LMFDB - Embedded newform 414.6.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,6,Mod(1,414)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("414.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	414.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:	$66.3989014026$
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} - 8369x^{2} - 182616x - 370980 $ x^4 - x^3 - 8369x^2 - 182616x - 370980
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 138)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-20.8222$ of defining polynomial
Character	$\chi$	$=$	414.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} -93.5694 q^{5} +209.214 q^{7} -64.0000 q^{8} +O(q^{10})$	\(q-4.00000 q^{2} +16.0000 q^{4} -93.5694 q^{5} +209.214 q^{7} -64.0000 q^{8} +374.278 q^{10} +636.683 q^{11} +505.046 q^{13} -836.855 q^{14} +256.000 q^{16} +2159.05 q^{17} -2325.61 q^{19} -1497.11 q^{20} -2546.73 q^{22} -529.000 q^{23} +5630.23 q^{25} -2020.19 q^{26} +3347.42 q^{28} -4697.53 q^{29} -6758.81 q^{31} -1024.00 q^{32} -8636.18 q^{34} -19576.0 q^{35} -4269.37 q^{37} +9302.42 q^{38} +5988.44 q^{40} +7067.43 q^{41} +17047.8 q^{43} +10186.9 q^{44} +2116.00 q^{46} +7922.73 q^{47} +26963.4 q^{49} -22520.9 q^{50} +8080.74 q^{52} -8676.81 q^{53} -59574.0 q^{55} -13389.7 q^{56} +18790.1 q^{58} +31637.8 q^{59} -17881.9 q^{61} +27035.2 q^{62} +4096.00 q^{64} -47256.9 q^{65} +26682.9 q^{67} +34544.7 q^{68} +78304.0 q^{70} -33669.5 q^{71} +45101.0 q^{73} +17077.5 q^{74} -37209.7 q^{76} +133203. q^{77} -2634.20 q^{79} -23953.8 q^{80} -28269.7 q^{82} +42555.0 q^{83} -202021. q^{85} -68191.2 q^{86} -40747.7 q^{88} -85717.5 q^{89} +105663. q^{91} -8464.00 q^{92} -31690.9 q^{94} +217606. q^{95} -7080.31 q^{97} -107854. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8}+O(q^{10}) $ 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8	$ 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8} + 216 q^{10} + 6 q^{11} + 1248 q^{13} - 1392 q^{14} + 1024 q^{16} + 56 q^{17} + 1530 q^{19} - 864 q^{20} - 24 q^{22} - 2116 q^{23} + 8852 q^{25} - 4992 q^{26} + 5568 q^{28} - 2292 q^{29} + 1556 q^{31} - 4096 q^{32} - 224 q^{34} - 5688 q^{35} - 9586 q^{37} - 6120 q^{38} + 3456 q^{40} + 11768 q^{41} + 13758 q^{43} + 96 q^{44} + 8464 q^{46} - 10636 q^{47} + 11360 q^{49} - 35408 q^{50} + 19968 q^{52} - 26686 q^{53} - 28900 q^{55} - 22272 q^{56} + 9168 q^{58} + 9108 q^{59} - 37878 q^{61} - 6224 q^{62} + 16384 q^{64} + 72212 q^{65} - 23302 q^{67} + 896 q^{68} + 22752 q^{70} - 31728 q^{71} - 28340 q^{73} + 38344 q^{74} + 24480 q^{76} + 121276 q^{77} - 26668 q^{79} - 13824 q^{80} - 47072 q^{82} + 119026 q^{83} - 217876 q^{85} - 55032 q^{86} - 384 q^{88} + 148236 q^{89} + 89008 q^{91} - 33856 q^{92} + 42544 q^{94} + 399292 q^{95} - 16092 q^{97} - 45440 q^{98}+O(q^{100}) $ 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8 + 216 * q^10 + 6 * q^11 + 1248 * q^13 - 1392 * q^14 + 1024 * q^16 + 56 * q^17 + 1530 * q^19 - 864 * q^20 - 24 * q^22 - 2116 * q^23 + 8852 * q^25 - 4992 * q^26 + 5568 * q^28 - 2292 * q^29 + 1556 * q^31 - 4096 * q^32 - 224 * q^34 - 5688 * q^35 - 9586 * q^37 - 6120 * q^38 + 3456 * q^40 + 11768 * q^41 + 13758 * q^43 + 96 * q^44 + 8464 * q^46 - 10636 * q^47 + 11360 * q^49 - 35408 * q^50 + 19968 * q^52 - 26686 * q^53 - 28900 * q^55 - 22272 * q^56 + 9168 * q^58 + 9108 * q^59 - 37878 * q^61 - 6224 * q^62 + 16384 * q^64 + 72212 * q^65 - 23302 * q^67 + 896 * q^68 + 22752 * q^70 - 31728 * q^71 - 28340 * q^73 + 38344 * q^74 + 24480 * q^76 + 121276 * q^77 - 26668 * q^79 - 13824 * q^80 - 47072 * q^82 + 119026 * q^83 - 217876 * q^85 - 55032 * q^86 - 384 * q^88 + 148236 * q^89 + 89008 * q^91 - 33856 * q^92 + 42544 * q^94 + 399292 * q^95 - 16092 * q^97 - 45440 * q^98

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−93.5694	−1.67382	−0.836910	−	0.547340i	$-0.815640\pi$
$5$	−93.5694	−1.67382	−0.836910	+	0.547340i	$0.815640\pi$
$6$	0	0
$7$	209.214	1.61378	0.806891	−	0.590700i	$-0.201148\pi$
$7$	209.214	1.61378	0.806891	+	0.590700i	$0.201148\pi$
$8$	−64.0000	−0.353553
$9$	0	0
$10$	374.278	1.18357
$11$	636.683	1.58650	0.793252	−	0.608893i	$-0.208386\pi$
$11$	636.683	1.58650	0.793252	+	0.608893i	$0.208386\pi$
$12$	0	0
$13$	505.046	0.828844	0.414422	−	0.910085i	$-0.363984\pi$
$13$	505.046	0.828844	0.414422	+	0.910085i	$0.363984\pi$
$14$	−836.855	−1.14112
$15$	0	0
$16$	256.000	0.250000
$17$	2159.05	1.81192	0.905961	−	0.423361i	$-0.139150\pi$
$17$	2159.05	1.81192	0.905961	+	0.423361i	$0.139150\pi$
$18$	0	0
$19$	−2325.61	−1.47792	−0.738962	−	0.673747i	$-0.764684\pi$
$19$	−2325.61	−1.47792	−0.738962	+	0.673747i	$0.764684\pi$
$20$	−1497.11	−0.836910
$21$	0	0
$22$	−2546.73	−1.12183
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	5630.23	1.80168
$26$	−2020.19	−0.586081
$27$	0	0
$28$	3347.42	0.806891
$29$	−4697.53	−1.03723	−0.518614	−	0.855008i	$-0.673552\pi$
$29$	−4697.53	−1.03723	−0.518614	+	0.855008i	$0.673552\pi$
$30$	0	0
$31$	−6758.81	−1.26318	−0.631591	−	0.775302i	$-0.717598\pi$
$31$	−6758.81	−1.26318	−0.631591	+	0.775302i	$0.717598\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−8636.18	−1.28122
$35$	−19576.0	−2.70118
$36$	0	0
$37$	−4269.37	−0.512696	−0.256348	−	0.966585i	$-0.582519\pi$
$37$	−4269.37	−0.512696	−0.256348	+	0.966585i	$0.582519\pi$
$38$	9302.42	1.04505
$39$	0	0
$40$	5988.44	0.591785
$41$	7067.43	0.656602	0.328301	−	0.944573i	$-0.393524\pi$
$41$	7067.43	0.656602	0.328301	+	0.944573i	$0.393524\pi$
$42$	0	0
$43$	17047.8	1.40604	0.703020	−	0.711171i	$-0.251835\pi$
$43$	17047.8	1.40604	0.703020	+	0.711171i	$0.251835\pi$
$44$	10186.9	0.793252
$45$	0	0
$46$	2116.00	0.147442
$47$	7922.73	0.523155	0.261577	−	0.965183i	$-0.415757\pi$
$47$	7922.73	0.523155	0.261577	+	0.965183i	$0.415757\pi$
$48$	0	0
$49$	26963.4	1.60430
$50$	−22520.9	−1.27398
$51$	0	0
$52$	8080.74	0.414422
$53$	−8676.81	−0.424298	−0.212149	−	0.977237i	$-0.568046\pi$
$53$	−8676.81	−0.424298	−0.212149	+	0.977237i	$0.568046\pi$
$54$	0	0
$55$	−59574.0	−2.65552
$56$	−13389.7	−0.570558
$57$	0	0
$58$	18790.1	0.733432
$59$	31637.8	1.18325	0.591625	−	0.806214i	$-0.298487\pi$
$59$	31637.8	1.18325	0.591625	+	0.806214i	$0.298487\pi$
$60$	0	0
$61$	−17881.9	−0.615302	−0.307651	−	0.951499i	$-0.599543\pi$
$61$	−17881.9	−0.615302	−0.307651	+	0.951499i	$0.599543\pi$
$62$	27035.2	0.893205
$63$	0	0
$64$	4096.00	0.125000
$65$	−47256.9	−1.38734
$66$	0	0
$67$	26682.9	0.726184	0.363092	−	0.931753i	$-0.381721\pi$
$67$	26682.9	0.726184	0.363092	+	0.931753i	$0.381721\pi$
$68$	34544.7	0.905961
$69$	0	0
$70$	78304.0	1.91002
$71$	−33669.5	−0.792667	−0.396333	−	0.918107i	$-0.629718\pi$
$71$	−33669.5	−0.792667	−0.396333	+	0.918107i	$0.629718\pi$
$72$	0	0
$73$	45101.0	0.990556	0.495278	−	0.868734i	$-0.335066\pi$
$73$	45101.0	0.990556	0.495278	+	0.868734i	$0.335066\pi$
$74$	17077.5	0.362531
$75$	0	0
$76$	−37209.7	−0.738962
$77$	133203.	2.56027
$78$	0	0
$79$	−2634.20	−0.0474877	−0.0237438	−	0.999718i	$-0.507559\pi$
$79$	−2634.20	−0.0474877	−0.0237438	+	0.999718i	$0.507559\pi$
$80$	−23953.8	−0.418455
$81$	0	0
$82$	−28269.7	−0.464288
$83$	42555.0	0.678039	0.339020	−	0.940779i	$-0.389905\pi$
$83$	42555.0	0.678039	0.339020	+	0.940779i	$0.389905\pi$
$84$	0	0
$85$	−202021.	−3.03283
$86$	−68191.2	−0.994220
$87$	0	0
$88$	−40747.7	−0.560914
$89$	−85717.5	−1.14708	−0.573541	−	0.819177i	$-0.694431\pi$
$89$	−85717.5	−1.14708	−0.573541	+	0.819177i	$0.694431\pi$
$90$	0	0
$91$	105663.	1.33757
$92$	−8464.00	−0.104257
$93$	0	0
$94$	−31690.9	−0.369926
$95$	217606.	2.47378
$96$	0	0
$97$	−7080.31	−0.0764052	−0.0382026	−	0.999270i	$-0.512163\pi$
$97$	−7080.31	−0.0764052	−0.0382026	+	0.999270i	$0.512163\pi$
$98$	−107854.	−1.13441
$99$	0	0
$100$	90083.8	0.900838
$101$	−17026.2	−0.166079	−0.0830396	−	0.996546i	$-0.526463\pi$
$101$	−17026.2	−0.166079	−0.0830396	+	0.996546i	$0.526463\pi$
$102$	0	0
$103$	−58945.9	−0.547471	−0.273735	−	0.961805i	$-0.588259\pi$
$103$	−58945.9	−0.547471	−0.273735	+	0.961805i	$0.588259\pi$
$104$	−32323.0	−0.293041
$105$	0	0
$106$	34707.3	0.300024
$107$	77106.0	0.651071	0.325536	−	0.945530i	$-0.394455\pi$
$107$	77106.0	0.651071	0.325536	+	0.945530i	$0.394455\pi$
$108$	0	0
$109$	−143714.	−1.15859	−0.579297	−	0.815116i	$-0.696673\pi$
$109$	−143714.	−1.15859	−0.579297	+	0.815116i	$0.696673\pi$
$110$	238296.	1.87774
$111$	0	0
$112$	53558.7	0.403446
$113$	93719.1	0.690449	0.345225	−	0.938520i	$-0.387803\pi$
$113$	93719.1	0.690449	0.345225	+	0.938520i	$0.387803\pi$
$114$	0	0
$115$	49498.2	0.349016
$116$	−75160.5	−0.518614
$117$	0	0
$118$	−126551.	−0.836684
$119$	451702.	2.92405
$120$	0	0
$121$	244314.	1.51700
$122$	71527.5	0.435084
$123$	0	0
$124$	−108141.	−0.631591
$125$	−234413.	−1.34186
$126$	0	0
$127$	107601.	0.591978	0.295989	−	0.955191i	$-0.404351\pi$
$127$	107601.	0.591978	0.295989	+	0.955191i	$0.404351\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	189028.	0.980995
$131$	−21595.8	−0.109949	−0.0549746	−	0.998488i	$-0.517508\pi$
$131$	−21595.8	−0.109949	−0.0549746	+	0.998488i	$0.517508\pi$
$132$	0	0
$133$	−486549.	−2.38505
$134$	−106732.	−0.513489
$135$	0	0
$136$	−138179.	−0.640611
$137$	349995.	1.59317	0.796583	−	0.604530i	$-0.206639\pi$
$137$	349995.	1.59317	0.796583	+	0.604530i	$0.206639\pi$
$138$	0	0
$139$	57776.5	0.253638	0.126819	−	0.991926i	$-0.459523\pi$
$139$	57776.5	0.253638	0.126819	+	0.991926i	$0.459523\pi$
$140$	−313216.	−1.35059
$141$	0	0
$142$	134678.	0.560500
$143$	321554.	1.31496
$144$	0	0
$145$	439545.	1.73614
$146$	−180404.	−0.700429
$147$	0	0
$148$	−68309.9	−0.256348
$149$	−370736.	−1.36804	−0.684021	−	0.729462i	$-0.739771\pi$
$149$	−370736.	−1.36804	−0.684021	+	0.729462i	$0.739771\pi$
$150$	0	0
$151$	525493.	1.87553	0.937765	−	0.347269i	$-0.112891\pi$
$151$	525493.	1.87553	0.937765	+	0.347269i	$0.112891\pi$
$152$	148839.	0.522525
$153$	0	0
$154$	−532811.	−1.81039
$155$	632418.	2.11434
$156$	0	0
$157$	300900.	0.974256	0.487128	−	0.873331i	$-0.338045\pi$
$157$	300900.	0.974256	0.487128	+	0.873331i	$0.338045\pi$
$158$	10536.8	0.0335789
$159$	0	0
$160$	95815.1	0.295892
$161$	−110674.	−0.336497
$162$	0	0
$163$	202799.	0.597857	0.298929	−	0.954275i	$-0.403371\pi$
$163$	202799.	0.597857	0.298929	+	0.954275i	$0.403371\pi$
$164$	113079.	0.328301
$165$	0	0
$166$	−170220.	−0.479446
$167$	638454.	1.77149	0.885745	−	0.464173i	$-0.153648\pi$
$167$	638454.	1.77149	0.885745	+	0.464173i	$0.153648\pi$
$168$	0	0
$169$	−116221.	−0.313017
$170$	808082.	2.14454
$171$	0	0
$172$	272765.	0.703020
$173$	196106.	0.498167	0.249084	−	0.968482i	$-0.419871\pi$
$173$	196106.	0.498167	0.249084	+	0.968482i	$0.419871\pi$
$174$	0	0
$175$	1.17792e6	2.90751
$176$	162991.	0.396626
$177$	0	0
$178$	342870.	0.811110
$179$	−485142.	−1.13171	−0.565857	−	0.824503i	$-0.691455\pi$
$179$	−485142.	−1.13171	−0.565857	+	0.824503i	$0.691455\pi$
$180$	0	0
$181$	533145.	1.20962	0.604810	−	0.796370i	$-0.293249\pi$
$181$	533145.	1.20962	0.604810	+	0.796370i	$0.293249\pi$
$182$	−422651.	−0.945808
$183$	0	0
$184$	33856.0	0.0737210
$185$	399483.	0.858161
$186$	0	0
$187$	1.37463e6	2.87462
$188$	126764.	0.261577
$189$	0	0
$190$	−870422.	−1.74923
$191$	−362661.	−0.719313	−0.359656	−	0.933085i	$-0.617106\pi$
$191$	−362661.	−0.719313	−0.359656	+	0.933085i	$0.617106\pi$
$192$	0	0
$193$	393545.	0.760503	0.380252	−	0.924883i	$-0.375837\pi$
$193$	393545.	0.760503	0.380252	+	0.924883i	$0.375837\pi$
$194$	28321.2	0.0540266
$195$	0	0
$196$	431414.	0.802148
$197$	−571241.	−1.04871	−0.524353	−	0.851501i	$-0.675693\pi$
$197$	−571241.	−1.04871	−0.524353	+	0.851501i	$0.675693\pi$
$198$	0	0
$199$	−21096.9	−0.0377646	−0.0188823	−	0.999822i	$-0.506011\pi$
$199$	−21096.9	−0.0377646	−0.0188823	+	0.999822i	$0.506011\pi$
$200$	−360335.	−0.636988
$201$	0	0
$202$	68105.0	0.117436
$203$	−982788.	−1.67386
$204$	0	0
$205$	−661296.	−1.09903
$206$	235784.	0.387120
$207$	0	0
$208$	129292.	0.207211
$209$	−1.48067e6	−2.34473
$210$	0	0
$211$	−1.25351e6	−1.93830	−0.969149	−	0.246476i	$-0.920727\pi$
$211$	−1.25351e6	−1.93830	−0.969149	+	0.246476i	$0.920727\pi$
$212$	−138829.	−0.212149
$213$	0	0
$214$	−308424.	−0.460377
$215$	−1.59515e6	−2.35346
$216$	0	0
$217$	−1.41404e6	−2.03850
$218$	574854.	0.819250
$219$	0	0
$220$	−953184.	−1.32776
$221$	1.09042e6	1.50180
$222$	0	0
$223$	94483.5	0.127231	0.0636156	−	0.997974i	$-0.479737\pi$
$223$	94483.5	0.127231	0.0636156	+	0.997974i	$0.479737\pi$
$224$	−214235.	−0.285279
$225$	0	0
$226$	−374876.	−0.488221
$227$	433733.	0.558674	0.279337	−	0.960193i	$-0.409885\pi$
$227$	433733.	0.558674	0.279337	+	0.960193i	$0.409885\pi$
$228$	0	0
$229$	413501.	0.521061	0.260530	−	0.965466i	$-0.416103\pi$
$229$	413501.	0.521061	0.260530	+	0.965466i	$0.416103\pi$
$230$	−197993.	−0.246791
$231$	0	0
$232$	300642.	0.366716
$233$	560757.	0.676683	0.338341	−	0.941023i	$-0.390134\pi$
$233$	560757.	0.676683	0.338341	+	0.941023i	$0.390134\pi$
$234$	0	0
$235$	−741325.	−0.875667
$236$	506205.	0.591625
$237$	0	0
$238$	−1.80681e6	−2.06761
$239$	355253.	0.402294	0.201147	−	0.979561i	$-0.435533\pi$
$239$	355253.	0.402294	0.201147	+	0.979561i	$0.435533\pi$
$240$	0	0
$241$	798917.	0.886052	0.443026	−	0.896509i	$-0.353905\pi$
$241$	798917.	0.886052	0.443026	+	0.896509i	$0.353905\pi$
$242$	−977254.	−1.07268
$243$	0	0
$244$	−286110.	−0.307651
$245$	−2.52295e6	−2.68530
$246$	0	0
$247$	−1.17454e6	−1.22497
$248$	432564.	0.446602
$249$	0	0
$250$	937653.	0.948838
$251$	1.19159e6	1.19383	0.596914	−	0.802306i	$-0.296394\pi$
$251$	1.19159e6	1.19383	0.596914	+	0.802306i	$0.296394\pi$
$252$	0	0
$253$	−336805.	−0.330809
$254$	−430403.	−0.418592
$255$	0	0
$256$	65536.0	0.0625000
$257$	−398929.	−0.376758	−0.188379	−	0.982096i	$-0.560323\pi$
$257$	−398929.	−0.376758	−0.188379	+	0.982096i	$0.560323\pi$
$258$	0	0
$259$	−893211.	−0.827380
$260$	−756110.	−0.693668
$261$	0	0
$262$	86383.4	0.0777458
$263$	−98496.8	−0.0878077	−0.0439039	−	0.999036i	$-0.513980\pi$
$263$	−98496.8	−0.0878077	−0.0439039	+	0.999036i	$0.513980\pi$
$264$	0	0
$265$	811884.	0.710198
$266$	1.94619e6	1.68648
$267$	0	0
$268$	426927.	0.363092
$269$	879912.	0.741410	0.370705	−	0.928751i	$-0.379116\pi$
$269$	879912.	0.741410	0.370705	+	0.928751i	$0.379116\pi$
$270$	0	0
$271$	−881037.	−0.728737	−0.364368	−	0.931255i	$-0.618715\pi$
$271$	−881037.	−0.728737	−0.364368	+	0.931255i	$0.618715\pi$
$272$	552716.	0.452980
$273$	0	0
$274$	−1.39998e6	−1.12654
$275$	3.58467e6	2.85836
$276$	0	0
$277$	−258437.	−0.202374	−0.101187	−	0.994867i	$-0.532264\pi$
$277$	−258437.	−0.202374	−0.101187	+	0.994867i	$0.532264\pi$
$278$	−231106.	−0.179349
$279$	0	0
$280$	1.25286e6	0.955012
$281$	−1.26172e6	−0.953226	−0.476613	−	0.879113i	$-0.658136\pi$
$281$	−1.26172e6	−0.953226	−0.476613	+	0.879113i	$0.658136\pi$
$282$	0	0
$283$	−2.31710e6	−1.71981	−0.859903	−	0.510457i	$-0.829476\pi$
$283$	−2.31710e6	−1.71981	−0.859903	+	0.510457i	$0.829476\pi$
$284$	−538712.	−0.396333
$285$	0	0
$286$	−1.28622e6	−0.929820
$287$	1.47860e6	1.05961
$288$	0	0
$289$	3.24162e6	2.28306
$290$	−1.75818e6	−1.22763
$291$	0	0
$292$	721616.	0.495278
$293$	−1.88310e6	−1.28146	−0.640728	−	0.767768i	$-0.721367\pi$
$293$	−1.88310e6	−1.28146	−0.640728	+	0.767768i	$0.721367\pi$
$294$	0	0
$295$	−2.96033e6	−1.98055
$296$	273240.	0.181265
$297$	0	0
$298$	1.48295e6	0.967352
$299$	−267170.	−0.172826
$300$	0	0
$301$	3.56664e6	2.26904
$302$	−2.10197e6	−1.32620
$303$	0	0
$304$	−595355.	−0.369481
$305$	1.67320e6	1.02991
$306$	0	0
$307$	409511.	0.247982	0.123991	−	0.992283i	$-0.460431\pi$
$307$	409511.	0.247982	0.123991	+	0.992283i	$0.460431\pi$
$308$	2.13124e6	1.28014
$309$	0	0
$310$	−2.52967e6	−1.49506
$311$	2.30485e6	1.35127	0.675634	−	0.737237i	$-0.263870\pi$
$311$	2.30485e6	1.35127	0.675634	+	0.737237i	$0.263870\pi$
$312$	0	0
$313$	−1.04868e6	−0.605037	−0.302519	−	0.953143i	$-0.597827\pi$
$313$	−1.04868e6	−0.605037	−0.302519	+	0.953143i	$0.597827\pi$
$314$	−1.20360e6	−0.688903
$315$	0	0
$316$	−42147.2	−0.0237438
$317$	1.74434e6	0.974951	0.487476	−	0.873137i	$-0.337918\pi$
$317$	1.74434e6	0.974951	0.487476	+	0.873137i	$0.337918\pi$
$318$	0	0
$319$	−2.99084e6	−1.64557
$320$	−383260.	−0.209228
$321$	0	0
$322$	442696.	0.237939
$323$	−5.02109e6	−2.67788
$324$	0	0
$325$	2.84353e6	1.49331
$326$	−811198.	−0.422749
$327$	0	0
$328$	−452316.	−0.232144
$329$	1.65754e6	0.844258
$330$	0	0
$331$	3.16113e6	1.58589	0.792945	−	0.609294i	$-0.208547\pi$
$331$	3.16113e6	1.58589	0.792945	+	0.609294i	$0.208547\pi$
$332$	680879.	0.339020
$333$	0	0
$334$	−2.55382e6	−1.25263
$335$	−2.49671e6	−1.21550
$336$	0	0
$337$	890560.	0.427158	0.213579	−	0.976926i	$-0.431488\pi$
$337$	890560.	0.427158	0.213579	+	0.976926i	$0.431488\pi$
$338$	464885.	0.221337
$339$	0	0
$340$	−3.23233e6	−1.51642
$341$	−4.30322e6	−2.00404
$342$	0	0
$343$	2.12486e6	0.975201
$344$	−1.09106e6	−0.497110
$345$	0	0
$346$	−784424.	−0.352258
$347$	3.93877e6	1.75605	0.878026	−	0.478613i	$-0.158860\pi$
$347$	3.93877e6	1.75605	0.878026	+	0.478613i	$0.158860\pi$
$348$	0	0
$349$	1.70283e6	0.748354	0.374177	−	0.927357i	$-0.377925\pi$
$349$	1.70283e6	0.748354	0.374177	+	0.927357i	$0.377925\pi$
$350$	−4.71169e6	−2.05592
$351$	0	0
$352$	−651963.	−0.280457
$353$	−2.21690e6	−0.946912	−0.473456	−	0.880817i	$-0.656994\pi$
$353$	−2.21690e6	−0.946912	−0.473456	+	0.880817i	$0.656994\pi$
$354$	0	0
$355$	3.15043e6	1.32678
$356$	−1.37148e6	−0.573541
$357$	0	0
$358$	1.94057e6	0.800243
$359$	4.05106e6	1.65895	0.829474	−	0.558546i	$-0.188641\pi$
$359$	4.05106e6	1.65895	0.829474	+	0.558546i	$0.188641\pi$
$360$	0	0
$361$	2.93234e6	1.18426
$362$	−2.13258e6	−0.855330
$363$	0	0
$364$	1.69060e6	0.668787
$365$	−4.22008e6	−1.65801
$366$	0	0
$367$	−1.40886e6	−0.546013	−0.273006	−	0.962012i	$-0.588018\pi$
$367$	−1.40886e6	−0.546013	−0.273006	+	0.962012i	$0.588018\pi$
$368$	−135424.	−0.0521286
$369$	0	0
$370$	−1.59793e6	−0.606811
$371$	−1.81531e6	−0.684724
$372$	0	0
$373$	−667663.	−0.248476	−0.124238	−	0.992252i	$-0.539649\pi$
$373$	−667663.	−0.248476	−0.124238	+	0.992252i	$0.539649\pi$
$374$	−5.49851e6	−2.03266
$375$	0	0
$376$	−507055.	−0.184963
$377$	−2.37247e6	−0.859701
$378$	0	0
$379$	−1.44604e6	−0.517110	−0.258555	−	0.965997i	$-0.583246\pi$
$379$	−1.44604e6	−0.517110	−0.258555	+	0.965997i	$0.583246\pi$
$380$	3.48169e6	1.23689
$381$	0	0
$382$	1.45065e6	0.508631
$383$	3.82328e6	1.33180	0.665901	−	0.746040i	$-0.268047\pi$
$383$	3.82328e6	1.33180	0.665901	+	0.746040i	$0.268047\pi$
$384$	0	0
$385$	−1.24637e7	−4.28544
$386$	−1.57418e6	−0.537757
$387$	0	0
$388$	−113285.	−0.0382026
$389$	−4.30704e6	−1.44313	−0.721565	−	0.692347i	$-0.756577\pi$
$389$	−4.30704e6	−1.44313	−0.721565	+	0.692347i	$0.756577\pi$
$390$	0	0
$391$	−1.14213e6	−0.377812
$392$	−1.72566e6	−0.567204
$393$	0	0
$394$	2.28497e6	0.741548
$395$	246481.	0.0794859
$396$	0	0
$397$	2.83356e6	0.902312	0.451156	−	0.892445i	$-0.351012\pi$
$397$	2.83356e6	0.902312	0.451156	+	0.892445i	$0.351012\pi$
$398$	84387.5	0.0267036
$399$	0	0
$400$	1.44134e6	0.450419
$401$	1.66982e6	0.518573	0.259286	−	0.965800i	$-0.416513\pi$
$401$	1.66982e6	0.518573	0.259286	+	0.965800i	$0.416513\pi$
$402$	0	0
$403$	−3.41351e6	−1.04698
$404$	−272420.	−0.0830396
$405$	0	0
$406$	3.93115e6	1.18360
$407$	−2.71823e6	−0.813394
$408$	0	0
$409$	2.55146e6	0.754188	0.377094	−	0.926175i	$-0.376923\pi$
$409$	2.55146e6	0.754188	0.377094	+	0.926175i	$0.376923\pi$
$410$	2.64518e6	0.777134
$411$	0	0
$412$	−943135.	−0.273735
$413$	6.61906e6	1.90951
$414$	0	0
$415$	−3.98184e6	−1.13492
$416$	−517168.	−0.146520
$417$	0	0
$418$	5.92269e6	1.65798
$419$	−1.00113e6	−0.278582	−0.139291	−	0.990251i	$-0.544482\pi$
$419$	−1.00113e6	−0.278582	−0.139291	+	0.990251i	$0.544482\pi$
$420$	0	0
$421$	−2.94280e6	−0.809198	−0.404599	−	0.914494i	$-0.632589\pi$
$421$	−2.94280e6	−0.809198	−0.404599	+	0.914494i	$0.632589\pi$
$422$	5.01403e6	1.37058
$423$	0	0
$424$	555316.	0.150012
$425$	1.21559e7	3.26449
$426$	0	0
$427$	−3.74113e6	−0.992964
$428$	1.23370e6	0.325536
$429$	0	0
$430$	6.38061e6	1.66415
$431$	−437513.	−0.113448	−0.0567242	−	0.998390i	$-0.518066\pi$
$431$	−437513.	−0.113448	−0.0567242	+	0.998390i	$0.518066\pi$
$432$	0	0
$433$	−882971.	−0.226322	−0.113161	−	0.993577i	$-0.536098\pi$
$433$	−882971.	−0.226322	−0.113161	+	0.993577i	$0.536098\pi$
$434$	5.65614e6	1.44144
$435$	0	0
$436$	−2.29942e6	−0.579297
$437$	1.23025e6	0.308168
$438$	0	0
$439$	3.48615e6	0.863345	0.431672	−	0.902030i	$-0.357924\pi$
$439$	3.48615e6	0.863345	0.431672	+	0.902030i	$0.357924\pi$
$440$	3.81274e6	0.938869
$441$	0	0
$442$	−4.36167e6	−1.06193
$443$	−5.44619e6	−1.31851	−0.659255	−	0.751920i	$-0.729128\pi$
$443$	−5.44619e6	−1.31851	−0.659255	+	0.751920i	$0.729128\pi$
$444$	0	0
$445$	8.02054e6	1.92001
$446$	−377934.	−0.0899661
$447$	0	0
$448$	856939.	0.201723
$449$	−5.02197e6	−1.17560	−0.587798	−	0.809007i	$-0.700005\pi$
$449$	−5.02197e6	−1.17560	−0.587798	+	0.809007i	$0.700005\pi$
$450$	0	0
$451$	4.49971e6	1.04170
$452$	1.49950e6	0.345225
$453$	0	0
$454$	−1.73493e6	−0.395042
$455$	−9.88679e6	−2.23886
$456$	0	0
$457$	7.05819e6	1.58090	0.790448	−	0.612530i	$-0.209848\pi$
$457$	7.05819e6	1.58090	0.790448	+	0.612530i	$0.209848\pi$
$458$	−1.65401e6	−0.368446
$459$	0	0
$460$	791971.	0.174508
$461$	4.77398e6	1.04623	0.523116	−	0.852262i	$-0.324769\pi$
$461$	4.77398e6	1.04623	0.523116	+	0.852262i	$0.324769\pi$
$462$	0	0
$463$	2.52663e6	0.547758	0.273879	−	0.961764i	$-0.411693\pi$
$463$	2.52663e6	0.547758	0.273879	+	0.961764i	$0.411693\pi$
$464$	−1.20257e6	−0.259307
$465$	0	0
$466$	−2.24303e6	−0.478487
$467$	1.84535e6	0.391550	0.195775	−	0.980649i	$-0.437278\pi$
$467$	1.84535e6	0.391550	0.195775	+	0.980649i	$0.437278\pi$
$468$	0	0
$469$	5.58243e6	1.17190
$470$	2.96530e6	0.619190
$471$	0	0
$472$	−2.02482e6	−0.418342
$473$	1.08540e7	2.23069
$474$	0	0
$475$	−1.30937e7	−2.66274
$476$	7.22723e6	1.46202
$477$	0	0
$478$	−1.42101e6	−0.284465
$479$	−2.77301e6	−0.552221	−0.276110	−	0.961126i	$-0.589046\pi$
$479$	−2.77301e6	−0.552221	−0.276110	+	0.961126i	$0.589046\pi$
$480$	0	0
$481$	−2.15623e6	−0.424945
$482$	−3.19567e6	−0.626533
$483$	0	0
$484$	3.90902e6	0.758498
$485$	662500.	0.127889
$486$	0	0
$487$	5.30448e6	1.01349	0.506746	−	0.862095i	$-0.330848\pi$
$487$	5.30448e6	1.01349	0.506746	+	0.862095i	$0.330848\pi$
$488$	1.14444e6	0.217542
$489$	0	0
$490$	1.00918e7	1.89880
$491$	−9.27551e6	−1.73634	−0.868169	−	0.496269i	$-0.834703\pi$
$491$	−9.27551e6	−1.73634	−0.868169	+	0.496269i	$0.834703\pi$
$492$	0	0
$493$	−1.01422e7	−1.87938
$494$	4.69816e6	0.866184
$495$	0	0
$496$	−1.73026e6	−0.315796
$497$	−7.04412e6	−1.27919
$498$	0	0
$499$	−1.92766e6	−0.346561	−0.173281	−	0.984872i	$-0.555437\pi$
$499$	−1.92766e6	−0.346561	−0.173281	+	0.984872i	$0.555437\pi$
$500$	−3.75061e6	−0.670930
$501$	0	0
$502$	−4.76635e6	−0.844163
$503$	7.46418e6	1.31541	0.657706	−	0.753274i	$-0.271527\pi$
$503$	7.46418e6	1.31541	0.657706	+	0.753274i	$0.271527\pi$
$504$	0	0
$505$	1.59314e6	0.277987
$506$	1.34722e6	0.233917
$507$	0	0
$508$	1.72161e6	0.295989
$509$	−9.77239e6	−1.67188	−0.835942	−	0.548817i	$-0.815078\pi$
$509$	−9.77239e6	−1.67188	−0.835942	+	0.548817i	$0.815078\pi$
$510$	0	0
$511$	9.43575e6	1.59854
$512$	−262144.	−0.0441942
$513$	0	0
$514$	1.59572e6	0.266408
$515$	5.51554e6	0.916367
$516$	0	0
$517$	5.04426e6	0.829987
$518$	3.57284e6	0.585046
$519$	0	0
$520$	3.02444e6	0.490497
$521$	3.61522e6	0.583499	0.291750	−	0.956495i	$-0.405763\pi$
$521$	3.61522e6	0.583499	0.291750	+	0.956495i	$0.405763\pi$
$522$	0	0
$523$	5.00023e6	0.799348	0.399674	−	0.916657i	$-0.369123\pi$
$523$	5.00023e6	0.799348	0.399674	+	0.916657i	$0.369123\pi$
$524$	−345534.	−0.0549746
$525$	0	0
$526$	393987.	0.0620894
$527$	−1.45926e7	−2.28879
$528$	0	0
$529$	279841.	0.0434783
$530$	−3.24754e6	−0.502186
$531$	0	0
$532$	−7.78478e6	−1.19252
$533$	3.56938e6	0.544221
$534$	0	0
$535$	−7.21476e6	−1.08978
$536$	−1.70771e6	−0.256745
$537$	0	0
$538$	−3.51965e6	−0.524256
$539$	1.71671e7	2.54522
$540$	0	0
$541$	−1.06059e6	−0.155795	−0.0778974	−	0.996961i	$-0.524821\pi$
$541$	−1.06059e6	−0.155795	−0.0778974	+	0.996961i	$0.524821\pi$
$542$	3.52415e6	0.515295
$543$	0	0
$544$	−2.21086e6	−0.320306
$545$	1.34472e7	1.93928
$546$	0	0
$547$	−4.23924e6	−0.605787	−0.302893	−	0.953024i	$-0.597953\pi$
$547$	−4.23924e6	−0.605787	−0.302893	+	0.953024i	$0.597953\pi$
$548$	5.59992e6	0.796583
$549$	0	0
$550$	−1.43387e7	−2.02117
$551$	1.09246e7	1.53295
$552$	0	0
$553$	−551111.	−0.0766348
$554$	1.03375e6	0.143100
$555$	0	0
$556$	924424.	0.126819
$557$	1.09013e7	1.48882	0.744409	−	0.667724i	$-0.232731\pi$
$557$	1.09013e7	1.48882	0.744409	+	0.667724i	$0.232731\pi$
$558$	0	0
$559$	8.60993e6	1.16539
$560$	−5.01146e6	−0.675296
$561$	0	0
$562$	5.04687e6	0.674033
$563$	7.31747e6	0.972949	0.486475	−	0.873695i	$-0.338283\pi$
$563$	7.31747e6	0.972949	0.486475	+	0.873695i	$0.338283\pi$
$564$	0	0
$565$	−8.76924e6	−1.15569
$566$	9.26842e6	1.21609
$567$	0	0
$568$	2.15485e6	0.280250
$569$	−6.00142e6	−0.777094	−0.388547	−	0.921429i	$-0.627023\pi$
$569$	−6.00142e6	−0.777094	−0.388547	+	0.921429i	$0.627023\pi$
$570$	0	0
$571$	−622385.	−0.0798857	−0.0399429	−	0.999202i	$-0.512718\pi$
$571$	−622385.	−0.0798857	−0.0399429	+	0.999202i	$0.512718\pi$
$572$	5.14487e6	0.657482
$573$	0	0
$574$	−5.91442e6	−0.749259
$575$	−2.97839e6	−0.375675
$576$	0	0
$577$	−9.67960e6	−1.21037	−0.605185	−	0.796085i	$-0.706901\pi$
$577$	−9.67960e6	−1.21037	−0.605185	+	0.796085i	$0.706901\pi$
$578$	−1.29665e7	−1.61437
$579$	0	0
$580$	7.03272e6	0.868068
$581$	8.90308e6	1.09421
$582$	0	0
$583$	−5.52437e6	−0.673150
$584$	−2.88647e6	−0.350215
$585$	0	0
$586$	7.53238e6	0.906126
$587$	−1.12202e7	−1.34402	−0.672012	−	0.740541i	$-0.734570\pi$
$587$	−1.12202e7	−1.34402	−0.672012	+	0.740541i	$0.734570\pi$
$588$	0	0
$589$	1.57183e7	1.86689
$590$	1.18413e7	1.40046
$591$	0	0
$592$	−1.09296e6	−0.128174
$593$	7.38730e6	0.862678	0.431339	−	0.902190i	$-0.358041\pi$
$593$	7.38730e6	0.862678	0.431339	+	0.902190i	$0.358041\pi$
$594$	0	0
$595$	−4.22655e7	−4.89433
$596$	−5.93178e6	−0.684021
$597$	0	0
$598$	1.06868e6	0.122206
$599$	−217174.	−0.0247309	−0.0123655	−	0.999924i	$-0.503936\pi$
$599$	−217174.	−0.0247309	−0.0123655	+	0.999924i	$0.503936\pi$
$600$	0	0
$601$	1.80786e6	0.204164	0.102082	−	0.994776i	$-0.467450\pi$
$601$	1.80786e6	0.204164	0.102082	+	0.994776i	$0.467450\pi$
$602$	−1.42665e7	−1.60445
$603$	0	0
$604$	8.40788e6	0.937765
$605$	−2.28603e7	−2.53918
$606$	0	0
$607$	2.34258e6	0.258062	0.129031	−	0.991641i	$-0.458813\pi$
$607$	2.34258e6	0.258062	0.129031	+	0.991641i	$0.458813\pi$
$608$	2.38142e6	0.261263
$609$	0	0
$610$	−6.69278e6	−0.728253
$611$	4.00135e6	0.433614
$612$	0	0
$613$	1.53780e7	1.65291	0.826453	−	0.563005i	$-0.190355\pi$
$613$	1.53780e7	1.65291	0.826453	+	0.563005i	$0.190355\pi$
$614$	−1.63805e6	−0.175350
$615$	0	0
$616$	−8.52497e6	−0.905193
$617$	−1.34656e7	−1.42401	−0.712005	−	0.702174i	$-0.752213\pi$
$617$	−1.34656e7	−1.42401	−0.712005	+	0.702174i	$0.752213\pi$
$618$	0	0
$619$	5.38044e6	0.564405	0.282203	−	0.959355i	$-0.408935\pi$
$619$	5.38044e6	0.564405	0.282203	+	0.959355i	$0.408935\pi$
$620$	1.01187e7	1.05717
$621$	0	0
$622$	−9.21940e6	−0.955491
$623$	−1.79333e7	−1.85114
$624$	0	0
$625$	4.33943e6	0.444358
$626$	4.19472e6	0.427826
$627$	0	0
$628$	4.81440e6	0.487128
$629$	−9.21777e6	−0.928964
$630$	0	0
$631$	−8.74264e6	−0.874117	−0.437058	−	0.899433i	$-0.643980\pi$
$631$	−8.74264e6	−0.874117	−0.437058	+	0.899433i	$0.643980\pi$
$632$	168589.	0.0167894
$633$	0	0
$634$	−6.97736e6	−0.689395
$635$	−1.00681e7	−0.990866
$636$	0	0
$637$	1.36178e7	1.32971
$638$	1.19633e7	1.16359
$639$	0	0
$640$	1.53304e6	0.147946
$641$	−1.14883e7	−1.10436	−0.552180	−	0.833725i	$-0.686204\pi$
$641$	−1.14883e7	−1.10436	−0.552180	+	0.833725i	$0.686204\pi$
$642$	0	0
$643$	1.56203e7	1.48992	0.744959	−	0.667111i	$-0.232469\pi$
$643$	1.56203e7	1.48992	0.744959	+	0.667111i	$0.232469\pi$
$644$	−1.77079e6	−0.168248
$645$	0	0
$646$	2.00844e7	1.89355
$647$	−733485.	−0.0688860	−0.0344430	−	0.999407i	$-0.510966\pi$
$647$	−733485.	−0.0688860	−0.0344430	+	0.999407i	$0.510966\pi$
$648$	0	0
$649$	2.01432e7	1.87723
$650$	−1.13741e7	−1.05593
$651$	0	0
$652$	3.24479e6	0.298929
$653$	9.92739e6	0.911071	0.455536	−	0.890218i	$-0.349448\pi$
$653$	9.92739e6	0.911071	0.455536	+	0.890218i	$0.349448\pi$
$654$	0	0
$655$	2.02071e6	0.184035
$656$	1.80926e6	0.164150
$657$	0	0
$658$	−6.63017e6	−0.596981
$659$	1.63658e7	1.46800	0.733998	−	0.679152i	$-0.237652\pi$
$659$	1.63658e7	1.46800	0.733998	+	0.679152i	$0.237652\pi$
$660$	0	0
$661$	−1.80214e7	−1.60430	−0.802149	−	0.597124i	$-0.796310\pi$
$661$	−1.80214e7	−1.60430	−0.802149	+	0.597124i	$0.796310\pi$
$662$	−1.26445e7	−1.12139
$663$	0	0
$664$	−2.72352e6	−0.239723
$665$	4.55261e7	3.99214
$666$	0	0
$667$	2.48499e6	0.216277
$668$	1.02153e7	0.885745
$669$	0	0
$670$	9.98682e6	0.859489
$671$	−1.13851e7	−0.976179
$672$	0	0
$673$	−4.82777e6	−0.410874	−0.205437	−	0.978670i	$-0.565862\pi$
$673$	−4.82777e6	−0.410874	−0.205437	+	0.978670i	$0.565862\pi$
$674$	−3.56224e6	−0.302046
$675$	0	0
$676$	−1.85954e6	−0.156509
$677$	−1.40571e7	−1.17876	−0.589379	−	0.807856i	$-0.700628\pi$
$677$	−1.40571e7	−1.17876	−0.589379	+	0.807856i	$0.700628\pi$
$678$	0	0
$679$	−1.48130e6	−0.123301
$680$	1.29293e7	1.07227
$681$	0	0
$682$	1.72129e7	1.41707
$683$	−7.56353e6	−0.620401	−0.310201	−	0.950671i	$-0.600396\pi$
$683$	−7.56353e6	−0.620401	−0.310201	+	0.950671i	$0.600396\pi$
$684$	0	0
$685$	−3.27489e7	−2.66667
$686$	−8.49942e6	−0.689571
$687$	0	0
$688$	4.36424e6	0.351510
$689$	−4.38219e6	−0.351677
$690$	0	0
$691$	7.54802e6	0.601365	0.300682	−	0.953724i	$-0.402786\pi$
$691$	7.54802e6	0.601365	0.300682	+	0.953724i	$0.402786\pi$
$692$	3.13769e6	0.249084
$693$	0	0
$694$	−1.57551e7	−1.24172
$695$	−5.40611e6	−0.424544
$696$	0	0
$697$	1.52589e7	1.18971
$698$	−6.81131e6	−0.529166
$699$	0	0
$700$	1.88468e7	1.45376
$701$	2.51426e6	0.193248	0.0966240	−	0.995321i	$-0.469196\pi$
$701$	2.51426e6	0.193248	0.0966240	+	0.995321i	$0.469196\pi$
$702$	0	0
$703$	9.92888e6	0.757725
$704$	2.60785e6	0.198313
$705$	0	0
$706$	8.86761e6	0.669568
$707$	−3.56212e6	−0.268016
$708$	0	0
$709$	−2.49752e7	−1.86592	−0.932959	−	0.359982i	$-0.882783\pi$
$709$	−2.49752e7	−1.86592	−0.932959	+	0.359982i	$0.882783\pi$
$710$	−1.26017e7	−0.938176
$711$	0	0
$712$	5.48592e6	0.405555
$713$	3.57541e6	0.263392
$714$	0	0
$715$	−3.00876e7	−2.20101
$716$	−7.76228e6	−0.565857
$717$	0	0
$718$	−1.62042e7	−1.17305
$719$	−1.69567e7	−1.22326	−0.611631	−	0.791143i	$-0.709486\pi$
$719$	−1.69567e7	−1.22326	−0.611631	+	0.791143i	$0.709486\pi$
$720$	0	0
$721$	−1.23323e7	−0.883499
$722$	−1.17294e7	−0.837398
$723$	0	0
$724$	8.53031e6	0.604810
$725$	−2.64482e7	−1.86875
$726$	0	0
$727$	2.44848e7	1.71815	0.859075	−	0.511850i	$-0.171040\pi$
$727$	2.44848e7	1.71815	0.859075	+	0.511850i	$0.171040\pi$
$728$	−6.76241e6	−0.472904
$729$	0	0
$730$	1.68803e7	1.17239
$731$	3.68070e7	2.54763
$732$	0	0
$733$	−1.52880e7	−1.05097	−0.525486	−	0.850802i	$-0.676117\pi$
$733$	−1.52880e7	−1.05097	−0.525486	+	0.850802i	$0.676117\pi$
$734$	5.63544e6	0.386089
$735$	0	0
$736$	541696.	0.0368605
$737$	1.69886e7	1.15209
$738$	0	0
$739$	1.40358e7	0.945421	0.472710	−	0.881218i	$-0.343276\pi$
$739$	1.40358e7	0.945421	0.472710	+	0.881218i	$0.343276\pi$
$740$	6.39172e6	0.429080
$741$	0	0
$742$	7.26123e6	0.484173
$743$	1.43183e7	0.951526	0.475763	−	0.879573i	$-0.342172\pi$
$743$	1.43183e7	0.951526	0.475763	+	0.879573i	$0.342172\pi$
$744$	0	0
$745$	3.46896e7	2.28986
$746$	2.67065e6	0.175699
$747$	0	0
$748$	2.19940e7	1.43731
$749$	1.61316e7	1.05069
$750$	0	0
$751$	2.11750e7	1.37001	0.685003	−	0.728540i	$-0.259801\pi$
$751$	2.11750e7	1.37001	0.685003	+	0.728540i	$0.259801\pi$
$752$	2.02822e6	0.130789
$753$	0	0
$754$	9.48988e6	0.607901
$755$	−4.91700e7	−3.13930
$756$	0	0
$757$	2.10506e6	0.133513	0.0667567	−	0.997769i	$-0.478735\pi$
$757$	2.10506e6	0.133513	0.0667567	+	0.997769i	$0.478735\pi$
$758$	5.78417e6	0.365652
$759$	0	0
$760$	−1.39268e7	−0.874613
$761$	−4.70786e6	−0.294688	−0.147344	−	0.989085i	$-0.547072\pi$
$761$	−4.70786e6	−0.294688	−0.147344	+	0.989085i	$0.547072\pi$
$762$	0	0
$763$	−3.00668e7	−1.86972
$764$	−5.80258e6	−0.359656
$765$	0	0
$766$	−1.52931e7	−0.941726
$767$	1.59786e7	0.980729
$768$	0	0
$769$	1.58008e7	0.963524	0.481762	−	0.876302i	$-0.339997\pi$
$769$	1.58008e7	0.963524	0.481762	+	0.876302i	$0.339997\pi$
$770$	4.98548e7	3.03026
$771$	0	0
$772$	6.29672e6	0.380252
$773$	−8.18693e6	−0.492802	−0.246401	−	0.969168i	$-0.579248\pi$
$773$	−8.18693e6	−0.492802	−0.246401	+	0.969168i	$0.579248\pi$
$774$	0	0
$775$	−3.80537e7	−2.27584
$776$	453140.	0.0270133
$777$	0	0
$778$	1.72282e7	1.02045
$779$	−1.64361e7	−0.970408
$780$	0	0
$781$	−2.14368e7	−1.25757
$782$	4.56854e6	0.267153
$783$	0	0
$784$	6.90263e6	0.401074
$785$	−2.81550e7	−1.63073
$786$	0	0
$787$	−1.50016e7	−0.863375	−0.431688	−	0.902023i	$-0.642082\pi$
$787$	−1.50016e7	−0.863375	−0.431688	+	0.902023i	$0.642082\pi$
$788$	−9.13986e6	−0.524353
$789$	0	0
$790$	−985922.	−0.0562050
$791$	1.96073e7	1.11424
$792$	0	0
$793$	−9.03117e6	−0.509990
$794$	−1.13343e7	−0.638031
$795$	0	0
$796$	−337550.	−0.0188823
$797$	−1.70925e7	−0.953149	−0.476574	−	0.879134i	$-0.658122\pi$
$797$	−1.70925e7	−0.953149	−0.476574	+	0.879134i	$0.658122\pi$
$798$	0	0
$799$	1.71055e7	0.947915
$800$	−5.76536e6	−0.318494
$801$	0	0
$802$	−6.67930e6	−0.366686
$803$	2.87150e7	1.57152
$804$	0	0
$805$	1.03557e7	0.563236
$806$	1.36540e7	0.740327
$807$	0	0
$808$	1.08968e6	0.0587179
$809$	−1.68487e6	−0.0905099	−0.0452549	−	0.998975i	$-0.514410\pi$
$809$	−1.68487e6	−0.0905099	−0.0452549	+	0.998975i	$0.514410\pi$
$810$	0	0
$811$	9.98251e6	0.532951	0.266476	−	0.963842i	$-0.414141\pi$
$811$	9.98251e6	0.532951	0.266476	+	0.963842i	$0.414141\pi$
$812$	−1.57246e7	−0.836931
$813$	0	0
$814$	1.08729e7	0.575156
$815$	−1.89758e7	−1.00071
$816$	0	0
$817$	−3.96465e7	−2.07802
$818$	−1.02058e7	−0.533292
$819$	0	0
$820$	−1.05807e7	−0.549517
$821$	9.39308e6	0.486352	0.243176	−	0.969982i	$-0.421811\pi$
$821$	9.39308e6	0.486352	0.243176	+	0.969982i	$0.421811\pi$
$822$	0	0
$823$	9.94964e6	0.512045	0.256022	−	0.966671i	$-0.417588\pi$
$823$	9.94964e6	0.512045	0.256022	+	0.966671i	$0.417588\pi$
$824$	3.77254e6	0.193560
$825$	0	0
$826$	−2.64763e7	−1.35023
$827$	2.37179e7	1.20590	0.602952	−	0.797777i	$-0.293991\pi$
$827$	2.37179e7	1.20590	0.602952	+	0.797777i	$0.293991\pi$
$828$	0	0
$829$	−3.28448e7	−1.65989	−0.829947	−	0.557842i	$-0.811629\pi$
$829$	−3.28448e7	−1.65989	−0.829947	+	0.557842i	$0.811629\pi$
$830$	1.59274e7	0.802507
$831$	0	0
$832$	2.06867e6	0.103606
$833$	5.82152e7	2.90686
$834$	0	0
$835$	−5.97398e7	−2.96515
$836$	−2.36908e7	−1.17237
$837$	0	0
$838$	4.00450e6	0.196987
$839$	−1.57665e7	−0.773266	−0.386633	−	0.922234i	$-0.626362\pi$
$839$	−1.57665e7	−0.773266	−0.386633	+	0.922234i	$0.626362\pi$
$840$	0	0
$841$	1.55565e6	0.0758439
$842$	1.17712e7	0.572189
$843$	0	0
$844$	−2.00561e7	−0.969149
$845$	1.08747e7	0.523935
$846$	0	0
$847$	5.11138e7	2.44810
$848$	−2.22126e6	−0.106074
$849$	0	0
$850$	−4.86237e7	−2.30835
$851$	2.25850e6	0.106904
$852$	0	0
$853$	−6.96223e6	−0.327624	−0.163812	−	0.986492i	$-0.552379\pi$
$853$	−6.96223e6	−0.327624	−0.163812	+	0.986492i	$0.552379\pi$
$854$	1.49645e7	0.702132
$855$	0	0
$856$	−4.93478e6	−0.230188
$857$	−5.00325e6	−0.232702	−0.116351	−	0.993208i	$-0.537120\pi$
$857$	−5.00325e6	−0.232702	−0.116351	+	0.993208i	$0.537120\pi$
$858$	0	0
$859$	1.23328e7	0.570269	0.285134	−	0.958488i	$-0.407962\pi$
$859$	1.23328e7	0.570269	0.285134	+	0.958488i	$0.407962\pi$
$860$	−2.55225e7	−1.17673
$861$	0	0
$862$	1.75005e6	0.0802201
$863$	−1.79095e7	−0.818572	−0.409286	−	0.912406i	$-0.634222\pi$
$863$	−1.79095e7	−0.818572	−0.409286	+	0.912406i	$0.634222\pi$
$864$	0	0
$865$	−1.83495e7	−0.833843
$866$	3.53188e6	0.160034
$867$	0	0
$868$	−2.26246e7	−1.01925
$869$	−1.67715e6	−0.0753394
$870$	0	0
$871$	1.34761e7	0.601893
$872$	9.19767e6	0.409625
$873$	0	0
$874$	−4.92098e6	−0.217908
$875$	−4.90425e7	−2.16547
$876$	0	0
$877$	3.29972e7	1.44870	0.724350	−	0.689433i	$-0.242140\pi$
$877$	3.29972e7	1.44870	0.724350	+	0.689433i	$0.242140\pi$
$878$	−1.39446e7	−0.610477
$879$	0	0
$880$	−1.52509e7	−0.663881
$881$	7.15114e6	0.310410	0.155205	−	0.987882i	$-0.450396\pi$
$881$	7.15114e6	0.310410	0.155205	+	0.987882i	$0.450396\pi$
$882$	0	0
$883$	−2.99398e7	−1.29225	−0.646126	−	0.763231i	$-0.723612\pi$
$883$	−2.99398e7	−1.29225	−0.646126	+	0.763231i	$0.723612\pi$
$884$	1.74467e7	0.750900
$885$	0	0
$886$	2.17847e7	0.932327
$887$	7.12386e6	0.304023	0.152011	−	0.988379i	$-0.451425\pi$
$887$	7.12386e6	0.304023	0.152011	+	0.988379i	$0.451425\pi$
$888$	0	0
$889$	2.25116e7	0.955325
$890$	−3.20822e7	−1.35765
$891$	0	0
$892$	1.51174e6	0.0636156
$893$	−1.84251e7	−0.773183
$894$	0	0
$895$	4.53945e7	1.89429
$896$	−3.42776e6	−0.142640
$897$	0	0
$898$	2.00879e7	0.831272
$899$	3.17497e7	1.31021
$900$	0	0
$901$	−1.87336e7	−0.768794
$902$	−1.79988e7	−0.736594
$903$	0	0
$904$	−5.99802e6	−0.244111
$905$	−4.98860e7	−2.02469
$906$	0	0
$907$	−1.22527e6	−0.0494553	−0.0247276	−	0.999694i	$-0.507872\pi$
$907$	−1.22527e6	−0.0494553	−0.0247276	+	0.999694i	$0.507872\pi$
$908$	6.93973e6	0.279337
$909$	0	0
$910$	3.95472e7	1.58311
$911$	1.03752e7	0.414189	0.207095	−	0.978321i	$-0.433599\pi$
$911$	1.03752e7	0.414189	0.207095	+	0.978321i	$0.433599\pi$
$912$	0	0
$913$	2.70940e7	1.07571
$914$	−2.82328e7	−1.11786
$915$	0	0
$916$	6.61602e6	0.260530
$917$	−4.51815e6	−0.177434
$918$	0	0
$919$	−3.50572e7	−1.36927	−0.684634	−	0.728887i	$-0.740038\pi$
$919$	−3.50572e7	−1.36927	−0.684634	+	0.728887i	$0.740038\pi$
$920$	−3.16789e6	−0.123396
$921$	0	0
$922$	−1.90959e7	−0.739798
$923$	−1.70047e7	−0.656997
$924$	0	0
$925$	−2.40376e7	−0.923711
$926$	−1.01065e7	−0.387324
$927$	0	0
$928$	4.81027e6	0.183358
$929$	1.80076e7	0.684569	0.342285	−	0.939596i	$-0.388799\pi$
$929$	1.80076e7	0.684569	0.342285	+	0.939596i	$0.388799\pi$
$930$	0	0
$931$	−6.27062e7	−2.37103
$932$	8.97212e6	0.338341
$933$	0	0
$934$	−7.38141e6	−0.276868
$935$	−1.28623e8	−4.81160
$936$	0	0
$937$	3.03437e7	1.12907	0.564533	−	0.825410i	$-0.309056\pi$
$937$	3.03437e7	1.12907	0.564533	+	0.825410i	$0.309056\pi$
$938$	−2.23297e7	−0.828660
$939$	0	0
$940$	−1.18612e7	−0.437834
$941$	3.57861e7	1.31747	0.658734	−	0.752376i	$-0.271092\pi$
$941$	3.57861e7	1.31747	0.658734	+	0.752376i	$0.271092\pi$
$942$	0	0
$943$	−3.73867e6	−0.136911
$944$	8.09928e6	0.295812
$945$	0	0
$946$	−4.34162e7	−1.57733
$947$	−2.91746e7	−1.05713	−0.528567	−	0.848891i	$-0.677271\pi$
$947$	−2.91746e7	−1.05713	−0.528567	+	0.848891i	$0.677271\pi$
$948$	0	0
$949$	2.27781e7	0.821017
$950$	5.23748e7	1.88284
$951$	0	0
$952$	−2.89089e7	−1.03381
$953$	1.03849e7	0.370400	0.185200	−	0.982701i	$-0.440707\pi$
$953$	1.03849e7	0.370400	0.185200	+	0.982701i	$0.440707\pi$
$954$	0	0
$955$	3.39340e7	1.20400
$956$	5.68405e6	0.201147
$957$	0	0
$958$	1.10920e7	0.390479
$959$	7.32238e7	2.57102
$960$	0	0
$961$	1.70523e7	0.595629
$962$	8.62492e6	0.300481
$963$	0	0
$964$	1.27827e7	0.443026
$965$	−3.68238e7	−1.27295
$966$	0	0
$967$	−1.14600e7	−0.394109	−0.197055	−	0.980392i	$-0.563138\pi$
$967$	−1.14600e7	−0.394109	−0.197055	+	0.980392i	$0.563138\pi$
$968$	−1.56361e7	−0.536339
$969$	0	0
$970$	−2.65000e6	−0.0904309
$971$	−3.72049e7	−1.26635	−0.633173	−	0.774011i	$-0.718248\pi$
$971$	−3.72049e7	−1.26635	−0.633173	+	0.774011i	$0.718248\pi$
$972$	0	0
$973$	1.20876e7	0.409316
$974$	−2.12179e7	−0.716647
$975$	0	0
$976$	−4.57776e6	−0.153826
$977$	−4.05876e6	−0.136037	−0.0680185	−	0.997684i	$-0.521668\pi$
$977$	−4.05876e6	−0.136037	−0.0680185	+	0.997684i	$0.521668\pi$
$978$	0	0
$979$	−5.45749e7	−1.81985
$980$	−4.03672e7	−1.34265
$981$	0	0
$982$	3.71021e7	1.22778
$983$	5.22465e7	1.72454	0.862270	−	0.506449i	$-0.169042\pi$
$983$	5.22465e7	1.72454	0.862270	+	0.506449i	$0.169042\pi$
$984$	0	0
$985$	5.34507e7	1.75535
$986$	4.05687e7	1.32892
$987$	0	0
$988$	−1.87926e7	−0.612484
$989$	−9.01829e6	−0.293179
$990$	0	0
$991$	−4.96881e7	−1.60719	−0.803596	−	0.595175i	$-0.797083\pi$
$991$	−4.96881e7	−1.60719	−0.803596	+	0.595175i	$0.797083\pi$
$992$	6.92102e6	0.223301
$993$	0	0
$994$	2.81765e7	0.904525
$995$	1.97402e6	0.0632112
$996$	0	0
$997$	9.60828e6	0.306131	0.153066	−	0.988216i	$-0.451085\pi$
$997$	9.60828e6	0.306131	0.153066	+	0.988216i	$0.451085\pi$
$998$	7.71066e6	0.245056
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.6.a.o.1.1		4
3.2	odd	2		138.6.a.i.1.4	✓	4
12.11	even	2		1104.6.a.m.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.6.a.i.1.4	✓	4	3.2	odd	2
414.6.a.o.1.1		4	1.1	even	1	trivial
1104.6.a.m.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	414.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} -93.5694 q^{5} +209.214 q^{7} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} -93.5694 q^{5} +209.214 q^{7} -64.0000 q^{8} +374.278 q^{10} +636.683 q^{11} +505.046 q^{13} -836.855 q^{14} +256.000 q^{16} +2159.05 q^{17} -2325.61 q^{19} -1497.11 q^{20} -2546.73 q^{22} -529.000 q^{23} +5630.23 q^{25} -2020.19 q^{26} +3347.42 q^{28} -4697.53 q^{29} -6758.81 q^{31} -1024.00 q^{32} -8636.18 q^{34} -19576.0 q^{35} -4269.37 q^{37} +9302.42 q^{38} +5988.44 q^{40} +7067.43 q^{41} +17047.8 q^{43} +10186.9 q^{44} +2116.00 q^{46} +7922.73 q^{47} +26963.4 q^{49} -22520.9 q^{50} +8080.74 q^{52} -8676.81 q^{53} -59574.0 q^{55} -13389.7 q^{56} +18790.1 q^{58} +31637.8 q^{59} -17881.9 q^{61} +27035.2 q^{62} +4096.00 q^{64} -47256.9 q^{65} +26682.9 q^{67} +34544.7 q^{68} +78304.0 q^{70} -33669.5 q^{71} +45101.0 q^{73} +17077.5 q^{74} -37209.7 q^{76} +133203. q^{77} -2634.20 q^{79} -23953.8 q^{80} -28269.7 q^{82} +42555.0 q^{83} -202021. q^{85} -68191.2 q^{86} -40747.7 q^{88} -85717.5 q^{89} +105663. q^{91} -8464.00 q^{92} -31690.9 q^{94} +217606. q^{95} -7080.31 q^{97} -107854. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8}+O(q^{10}) \) 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8	\( 4 q - 16 q^{2} + 64 q^{4} - 54 q^{5} + 348 q^{7} - 256 q^{8} + 216 q^{10} + 6 q^{11} + 1248 q^{13} - 1392 q^{14} + 1024 q^{16} + 56 q^{17} + 1530 q^{19} - 864 q^{20} - 24 q^{22} - 2116 q^{23} + 8852 q^{25} - 4992 q^{26} + 5568 q^{28} - 2292 q^{29} + 1556 q^{31} - 4096 q^{32} - 224 q^{34} - 5688 q^{35} - 9586 q^{37} - 6120 q^{38} + 3456 q^{40} + 11768 q^{41} + 13758 q^{43} + 96 q^{44} + 8464 q^{46} - 10636 q^{47} + 11360 q^{49} - 35408 q^{50} + 19968 q^{52} - 26686 q^{53} - 28900 q^{55} - 22272 q^{56} + 9168 q^{58} + 9108 q^{59} - 37878 q^{61} - 6224 q^{62} + 16384 q^{64} + 72212 q^{65} - 23302 q^{67} + 896 q^{68} + 22752 q^{70} - 31728 q^{71} - 28340 q^{73} + 38344 q^{74} + 24480 q^{76} + 121276 q^{77} - 26668 q^{79} - 13824 q^{80} - 47072 q^{82} + 119026 q^{83} - 217876 q^{85} - 55032 q^{86} - 384 q^{88} + 148236 q^{89} + 89008 q^{91} - 33856 q^{92} + 42544 q^{94} + 399292 q^{95} - 16092 q^{97} - 45440 q^{98}+O(q^{100}) \) 4 * q - 16 * q^2 + 64 * q^4 - 54 * q^5 + 348 * q^7 - 256 * q^8 + 216 * q^10 + 6 * q^11 + 1248 * q^13 - 1392 * q^14 + 1024 * q^16 + 56 * q^17 + 1530 * q^19 - 864 * q^20 - 24 * q^22 - 2116 * q^23 + 8852 * q^25 - 4992 * q^26 + 5568 * q^28 - 2292 * q^29 + 1556 * q^31 - 4096 * q^32 - 224 * q^34 - 5688 * q^35 - 9586 * q^37 - 6120 * q^38 + 3456 * q^40 + 11768 * q^41 + 13758 * q^43 + 96 * q^44 + 8464 * q^46 - 10636 * q^47 + 11360 * q^49 - 35408 * q^50 + 19968 * q^52 - 26686 * q^53 - 28900 * q^55 - 22272 * q^56 + 9168 * q^58 + 9108 * q^59 - 37878 * q^61 - 6224 * q^62 + 16384 * q^64 + 72212 * q^65 - 23302 * q^67 + 896 * q^68 + 22752 * q^70 - 31728 * q^71 - 28340 * q^73 + 38344 * q^74 + 24480 * q^76 + 121276 * q^77 - 26668 * q^79 - 13824 * q^80 - 47072 * q^82 + 119026 * q^83 - 217876 * q^85 - 55032 * q^86 - 384 * q^88 + 148236 * q^89 + 89008 * q^91 - 33856 * q^92 + 42544 * q^94 + 399292 * q^95 - 16092 * q^97 - 45440 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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