i	0.365036i
$659$	6.02458e6	0.540397	0.270199	−	0.962805i	$-0.412911\pi$
$659$	6.02458e6	0.540397	0.270199	+	0.962805i	$0.412911\pi$
$660$	0	0
$661$	−2.00705e7	−1.78671	−0.893355	−	0.449352i	$-0.851655\pi$
$661$	−2.00705e7	−1.78671	−0.893355	+	0.449352i	$0.851655\pi$
$662$	8.22911e6i	0.729807i
$663$	0	0
$664$	−5.59603e6	−0.492561
$665$	0	0
$666$	1.06784e7	0.932868
$667$	4.06293e6i	0.353610i
$668$	− 8.52682e6i	− 0.739343i
$669$	0	0
$670$	0	0
$671$	8.22050e6	0.704842
$672$	0	0
$673$	5.48575e6i	0.466873i	0.972372	+	0.233436i	$0.0749971\pi$
$673$	5.48575e6i	0.466873i	−0.972372	+	0.233436i	$0.925003\pi$
$674$	1.81359e6	0.153776
$675$	0	0
$676$	456976.	0.0384615
$677$	− 4.74926e6i	− 0.398248i	−0.979974	−	0.199124i	$-0.936190\pi$
$677$	− 4.74926e6i	− 0.398248i	0.979974	−	0.199124i	$-0.0638097\pi$
$678$	0	0
$679$	3.10736e7	2.58653
$680$	0	0
$681$	0	0
$682$	− 8.65400e6i	− 0.712453i
$683$	6.13964e6i	0.503606i	0.967778	+	0.251803i	$0.0810236\pi$
$683$	6.13964e6i	0.503606i	−0.967778	+	0.251803i	$0.918976\pi$
$684$	−303264.	−0.0247845
$685$	0	0
$686$	−3.20552e6	−0.260069
$687$	0	0
$688$	1.51040e6i	0.121652i
$689$	4.90877e6	0.393935
$690$	0	0
$691$	1.57617e7	1.25577	0.627883	−	0.778308i	$-0.283922\pi$
$691$	1.57617e7	1.25577	0.627883	+	0.778308i	$0.283922\pi$
$692$	− 1.00873e7i	− 0.800776i
$693$	1.03275e7i	0.816887i
$694$	−4.92259e6	−0.387967
$695$	0	0
$696$	0	0
$697$	1.11510e6i	0.0869424i
$698$	− 9.75298e6i	− 0.757703i
$699$	0	0
$700$	0	0
$701$	−1.42036e7	−1.09170	−0.545851	−	0.837882i	$-0.683793\pi$
$701$	−1.42036e7	−1.09170	−0.545851	+	0.837882i	$0.683793\pi$
$702$	0	0
$703$	856908.i	0.0653952i
$704$	1.02400e6	0.0778697
$705$	0	0
$706$	1.07326e7	0.810388
$707$	3.14738e6i	0.236810i
$708$	0	0
$709$	−1.60718e7	−1.20074	−0.600369	−	0.799723i	$-0.704980\pi$
$709$	−1.60718e7	−1.20074	−0.600369	+	0.799723i	$0.704980\pi$
$710$	0	0
$711$	4.70156e6	0.348793
$712$	− 6.02688e6i	− 0.445546i
$713$	1.36387e7i	1.00473i
$714$	0	0
$715$	0	0
$716$	−1.07987e7	−0.787204
$717$	0	0
$718$	6.34774e6i	0.459524i
$719$	2.07078e7	1.49387	0.746933	−	0.664900i	$-0.231526\pi$
$719$	2.07078e7	1.49387	0.746933	+	0.664900i	$0.231526\pi$
$720$	0	0
$721$	−1.97295e7	−1.41344
$722$	9.88006e6i	0.705369i
$723$	0	0
$724$	−2.98051e6	−0.211322
$725$	0	0
$726$	0	0
$727$	5.04803e6i	0.354231i	0.984190	+	0.177115i	$0.0566766\pi$
$727$	5.04803e6i	0.354231i	−0.984190	+	0.177115i	$0.943323\pi$
$728$	1.83872e6i	0.128584i
$729$	1.43489e7	1.00000
$730$	0	0
$731$	−6.26580e6	−0.433694
$732$	0	0
$733$	2.10377e7i	1.44623i	0.690728	+	0.723115i	$0.257290\pi$
$733$	2.10377e7i	1.44623i	−0.690728	+	0.723115i	$0.742710\pi$
$734$	−240208.	−0.0164569
$735$	0	0
$736$	−1.61382e6	−0.109815
$737$	1.73915e7i	1.17942i
$738$	− 1.02060e6i	− 0.0689786i
$739$	−1.38992e7	−0.936218	−0.468109	−	0.883671i	$-0.655065\pi$
$739$	−1.38992e7	−0.936218	−0.468109	+	0.883671i	$0.655065\pi$
$740$	0	0
$741$	0	0
$742$	1.97513e7i	1.31700i
$743$	− 1.23267e6i	− 0.0819169i	−0.999161	−	0.0409584i	$-0.986959\pi$
$743$	− 1.23267e6i	− 0.0819169i	0.999161	−	0.0409584i	$-0.0130411\pi$
$744$	0	0
$745$	0	0
$746$	1.60741e7	1.05750
$747$	2.12474e7i	1.39317i
$748$	4.24800e6i	0.277607i
$749$	2.60984e7	1.69985
$750$	0	0
$751$	−1.62624e6	−0.105217	−0.0526084	−	0.998615i	$-0.516753\pi$
$751$	−1.62624e6	−0.105217	−0.0526084	+	0.998615i	$0.516753\pi$
$752$	− 1.52627e6i	− 0.0984209i
$753$	0	0
$754$	−1.74273e6	−0.111635
$755$	0	0
$756$	0	0
$757$	3.49882e6i	0.221913i	0.993825	+	0.110956i	$0.0353914\pi$
$757$	3.49882e6i	0.221913i	−0.993825	+	0.110956i	$0.964609\pi$
$758$	6.70324e6i	0.423752i
$759$	0	0
$760$	0	0
$761$	2.21713e7	1.38781	0.693905	−	0.720067i	$-0.255889\pi$
$761$	2.21713e7	1.38781	0.693905	+	0.720067i	$0.255889\pi$
$762$	0	0
$763$	− 3.03657e7i	− 1.88831i
$764$	−1.29949e7	−0.805451
$765$	0	0
$766$	−2.74903e6	−0.169281
$767$	2.35282e6i	0.144411i
$768$	0	0
$769$	−1.08955e6	−0.0664400	−0.0332200	−	0.999448i	$-0.510576\pi$
$769$	−1.08955e6	−0.0664400	−0.0332200	+	0.999448i	$0.510576\pi$
$770$	0	0
$771$	0	0
$772$	− 2.40227e6i	− 0.145070i
$773$	− 1.95219e6i	− 0.117510i	−0.998272	−	0.0587549i	$-0.981287\pi$
$773$	− 1.95219e6i	− 0.117510i	0.998272	−	0.0587549i	$-0.0187130\pi$
$774$	5.73480e6	0.344085
$775$	0	0
$776$	−1.16983e7	−0.697379
$777$	0	0
$778$	5.50442e6i	0.326034i
$779$	81900.0	0.00483549
$780$	0	0
$781$	1.26355e7	0.741250
$782$	− 6.69485e6i	− 0.391493i
$783$	0	0
$784$	−3.09581e6	−0.179880
$785$	0	0
$786$	0	0
$787$	− 1.44531e7i	− 0.831809i	−0.909408	−	0.415904i	$-0.863465\pi$
$787$	− 1.44531e7i	− 0.831809i	0.909408	−	0.415904i	$-0.136535\pi$
$788$	− 3.78230e6i	− 0.216991i
$789$	0	0
$790$	0	0
$791$	4.16082e7	2.36449
$792$	− 3.88800e6i	− 0.220249i
$793$	− 5.55706e6i	− 0.313807i
$794$	−2.88479e6	−0.162391
$795$	0	0
$796$	−630016.	−0.0352427
$797$	1.23500e7i	0.688685i	0.938844	+	0.344343i	$0.111898\pi$
$797$	1.23500e7i	0.688685i	−0.938844	+	0.344343i	$0.888102\pi$
$798$	0	0
$799$	6.33164e6	0.350873
$800$	0	0
$801$	−2.28833e7	−1.26019
$802$	− 8.90724e6i	− 0.488998i
$803$	− 1.16875e7i	− 0.639636i
$804$	0	0
$805$	0	0
$806$	−5.85010e6	−0.317195
$807$	0	0
$808$	− 1.18490e6i	− 0.0638487i
$809$	−1.15968e7	−0.622970	−0.311485	−	0.950251i	$-0.600826\pi$
$809$	−1.15968e7	−0.622970	−0.311485	+	0.950251i	$0.600826\pi$
$810$	0	0
$811$	−2.47534e7	−1.32155	−0.660774	−	0.750585i	$-0.729772\pi$
$811$	−2.47534e7	−1.32155	−0.660774	+	0.750585i	$0.729772\pi$
$812$	− 7.01216e6i	− 0.373217i
$813$	0	0
$814$	−1.09860e7	−0.581137
$815$	0	0
$816$	0	0
$817$	460200.i	0.0241208i
$818$	8.03130e6i	0.419665i
$819$	6.98139e6	0.363691
$820$	0	0
$821$	−2.47470e6	−0.128134	−0.0640671	−	0.997946i	$-0.520407\pi$
$821$	−2.47470e6	−0.128134	−0.0640671	+	0.997946i	$0.520407\pi$
$822$	0	0
$823$	7.84754e6i	0.403863i	0.979400	+	0.201932i	$0.0647219\pi$
$823$	7.84754e6i	0.403863i	−0.979400	+	0.201932i	$0.935278\pi$
$824$	7.42758e6	0.381092
$825$	0	0
$826$	−9.46696e6	−0.482792
$827$	− 2.26192e7i	− 1.15004i	−0.818140	−	0.575020i	$-0.804994\pi$
$827$	− 2.26192e7i	− 1.15004i	0.818140	−	0.575020i	$-0.195006\pi$
$828$	6.12749e6i	0.310604i
$829$	1.73912e7	0.878907	0.439454	−	0.898265i	$-0.355172\pi$
$829$	1.73912e7	0.878907	0.439454	+	0.898265i	$0.355172\pi$
$830$	0	0
$831$	0	0
$832$	− 692224.i	− 0.0346688i
$833$	− 1.28428e7i	− 0.641278i
$834$	0	0
$835$	0	0
$836$	312000.	0.0154397
$837$	0	0
$838$	2.39751e7i	1.17937i
$839$	3.43825e7	1.68629	0.843147	−	0.537684i	$-0.180701\pi$
$839$	3.43825e7	1.68629	0.843147	+	0.537684i	$0.180701\pi$
$840$	0	0
$841$	−1.38651e7	−0.675977
$842$	2.13095e7i	1.03584i
$843$	0	0
$844$	6.57242e6	0.317592
$845$	0	0
$846$	−5.79506e6	−0.278376
$847$	1.67537e7i	0.802419i
$848$	− 7.43578e6i	− 0.355089i
$849$	0	0
$850$	0	0
$851$	1.73139e7	0.819543
$852$	0	0
$853$	− 2.31007e7i	− 1.08706i	−0.839391	−	0.543528i	$-0.817088\pi$
$853$	− 2.31007e7i	− 1.08706i	0.839391	−	0.543528i	$-0.182912\pi$
$854$	2.23598e7	1.04911
$855$	0	0
$856$	−9.82528e6	−0.458311
$857$	− 7.02305e6i	− 0.326643i	−0.986573	−	0.163322i	$-0.947779\pi$
$857$	− 7.02305e6i	− 0.326643i	0.986573	−	0.163322i	$-0.0522208\pi$
$858$	0	0
$859$	−8.82135e6	−0.407899	−0.203949	−	0.978981i	$-0.565378\pi$
$859$	−8.82135e6	−0.407899	−0.203949	+	0.978981i	$0.565378\pi$
$860$	0	0
$861$	0	0
$862$	2.17189e7i	0.995564i
$863$	− 2.39560e7i	− 1.09493i	−0.836828	−	0.547466i	$-0.815592\pi$
$863$	− 2.39560e7i	− 1.09493i	0.836828	−	0.547466i	$-0.184408\pi$
$864$	0	0
$865$	0	0
$866$	−2.97592e7	−1.34842
$867$	0	0
$868$	− 2.35389e7i	− 1.06044i
$869$	−4.83700e6	−0.217283
$870$	0	0
$871$	1.17567e7	0.525096
$872$	1.14318e7i	0.509124i
$873$	4.44170e7i	1.97248i
$874$	−491712.	−0.0217737
$875$	0	0
$876$	0	0
$877$	− 5.79805e6i	− 0.254556i	−0.991867	−	0.127278i	$-0.959376\pi$
$877$	− 5.79805e6i	− 0.254556i	0.991867	−	0.127278i	$-0.0406240\pi$
$878$	2.74567e7i	1.20202i
$879$	0	0
$880$	0	0
$881$	−1.30527e7	−0.566580	−0.283290	−	0.959034i	$-0.591426\pi$
$881$	−1.30527e7	−0.566580	−0.283290	+	0.959034i	$0.591426\pi$
$882$	1.17544e7i	0.508779i
$883$	− 4.73009e6i	− 0.204159i	−0.994776	−	0.102079i	$-0.967450\pi$
$883$	− 4.73009e6i	− 0.204159i	0.994776	−	0.102079i	$-0.0325496\pi$
$884$	2.87165e6	0.123595
$885$	0	0
$886$	1.38652e7	0.593392
$887$	2.80737e7i	1.19809i	0.800714	+	0.599046i	$0.204453\pi$
$887$	2.80737e7i	1.19809i	−0.800714	+	0.599046i	$0.795547\pi$
$888$	0	0
$889$	4.36220e7	1.85119
$890$	0	0
$891$	−1.47622e7	−0.622957
$892$	1.73901e7i	0.731796i
$893$	− 465036.i	− 0.0195145i
$894$	0	0
$895$	0	0
$896$	2.78528e6	0.115904
$897$	0	0
$898$	− 5.61705e6i	− 0.232443i
$899$	2.23100e7	0.920663
$900$	0	0
$901$	3.08469e7	1.26590
$902$	1.05000e6i	0.0429708i
$903$	0	0
$904$	−1.56643e7	−0.637512
$905$	0	0
$906$	0	0
$907$	2.28552e7i	0.922500i	0.887270	+	0.461250i	$0.152599\pi$
$907$	2.28552e7i	0.922500i	−0.887270	+	0.461250i	$0.847401\pi$
$908$	4.10352e6i	0.165174i
$909$	−4.49890e6	−0.180591
$910$	0	0
$911$	−3.27335e7	−1.30676	−0.653381	−	0.757029i	$-0.726650\pi$
$911$	−3.27335e7	−1.30676	−0.653381	+	0.757029i	$0.726650\pi$
$912$	0	0
$913$	− 2.18595e7i	− 0.867887i
$914$	−2.38029e7	−0.942462
$915$	0	0
$916$	−4.76976e6	−0.187827
$917$	4.46651e7i	1.75406i
$918$	0	0
$919$	1.27717e7	0.498839	0.249419	−	0.968396i	$-0.419760\pi$
$919$	1.27717e7	0.498839	0.249419	+	0.968396i	$0.419760\pi$
$920$	0	0
$921$	0	0
$922$	2.50186e7i	0.969249i
$923$	− 8.54160e6i	− 0.330016i
$924$	0	0
$925$	0	0
$926$	6.20218e6	0.237693
$927$	− 2.82016e7i	− 1.07789i
$928$	2.63987e6i	0.100627i
$929$	−3.48297e7	−1.32407	−0.662034	−	0.749473i	$-0.730307\pi$
$929$	−3.48297e7	−1.32407	−0.662034	+	0.749473i	$0.730307\pi$
$930$	0	0
$931$	−943254.	−0.0356660
$932$	− 9.79082e6i	− 0.369215i
$933$	0	0
$934$	−7.21918e6	−0.270783
$935$	0	0
$936$	−2.62829e6	−0.0980581
$937$	3.00172e7i	1.11692i	0.829532	+	0.558459i	$0.188607\pi$
$937$	3.00172e7i	1.11692i	−0.829532	+	0.558459i	$0.811393\pi$
$938$	4.73049e7i	1.75549i
$939$	0	0
$940$	0	0
$941$	−4.50649e7	−1.65907	−0.829534	−	0.558457i	$-0.811394\pi$
$941$	−4.50649e7	−1.65907	−0.829534	+	0.558457i	$0.811394\pi$
$942$	0	0
$943$	− 1.65480e6i	− 0.0605991i
$944$	3.56403e6	0.130170
$945$	0	0
$946$	−5.90000e6	−0.214350
$947$	2.99276e7i	1.08442i	0.840243	+	0.542210i	$0.182412\pi$
$947$	2.99276e7i	1.08442i	−0.840243	+	0.542210i	$0.817588\pi$
$948$	0	0
$949$	−7.90075e6	−0.284776
$950$	0	0
$951$	0	0
$952$	1.15546e7i	0.413201i
$953$	4.25147e7i	1.51638i	0.652036	+	0.758188i	$0.273915\pi$
$953$	4.25147e7i	1.51638i	−0.652036	+	0.758188i	$0.726085\pi$
$954$	−2.82327e7	−1.00434
$955$	0	0
$956$	585120.	0.0207062
$957$	0	0
$958$	8.87234e6i	0.312338i
$959$	−6.50862e6	−0.228530
$960$	0	0
$961$	4.62626e7	1.61593
$962$	7.42654e6i	0.258731i
$963$	3.73054e7i	1.29630i
$964$	−6.09475e6	−0.211234
$965$	0	0
$966$	0	0
$967$	− 3.00251e7i	− 1.03257i	−0.856417	−	0.516284i	$-0.827315\pi$
$967$	− 3.00251e7i	− 1.03257i	0.856417	−	0.516284i	$-0.172685\pi$
$968$	− 6.30726e6i	− 0.216348i
$969$	0	0
$970$	0	0
$971$	−4.00864e7	−1.36442	−0.682211	−	0.731155i	$-0.738982\pi$
$971$	−4.00864e7	−1.36442	−0.682211	+	0.731155i	$0.738982\pi$
$972$	0	0
$973$	− 9.82192e6i	− 0.332594i
$974$	−2.45707e7	−0.829890
$975$	0	0
$976$	−8.41779e6	−0.282861
$977$	5.12151e7i	1.71657i	0.513174	+	0.858284i	$0.328469\pi$
$977$	5.12151e7i	1.71657i	−0.513174	+	0.858284i	$0.671531\pi$
$978$	0	0
$979$	2.35425e7	0.785047
$980$	0	0
$981$	4.34051e7	1.44002
$982$	− 2.57794e7i	− 0.853090i
$983$	− 1.82382e7i	− 0.602004i	−0.953624	−	0.301002i	$-0.902679\pi$
$983$	− 1.82382e7i	− 0.602004i	0.953624	−	0.301002i	$-0.0973211\pi$
$984$	0	0
$985$	0	0
$986$	−1.09513e7	−0.358736
$987$	0	0
$988$	− 210912.i	− 0.00687399i
$989$	9.29840e6	0.302286
$990$	0	0
$991$	−3.24103e7	−1.04833	−0.524166	−	0.851616i	$-0.675623\pi$
$991$	−3.24103e7	−1.04833	−0.524166	+	0.851616i	$0.675623\pi$
$992$	8.86170e6i	0.285915i
$993$	0	0
$994$	3.43686e7	1.10330
$995$	0	0
$996$	0	0
$997$	− 2.07867e7i	− 0.662289i	−0.943580	−	0.331145i	$-0.892565\pi$
$997$	− 2.07867e7i	− 0.662289i	0.943580	−	0.331145i	$-0.107435\pi$
$998$	1.70335e7i	0.541350i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	650.6.b.a.599.1		2
5.2	odd	4		650.6.a.a.1.1		1
5.3	odd	4		26.6.a.a.1.1	✓	1
5.4	even	2	inner	650.6.b.a.599.2		2
15.8	even	4		234.6.a.g.1.1		1
20.3	even	4		208.6.a.b.1.1		1
40.3	even	4		832.6.a.e.1.1		1
40.13	odd	4		832.6.a.d.1.1		1
65.8	even	4		338.6.b.a.337.1		2
65.18	even	4		338.6.b.a.337.2		2
65.38	odd	4		338.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.6.a.a.1.1	✓	1	5.3	odd	4
208.6.a.b.1.1		1	20.3	even	4
234.6.a.g.1.1		1	15.8	even	4
338.6.a.d.1.1		1	65.38	odd	4
338.6.b.a.337.1		2	65.8	even	4
338.6.b.a.337.2		2	65.18	even	4
650.6.a.a.1.1		1	5.2	odd	4
650.6.b.a.599.1		2	1.1	even	1	trivial
650.6.b.a.599.2		2	5.4	even	2	inner
832.6.a.d.1.1		1	40.13	odd	4
832.6.a.e.1.1		1	40.3	even	4

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:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.00000i q^{2} -16.0000 q^{4} -170.000i q^{7} +64.0000i q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-4.00000i q^{2} -16.0000 q^{4} -170.000i q^{7} +64.0000i q^{8} +243.000 q^{9} -250.000 q^{11} +169.000i q^{13} -680.000 q^{14} +256.000 q^{16} +1062.00i q^{17} -972.000i q^{18} +78.0000 q^{19} +1000.00i q^{22} -1576.00i q^{23} +676.000 q^{26} +2720.00i q^{28} -2578.00 q^{29} -8654.00 q^{31} -1024.00i q^{32} +4248.00 q^{34} -3888.00 q^{36} +10986.0i q^{37} -312.000i q^{38} +1050.00 q^{41} +5900.00i q^{43} +4000.00 q^{44} -6304.00 q^{46} -5962.00i q^{47} -12093.0 q^{49} -2704.00i q^{52} -29046.0i q^{53} +10880.0 q^{56} +10312.0i q^{58} +13922.0 q^{59} -32882.0 q^{61} +34616.0i q^{62} -41310.0i q^{63} -4096.00 q^{64} -69566.0i q^{67} -16992.0i q^{68} -50542.0 q^{71} +15552.0i q^{72} +46750.0i q^{73} +43944.0 q^{74} -1248.00 q^{76} +42500.0i q^{77} +19348.0 q^{79} +59049.0 q^{81} -4200.00i q^{82} +87438.0i q^{83} +23600.0 q^{86} -16000.0i q^{88} -94170.0 q^{89} +28730.0 q^{91} +25216.0i q^{92} -23848.0 q^{94} +182786. i q^{97} +48372.0i q^{98} -60750.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{4} + 486 q^{9}+O(q^{10}) \) 2 * q - 32 * q^4 + 486 * q^9	\( 2 q - 32 q^{4} + 486 q^{9} - 500 q^{11} - 1360 q^{14} + 512 q^{16} + 156 q^{19} + 1352 q^{26} - 5156 q^{29} - 17308 q^{31} + 8496 q^{34} - 7776 q^{36} + 2100 q^{41} + 8000 q^{44} - 12608 q^{46} - 24186 q^{49} + 21760 q^{56} + 27844 q^{59} - 65764 q^{61} - 8192 q^{64} - 101084 q^{71} + 87888 q^{74} - 2496 q^{76} + 38696 q^{79} + 118098 q^{81} + 47200 q^{86} - 188340 q^{89} + 57460 q^{91} - 47696 q^{94} - 121500 q^{99}+O(q^{100}) \) 2 * q - 32 * q^4 + 486 * q^9 - 500 * q^11 - 1360 * q^14 + 512 * q^16 + 156 * q^19 + 1352 * q^26 - 5156 * q^29 - 17308 * q^31 + 8496 * q^34 - 7776 * q^36 + 2100 * q^41 + 8000 * q^44 - 12608 * q^46 - 24186 * q^49 + 21760 * q^56 + 27844 * q^59 - 65764 * q^61 - 8192 * q^64 - 101084 * q^71 + 87888 * q^74 - 2496 * q^76 + 38696 * q^79 + 118098 * q^81 + 47200 * q^86 - 188340 * q^89 + 57460 * q^91 - 47696 * q^94 - 121500 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more