LMFDB - Embedded newform 414.3.c.b.323.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,3,Mod(323,414)]

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([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("414.323");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	414.c (of order $2$, degree $1$, 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:	$11.2806829445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 19x^{6} - 88x^{5} + 301x^{4} - 1010x^{3} + 2713x^{2} - 7044x + 9558 $ x^8 - 2x^7 + 19x^6 - 88x^5 + 301x^4 - 1010x^3 + 2713x^2 - 7044*x + 9558
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			323.8
Root			$2.43130 - 0.475563i$ of defining polynomial
Character	$\chi$	$=$	414.323
Dual form			414.3.c.b.323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +6.87676i q^{5} -5.43723 q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+1.41421i q^{2} -2.00000 q^{4} +6.87676i q^{5} -5.43723 q^{7} -2.82843i q^{8} -9.72521 q^{10} +7.68941i q^{11} -5.83945 q^{13} -7.68941i q^{14} +4.00000 q^{16} -17.8019i q^{17} -10.8176 q^{19} -13.7535i q^{20} -10.8745 q^{22} -4.79583i q^{23} -22.2899 q^{25} -8.25823i q^{26} +10.8745 q^{28} +4.08109i q^{29} -16.2549 q^{31} +5.65685i q^{32} +25.1756 q^{34} -37.3906i q^{35} -5.67890 q^{37} -15.2984i q^{38} +19.4504 q^{40} -54.0221i q^{41} -55.7885 q^{43} -15.3788i q^{44} +6.78233 q^{46} +36.2529i q^{47} -19.4365 q^{49} -31.5226i q^{50} +11.6789 q^{52} -46.4465i q^{53} -52.8783 q^{55} +15.3788i q^{56} -5.77153 q^{58} +83.4972i q^{59} -4.39115 q^{61} -22.9878i q^{62} -8.00000 q^{64} -40.1565i q^{65} +37.8058 q^{67} +35.6037i q^{68} +52.8783 q^{70} +30.7203i q^{71} -141.252 q^{73} -8.03117i q^{74} +21.6352 q^{76} -41.8091i q^{77} +69.9060 q^{79} +27.5071i q^{80} +76.3987 q^{82} +69.1749i q^{83} +122.419 q^{85} -78.8969i q^{86} +21.7489 q^{88} -101.485i q^{89} +31.7505 q^{91} +9.59166i q^{92} -51.2694 q^{94} -74.3902i q^{95} +67.1358 q^{97} -27.4873i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 + 16 * q^7	$ 8 q - 16 q^{4} + 16 q^{7} - 8 q^{10} - 8 q^{13} + 32 q^{16} - 48 q^{19} + 32 q^{22} - 32 q^{28} - 32 q^{31} - 8 q^{34} + 32 q^{37} + 16 q^{40} + 32 q^{43} + 80 q^{49} + 16 q^{52} + 32 q^{55} + 16 q^{58} + 48 q^{61} - 64 q^{64} - 16 q^{67} - 32 q^{70} - 432 q^{73} + 96 q^{76} - 416 q^{79} - 144 q^{82} + 584 q^{85} - 64 q^{88} + 368 q^{91} + 128 q^{94} - 8 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 16 * q^7 - 8 * q^10 - 8 * q^13 + 32 * q^16 - 48 * q^19 + 32 * q^22 - 32 * q^28 - 32 * q^31 - 8 * q^34 + 32 * q^37 + 16 * q^40 + 32 * q^43 + 80 * q^49 + 16 * q^52 + 32 * q^55 + 16 * q^58 + 48 * q^61 - 64 * q^64 - 16 * q^67 - 32 * q^70 - 432 * q^73 + 96 * q^76 - 416 * q^79 - 144 * q^82 + 584 * q^85 - 64 * q^88 + 368 * q^91 + 128 * q^94 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	6.87676i	1.37535i	0.726017	+	0.687676i	$0.241369\pi$
$5$	6.87676i	1.37535i	−0.726017	+	0.687676i	$0.758631\pi$
$6$	0	0
$7$	−5.43723	−0.776748	−0.388374	−	0.921502i	$-0.626963\pi$
$7$	−5.43723	−0.776748	−0.388374	+	0.921502i	$0.626963\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	−9.72521	−0.972521
$11$	7.68941i	0.699037i	0.936929	+	0.349519i	$0.113655\pi$
$11$	7.68941i	0.699037i	−0.936929	+	0.349519i	$0.886345\pi$
$12$	0	0
$13$	−5.83945	−0.449188	−0.224594	−	0.974452i	$-0.572106\pi$
$13$	−5.83945	−0.449188	−0.224594	+	0.974452i	$0.572106\pi$
$14$	− 7.68941i	− 0.549244i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 17.8019i	− 1.04717i	−0.851974	−	0.523584i	$-0.824595\pi$
$17$	− 17.8019i	− 1.04717i	0.851974	−	0.523584i	$-0.175405\pi$
$18$	0	0
$19$	−10.8176	−0.569348	−0.284674	−	0.958624i	$-0.591885\pi$
$19$	−10.8176	−0.569348	−0.284674	+	0.958624i	$0.591885\pi$
$20$	− 13.7535i	− 0.687676i
$21$	0	0
$22$	−10.8745	−0.494294
$23$	− 4.79583i	− 0.208514i
$23$	− 4.79583i	− 0.208514i
$24$	0	0
$25$	−22.2899	−0.891595
$26$	− 8.25823i	− 0.317624i
$27$	0	0
$28$	10.8745	0.388374
$29$	4.08109i	0.140727i	0.997521	+	0.0703636i	$0.0224159\pi$
$29$	4.08109i	0.140727i	−0.997521	+	0.0703636i	$0.977584\pi$
$30$	0	0
$31$	−16.2549	−0.524350	−0.262175	−	0.965020i	$-0.584440\pi$
$31$	−16.2549	−0.524350	−0.262175	+	0.965020i	$0.584440\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	25.1756	0.740460
$35$	− 37.3906i	− 1.06830i
$36$	0	0
$37$	−5.67890	−0.153484	−0.0767418	−	0.997051i	$-0.524452\pi$
$37$	−5.67890	−0.153484	−0.0767418	+	0.997051i	$0.524452\pi$
$38$	− 15.2984i	− 0.402590i
$39$	0	0
$40$	19.4504	0.486261
$41$	− 54.0221i	− 1.31761i	−0.752313	−	0.658806i	$-0.771062\pi$
$41$	− 54.0221i	− 1.31761i	0.752313	−	0.658806i	$-0.228938\pi$
$42$	0	0
$43$	−55.7885	−1.29741	−0.648704	−	0.761041i	$-0.724688\pi$
$43$	−55.7885	−1.29741	−0.648704	+	0.761041i	$0.724688\pi$
$44$	− 15.3788i	− 0.349519i
$45$	0	0
$46$	6.78233	0.147442
$47$	36.2529i	0.771339i	0.922637	+	0.385670i	$0.126030\pi$
$47$	36.2529i	0.771339i	−0.922637	+	0.385670i	$0.873970\pi$
$48$	0	0
$49$	−19.4365	−0.396663
$50$	− 31.5226i	− 0.630453i
$51$	0	0
$52$	11.6789	0.224594
$53$	− 46.4465i	− 0.876350i	−0.898890	−	0.438175i	$-0.855625\pi$
$53$	− 46.4465i	− 0.876350i	0.898890	−	0.438175i	$-0.144375\pi$
$54$	0	0
$55$	−52.8783	−0.961423
$56$	15.3788i	0.274622i
$57$	0	0
$58$	−5.77153	−0.0995091
$59$	83.4972i	1.41521i	0.706610	+	0.707604i	$0.250224\pi$
$59$	83.4972i	1.41521i	−0.706610	+	0.707604i	$0.749776\pi$
$60$	0	0
$61$	−4.39115	−0.0719860	−0.0359930	−	0.999352i	$-0.511459\pi$
$61$	−4.39115	−0.0719860	−0.0359930	+	0.999352i	$0.511459\pi$
$62$	− 22.9878i	− 0.370771i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 40.1565i	− 0.617792i
$66$	0	0
$67$	37.8058	0.564266	0.282133	−	0.959375i	$-0.408958\pi$
$67$	37.8058	0.564266	0.282133	+	0.959375i	$0.408958\pi$
$68$	35.6037i	0.523584i
$69$	0	0
$70$	52.8783	0.755404
$71$	30.7203i	0.432681i	0.976318	+	0.216340i	$0.0694121\pi$
$71$	30.7203i	0.432681i	−0.976318	+	0.216340i	$0.930588\pi$
$72$	0	0
$73$	−141.252	−1.93496	−0.967480	−	0.252947i	$-0.918600\pi$
$73$	−141.252	−1.93496	−0.967480	+	0.252947i	$0.918600\pi$
$74$	− 8.03117i	− 0.108529i
$75$	0	0
$76$	21.6352	0.284674
$77$	− 41.8091i	− 0.542976i
$78$	0	0
$79$	69.9060	0.884886	0.442443	−	0.896797i	$-0.354112\pi$
$79$	69.9060	0.884886	0.442443	+	0.896797i	$0.354112\pi$
$80$	27.5071i	0.343838i
$81$	0	0
$82$	76.3987	0.931692
$83$	69.1749i	0.833432i	0.909037	+	0.416716i	$0.136819\pi$
$83$	69.1749i	0.833432i	−0.909037	+	0.416716i	$0.863181\pi$
$84$	0	0
$85$	122.419	1.44023
$86$	− 78.8969i	− 0.917405i
$87$	0	0
$88$	21.7489	0.247147
$89$	− 101.485i	− 1.14028i	−0.821547	−	0.570141i	$-0.806889\pi$
$89$	− 101.485i	− 1.14028i	0.821547	−	0.570141i	$-0.193111\pi$
$90$	0	0
$91$	31.7505	0.348906
$92$	9.59166i	0.104257i
$93$	0	0
$94$	−51.2694	−0.545419
$95$	− 74.3902i	− 0.783055i
$96$	0	0
$97$	67.1358	0.692121	0.346061	−	0.938212i	$-0.387519\pi$
$97$	67.1358	0.692121	0.346061	+	0.938212i	$0.387519\pi$
$98$	− 27.4873i	− 0.280483i
$99$	0	0
$100$	44.5797	0.445797
$101$	168.433i	1.66765i	0.552025	+	0.833827i	$0.313855\pi$
$101$	168.433i	1.66765i	−0.552025	+	0.833827i	$0.686145\pi$
$102$	0	0
$103$	16.6856	0.161996	0.0809979	−	0.996714i	$-0.474189\pi$
$103$	16.6856	0.161996	0.0809979	+	0.996714i	$0.474189\pi$
$104$	16.5165i	0.158812i
$105$	0	0
$106$	65.6853	0.619673
$107$	141.806i	1.32529i	0.748935	+	0.662644i	$0.230566\pi$
$107$	141.806i	1.32529i	−0.748935	+	0.662644i	$0.769434\pi$
$108$	0	0
$109$	4.31466	0.0395841	0.0197920	−	0.999804i	$-0.493700\pi$
$109$	4.31466	0.0395841	0.0197920	+	0.999804i	$0.493700\pi$
$110$	− 74.7812i	− 0.679829i
$111$	0	0
$112$	−21.7489	−0.194187
$113$	23.3559i	0.206689i	0.994646	+	0.103345i	$0.0329545\pi$
$113$	23.3559i	0.206689i	−0.994646	+	0.103345i	$0.967046\pi$
$114$	0	0
$115$	32.9798	0.286781
$116$	− 8.16217i	− 0.0703636i
$117$	0	0
$118$	−118.083	−1.00070
$119$	96.7929i	0.813386i
$120$	0	0
$121$	61.8730	0.511347
$122$	− 6.21002i	− 0.0509018i
$123$	0	0
$124$	32.5097	0.262175
$125$	18.6369i	0.149095i
$126$	0	0
$127$	−217.702	−1.71419	−0.857095	−	0.515159i	$-0.827733\pi$
$127$	−217.702	−1.71419	−0.857095	+	0.515159i	$0.827733\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	56.7899	0.436845
$131$	202.955i	1.54927i	0.632406	+	0.774637i	$0.282067\pi$
$131$	202.955i	1.54927i	−0.632406	+	0.774637i	$0.717933\pi$
$132$	0	0
$133$	58.8179	0.442240
$134$	53.4655i	0.398996i
$135$	0	0
$136$	−50.3513	−0.370230
$137$	196.599i	1.43503i	0.696542	+	0.717516i	$0.254721\pi$
$137$	196.599i	1.43503i	−0.696542	+	0.717516i	$0.745279\pi$
$138$	0	0
$139$	−196.342	−1.41253	−0.706265	−	0.707948i	$-0.749621\pi$
$139$	−196.342	−1.41253	−0.706265	+	0.707948i	$0.749621\pi$
$140$	74.7812i	0.534151i
$141$	0	0
$142$	−43.4451	−0.305952
$143$	− 44.9019i	− 0.313999i
$144$	0	0
$145$	−28.0647	−0.193549
$146$	− 199.761i	− 1.36822i
$147$	0	0
$148$	11.3578	0.0767418
$149$	257.166i	1.72595i	0.505251	+	0.862973i	$0.331400\pi$
$149$	257.166i	1.72595i	−0.505251	+	0.862973i	$0.668600\pi$
$150$	0	0
$151$	94.8135	0.627904	0.313952	−	0.949439i	$-0.398347\pi$
$151$	94.8135	0.627904	0.313952	+	0.949439i	$0.398347\pi$
$152$	30.5968i	0.201295i
$153$	0	0
$154$	59.1270	0.383942
$155$	− 111.781i	− 0.721166i
$156$	0	0
$157$	37.0443	0.235951	0.117975	−	0.993017i	$-0.462360\pi$
$157$	37.0443	0.235951	0.117975	+	0.993017i	$0.462360\pi$
$158$	98.8620i	0.625709i
$159$	0	0
$160$	−38.9008	−0.243130
$161$	26.0761i	0.161963i
$162$	0	0
$163$	−115.075	−0.705981	−0.352990	−	0.935627i	$-0.614835\pi$
$163$	−115.075	−0.705981	−0.352990	+	0.935627i	$0.614835\pi$
$164$	108.044i	0.658806i
$165$	0	0
$166$	−97.8281	−0.589326
$167$	− 5.06703i	− 0.0303415i	−0.999885	−	0.0151707i	$-0.995171\pi$
$167$	− 5.06703i	− 0.0303415i	0.999885	−	0.0151707i	$-0.00482918\pi$
$168$	0	0
$169$	−134.901	−0.798230
$170$	173.127i	1.01839i
$171$	0	0
$172$	111.577	0.648704
$173$	− 257.245i	− 1.48697i	−0.668755	−	0.743483i	$-0.733173\pi$
$173$	− 257.245i	− 1.48697i	0.668755	−	0.743483i	$-0.266827\pi$
$174$	0	0
$175$	121.195	0.692544
$176$	30.7576i	0.174759i
$177$	0	0
$178$	143.522	0.806301
$179$	32.7967i	0.183222i	0.995795	+	0.0916110i	$0.0292016\pi$
$179$	32.7967i	0.183222i	−0.995795	+	0.0916110i	$0.970798\pi$
$180$	0	0
$181$	183.605	1.01439	0.507195	−	0.861831i	$-0.330682\pi$
$181$	183.605	1.01439	0.507195	+	0.861831i	$0.330682\pi$
$182$	44.9019i	0.246714i
$183$	0	0
$184$	−13.5647	−0.0737210
$185$	− 39.0524i	− 0.211094i
$186$	0	0
$187$	136.886	0.732010
$188$	− 72.5059i	− 0.385670i
$189$	0	0
$190$	105.204	0.553703
$191$	− 13.7258i	− 0.0718627i	−0.999354	−	0.0359313i	$-0.988560\pi$
$191$	− 13.7258i	− 0.0718627i	0.999354	−	0.0359313i	$-0.0114398\pi$
$192$	0	0
$193$	106.204	0.550278	0.275139	−	0.961404i	$-0.411276\pi$
$193$	106.204	0.550278	0.275139	+	0.961404i	$0.411276\pi$
$194$	94.9443i	0.489404i
$195$	0	0
$196$	38.8730	0.198331
$197$	− 35.3539i	− 0.179462i	−0.995966	−	0.0897308i	$-0.971399\pi$
$197$	− 35.3539i	− 0.179462i	0.995966	−	0.0897308i	$-0.0286007\pi$
$198$	0	0
$199$	−268.863	−1.35107	−0.675535	−	0.737328i	$-0.736087\pi$
$199$	−268.863	−1.35107	−0.675535	+	0.737328i	$0.736087\pi$
$200$	63.0453i	0.315226i
$201$	0	0
$202$	−238.200	−1.17921
$203$	− 22.1898i	− 0.109309i
$204$	0	0
$205$	371.497	1.81218
$206$	23.5969i	0.114548i
$207$	0	0
$208$	−23.3578	−0.112297
$209$	− 83.1811i	− 0.397996i
$210$	0	0
$211$	−129.399	−0.613266	−0.306633	−	0.951828i	$-0.599202\pi$
$211$	−129.399	−0.613266	−0.306633	+	0.951828i	$0.599202\pi$
$212$	92.8931i	0.438175i
$213$	0	0
$214$	−200.544	−0.937120
$215$	− 383.644i	− 1.78439i
$216$	0	0
$217$	88.3814	0.407288
$218$	6.10186i	0.0279902i
$219$	0	0
$220$	105.757	0.480711
$221$	103.953i	0.470376i
$222$	0	0
$223$	389.713	1.74759	0.873797	−	0.486291i	$-0.161651\pi$
$223$	389.713	1.74759	0.873797	+	0.486291i	$0.161651\pi$
$224$	− 30.7576i	− 0.137311i
$225$	0	0
$226$	−33.0302	−0.146152
$227$	− 306.520i	− 1.35031i	−0.737677	−	0.675154i	$-0.764077\pi$
$227$	− 306.520i	− 1.35031i	0.737677	−	0.675154i	$-0.235923\pi$
$228$	0	0
$229$	366.794	1.60172	0.800861	−	0.598851i	$-0.204376\pi$
$229$	366.794	1.60172	0.800861	+	0.598851i	$0.204376\pi$
$230$	46.6405i	0.202785i
$231$	0	0
$232$	11.5431	0.0497545
$233$	− 241.335i	− 1.03577i	−0.855449	−	0.517887i	$-0.826719\pi$
$233$	− 241.335i	− 1.03577i	0.855449	−	0.517887i	$-0.173281\pi$
$234$	0	0
$235$	−249.303	−1.06086
$236$	− 166.994i	− 0.707604i
$237$	0	0
$238$	−136.886	−0.575151
$239$	183.710i	0.768661i	0.923196	+	0.384330i	$0.125568\pi$
$239$	183.710i	0.768661i	−0.923196	+	0.384330i	$0.874432\pi$
$240$	0	0
$241$	−326.918	−1.35651	−0.678253	−	0.734828i	$-0.737263\pi$
$241$	−326.918	−1.35651	−0.678253	+	0.734828i	$0.737263\pi$
$242$	87.5016i	0.361577i
$243$	0	0
$244$	8.78229	0.0359930
$245$	− 133.660i	− 0.545551i
$246$	0	0
$247$	63.1689	0.255745
$248$	45.9757i	0.185386i
$249$	0	0
$250$	−26.3566	−0.105426
$251$	− 70.7716i	− 0.281958i	−0.990013	−	0.140979i	$-0.954975\pi$
$251$	− 70.7716i	− 0.281958i	0.990013	−	0.140979i	$-0.0450251\pi$
$252$	0	0
$253$	36.8771	0.145759
$254$	− 307.877i	− 1.21212i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 88.0745i	− 0.342702i	−0.985210	−	0.171351i	$-0.945187\pi$
$257$	− 88.0745i	− 0.342702i	0.985210	−	0.171351i	$-0.0548132\pi$
$258$	0	0
$259$	30.8775	0.119218
$260$	80.3130i	0.308896i
$261$	0	0
$262$	−287.022	−1.09550
$263$	− 235.795i	− 0.896561i	−0.893893	−	0.448280i	$-0.852037\pi$
$263$	− 235.795i	− 0.896561i	0.893893	−	0.448280i	$-0.147963\pi$
$264$	0	0
$265$	319.402	1.20529
$266$	83.1811i	0.312711i
$267$	0	0
$268$	−75.6116	−0.282133
$269$	4.44214i	0.0165135i	0.999966	+	0.00825677i	$0.00262824\pi$
$269$	4.44214i	0.0165135i	−0.999966	+	0.00825677i	$0.997372\pi$
$270$	0	0
$271$	−258.331	−0.953252	−0.476626	−	0.879106i	$-0.658140\pi$
$271$	−258.331	−0.953252	−0.476626	+	0.879106i	$0.658140\pi$
$272$	− 71.2074i	− 0.261792i
$273$	0	0
$274$	−278.033	−1.01472
$275$	− 171.396i	− 0.623258i
$276$	0	0
$277$	−150.711	−0.544084	−0.272042	−	0.962285i	$-0.587699\pi$
$277$	−150.711	−0.544084	−0.272042	+	0.962285i	$0.587699\pi$
$278$	− 277.669i	− 0.998809i
$279$	0	0
$280$	−105.757	−0.377702
$281$	− 461.687i	− 1.64301i	−0.570199	−	0.821507i	$-0.693134\pi$
$281$	− 461.687i	− 1.64301i	0.570199	−	0.821507i	$-0.306866\pi$
$282$	0	0
$283$	−335.124	−1.18418	−0.592092	−	0.805870i	$-0.701698\pi$
$283$	−335.124	−1.18418	−0.592092	+	0.805870i	$0.701698\pi$
$284$	− 61.4407i	− 0.216340i
$285$	0	0
$286$	63.5009	0.222031
$287$	293.731i	1.02345i
$288$	0	0
$289$	−27.9063	−0.0965616
$290$	− 39.6894i	− 0.136860i
$291$	0	0
$292$	282.504	0.967480
$293$	− 331.756i	− 1.13227i	−0.824312	−	0.566136i	$-0.808438\pi$
$293$	− 331.756i	− 1.13227i	0.824312	−	0.566136i	$-0.191562\pi$
$294$	0	0
$295$	−574.191	−1.94641
$296$	16.0623i	0.0542647i
$297$	0	0
$298$	−363.687	−1.22043
$299$	28.0050i	0.0936622i
$300$	0	0
$301$	303.335	1.00776
$302$	134.087i	0.443995i
$303$	0	0
$304$	−43.2705	−0.142337
$305$	− 30.1969i	− 0.0990061i
$306$	0	0
$307$	−246.452	−0.802775	−0.401387	−	0.915908i	$-0.631472\pi$
$307$	−246.452	−0.802775	−0.401387	+	0.915908i	$0.631472\pi$
$308$	83.6183i	0.271488i
$309$	0	0
$310$	158.082	0.509941
$311$	347.854i	1.11850i	0.828999	+	0.559250i	$0.188911\pi$
$311$	347.854i	1.11850i	−0.828999	+	0.559250i	$0.811089\pi$
$312$	0	0
$313$	262.326	0.838101	0.419051	−	0.907963i	$-0.362363\pi$
$313$	262.326	0.838101	0.419051	+	0.907963i	$0.362363\pi$
$314$	52.3885i	0.166842i
$315$	0	0
$316$	−139.812	−0.442443
$317$	496.740i	1.56700i	0.621391	+	0.783501i	$0.286568\pi$
$317$	496.740i	1.56700i	−0.621391	+	0.783501i	$0.713432\pi$
$318$	0	0
$319$	−31.3811	−0.0983735
$320$	− 55.0141i	− 0.171919i
$321$	0	0
$322$	−36.8771	−0.114525
$323$	192.574i	0.596203i
$324$	0	0
$325$	130.161	0.400494
$326$	− 162.740i	− 0.499204i
$327$	0	0
$328$	−152.797	−0.465846
$329$	− 197.116i	− 0.599136i
$330$	0	0
$331$	157.807	0.476757	0.238378	−	0.971172i	$-0.423384\pi$
$331$	157.807	0.476757	0.238378	+	0.971172i	$0.423384\pi$
$332$	− 138.350i	− 0.416716i
$333$	0	0
$334$	7.16586	0.0214547
$335$	259.981i	0.776064i
$336$	0	0
$337$	387.069	1.14857	0.574286	−	0.818655i	$-0.305280\pi$
$337$	387.069	1.14857	0.574286	+	0.818655i	$0.305280\pi$
$338$	− 190.779i	− 0.564434i
$339$	0	0
$340$	−244.838	−0.720113
$341$	− 124.990i	− 0.366540i
$342$	0	0
$343$	372.105	1.08485
$344$	157.794i	0.458703i
$345$	0	0
$346$	363.799	1.05144
$347$	226.549i	0.652879i	0.945218	+	0.326440i	$0.105849\pi$
$347$	226.549i	0.652879i	−0.945218	+	0.326440i	$0.894151\pi$
$348$	0	0
$349$	−85.2057	−0.244142	−0.122071	−	0.992521i	$-0.538954\pi$
$349$	−85.2057	−0.244142	−0.122071	+	0.992521i	$0.538954\pi$
$350$	171.396i	0.489703i
$351$	0	0
$352$	−43.4979	−0.123574
$353$	− 101.019i	− 0.286172i	−0.989710	−	0.143086i	$-0.954297\pi$
$353$	− 101.019i	− 0.286172i	0.989710	−	0.143086i	$-0.0457026\pi$
$354$	0	0
$355$	−211.257	−0.595089
$356$	202.970i	0.570141i
$357$	0	0
$358$	−46.3816	−0.129558
$359$	− 473.121i	− 1.31789i	−0.752193	−	0.658943i	$-0.771004\pi$
$359$	− 473.121i	− 1.31789i	0.752193	−	0.658943i	$-0.228996\pi$
$360$	0	0
$361$	−243.979	−0.675843
$362$	259.656i	0.717282i
$363$	0	0
$364$	−63.5009	−0.174453
$365$	− 971.357i	− 2.66125i
$366$	0	0
$367$	378.938	1.03253	0.516264	−	0.856429i	$-0.327322\pi$
$367$	378.938	1.03253	0.516264	+	0.856429i	$0.327322\pi$
$368$	− 19.1833i	− 0.0521286i
$369$	0	0
$370$	55.2285	0.149266
$371$	252.541i	0.680703i
$372$	0	0
$373$	21.7047	0.0581894	0.0290947	−	0.999577i	$-0.490738\pi$
$373$	21.7047	0.0581894	0.0290947	+	0.999577i	$0.490738\pi$
$374$	193.586i	0.517609i
$375$	0	0
$376$	102.539	0.272710
$377$	− 23.8313i	− 0.0632130i
$378$	0	0
$379$	−379.508	−1.00134	−0.500670	−	0.865638i	$-0.666913\pi$
$379$	−379.508	−1.00134	−0.500670	+	0.865638i	$0.666913\pi$
$380$	148.780i	0.391527i
$381$	0	0
$382$	19.4112	0.0508146
$383$	− 198.586i	− 0.518502i	−0.965810	−	0.259251i	$-0.916524\pi$
$383$	− 198.586i	− 0.518502i	0.965810	−	0.259251i	$-0.0834756\pi$
$384$	0	0
$385$	287.512	0.746783
$386$	150.195i	0.389105i
$387$	0	0
$388$	−134.272	−0.346061
$389$	336.782i	0.865763i	0.901451	+	0.432881i	$0.142503\pi$
$389$	336.782i	0.865763i	−0.901451	+	0.432881i	$0.857497\pi$
$390$	0	0
$391$	−85.3747	−0.218350
$392$	54.9747i	0.140241i
$393$	0	0
$394$	49.9980	0.126899
$395$	480.727i	1.21703i
$396$	0	0
$397$	233.940	0.589269	0.294634	−	0.955610i	$-0.404802\pi$
$397$	233.940	0.589269	0.294634	+	0.955610i	$0.404802\pi$
$398$	− 380.230i	− 0.955351i
$399$	0	0
$400$	−89.1595	−0.222899
$401$	− 49.1903i	− 0.122669i	−0.998117	−	0.0613345i	$-0.980464\pi$
$401$	− 49.1903i	− 0.122669i	0.998117	−	0.0613345i	$-0.0195356\pi$
$402$	0	0
$403$	94.9194	0.235532
$404$	− 336.866i	− 0.833827i
$405$	0	0
$406$	31.3811	0.0772935
$407$	− 43.6674i	− 0.107291i
$408$	0	0
$409$	−615.803	−1.50563	−0.752815	−	0.658232i	$-0.771305\pi$
$409$	−615.803	−1.50563	−0.752815	+	0.658232i	$0.771305\pi$
$410$	525.376i	1.28140i
$411$	0	0
$412$	−33.3711	−0.0809979
$413$	− 453.994i	− 1.09926i
$414$	0	0
$415$	−475.699	−1.14626
$416$	− 33.0329i	− 0.0794060i
$417$	0	0
$418$	117.636	0.281425
$419$	− 274.066i	− 0.654096i	−0.945008	−	0.327048i	$-0.893946\pi$
$419$	− 274.066i	− 0.654096i	0.945008	−	0.327048i	$-0.106054\pi$
$420$	0	0
$421$	−256.187	−0.608521	−0.304260	−	0.952589i	$-0.598409\pi$
$421$	−256.187	−0.608521	−0.304260	+	0.952589i	$0.598409\pi$
$422$	− 182.998i	− 0.433645i
$423$	0	0
$424$	−131.371	−0.309836
$425$	396.801i	0.933650i
$426$	0	0
$427$	23.8757	0.0559150
$428$	− 283.612i	− 0.662644i
$429$	0	0
$430$	542.555	1.26176
$431$	651.526i	1.51166i	0.654768	+	0.755830i	$0.272767\pi$
$431$	651.526i	1.51166i	−0.654768	+	0.755830i	$0.727233\pi$
$432$	0	0
$433$	−717.465	−1.65696	−0.828481	−	0.560017i	$-0.810795\pi$
$433$	−717.465	−1.65696	−0.828481	+	0.560017i	$0.810795\pi$
$434$	124.990i	0.287996i
$435$	0	0
$436$	−8.62933	−0.0197920
$437$	51.8795i	0.118717i
$438$	0	0
$439$	−273.747	−0.623569	−0.311784	−	0.950153i	$-0.600927\pi$
$439$	−273.747	−0.623569	−0.311784	+	0.950153i	$0.600927\pi$
$440$	149.562i	0.339914i
$441$	0	0
$442$	−147.012	−0.332606
$443$	219.891i	0.496369i	0.968713	+	0.248184i	$0.0798339\pi$
$443$	219.891i	0.496369i	−0.968713	+	0.248184i	$0.920166\pi$
$444$	0	0
$445$	697.889	1.56829
$446$	551.138i	1.23574i
$447$	0	0
$448$	43.4979	0.0970935
$449$	235.399i	0.524275i	0.965031	+	0.262137i	$0.0844274\pi$
$449$	235.399i	0.524275i	−0.965031	+	0.262137i	$0.915573\pi$
$450$	0	0
$451$	415.398	0.921059
$452$	− 46.7118i	− 0.103345i
$453$	0	0
$454$	433.485	0.954812
$455$	218.340i	0.479869i
$456$	0	0
$457$	464.879	1.01724	0.508621	−	0.860991i	$-0.330156\pi$
$457$	464.879	1.01724	0.508621	+	0.860991i	$0.330156\pi$
$458$	518.725i	1.13259i
$459$	0	0
$460$	−65.9596	−0.143390
$461$	705.324i	1.52999i	0.644038	+	0.764993i	$0.277258\pi$
$461$	705.324i	1.52999i	−0.644038	+	0.764993i	$0.722742\pi$
$462$	0	0
$463$	249.636	0.539170	0.269585	−	0.962977i	$-0.413114\pi$
$463$	249.636	0.539170	0.269585	+	0.962977i	$0.413114\pi$
$464$	16.3243i	0.0351818i
$465$	0	0
$466$	341.300	0.732402
$467$	− 342.795i	− 0.734035i	−0.930214	−	0.367018i	$-0.880379\pi$
$467$	− 342.795i	− 0.734035i	0.930214	−	0.367018i	$-0.119621\pi$
$468$	0	0
$469$	−205.559	−0.438292
$470$	− 352.568i	− 0.750144i
$471$	0	0
$472$	236.166	0.500351
$473$	− 428.981i	− 0.906936i
$474$	0	0
$475$	241.123	0.507628
$476$	− 193.586i	− 0.406693i
$477$	0	0
$478$	−259.805	−0.543525
$479$	710.409i	1.48311i	0.670892	+	0.741555i	$0.265911\pi$
$479$	710.409i	1.48311i	−0.670892	+	0.741555i	$0.734089\pi$
$480$	0	0
$481$	33.1616	0.0689431
$482$	− 462.332i	− 0.959195i
$483$	0	0
$484$	−123.746	−0.255673
$485$	461.677i	0.951911i
$486$	0	0
$487$	−332.013	−0.681752	−0.340876	−	0.940108i	$-0.610724\pi$
$487$	−332.013	−0.681752	−0.340876	+	0.940108i	$0.610724\pi$
$488$	12.4200i	0.0254509i
$489$	0	0
$490$	189.024	0.385763
$491$	475.304i	0.968034i	0.875059	+	0.484017i	$0.160823\pi$
$491$	475.304i	0.968034i	−0.875059	+	0.484017i	$0.839177\pi$
$492$	0	0
$493$	72.6509	0.147365
$494$	89.3343i	0.180839i
$495$	0	0
$496$	−65.0194	−0.131088
$497$	− 167.034i	− 0.336084i
$498$	0	0
$499$	764.152	1.53137	0.765683	−	0.643218i	$-0.222401\pi$
$499$	764.152	1.53137	0.765683	+	0.643218i	$0.222401\pi$
$500$	− 37.2738i	− 0.0745476i
$501$	0	0
$502$	100.086	0.199375
$503$	209.165i	0.415834i	0.978146	+	0.207917i	$0.0666684\pi$
$503$	209.165i	0.415834i	−0.978146	+	0.207917i	$0.933332\pi$
$504$	0	0
$505$	−1158.27	−2.29361
$506$	52.1521i	0.103067i
$507$	0	0
$508$	435.404	0.857095
$509$	− 784.506i	− 1.54127i	−0.637277	−	0.770635i	$-0.719939\pi$
$509$	− 784.506i	− 1.54127i	0.637277	−	0.770635i	$-0.280061\pi$
$510$	0	0
$511$	768.021	1.50298
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	124.556	0.242327
$515$	114.743i	0.222801i
$516$	0	0
$517$	−278.764	−0.539195
$518$	43.6674i	0.0842999i
$519$	0	0
$520$	−113.580	−0.218423
$521$	473.326i	0.908495i	0.890876	+	0.454247i	$0.150092\pi$
$521$	473.326i	0.908495i	−0.890876	+	0.454247i	$0.849908\pi$
$522$	0	0
$523$	635.254	1.21463	0.607317	−	0.794459i	$-0.292246\pi$
$523$	635.254	1.21463	0.607317	+	0.794459i	$0.292246\pi$
$524$	− 405.910i	− 0.774637i
$525$	0	0
$526$	333.465	0.633964
$527$	289.367i	0.549083i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	451.703i	0.852269i
$531$	0	0
$532$	−117.636	−0.221120
$533$	315.459i	0.591856i
$534$	0	0
$535$	−975.165	−1.82274
$536$	− 106.931i	− 0.199498i
$537$	0	0
$538$	−6.28213	−0.0116768
$539$	− 149.455i	− 0.277282i
$540$	0	0
$541$	1011.63	1.86993	0.934967	−	0.354734i	$-0.115429\pi$
$541$	1011.63	1.86993	0.934967	+	0.354734i	$0.115429\pi$
$542$	− 365.336i	− 0.674051i
$543$	0	0
$544$	100.703	0.185115
$545$	29.6709i	0.0544421i
$546$	0	0
$547$	567.388	1.03727	0.518636	−	0.854995i	$-0.326440\pi$
$547$	567.388	1.03727	0.518636	+	0.854995i	$0.326440\pi$
$548$	− 393.199i	− 0.717516i
$549$	0	0
$550$	242.391	0.440710
$551$	− 44.1476i	− 0.0801227i
$552$	0	0
$553$	−380.095	−0.687333
$554$	− 213.138i	− 0.384725i
$555$	0	0
$556$	392.683	0.706265
$557$	− 449.285i	− 0.806615i	−0.915064	−	0.403308i	$-0.867860\pi$
$557$	− 449.285i	− 0.806615i	0.915064	−	0.403308i	$-0.132140\pi$
$558$	0	0
$559$	325.774	0.582780
$560$	− 149.562i	− 0.267076i
$561$	0	0
$562$	652.924	1.16179
$563$	− 762.535i	− 1.35441i	−0.735792	−	0.677207i	$-0.763190\pi$
$563$	− 762.535i	− 1.35441i	0.735792	−	0.677207i	$-0.236810\pi$
$564$	0	0
$565$	−160.613	−0.284271
$566$	− 473.937i	− 0.837345i
$567$	0	0
$568$	86.8902	0.152976
$569$	504.828i	0.887221i	0.896220	+	0.443610i	$0.146303\pi$
$569$	504.828i	0.887221i	−0.896220	+	0.443610i	$0.853697\pi$
$570$	0	0
$571$	−1046.44	−1.83264	−0.916320	−	0.400447i	$-0.868855\pi$
$571$	−1046.44	−1.83264	−0.916320	+	0.400447i	$0.868855\pi$
$572$	89.8038i	0.157000i
$573$	0	0
$574$	−415.398	−0.723690
$575$	106.898i	0.185910i
$576$	0	0
$577$	1088.56	1.88659	0.943296	−	0.331953i	$-0.107708\pi$
$577$	1088.56	1.88659	0.943296	+	0.331953i	$0.107708\pi$
$578$	− 39.4655i	− 0.0682794i
$579$	0	0
$580$	56.1293	0.0967747
$581$	− 376.120i	− 0.647367i
$582$	0	0
$583$	357.147	0.612601
$584$	399.521i	0.684112i
$585$	0	0
$586$	469.173	0.800637
$587$	720.312i	1.22711i	0.789653	+	0.613553i	$0.210260\pi$
$587$	720.312i	1.22711i	−0.789653	+	0.613553i	$0.789740\pi$
$588$	0	0
$589$	175.839	0.298538
$590$	− 812.028i	− 1.37632i
$591$	0	0
$592$	−22.7156	−0.0383709
$593$	132.133i	0.222821i	0.993774	+	0.111411i	$0.0355369\pi$
$593$	132.133i	0.222821i	−0.993774	+	0.111411i	$0.964463\pi$
$594$	0	0
$595$	−665.622	−1.11869
$596$	− 514.332i	− 0.862973i
$597$	0	0
$598$	−39.6051	−0.0662292
$599$	506.024i	0.844782i	0.906414	+	0.422391i	$0.138809\pi$
$599$	506.024i	0.844782i	−0.906414	+	0.422391i	$0.861191\pi$
$600$	0	0
$601$	511.932	0.851800	0.425900	−	0.904770i	$-0.359958\pi$
$601$	511.932	0.851800	0.425900	+	0.904770i	$0.359958\pi$
$602$	428.981i	0.712593i
$603$	0	0
$604$	−189.627	−0.313952
$605$	425.486i	0.703282i
$606$	0	0
$607$	−515.170	−0.848716	−0.424358	−	0.905495i	$-0.639500\pi$
$607$	−515.170	−0.848716	−0.424358	+	0.905495i	$0.639500\pi$
$608$	− 61.1937i	− 0.100647i
$609$	0	0
$610$	42.7048	0.0700079
$611$	− 211.697i	− 0.346477i
$612$	0	0
$613$	261.653	0.426840	0.213420	−	0.976961i	$-0.431540\pi$
$613$	261.653	0.426840	0.213420	+	0.976961i	$0.431540\pi$
$614$	− 348.536i	− 0.567648i
$615$	0	0
$616$	−118.254	−0.191971
$617$	619.954i	1.00479i	0.864639	+	0.502394i	$0.167547\pi$
$617$	619.954i	1.00479i	−0.864639	+	0.502394i	$0.832453\pi$
$618$	0	0
$619$	775.178	1.25231	0.626154	−	0.779700i	$-0.284628\pi$
$619$	775.178	1.25231	0.626154	+	0.779700i	$0.284628\pi$
$620$	223.562i	0.360583i
$621$	0	0
$622$	−491.939	−0.790899
$623$	551.798i	0.885712i
$624$	0	0
$625$	−685.408	−1.09665
$626$	370.985i	0.592627i
$627$	0	0
$628$	−74.0885	−0.117975
$629$	101.095i	0.160723i
$630$	0	0
$631$	175.311	0.277831	0.138915	−	0.990304i	$-0.455638\pi$
$631$	175.311	0.277831	0.138915	+	0.990304i	$0.455638\pi$
$632$	− 197.724i	− 0.312854i
$633$	0	0
$634$	−702.496	−1.10804
$635$	− 1497.09i	− 2.35762i
$636$	0	0
$637$	113.498	0.178176
$638$	− 44.3796i	− 0.0695606i
$639$	0	0
$640$	77.8017	0.121565
$641$	205.237i	0.320183i	0.987102	+	0.160091i	$0.0511789\pi$
$641$	205.237i	0.320183i	−0.987102	+	0.160091i	$0.948821\pi$
$642$	0	0
$643$	−614.006	−0.954908	−0.477454	−	0.878657i	$-0.658440\pi$
$643$	−614.006	−0.954908	−0.477454	+	0.878657i	$0.658440\pi$
$644$	− 52.1521i	− 0.0809816i
$645$	0	0
$646$	−272.340	−0.421579
$647$	10.3908i	0.0160599i	0.999968	+	0.00802997i	$0.00255605\pi$
$647$	10.3908i	0.0160599i	−0.999968	+	0.00802997i	$0.997444\pi$
$648$	0	0
$649$	−642.045	−0.989283
$650$	184.075i	0.283192i
$651$	0	0
$652$	230.150	0.352990
$653$	435.069i	0.666261i	0.942881	+	0.333131i	$0.108105\pi$
$653$	435.069i	0.666261i	−0.942881	+	0.333131i	$0.891895\pi$
$654$	0	0
$655$	−1395.67	−2.13080
$656$	− 216.088i	− 0.329403i
$657$	0	0
$658$	278.764	0.423653
$659$	907.782i	1.37751i	0.724992	+	0.688757i	$0.241843\pi$
$659$	907.782i	1.37751i	−0.724992	+	0.688757i	$0.758157\pi$
$660$	0	0
$661$	856.923	1.29640	0.648202	−	0.761468i	$-0.275521\pi$
$661$	856.923	1.29640	0.648202	+	0.761468i	$0.275521\pi$
$662$	223.172i	0.337118i
$663$	0	0
$664$	195.656	0.294663
$665$	404.477i	0.608236i
$666$	0	0
$667$	19.5722	0.0293436
$668$	10.1341i	0.0151707i
$669$	0	0
$670$	−367.669	−0.548760
$671$	− 33.7653i	− 0.0503209i
$672$	0	0
$673$	−958.152	−1.42370	−0.711851	−	0.702330i	$-0.752143\pi$
$673$	−958.152	−1.42370	−0.711851	+	0.702330i	$0.752143\pi$
$674$	547.398i	0.812163i
$675$	0	0
$676$	269.802	0.399115
$677$	− 848.650i	− 1.25355i	−0.779202	−	0.626773i	$-0.784375\pi$
$677$	− 848.650i	− 1.25355i	0.779202	−	0.626773i	$-0.215625\pi$
$678$	0	0
$679$	−365.033	−0.537604
$680$	− 346.254i	− 0.509197i
$681$	0	0
$682$	176.763	0.259183
$683$	− 1140.08i	− 1.66923i	−0.550837	−	0.834613i	$-0.685691\pi$
$683$	− 1140.08i	− 1.66923i	0.550837	−	0.834613i	$-0.314309\pi$
$684$	0	0
$685$	−1351.97	−1.97367
$686$	526.236i	0.767108i
$687$	0	0
$688$	−223.154	−0.324352
$689$	271.222i	0.393646i
$690$	0	0
$691$	−905.691	−1.31070	−0.655348	−	0.755327i	$-0.727478\pi$
$691$	−905.691	−1.31070	−0.655348	+	0.755327i	$0.727478\pi$
$692$	514.490i	0.743483i
$693$	0	0
$694$	−320.389	−0.461655
$695$	− 1350.19i	− 1.94273i
$696$	0	0
$697$	−961.693	−1.37976
$698$	− 120.499i	− 0.172635i
$699$	0	0
$700$	−242.391	−0.346272
$701$	− 493.120i	− 0.703453i	−0.936103	−	0.351726i	$-0.885595\pi$
$701$	− 493.120i	− 0.703453i	0.936103	−	0.351726i	$-0.114405\pi$
$702$	0	0
$703$	61.4321	0.0873857
$704$	− 61.5153i	− 0.0873797i
$705$	0	0
$706$	142.862	0.202354
$707$	− 915.810i	− 1.29535i
$708$	0	0
$709$	283.873	0.400385	0.200192	−	0.979757i	$-0.435843\pi$
$709$	283.873	0.400385	0.200192	+	0.979757i	$0.435843\pi$
$710$	− 298.762i	− 0.420791i
$711$	0	0
$712$	−287.043	−0.403151
$713$	77.9555i	0.109335i
$714$	0	0
$715$	308.780	0.431860
$716$	− 65.5935i	− 0.0916110i
$717$	0	0
$718$	669.094	0.931886
$719$	− 646.489i	− 0.899150i	−0.893243	−	0.449575i	$-0.851575\pi$
$719$	− 646.489i	− 0.899150i	0.893243	−	0.449575i	$-0.148425\pi$
$720$	0	0
$721$	−90.7233	−0.125830
$722$	− 345.039i	− 0.477893i
$723$	0	0
$724$	−367.209	−0.507195
$725$	− 90.9669i	− 0.125472i
$726$	0	0
$727$	−371.314	−0.510748	−0.255374	−	0.966842i	$-0.582199\pi$
$727$	−371.314	−0.510748	−0.255374	+	0.966842i	$0.582199\pi$
$728$	− 89.8038i	− 0.123357i
$729$	0	0
$730$	1373.71	1.88179
$731$	993.139i	1.35860i
$732$	0	0
$733$	−290.264	−0.395995	−0.197997	−	0.980203i	$-0.563444\pi$
$733$	−290.264	−0.395995	−0.197997	+	0.980203i	$0.563444\pi$
$734$	535.899i	0.730108i
$735$	0	0
$736$	27.1293	0.0368605
$737$	290.704i	0.394443i
$738$	0	0
$739$	−532.129	−0.720066	−0.360033	−	0.932940i	$-0.617235\pi$
$739$	−532.129	−0.720066	−0.360033	+	0.932940i	$0.617235\pi$
$740$	78.1049i	0.105547i
$741$	0	0
$742$	−357.147	−0.481330
$743$	− 1288.86i	− 1.73466i	−0.497730	−	0.867332i	$-0.665833\pi$
$743$	− 1288.86i	− 1.73466i	0.497730	−	0.867332i	$-0.334167\pi$
$744$	0	0
$745$	−1768.47	−2.37378
$746$	30.6950i	0.0411461i
$747$	0	0
$748$	−273.772	−0.366005
$749$	− 771.031i	− 1.02941i
$750$	0	0
$751$	578.243	0.769964	0.384982	−	0.922924i	$-0.374208\pi$
$751$	578.243	0.769964	0.384982	+	0.922924i	$0.374208\pi$
$752$	145.012i	0.192835i
$753$	0	0
$754$	33.7025	0.0446983
$755$	652.010i	0.863590i
$756$	0	0
$757$	903.264	1.19322	0.596608	−	0.802533i	$-0.296515\pi$
$757$	903.264	1.19322	0.596608	+	0.802533i	$0.296515\pi$
$758$	− 536.705i	− 0.708054i
$759$	0	0
$760$	−210.407	−0.276852
$761$	771.153i	1.01334i	0.862139	+	0.506671i	$0.169124\pi$
$761$	771.153i	1.01334i	−0.862139	+	0.506671i	$0.830876\pi$
$762$	0	0
$763$	−23.4598	−0.0307468
$764$	27.4515i	0.0359313i
$765$	0	0
$766$	280.843	0.366636
$767$	− 487.578i	− 0.635695i
$768$	0	0
$769$	282.553	0.367429	0.183714	−	0.982980i	$-0.441188\pi$
$769$	282.553	0.367429	0.183714	+	0.982980i	$0.441188\pi$
$770$	406.603i	0.528055i
$771$	0	0
$772$	−212.407	−0.275139
$773$	− 871.210i	− 1.12705i	−0.826099	−	0.563525i	$-0.809445\pi$
$773$	− 871.210i	− 1.12705i	0.826099	−	0.563525i	$-0.190555\pi$
$774$	0	0
$775$	362.319	0.467508
$776$	− 189.889i	− 0.244702i
$777$	0	0
$778$	−476.281	−0.612187
$779$	584.390i	0.750180i
$780$	0	0
$781$	−236.221	−0.302460
$782$	− 120.738i	− 0.154397i
$783$	0	0
$784$	−77.7459	−0.0991657
$785$	254.745i	0.324515i
$786$	0	0
$787$	485.671	0.617117	0.308559	−	0.951205i	$-0.400153\pi$
$787$	485.671	0.617117	0.308559	+	0.951205i	$0.400153\pi$
$788$	70.7079i	0.0897308i
$789$	0	0
$790$	−679.851	−0.860570
$791$	− 126.992i	− 0.160546i
$792$	0	0
$793$	25.6419	0.0323353
$794$	330.841i	0.416676i
$795$	0	0
$796$	537.726	0.675535
$797$	− 1068.29i	− 1.34038i	−0.742188	−	0.670192i	$-0.766212\pi$
$797$	− 1068.29i	− 1.34038i	0.742188	−	0.670192i	$-0.233788\pi$
$798$	0	0
$799$	645.370	0.807722
$800$	− 126.091i	− 0.157613i
$801$	0	0
$802$	69.5656	0.0867401
$803$	− 1086.15i	− 1.35261i
$804$	0	0
$805$	−179.319	−0.222756
$806$	134.236i	0.166546i
$807$	0	0
$808$	476.401	0.589605
$809$	44.3081i	0.0547690i	0.999625	+	0.0273845i	$0.00871784\pi$
$809$	44.3081i	0.0547690i	−0.999625	+	0.0273845i	$0.991282\pi$
$810$	0	0
$811$	940.317	1.15945	0.579727	−	0.814811i	$-0.303159\pi$
$811$	940.317	1.15945	0.579727	+	0.814811i	$0.303159\pi$
$812$	44.3796i	0.0546547i
$813$	0	0
$814$	61.7550	0.0758661
$815$	− 791.343i	− 0.970973i
$816$	0	0
$817$	603.499	0.738676
$818$	− 870.877i	− 1.06464i
$819$	0	0
$820$	−742.994	−0.906090
$821$	− 159.843i	− 0.194693i	−0.995251	−	0.0973463i	$-0.968965\pi$
$821$	− 159.843i	− 0.194693i	0.995251	−	0.0973463i	$-0.0310354\pi$
$822$	0	0
$823$	757.411	0.920305	0.460153	−	0.887840i	$-0.347795\pi$
$823$	757.411	0.920305	0.460153	+	0.887840i	$0.347795\pi$
$824$	− 47.1939i	− 0.0572741i
$825$	0	0
$826$	642.045	0.777294
$827$	867.464i	1.04893i	0.851432	+	0.524464i	$0.175734\pi$
$827$	867.464i	1.04893i	−0.851432	+	0.524464i	$0.824266\pi$
$828$	0	0
$829$	−623.021	−0.751533	−0.375766	−	0.926714i	$-0.622620\pi$
$829$	−623.021	−0.751533	−0.375766	+	0.926714i	$0.622620\pi$
$830$	− 672.740i	− 0.810531i
$831$	0	0
$832$	46.7156	0.0561485
$833$	346.006i	0.415373i
$834$	0	0
$835$	34.8447	0.0417302
$836$	166.362i	0.198998i
$837$	0	0
$838$	387.588	0.462515
$839$	− 344.509i	− 0.410618i	−0.978697	−	0.205309i	$-0.934180\pi$
$839$	− 344.509i	− 0.410618i	0.978697	−	0.205309i	$-0.0658200\pi$
$840$	0	0
$841$	824.345	0.980196
$842$	− 362.304i	− 0.430289i
$843$	0	0
$844$	258.798	0.306633
$845$	− 927.681i	− 1.09785i
$846$	0	0
$847$	−336.418	−0.397187
$848$	− 185.786i	− 0.219087i
$849$	0	0
$850$	−561.162	−0.660190
$851$	27.2350i	0.0320036i
$852$	0	0
$853$	−570.907	−0.669294	−0.334647	−	0.942344i	$-0.608617\pi$
$853$	−570.907	−0.669294	−0.334647	+	0.942344i	$0.608617\pi$
$854$	33.7653i	0.0395379i
$855$	0	0
$856$	401.087	0.468560
$857$	1531.64i	1.78721i	0.448857	+	0.893603i	$0.351831\pi$
$857$	1531.64i	1.78721i	−0.448857	+	0.893603i	$0.648169\pi$
$858$	0	0
$859$	−245.268	−0.285528	−0.142764	−	0.989757i	$-0.545599\pi$
$859$	−245.268	−0.285528	−0.142764	+	0.989757i	$0.545599\pi$
$860$	767.289i	0.892196i
$861$	0	0
$862$	−921.397	−1.06891
$863$	258.768i	0.299847i	0.988698	+	0.149924i	$0.0479028\pi$
$863$	258.768i	0.299847i	−0.988698	+	0.149924i	$0.952097\pi$
$864$	0	0
$865$	1769.01	2.04510
$866$	− 1014.65i	− 1.17165i
$867$	0	0
$868$	−176.763	−0.203644
$869$	537.536i	0.618568i
$870$	0	0
$871$	−220.765	−0.253462
$872$	− 12.2037i	− 0.0139951i
$873$	0	0
$874$	−73.3686	−0.0839458
$875$	− 101.333i	− 0.115809i
$876$	0	0
$877$	986.840	1.12525	0.562623	−	0.826714i	$-0.309792\pi$
$877$	986.840	1.12525	0.562623	+	0.826714i	$0.309792\pi$
$878$	− 387.136i	− 0.440930i
$879$	0	0
$880$	−211.513	−0.240356
$881$	834.817i	0.947579i	0.880638	+	0.473789i	$0.157114\pi$
$881$	834.817i	0.947579i	−0.880638	+	0.473789i	$0.842886\pi$
$882$	0	0
$883$	−1217.61	−1.37894	−0.689472	−	0.724313i	$-0.742157\pi$
$883$	−1217.61	−1.37894	−0.689472	+	0.724313i	$0.742157\pi$
$884$	− 207.906i	− 0.235188i
$885$	0	0
$886$	−310.973	−0.350986
$887$	740.177i	0.834472i	0.908798	+	0.417236i	$0.137001\pi$
$887$	740.177i	0.834472i	−0.908798	+	0.417236i	$0.862999\pi$
$888$	0	0
$889$	1183.70	1.33149
$890$	986.964i	1.10895i
$891$	0	0
$892$	−779.427	−0.873797
$893$	− 392.170i	− 0.439161i
$894$	0	0
$895$	−225.535	−0.251995
$896$	61.5153i	0.0686555i
$897$	0	0
$898$	−332.905	−0.370718
$899$	− 66.3374i	− 0.0737903i
$900$	0	0
$901$	−826.835	−0.917686
$902$	587.461i	0.651287i
$903$	0	0
$904$	66.0605	0.0730758
$905$	1262.60i	1.39514i
$906$	0	0
$907$	1607.94	1.77282	0.886408	−	0.462905i	$-0.153193\pi$
$907$	1607.94	1.77282	0.886408	+	0.462905i	$0.153193\pi$
$908$	613.040i	0.675154i
$909$	0	0
$910$	−308.780	−0.339319
$911$	− 474.080i	− 0.520395i	−0.965555	−	0.260198i	$-0.916212\pi$
$911$	− 474.080i	− 0.520395i	0.965555	−	0.260198i	$-0.0837877\pi$
$912$	0	0
$913$	−531.914	−0.582600
$914$	657.439i	0.719298i
$915$	0	0
$916$	−733.588	−0.800861
$917$	− 1103.51i	− 1.20339i
$918$	0	0
$919$	253.043	0.275346	0.137673	−	0.990478i	$-0.456038\pi$
$919$	253.043	0.275346	0.137673	+	0.990478i	$0.456038\pi$
$920$	− 93.2810i	− 0.101392i
$921$	0	0
$922$	−997.479	−1.08186
$923$	− 179.390i	− 0.194355i
$924$	0	0
$925$	126.582	0.136845
$926$	353.038i	0.381250i
$927$	0	0
$928$	−23.0861	−0.0248773
$929$	867.351i	0.933640i	0.884352	+	0.466820i	$0.154600\pi$
$929$	867.351i	0.933640i	−0.884352	+	0.466820i	$0.845400\pi$
$930$	0	0
$931$	210.256	0.225839
$932$	482.670i	0.517887i
$933$	0	0
$934$	484.785	0.519041
$935$	941.331i	1.00677i
$936$	0	0
$937$	−225.411	−0.240567	−0.120283	−	0.992740i	$-0.538380\pi$
$937$	−225.411	−0.240567	−0.120283	+	0.992740i	$0.538380\pi$
$938$	− 290.704i	− 0.309919i
$939$	0	0
$940$	498.606	0.530432
$941$	− 85.7975i	− 0.0911769i	−0.998960	−	0.0455885i	$-0.985484\pi$
$941$	− 85.7975i	− 0.0911769i	0.998960	−	0.0455885i	$-0.0145163\pi$
$942$	0	0
$943$	−259.081	−0.274741
$944$	333.989i	0.353802i
$945$	0	0
$946$	606.670	0.641301
$947$	− 545.858i	− 0.576408i	−0.957569	−	0.288204i	$-0.906942\pi$
$947$	− 545.858i	− 0.576408i	0.957569	−	0.288204i	$-0.0930581\pi$
$948$	0	0
$949$	824.834	0.869162
$950$	341.000i	0.358947i
$951$	0	0
$952$	273.772	0.287575
$953$	− 1734.12i	− 1.81965i	−0.414996	−	0.909823i	$-0.636217\pi$
$953$	− 1734.12i	− 1.81965i	0.414996	−	0.909823i	$-0.363783\pi$
$954$	0	0
$955$	94.3889	0.0988365
$956$	− 367.420i	− 0.384330i
$957$	0	0
$958$	−1004.67	−1.04872
$959$	− 1068.96i	− 1.11466i
$960$	0	0
$961$	−696.780	−0.725057
$962$	46.8976i	0.0487501i
$963$	0	0
$964$	653.836	0.678253
$965$	730.337i	0.756826i
$966$	0	0
$967$	1020.38	1.05520	0.527598	−	0.849494i	$-0.323093\pi$
$967$	1020.38	1.05520	0.527598	+	0.849494i	$0.323093\pi$
$968$	− 175.003i	− 0.180788i
$969$	0	0
$970$	−652.909	−0.673103
$971$	927.378i	0.955075i	0.878611	+	0.477538i	$0.158471\pi$
$971$	927.378i	0.955075i	−0.878611	+	0.477538i	$0.841529\pi$
$972$	0	0
$973$	1067.56	1.09718
$974$	− 469.537i	− 0.482071i
$975$	0	0
$976$	−17.5646	−0.0179965
$977$	− 20.6108i	− 0.0210960i	−0.999944	−	0.0105480i	$-0.996642\pi$
$977$	− 20.6108i	− 0.0210960i	0.999944	−	0.0105480i	$-0.00335759\pi$
$978$	0	0
$979$	780.361	0.797100
$980$	267.320i	0.272776i
$981$	0	0
$982$	−672.182	−0.684503
$983$	− 489.658i	− 0.498126i	−0.968487	−	0.249063i	$-0.919877\pi$
$983$	− 489.658i	− 0.498126i	0.968487	−	0.249063i	$-0.0801226\pi$
$984$	0	0
$985$	243.121	0.246823
$986$	102.744i	0.104203i
$987$	0	0
$988$	−126.338	−0.127872
$989$	267.552i	0.270528i
$990$	0	0
$991$	1258.48	1.26991	0.634957	−	0.772548i	$-0.281018\pi$
$991$	1258.48	1.26991	0.634957	+	0.772548i	$0.281018\pi$
$992$	− 91.9513i	− 0.0926929i
$993$	0	0
$994$	236.221	0.237647
$995$	− 1848.91i	− 1.85820i
$996$	0	0
$997$	−628.704	−0.630595	−0.315298	−	0.948993i	$-0.602104\pi$
$997$	−628.704	−0.630595	−0.315298	+	0.948993i	$0.602104\pi$
$998$	1080.67i	1.08284i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.c.b.323.8	yes	8
3.2	odd	2	inner	414.3.c.b.323.1	✓	8
4.3	odd	2		3312.3.g.b.737.8		8
12.11	even	2		3312.3.g.b.737.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
414.3.c.b.323.1	✓	8	3.2	odd	2	inner
414.3.c.b.323.8	yes	8	1.1	even	1	trivial
3312.3.g.b.737.1		8	12.11	even	2
3312.3.g.b.737.8		8	4.3	odd	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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +6.87676i q^{5} -5.43723 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +6.87676i q^{5} -5.43723 q^{7} -2.82843i q^{8} -9.72521 q^{10} +7.68941i q^{11} -5.83945 q^{13} -7.68941i q^{14} +4.00000 q^{16} -17.8019i q^{17} -10.8176 q^{19} -13.7535i q^{20} -10.8745 q^{22} -4.79583i q^{23} -22.2899 q^{25} -8.25823i q^{26} +10.8745 q^{28} +4.08109i q^{29} -16.2549 q^{31} +5.65685i q^{32} +25.1756 q^{34} -37.3906i q^{35} -5.67890 q^{37} -15.2984i q^{38} +19.4504 q^{40} -54.0221i q^{41} -55.7885 q^{43} -15.3788i q^{44} +6.78233 q^{46} +36.2529i q^{47} -19.4365 q^{49} -31.5226i q^{50} +11.6789 q^{52} -46.4465i q^{53} -52.8783 q^{55} +15.3788i q^{56} -5.77153 q^{58} +83.4972i q^{59} -4.39115 q^{61} -22.9878i q^{62} -8.00000 q^{64} -40.1565i q^{65} +37.8058 q^{67} +35.6037i q^{68} +52.8783 q^{70} +30.7203i q^{71} -141.252 q^{73} -8.03117i q^{74} +21.6352 q^{76} -41.8091i q^{77} +69.9060 q^{79} +27.5071i q^{80} +76.3987 q^{82} +69.1749i q^{83} +122.419 q^{85} -78.8969i q^{86} +21.7489 q^{88} -101.485i q^{89} +31.7505 q^{91} +9.59166i q^{92} -51.2694 q^{94} -74.3902i q^{95} +67.1358 q^{97} -27.4873i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 16 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 + 16 * q^7	\( 8 q - 16 q^{4} + 16 q^{7} - 8 q^{10} - 8 q^{13} + 32 q^{16} - 48 q^{19} + 32 q^{22} - 32 q^{28} - 32 q^{31} - 8 q^{34} + 32 q^{37} + 16 q^{40} + 32 q^{43} + 80 q^{49} + 16 q^{52} + 32 q^{55} + 16 q^{58} + 48 q^{61} - 64 q^{64} - 16 q^{67} - 32 q^{70} - 432 q^{73} + 96 q^{76} - 416 q^{79} - 144 q^{82} + 584 q^{85} - 64 q^{88} + 368 q^{91} + 128 q^{94} - 8 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 16 * q^7 - 8 * q^10 - 8 * q^13 + 32 * q^16 - 48 * q^19 + 32 * q^22 - 32 * q^28 - 32 * q^31 - 8 * q^34 + 32 * q^37 + 16 * q^40 + 32 * q^43 + 80 * q^49 + 16 * q^52 + 32 * q^55 + 16 * q^58 + 48 * q^61 - 64 * q^64 - 16 * q^67 - 32 * q^70 - 432 * q^73 + 96 * q^76 - 416 * q^79 - 144 * q^82 + 584 * q^85 - 64 * q^88 + 368 * q^91 + 128 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

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