LMFDB - Embedded newform 476.4.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,4,Mod(1,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("476.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 476 = 2^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	476.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:	$28.0849091627$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 87x^{4} + 184x^{3} + 2031x^{2} - 4232x - 7516 $ x^6 - 2x^5 - 87x^4 + 184x^3 + 2031x^2 - 4232*x - 7516
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.21885$ of defining polynomial
Character	$\chi$	$=$	476.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.21885 q^{3} +3.60198 q^{5} -7.00000 q^{7} -9.20134 q^{9} +O(q^{10})$	$q+4.21885 q^{3} +3.60198 q^{5} -7.00000 q^{7} -9.20134 q^{9} -67.2337 q^{11} +73.9432 q^{13} +15.1962 q^{15} -17.0000 q^{17} -95.0370 q^{19} -29.5319 q^{21} -117.400 q^{23} -112.026 q^{25} -152.728 q^{27} +293.155 q^{29} -216.312 q^{31} -283.649 q^{33} -25.2139 q^{35} -169.165 q^{37} +311.955 q^{39} -230.478 q^{41} +372.876 q^{43} -33.1430 q^{45} +44.3943 q^{47} +49.0000 q^{49} -71.7204 q^{51} +568.830 q^{53} -242.175 q^{55} -400.947 q^{57} -171.608 q^{59} -323.941 q^{61} +64.4094 q^{63} +266.342 q^{65} -1035.53 q^{67} -495.294 q^{69} -158.364 q^{71} -632.455 q^{73} -472.619 q^{75} +470.636 q^{77} +613.395 q^{79} -395.899 q^{81} +508.574 q^{83} -61.2337 q^{85} +1236.78 q^{87} -346.953 q^{89} -517.602 q^{91} -912.587 q^{93} -342.322 q^{95} -535.844 q^{97} +618.640 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9}+O(q^{10}) $ 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9	$ 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9} - 20 q^{11} + 42 q^{13} - 136 q^{15} - 102 q^{17} - 104 q^{19} - 14 q^{21} - 230 q^{23} + 108 q^{25} - 130 q^{27} - 52 q^{29} - 564 q^{31} - 346 q^{33} - 70 q^{35} - 564 q^{37} - 626 q^{39} - 548 q^{41} - 648 q^{43} - 174 q^{45} - 366 q^{47} + 294 q^{49} - 34 q^{51} - 74 q^{53} - 1460 q^{55} - 316 q^{57} - 558 q^{59} - 620 q^{61} - 112 q^{63} - 1378 q^{65} - 164 q^{67} - 540 q^{69} - 822 q^{71} + 940 q^{73} - 2698 q^{75} + 140 q^{77} - 1838 q^{79} - 3094 q^{81} - 1118 q^{83} - 170 q^{85} - 1354 q^{87} - 1634 q^{89} - 294 q^{91} + 268 q^{93} - 2642 q^{95} + 182 q^{97} + 12 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9 - 20 * q^11 + 42 * q^13 - 136 * q^15 - 102 * q^17 - 104 * q^19 - 14 * q^21 - 230 * q^23 + 108 * q^25 - 130 * q^27 - 52 * q^29 - 564 * q^31 - 346 * q^33 - 70 * q^35 - 564 * q^37 - 626 * q^39 - 548 * q^41 - 648 * q^43 - 174 * q^45 - 366 * q^47 + 294 * q^49 - 34 * q^51 - 74 * q^53 - 1460 * q^55 - 316 * q^57 - 558 * q^59 - 620 * q^61 - 112 * q^63 - 1378 * q^65 - 164 * q^67 - 540 * q^69 - 822 * q^71 + 940 * q^73 - 2698 * q^75 + 140 * q^77 - 1838 * q^79 - 3094 * q^81 - 1118 * q^83 - 170 * q^85 - 1354 * q^87 - 1634 * q^89 - 294 * q^91 + 268 * q^93 - 2642 * q^95 + 182 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	4.21885	0.811917	0.405959	−	0.913891i	$-0.366938\pi$
$3$	4.21885	0.811917	0.405959	+	0.913891i	$0.366938\pi$
$4$	0	0
$5$	3.60198	0.322171	0.161086	−	0.986940i	$-0.448501\pi$
$5$	3.60198	0.322171	0.161086	+	0.986940i	$0.448501\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−9.20134	−0.340790
$10$	0	0
$11$	−67.2337	−1.84288	−0.921442	−	0.388516i	$-0.872988\pi$
$11$	−67.2337	−1.84288	−0.921442	+	0.388516i	$0.872988\pi$
$12$	0	0
$13$	73.9432	1.57755	0.788775	−	0.614682i	$-0.210716\pi$
$13$	73.9432	1.57755	0.788775	+	0.614682i	$0.210716\pi$
$14$	0	0
$15$	15.1962	0.261576
$16$	0	0
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	0	0
$19$	−95.0370	−1.14753	−0.573763	−	0.819021i	$-0.694517\pi$
$19$	−95.0370	−1.14753	−0.573763	+	0.819021i	$0.694517\pi$
$20$	0	0
$21$	−29.5319	−0.306876
$22$	0	0
$23$	−117.400	−1.06433	−0.532166	−	0.846640i	$-0.678622\pi$
$23$	−117.400	−1.06433	−0.532166	+	0.846640i	$0.678622\pi$
$24$	0	0
$25$	−112.026	−0.896206
$26$	0	0
$27$	−152.728	−1.08861
$28$	0	0
$29$	293.155	1.87716	0.938578	−	0.345067i	$-0.112144\pi$
$29$	293.155	1.87716	0.938578	+	0.345067i	$0.112144\pi$
$30$	0	0
$31$	−216.312	−1.25325	−0.626626	−	0.779320i	$-0.715564\pi$
$31$	−216.312	−1.25325	−0.626626	+	0.779320i	$0.715564\pi$
$32$	0	0
$33$	−283.649	−1.49627
$34$	0	0
$35$	−25.2139	−0.121769
$36$	0	0
$37$	−169.165	−0.751636	−0.375818	−	0.926693i	$-0.622638\pi$
$37$	−169.165	−0.751636	−0.375818	+	0.926693i	$0.622638\pi$
$38$	0	0
$39$	311.955	1.28084
$40$	0	0
$41$	−230.478	−0.877917	−0.438959	−	0.898507i	$-0.644653\pi$
$41$	−230.478	−0.877917	−0.438959	+	0.898507i	$0.644653\pi$
$42$	0	0
$43$	372.876	1.32240	0.661198	−	0.750211i	$-0.270048\pi$
$43$	372.876	1.32240	0.661198	+	0.750211i	$0.270048\pi$
$44$	0	0
$45$	−33.1430	−0.109793
$46$	0	0
$47$	44.3943	0.137778	0.0688890	−	0.997624i	$-0.478055\pi$
$47$	44.3943	0.137778	0.0688890	+	0.997624i	$0.478055\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−71.7204	−0.196919
$52$	0	0
$53$	568.830	1.47424	0.737120	−	0.675761i	$-0.236185\pi$
$53$	568.830	1.47424	0.737120	+	0.675761i	$0.236185\pi$
$54$	0	0
$55$	−242.175	−0.593724
$56$	0	0
$57$	−400.947	−0.931696
$58$	0	0
$59$	−171.608	−0.378669	−0.189334	−	0.981913i	$-0.560633\pi$
$59$	−171.608	−0.378669	−0.189334	+	0.981913i	$0.560633\pi$
$60$	0	0
$61$	−323.941	−0.679942	−0.339971	−	0.940436i	$-0.610417\pi$
$61$	−323.941	−0.679942	−0.339971	+	0.940436i	$0.610417\pi$
$62$	0	0
$63$	64.4094	0.128807
$64$	0	0
$65$	266.342	0.508241
$66$	0	0
$67$	−1035.53	−1.88820	−0.944102	−	0.329655i	$-0.893068\pi$
$67$	−1035.53	−1.88820	−0.944102	+	0.329655i	$0.893068\pi$
$68$	0	0
$69$	−495.294	−0.864150
$70$	0	0
$71$	−158.364	−0.264710	−0.132355	−	0.991202i	$-0.542254\pi$
$71$	−158.364	−0.264710	−0.132355	+	0.991202i	$0.542254\pi$
$72$	0	0
$73$	−632.455	−1.01402	−0.507008	−	0.861941i	$-0.669249\pi$
$73$	−632.455	−1.01402	−0.507008	+	0.861941i	$0.669249\pi$
$74$	0	0
$75$	−472.619	−0.727645
$76$	0	0
$77$	470.636	0.696545
$78$	0	0
$79$	613.395	0.873573	0.436787	−	0.899565i	$-0.356116\pi$
$79$	613.395	0.873573	0.436787	+	0.899565i	$0.356116\pi$
$80$	0	0
$81$	−395.899	−0.543072
$82$	0	0
$83$	508.574	0.672570	0.336285	−	0.941760i	$-0.390830\pi$
$83$	508.574	0.672570	0.336285	+	0.941760i	$0.390830\pi$
$84$	0	0
$85$	−61.2337	−0.0781380
$86$	0	0
$87$	1236.78	1.52410
$88$	0	0
$89$	−346.953	−0.413224	−0.206612	−	0.978423i	$-0.566244\pi$
$89$	−346.953	−0.413224	−0.206612	+	0.978423i	$0.566244\pi$
$90$	0	0
$91$	−517.602	−0.596258
$92$	0	0
$93$	−912.587	−1.01754
$94$	0	0
$95$	−342.322	−0.369700
$96$	0	0
$97$	−535.844	−0.560894	−0.280447	−	0.959870i	$-0.590483\pi$
$97$	−535.844	−0.560894	−0.280447	+	0.959870i	$0.590483\pi$
$98$	0	0
$99$	618.640	0.628037
$100$	0	0
$101$	815.789	0.803703	0.401852	−	0.915705i	$-0.368367\pi$
$101$	815.789	0.803703	0.401852	+	0.915705i	$0.368367\pi$
$102$	0	0
$103$	1129.36	1.08038	0.540189	−	0.841543i	$-0.318353\pi$
$103$	1129.36	1.08038	0.540189	+	0.841543i	$0.318353\pi$
$104$	0	0
$105$	−106.373	−0.0988665
$106$	0	0
$107$	−304.845	−0.275425	−0.137713	−	0.990472i	$-0.543975\pi$
$107$	−304.845	−0.275425	−0.137713	+	0.990472i	$0.543975\pi$
$108$	0	0
$109$	−505.895	−0.444550	−0.222275	−	0.974984i	$-0.571348\pi$
$109$	−505.895	−0.444550	−0.222275	+	0.974984i	$0.571348\pi$
$110$	0	0
$111$	−713.681	−0.610266
$112$	0	0
$113$	−1463.63	−1.21847	−0.609235	−	0.792990i	$-0.708523\pi$
$113$	−1463.63	−1.21847	−0.609235	+	0.792990i	$0.708523\pi$
$114$	0	0
$115$	−422.874	−0.342897
$116$	0	0
$117$	−680.376	−0.537614
$118$	0	0
$119$	119.000	0.0916698
$120$	0	0
$121$	3189.37	2.39622
$122$	0	0
$123$	−972.351	−0.712796
$124$	0	0
$125$	−853.762	−0.610903
$126$	0	0
$127$	−1345.46	−0.940080	−0.470040	−	0.882645i	$-0.655760\pi$
$127$	−1345.46	−0.940080	−0.470040	+	0.882645i	$0.655760\pi$
$128$	0	0
$129$	1573.11	1.07368
$130$	0	0
$131$	−524.583	−0.349871	−0.174935	−	0.984580i	$-0.555972\pi$
$131$	−524.583	−0.349871	−0.174935	+	0.984580i	$0.555972\pi$
$132$	0	0
$133$	665.259	0.433724
$134$	0	0
$135$	−550.123	−0.350719
$136$	0	0
$137$	−2462.60	−1.53572	−0.767862	−	0.640616i	$-0.778679\pi$
$137$	−2462.60	−1.53572	−0.767862	+	0.640616i	$0.778679\pi$
$138$	0	0
$139$	−371.843	−0.226902	−0.113451	−	0.993544i	$-0.536190\pi$
$139$	−371.843	−0.226902	−0.113451	+	0.993544i	$0.536190\pi$
$140$	0	0
$141$	187.293	0.111864
$142$	0	0
$143$	−4971.48	−2.90724
$144$	0	0
$145$	1055.94	0.604765
$146$	0	0
$147$	206.723	0.115988
$148$	0	0
$149$	1507.06	0.828614	0.414307	−	0.910137i	$-0.364024\pi$
$149$	1507.06	0.828614	0.414307	+	0.910137i	$0.364024\pi$
$150$	0	0
$151$	551.844	0.297407	0.148703	−	0.988882i	$-0.452490\pi$
$151$	551.844	0.297407	0.148703	+	0.988882i	$0.452490\pi$
$152$	0	0
$153$	156.423	0.0826538
$154$	0	0
$155$	−779.152	−0.403761
$156$	0	0
$157$	2832.75	1.43999	0.719994	−	0.693981i	$-0.244145\pi$
$157$	2832.75	1.43999	0.719994	+	0.693981i	$0.244145\pi$
$158$	0	0
$159$	2399.81	1.19696
$160$	0	0
$161$	821.802	0.402280
$162$	0	0
$163$	604.206	0.290338	0.145169	−	0.989407i	$-0.453627\pi$
$163$	604.206	0.290338	0.145169	+	0.989407i	$0.453627\pi$
$164$	0	0
$165$	−1021.70	−0.482055
$166$	0	0
$167$	1516.82	0.702845	0.351422	−	0.936217i	$-0.385698\pi$
$167$	1516.82	0.702845	0.351422	+	0.936217i	$0.385698\pi$
$168$	0	0
$169$	3270.60	1.48867
$170$	0	0
$171$	874.468	0.391066
$172$	0	0
$173$	2081.48	0.914749	0.457375	−	0.889274i	$-0.348790\pi$
$173$	2081.48	0.914749	0.457375	+	0.889274i	$0.348790\pi$
$174$	0	0
$175$	784.180	0.338734
$176$	0	0
$177$	−723.988	−0.307448
$178$	0	0
$179$	3326.70	1.38910	0.694551	−	0.719444i	$-0.255603\pi$
$179$	3326.70	1.38910	0.694551	+	0.719444i	$0.255603\pi$
$180$	0	0
$181$	−978.624	−0.401881	−0.200941	−	0.979603i	$-0.564400\pi$
$181$	−978.624	−0.401881	−0.200941	+	0.979603i	$0.564400\pi$
$182$	0	0
$183$	−1366.66	−0.552056
$184$	0	0
$185$	−609.329	−0.242155
$186$	0	0
$187$	1142.97	0.446965
$188$	0	0
$189$	1069.10	0.411456
$190$	0	0
$191$	−1227.80	−0.465132	−0.232566	−	0.972581i	$-0.574712\pi$
$191$	−1227.80	−0.465132	−0.232566	+	0.972581i	$0.574712\pi$
$192$	0	0
$193$	−4102.60	−1.53011	−0.765055	−	0.643965i	$-0.777288\pi$
$193$	−4102.60	−1.53011	−0.765055	+	0.643965i	$0.777288\pi$
$194$	0	0
$195$	1123.66	0.412650
$196$	0	0
$197$	2234.23	0.808033	0.404016	−	0.914752i	$-0.367614\pi$
$197$	2234.23	0.808033	0.404016	+	0.914752i	$0.367614\pi$
$198$	0	0
$199$	−531.272	−0.189251	−0.0946253	−	0.995513i	$-0.530165\pi$
$199$	−531.272	−0.189251	−0.0946253	+	0.995513i	$0.530165\pi$
$200$	0	0
$201$	−4368.72	−1.53306
$202$	0	0
$203$	−2052.09	−0.709498
$204$	0	0
$205$	−830.178	−0.282840
$206$	0	0
$207$	1080.24	0.362714
$208$	0	0
$209$	6389.69	2.11476
$210$	0	0
$211$	4240.30	1.38348	0.691740	−	0.722146i	$-0.256844\pi$
$211$	4240.30	1.38348	0.691740	+	0.722146i	$0.256844\pi$
$212$	0	0
$213$	−668.115	−0.214922
$214$	0	0
$215$	1343.09	0.426038
$216$	0	0
$217$	1514.18	0.473684
$218$	0	0
$219$	−2668.23	−0.823298
$220$	0	0
$221$	−1257.03	−0.382612
$222$	0	0
$223$	−6440.73	−1.93409	−0.967047	−	0.254597i	$-0.918057\pi$
$223$	−6440.73	−1.93409	−0.967047	+	0.254597i	$0.918057\pi$
$224$	0	0
$225$	1030.79	0.305418
$226$	0	0
$227$	609.298	0.178152	0.0890761	−	0.996025i	$-0.471609\pi$
$227$	609.298	0.178152	0.0890761	+	0.996025i	$0.471609\pi$
$228$	0	0
$229$	1553.90	0.448403	0.224202	−	0.974543i	$-0.428023\pi$
$229$	1553.90	0.448403	0.224202	+	0.974543i	$0.428023\pi$
$230$	0	0
$231$	1985.54	0.565537
$232$	0	0
$233$	6641.12	1.86727	0.933636	−	0.358223i	$-0.116617\pi$
$233$	6641.12	1.86727	0.933636	+	0.358223i	$0.116617\pi$
$234$	0	0
$235$	159.907	0.0443881
$236$	0	0
$237$	2587.82	0.709269
$238$	0	0
$239$	3764.25	1.01878	0.509392	−	0.860535i	$-0.329870\pi$
$239$	3764.25	1.01878	0.509392	+	0.860535i	$0.329870\pi$
$240$	0	0
$241$	4607.55	1.23153	0.615765	−	0.787930i	$-0.288847\pi$
$241$	4607.55	1.23153	0.615765	+	0.787930i	$0.288847\pi$
$242$	0	0
$243$	2453.41	0.647681
$244$	0	0
$245$	176.497	0.0460244
$246$	0	0
$247$	−7027.34	−1.81028
$248$	0	0
$249$	2145.60	0.546071
$250$	0	0
$251$	1640.44	0.412525	0.206263	−	0.978497i	$-0.433870\pi$
$251$	1640.44	0.412525	0.206263	+	0.978497i	$0.433870\pi$
$252$	0	0
$253$	7893.25	1.96144
$254$	0	0
$255$	−258.336	−0.0634416
$256$	0	0
$257$	−6185.11	−1.50123	−0.750615	−	0.660739i	$-0.770243\pi$
$257$	−6185.11	−1.50123	−0.750615	+	0.660739i	$0.770243\pi$
$258$	0	0
$259$	1184.15	0.284092
$260$	0	0
$261$	−2697.42	−0.639717
$262$	0	0
$263$	−195.361	−0.0458041	−0.0229020	−	0.999738i	$-0.507291\pi$
$263$	−195.361	−0.0458041	−0.0229020	+	0.999738i	$0.507291\pi$
$264$	0	0
$265$	2048.91	0.474958
$266$	0	0
$267$	−1463.74	−0.335504
$268$	0	0
$269$	5470.21	1.23987	0.619934	−	0.784654i	$-0.287159\pi$
$269$	5470.21	1.23987	0.619934	+	0.784654i	$0.287159\pi$
$270$	0	0
$271$	−5689.34	−1.27529	−0.637643	−	0.770332i	$-0.720091\pi$
$271$	−5689.34	−1.27529	−0.637643	+	0.770332i	$0.720091\pi$
$272$	0	0
$273$	−2183.69	−0.484112
$274$	0	0
$275$	7531.91	1.65160
$276$	0	0
$277$	−2224.84	−0.482590	−0.241295	−	0.970452i	$-0.577572\pi$
$277$	−2224.84	−0.482590	−0.241295	+	0.970452i	$0.577572\pi$
$278$	0	0
$279$	1990.36	0.427096
$280$	0	0
$281$	6220.59	1.32060	0.660302	−	0.751000i	$-0.270428\pi$
$281$	6220.59	1.32060	0.660302	+	0.751000i	$0.270428\pi$
$282$	0	0
$283$	6084.82	1.27811	0.639055	−	0.769161i	$-0.279326\pi$
$283$	6084.82	1.27811	0.639055	+	0.769161i	$0.279326\pi$
$284$	0	0
$285$	−1444.20	−0.300166
$286$	0	0
$287$	1613.35	0.331822
$288$	0	0
$289$	289.000	0.0588235
$290$	0	0
$291$	−2260.64	−0.455399
$292$	0	0
$293$	−3262.01	−0.650406	−0.325203	−	0.945644i	$-0.605433\pi$
$293$	−3262.01	−0.650406	−0.325203	+	0.945644i	$0.605433\pi$
$294$	0	0
$295$	−618.129	−0.121996
$296$	0	0
$297$	10268.5	2.00618
$298$	0	0
$299$	−8680.95	−1.67904
$300$	0	0
$301$	−2610.13	−0.499819
$302$	0	0
$303$	3441.69	0.652540
$304$	0	0
$305$	−1166.83	−0.219058
$306$	0	0
$307$	−4428.15	−0.823217	−0.411609	−	0.911361i	$-0.635033\pi$
$307$	−4428.15	−0.823217	−0.411609	+	0.911361i	$0.635033\pi$
$308$	0	0
$309$	4764.59	0.877178
$310$	0	0
$311$	−9301.30	−1.69591	−0.847956	−	0.530067i	$-0.822167\pi$
$311$	−9301.30	−1.69591	−0.847956	+	0.530067i	$0.822167\pi$
$312$	0	0
$313$	−967.734	−0.174759	−0.0873795	−	0.996175i	$-0.527849\pi$
$313$	−967.734	−0.174759	−0.0873795	+	0.996175i	$0.527849\pi$
$314$	0	0
$315$	232.001	0.0414978
$316$	0	0
$317$	−2928.17	−0.518808	−0.259404	−	0.965769i	$-0.583526\pi$
$317$	−2928.17	−0.518808	−0.259404	+	0.965769i	$0.583526\pi$
$318$	0	0
$319$	−19709.9	−3.45938
$320$	0	0
$321$	−1286.10	−0.223623
$322$	0	0
$323$	1615.63	0.278316
$324$	0	0
$325$	−8283.54	−1.41381
$326$	0	0
$327$	−2134.29	−0.360938
$328$	0	0
$329$	−310.760	−0.0520752
$330$	0	0
$331$	−2470.27	−0.410207	−0.205103	−	0.978740i	$-0.565753\pi$
$331$	−2470.27	−0.410207	−0.205103	+	0.978740i	$0.565753\pi$
$332$	0	0
$333$	1556.54	0.256150
$334$	0	0
$335$	−3729.94	−0.608324
$336$	0	0
$337$	−5597.14	−0.904734	−0.452367	−	0.891832i	$-0.649420\pi$
$337$	−5597.14	−0.904734	−0.452367	+	0.891832i	$0.649420\pi$
$338$	0	0
$339$	−6174.85	−0.989297
$340$	0	0
$341$	14543.5	2.30960
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−1784.04	−0.278404
$346$	0	0
$347$	182.773	0.0282760	0.0141380	−	0.999900i	$-0.495500\pi$
$347$	182.773	0.0282760	0.0141380	+	0.999900i	$0.495500\pi$
$348$	0	0
$349$	−4866.28	−0.746378	−0.373189	−	0.927755i	$-0.621736\pi$
$349$	−4866.28	−0.746378	−0.373189	+	0.927755i	$0.621736\pi$
$350$	0	0
$351$	−11293.2	−1.71734
$352$	0	0
$353$	5209.47	0.785474	0.392737	−	0.919651i	$-0.371528\pi$
$353$	5209.47	0.785474	0.392737	+	0.919651i	$0.371528\pi$
$354$	0	0
$355$	−570.426	−0.0852818
$356$	0	0
$357$	502.043	0.0744283
$358$	0	0
$359$	−328.896	−0.0483523	−0.0241762	−	0.999708i	$-0.507696\pi$
$359$	−328.896	−0.0483523	−0.0241762	+	0.999708i	$0.507696\pi$
$360$	0	0
$361$	2173.04	0.316815
$362$	0	0
$363$	13455.5	1.94553
$364$	0	0
$365$	−2278.09	−0.326687
$366$	0	0
$367$	−1028.41	−0.146274	−0.0731372	−	0.997322i	$-0.523301\pi$
$367$	−1028.41	−0.146274	−0.0731372	+	0.997322i	$0.523301\pi$
$368$	0	0
$369$	2120.71	0.299186
$370$	0	0
$371$	−3981.81	−0.557211
$372$	0	0
$373$	−1656.48	−0.229944	−0.114972	−	0.993369i	$-0.536678\pi$
$373$	−1656.48	−0.229944	−0.114972	+	0.993369i	$0.536678\pi$
$374$	0	0
$375$	−3601.89	−0.496002
$376$	0	0
$377$	21676.8	2.96131
$378$	0	0
$379$	−8733.74	−1.18370	−0.591850	−	0.806048i	$-0.701602\pi$
$379$	−8733.74	−1.18370	−0.591850	+	0.806048i	$0.701602\pi$
$380$	0	0
$381$	−5676.28	−0.763267
$382$	0	0
$383$	1060.20	0.141445	0.0707227	−	0.997496i	$-0.477469\pi$
$383$	1060.20	0.141445	0.0707227	+	0.997496i	$0.477469\pi$
$384$	0	0
$385$	1695.22	0.224407
$386$	0	0
$387$	−3430.96	−0.450660
$388$	0	0
$389$	−8042.65	−1.04827	−0.524137	−	0.851634i	$-0.675612\pi$
$389$	−8042.65	−1.04827	−0.524137	+	0.851634i	$0.675612\pi$
$390$	0	0
$391$	1995.80	0.258139
$392$	0	0
$393$	−2213.14	−0.284066
$394$	0	0
$395$	2209.44	0.281440
$396$	0	0
$397$	3134.30	0.396237	0.198119	−	0.980178i	$-0.436517\pi$
$397$	3134.30	0.396237	0.198119	+	0.980178i	$0.436517\pi$
$398$	0	0
$399$	2806.63	0.352148
$400$	0	0
$401$	−3479.86	−0.433356	−0.216678	−	0.976243i	$-0.569522\pi$
$401$	−3479.86	−0.433356	−0.216678	+	0.976243i	$0.569522\pi$
$402$	0	0
$403$	−15994.8	−1.97707
$404$	0	0
$405$	−1426.02	−0.174962
$406$	0	0
$407$	11373.6	1.38518
$408$	0	0
$409$	11857.6	1.43355	0.716774	−	0.697306i	$-0.245618\pi$
$409$	11857.6	1.43355	0.716774	+	0.697306i	$0.245618\pi$
$410$	0	0
$411$	−10389.3	−1.24688
$412$	0	0
$413$	1201.26	0.143123
$414$	0	0
$415$	1831.88	0.216682
$416$	0	0
$417$	−1568.75	−0.184225
$418$	0	0
$419$	9832.18	1.14638	0.573190	−	0.819422i	$-0.305706\pi$
$419$	9832.18	1.14638	0.573190	+	0.819422i	$0.305706\pi$
$420$	0	0
$421$	−15615.1	−1.80768	−0.903842	−	0.427866i	$-0.859266\pi$
$421$	−15615.1	−1.80768	−0.903842	+	0.427866i	$0.859266\pi$
$422$	0	0
$423$	−408.487	−0.0469534
$424$	0	0
$425$	1904.44	0.217362
$426$	0	0
$427$	2267.59	0.256994
$428$	0	0
$429$	−20973.9	−2.36044
$430$	0	0
$431$	−7706.49	−0.861273	−0.430636	−	0.902525i	$-0.641711\pi$
$431$	−7706.49	−0.861273	−0.430636	+	0.902525i	$0.641711\pi$
$432$	0	0
$433$	−8632.39	−0.958075	−0.479037	−	0.877794i	$-0.659014\pi$
$433$	−8632.39	−0.958075	−0.479037	+	0.877794i	$0.659014\pi$
$434$	0	0
$435$	4454.85	0.491020
$436$	0	0
$437$	11157.4	1.22135
$438$	0	0
$439$	−7631.97	−0.829736	−0.414868	−	0.909882i	$-0.636172\pi$
$439$	−7631.97	−0.829736	−0.414868	+	0.909882i	$0.636172\pi$
$440$	0	0
$441$	−450.865	−0.0486843
$442$	0	0
$443$	−7889.53	−0.846146	−0.423073	−	0.906096i	$-0.639049\pi$
$443$	−7889.53	−0.846146	−0.423073	+	0.906096i	$0.639049\pi$
$444$	0	0
$445$	−1249.72	−0.133129
$446$	0	0
$447$	6358.07	0.672766
$448$	0	0
$449$	−6466.32	−0.679654	−0.339827	−	0.940488i	$-0.610368\pi$
$449$	−6466.32	−0.679654	−0.339827	+	0.940488i	$0.610368\pi$
$450$	0	0
$451$	15495.9	1.61790
$452$	0	0
$453$	2328.14	0.241470
$454$	0	0
$455$	−1864.39	−0.192097
$456$	0	0
$457$	−3826.49	−0.391676	−0.195838	−	0.980636i	$-0.562743\pi$
$457$	−3826.49	−0.391676	−0.195838	+	0.980636i	$0.562743\pi$
$458$	0	0
$459$	2596.37	0.264027
$460$	0	0
$461$	13843.0	1.39855	0.699275	−	0.714853i	$-0.253506\pi$
$461$	13843.0	1.39855	0.699275	+	0.714853i	$0.253506\pi$
$462$	0	0
$463$	−2912.67	−0.292361	−0.146181	−	0.989258i	$-0.546698\pi$
$463$	−2912.67	−0.292361	−0.146181	+	0.989258i	$0.546698\pi$
$464$	0	0
$465$	−3287.12	−0.327821
$466$	0	0
$467$	−16765.7	−1.66129	−0.830646	−	0.556801i	$-0.812029\pi$
$467$	−16765.7	−1.66129	−0.830646	+	0.556801i	$0.812029\pi$
$468$	0	0
$469$	7248.68	0.713674
$470$	0	0
$471$	11950.9	1.16915
$472$	0	0
$473$	−25069.8	−2.43702
$474$	0	0
$475$	10646.6	1.02842
$476$	0	0
$477$	−5233.99	−0.502407
$478$	0	0
$479$	−486.715	−0.0464271	−0.0232136	−	0.999731i	$-0.507390\pi$
$479$	−486.715	−0.0464271	−0.0232136	+	0.999731i	$0.507390\pi$
$480$	0	0
$481$	−12508.6	−1.18574
$482$	0	0
$483$	3467.06	0.326618
$484$	0	0
$485$	−1930.10	−0.180704
$486$	0	0
$487$	10496.4	0.976671	0.488335	−	0.872656i	$-0.337604\pi$
$487$	10496.4	0.976671	0.488335	+	0.872656i	$0.337604\pi$
$488$	0	0
$489$	2549.05	0.235730
$490$	0	0
$491$	13211.2	1.21428	0.607142	−	0.794593i	$-0.292316\pi$
$491$	13211.2	1.21428	0.607142	+	0.794593i	$0.292316\pi$
$492$	0	0
$493$	−4983.64	−0.455277
$494$	0	0
$495$	2228.33	0.202335
$496$	0	0
$497$	1108.55	0.100051
$498$	0	0
$499$	−8909.12	−0.799252	−0.399626	−	0.916678i	$-0.630860\pi$
$499$	−8909.12	−0.799252	−0.399626	+	0.916678i	$0.630860\pi$
$500$	0	0
$501$	6399.23	0.570652
$502$	0	0
$503$	−1731.04	−0.153445	−0.0767227	−	0.997052i	$-0.524446\pi$
$503$	−1731.04	−0.153445	−0.0767227	+	0.997052i	$0.524446\pi$
$504$	0	0
$505$	2938.46	0.258930
$506$	0	0
$507$	13798.2	1.20867
$508$	0	0
$509$	−18149.0	−1.58044	−0.790218	−	0.612825i	$-0.790033\pi$
$509$	−18149.0	−1.58044	−0.790218	+	0.612825i	$0.790033\pi$
$510$	0	0
$511$	4427.18	0.383262
$512$	0	0
$513$	14514.8	1.24921
$514$	0	0
$515$	4067.93	0.348067
$516$	0	0
$517$	−2984.79	−0.253909
$518$	0	0
$519$	8781.43	0.742701
$520$	0	0
$521$	−2472.78	−0.207935	−0.103968	−	0.994581i	$-0.533154\pi$
$521$	−2472.78	−0.207935	−0.103968	+	0.994581i	$0.533154\pi$
$522$	0	0
$523$	18354.9	1.53462	0.767310	−	0.641277i	$-0.221595\pi$
$523$	18354.9	1.53462	0.767310	+	0.641277i	$0.221595\pi$
$524$	0	0
$525$	3308.34	0.275024
$526$	0	0
$527$	3677.30	0.303958
$528$	0	0
$529$	1615.82	0.132803
$530$	0	0
$531$	1579.02	0.129047
$532$	0	0
$533$	−17042.3	−1.38496
$534$	0	0
$535$	−1098.05	−0.0887341
$536$	0	0
$537$	14034.8	1.12784
$538$	0	0
$539$	−3294.45	−0.263269
$540$	0	0
$541$	17095.6	1.35859	0.679294	−	0.733866i	$-0.262286\pi$
$541$	17095.6	1.35859	0.679294	+	0.733866i	$0.262286\pi$
$542$	0	0
$543$	−4128.66	−0.326294
$544$	0	0
$545$	−1822.23	−0.143221
$546$	0	0
$547$	−17456.1	−1.36448	−0.682238	−	0.731130i	$-0.738993\pi$
$547$	−17456.1	−1.36448	−0.682238	+	0.731130i	$0.738993\pi$
$548$	0	0
$549$	2980.69	0.231717
$550$	0	0
$551$	−27860.6	−2.15409
$552$	0	0
$553$	−4293.76	−0.330180
$554$	0	0
$555$	−2570.67	−0.196610
$556$	0	0
$557$	16674.5	1.26844	0.634219	−	0.773153i	$-0.281322\pi$
$557$	16674.5	1.26844	0.634219	+	0.773153i	$0.281322\pi$
$558$	0	0
$559$	27571.6	2.08615
$560$	0	0
$561$	4822.03	0.362899
$562$	0	0
$563$	−25215.4	−1.88757	−0.943786	−	0.330557i	$-0.892763\pi$
$563$	−25215.4	−1.88757	−0.943786	+	0.330557i	$0.892763\pi$
$564$	0	0
$565$	−5271.98	−0.392556
$566$	0	0
$567$	2771.30	0.205262
$568$	0	0
$569$	−12318.8	−0.907612	−0.453806	−	0.891101i	$-0.649934\pi$
$569$	−12318.8	−0.907612	−0.453806	+	0.891101i	$0.649934\pi$
$570$	0	0
$571$	−9039.05	−0.662474	−0.331237	−	0.943548i	$-0.607466\pi$
$571$	−9039.05	−0.662474	−0.331237	+	0.943548i	$0.607466\pi$
$572$	0	0
$573$	−5179.88	−0.377648
$574$	0	0
$575$	13151.8	0.953861
$576$	0	0
$577$	−18526.7	−1.33670	−0.668352	−	0.743845i	$-0.733000\pi$
$577$	−18526.7	−1.33670	−0.668352	+	0.743845i	$0.733000\pi$
$578$	0	0
$579$	−17308.2	−1.24232
$580$	0	0
$581$	−3560.02	−0.254207
$582$	0	0
$583$	−38244.5	−2.71686
$584$	0	0
$585$	−2450.70	−0.173204
$586$	0	0
$587$	8485.07	0.596621	0.298310	−	0.954469i	$-0.403577\pi$
$587$	8485.07	0.596621	0.298310	+	0.954469i	$0.403577\pi$
$588$	0	0
$589$	20557.7	1.43814
$590$	0	0
$591$	9425.88	0.656056
$592$	0	0
$593$	4307.62	0.298302	0.149151	−	0.988814i	$-0.452346\pi$
$593$	4307.62	0.298302	0.149151	+	0.988814i	$0.452346\pi$
$594$	0	0
$595$	428.636	0.0295334
$596$	0	0
$597$	−2241.35	−0.153656
$598$	0	0
$599$	−960.377	−0.0655091	−0.0327545	−	0.999463i	$-0.510428\pi$
$599$	−960.377	−0.0655091	−0.0327545	+	0.999463i	$0.510428\pi$
$600$	0	0
$601$	−22031.9	−1.49534	−0.747671	−	0.664070i	$-0.768828\pi$
$601$	−22031.9	−1.49534	−0.747671	+	0.664070i	$0.768828\pi$
$602$	0	0
$603$	9528.22	0.643481
$604$	0	0
$605$	11488.1	0.771993
$606$	0	0
$607$	−24214.5	−1.61917	−0.809586	−	0.587002i	$-0.800308\pi$
$607$	−24214.5	−1.61917	−0.809586	+	0.587002i	$0.800308\pi$
$608$	0	0
$609$	−8657.43	−0.576054
$610$	0	0
$611$	3282.65	0.217352
$612$	0	0
$613$	−10600.0	−0.698420	−0.349210	−	0.937044i	$-0.613550\pi$
$613$	−10600.0	−0.698420	−0.349210	+	0.937044i	$0.613550\pi$
$614$	0	0
$615$	−3502.39	−0.229642
$616$	0	0
$617$	25638.6	1.67288	0.836442	−	0.548056i	$-0.184632\pi$
$617$	25638.6	1.67288	0.836442	+	0.548056i	$0.184632\pi$
$618$	0	0
$619$	−9832.18	−0.638431	−0.319215	−	0.947682i	$-0.603419\pi$
$619$	−9832.18	−0.638431	−0.319215	+	0.947682i	$0.603419\pi$
$620$	0	0
$621$	17930.3	1.15864
$622$	0	0
$623$	2428.67	0.156184
$624$	0	0
$625$	10928.0	0.699391
$626$	0	0
$627$	26957.1	1.71701
$628$	0	0
$629$	2875.80	0.182299
$630$	0	0
$631$	23322.9	1.47143	0.735714	−	0.677292i	$-0.236847\pi$
$631$	23322.9	1.47143	0.735714	+	0.677292i	$0.236847\pi$
$632$	0	0
$633$	17889.2	1.12327
$634$	0	0
$635$	−4846.32	−0.302866
$636$	0	0
$637$	3623.22	0.225364
$638$	0	0
$639$	1457.16	0.0902105
$640$	0	0
$641$	−8633.55	−0.531988	−0.265994	−	0.963975i	$-0.585700\pi$
$641$	−8633.55	−0.531988	−0.265994	+	0.963975i	$0.585700\pi$
$642$	0	0
$643$	18885.3	1.15826	0.579132	−	0.815234i	$-0.303392\pi$
$643$	18885.3	1.15826	0.579132	+	0.815234i	$0.303392\pi$
$644$	0	0
$645$	5666.30	0.345908
$646$	0	0
$647$	29050.7	1.76523	0.882614	−	0.470099i	$-0.155782\pi$
$647$	29050.7	1.76523	0.882614	+	0.470099i	$0.155782\pi$
$648$	0	0
$649$	11537.8	0.697843
$650$	0	0
$651$	6388.11	0.384593
$652$	0	0
$653$	−11808.6	−0.707668	−0.353834	−	0.935308i	$-0.615122\pi$
$653$	−11808.6	−0.707668	−0.353834	+	0.935308i	$0.615122\pi$
$654$	0	0
$655$	−1889.54	−0.112718
$656$	0	0
$657$	5819.43	0.345567
$658$	0	0
$659$	15754.9	0.931295	0.465647	−	0.884970i	$-0.345822\pi$
$659$	15754.9	0.931295	0.465647	+	0.884970i	$0.345822\pi$
$660$	0	0
$661$	−6425.37	−0.378090	−0.189045	−	0.981968i	$-0.560539\pi$
$661$	−6425.37	−0.378090	−0.189045	+	0.981968i	$0.560539\pi$
$662$	0	0
$663$	−5303.24	−0.310649
$664$	0	0
$665$	2396.25	0.139733
$666$	0	0
$667$	−34416.5	−1.99792
$668$	0	0
$669$	−27172.4	−1.57032
$670$	0	0
$671$	21779.8	1.25305
$672$	0	0
$673$	−27948.0	−1.60077	−0.800385	−	0.599487i	$-0.795371\pi$
$673$	−27948.0	−1.60077	−0.800385	+	0.599487i	$0.795371\pi$
$674$	0	0
$675$	17109.5	0.975619
$676$	0	0
$677$	755.984	0.0429170	0.0214585	−	0.999770i	$-0.493169\pi$
$677$	755.984	0.0429170	0.0214585	+	0.999770i	$0.493169\pi$
$678$	0	0
$679$	3750.91	0.211998
$680$	0	0
$681$	2570.54	0.144645
$682$	0	0
$683$	1116.65	0.0625587	0.0312793	−	0.999511i	$-0.490042\pi$
$683$	1116.65	0.0625587	0.0312793	+	0.999511i	$0.490042\pi$
$684$	0	0
$685$	−8870.24	−0.494766
$686$	0	0
$687$	6555.65	0.364067
$688$	0	0
$689$	42061.1	2.32569
$690$	0	0
$691$	12734.4	0.701068	0.350534	−	0.936550i	$-0.386000\pi$
$691$	12734.4	0.701068	0.350534	+	0.936550i	$0.386000\pi$
$692$	0	0
$693$	−4330.48	−0.237376
$694$	0	0
$695$	−1339.37	−0.0731012
$696$	0	0
$697$	3918.13	0.212926
$698$	0	0
$699$	28017.9	1.51607
$700$	0	0
$701$	−19453.6	−1.04815	−0.524073	−	0.851673i	$-0.675588\pi$
$701$	−19453.6	−1.04815	−0.524073	+	0.851673i	$0.675588\pi$
$702$	0	0
$703$	16076.9	0.862522
$704$	0	0
$705$	674.624	0.0360395
$706$	0	0
$707$	−5710.52	−0.303771
$708$	0	0
$709$	−27897.2	−1.47772	−0.738859	−	0.673860i	$-0.764635\pi$
$709$	−27897.2	−1.47772	−0.738859	+	0.673860i	$0.764635\pi$
$710$	0	0
$711$	−5644.05	−0.297705
$712$	0	0
$713$	25395.1	1.33388
$714$	0	0
$715$	−17907.2	−0.936630
$716$	0	0
$717$	15880.8	0.827168
$718$	0	0
$719$	1846.74	0.0957881	0.0478941	−	0.998852i	$-0.484749\pi$
$719$	1846.74	0.0957881	0.0478941	+	0.998852i	$0.484749\pi$
$720$	0	0
$721$	−7905.51	−0.408345
$722$	0	0
$723$	19438.6	0.999900
$724$	0	0
$725$	−32840.9	−1.68232
$726$	0	0
$727$	−8445.37	−0.430841	−0.215421	−	0.976521i	$-0.569112\pi$
$727$	−8445.37	−0.430841	−0.215421	+	0.976521i	$0.569112\pi$
$728$	0	0
$729$	21039.9	1.06894
$730$	0	0
$731$	−6338.89	−0.320728
$732$	0	0
$733$	3831.69	0.193078	0.0965392	−	0.995329i	$-0.469223\pi$
$733$	3831.69	0.193078	0.0965392	+	0.995329i	$0.469223\pi$
$734$	0	0
$735$	744.614	0.0373680
$736$	0	0
$737$	69622.2	3.47974
$738$	0	0
$739$	−12364.2	−0.615460	−0.307730	−	0.951474i	$-0.599569\pi$
$739$	−12364.2	−0.615460	−0.307730	+	0.951474i	$0.599569\pi$
$740$	0	0
$741$	−29647.3	−1.46980
$742$	0	0
$743$	13856.1	0.684160	0.342080	−	0.939671i	$-0.388869\pi$
$743$	13856.1	0.684160	0.342080	+	0.939671i	$0.388869\pi$
$744$	0	0
$745$	5428.42	0.266956
$746$	0	0
$747$	−4679.56	−0.229205
$748$	0	0
$749$	2133.92	0.104101
$750$	0	0
$751$	37444.2	1.81938	0.909691	−	0.415285i	$-0.136318\pi$
$751$	37444.2	1.81938	0.909691	+	0.415285i	$0.136318\pi$
$752$	0	0
$753$	6920.78	0.334937
$754$	0	0
$755$	1987.73	0.0958158
$756$	0	0
$757$	−20275.5	−0.973480	−0.486740	−	0.873547i	$-0.661814\pi$
$757$	−20275.5	−0.973480	−0.486740	+	0.873547i	$0.661814\pi$
$758$	0	0
$759$	33300.4	1.59253
$760$	0	0
$761$	−22107.6	−1.05309	−0.526543	−	0.850148i	$-0.676512\pi$
$761$	−22107.6	−1.05309	−0.526543	+	0.850148i	$0.676512\pi$
$762$	0	0
$763$	3541.27	0.168024
$764$	0	0
$765$	563.432	0.0266287
$766$	0	0
$767$	−12689.2	−0.597369
$768$	0	0
$769$	−6594.11	−0.309220	−0.154610	−	0.987976i	$-0.549412\pi$
$769$	−6594.11	−0.309220	−0.154610	+	0.987976i	$0.549412\pi$
$770$	0	0
$771$	−26094.0	−1.21888
$772$	0	0
$773$	−20302.3	−0.944660	−0.472330	−	0.881422i	$-0.656587\pi$
$773$	−20302.3	−0.944660	−0.472330	+	0.881422i	$0.656587\pi$
$774$	0	0
$775$	24232.5	1.12317
$776$	0	0
$777$	4995.77	0.230659
$778$	0	0
$779$	21903.9	1.00743
$780$	0	0
$781$	10647.4	0.487830
$782$	0	0
$783$	−44773.0	−2.04349
$784$	0	0
$785$	10203.5	0.463922
$786$	0	0
$787$	−42519.1	−1.92585	−0.962923	−	0.269775i	$-0.913051\pi$
$787$	−42519.1	−1.92585	−0.962923	+	0.269775i	$0.913051\pi$
$788$	0	0
$789$	−824.197	−0.0371891
$790$	0	0
$791$	10245.4	0.460538
$792$	0	0
$793$	−23953.3	−1.07264
$794$	0	0
$795$	8644.05	0.385626
$796$	0	0
$797$	−8286.20	−0.368271	−0.184136	−	0.982901i	$-0.558949\pi$
$797$	−8286.20	−0.368271	−0.184136	+	0.982901i	$0.558949\pi$
$798$	0	0
$799$	−754.702	−0.0334161
$800$	0	0
$801$	3192.43	0.140823
$802$	0	0
$803$	42522.3	1.86872
$804$	0	0
$805$	2960.12	0.129603
$806$	0	0
$807$	23078.0	1.00667
$808$	0	0
$809$	−12542.2	−0.545068	−0.272534	−	0.962146i	$-0.587862\pi$
$809$	−12542.2	−0.545068	−0.272534	+	0.962146i	$0.587862\pi$
$810$	0	0
$811$	−11606.9	−0.502556	−0.251278	−	0.967915i	$-0.580851\pi$
$811$	−11606.9	−0.502556	−0.251278	+	0.967915i	$0.580851\pi$
$812$	0	0
$813$	−24002.4	−1.03543
$814$	0	0
$815$	2176.34	0.0935384
$816$	0	0
$817$	−35437.0	−1.51748
$818$	0	0
$819$	4762.63	0.203199
$820$	0	0
$821$	−30147.0	−1.28153	−0.640766	−	0.767737i	$-0.721383\pi$
$821$	−30147.0	−1.28153	−0.640766	+	0.767737i	$0.721383\pi$
$822$	0	0
$823$	−16769.4	−0.710262	−0.355131	−	0.934817i	$-0.615564\pi$
$823$	−16769.4	−0.710262	−0.355131	+	0.934817i	$0.615564\pi$
$824$	0	0
$825$	31775.9	1.34097
$826$	0	0
$827$	33318.5	1.40096	0.700481	−	0.713671i	$-0.252969\pi$
$827$	33318.5	1.40096	0.700481	+	0.713671i	$0.252969\pi$
$828$	0	0
$829$	39886.8	1.67108	0.835540	−	0.549429i	$-0.185155\pi$
$829$	39886.8	1.67108	0.835540	+	0.549429i	$0.185155\pi$
$830$	0	0
$831$	−9386.24	−0.391823
$832$	0	0
$833$	−833.000	−0.0346479
$834$	0	0
$835$	5463.56	0.226436
$836$	0	0
$837$	33036.9	1.36430
$838$	0	0
$839$	18191.1	0.748542	0.374271	−	0.927319i	$-0.377893\pi$
$839$	18191.1	0.748542	0.374271	+	0.927319i	$0.377893\pi$
$840$	0	0
$841$	61550.9	2.52372
$842$	0	0
$843$	26243.7	1.07222
$844$	0	0
$845$	11780.6	0.479605
$846$	0	0
$847$	−22325.6	−0.905687
$848$	0	0
$849$	25670.9	1.03772
$850$	0	0
$851$	19860.0	0.799991
$852$	0	0
$853$	24488.6	0.982971	0.491485	−	0.870886i	$-0.336454\pi$
$853$	24488.6	0.982971	0.491485	+	0.870886i	$0.336454\pi$
$854$	0	0
$855$	3149.82	0.125990
$856$	0	0
$857$	−31421.3	−1.25243	−0.626214	−	0.779651i	$-0.715396\pi$
$857$	−31421.3	−1.25243	−0.626214	+	0.779651i	$0.715396\pi$
$858$	0	0
$859$	−13280.9	−0.527518	−0.263759	−	0.964589i	$-0.584962\pi$
$859$	−13280.9	−0.527518	−0.263759	+	0.964589i	$0.584962\pi$
$860$	0	0
$861$	6806.46	0.269412
$862$	0	0
$863$	7778.09	0.306801	0.153400	−	0.988164i	$-0.450978\pi$
$863$	7778.09	0.306801	0.153400	+	0.988164i	$0.450978\pi$
$864$	0	0
$865$	7497.44	0.294706
$866$	0	0
$867$	1219.25	0.0477598
$868$	0	0
$869$	−41240.8	−1.60989
$870$	0	0
$871$	−76570.1	−2.97874
$872$	0	0
$873$	4930.48	0.191147
$874$	0	0
$875$	5976.34	0.230899
$876$	0	0
$877$	−38888.7	−1.49735	−0.748675	−	0.662937i	$-0.769310\pi$
$877$	−38888.7	−1.49735	−0.748675	+	0.662937i	$0.769310\pi$
$878$	0	0
$879$	−13761.9	−0.528076
$880$	0	0
$881$	−35490.2	−1.35720	−0.678601	−	0.734507i	$-0.737413\pi$
$881$	−35490.2	−1.35720	−0.678601	+	0.734507i	$0.737413\pi$
$882$	0	0
$883$	10339.1	0.394043	0.197021	−	0.980399i	$-0.436873\pi$
$883$	10339.1	0.394043	0.197021	+	0.980399i	$0.436873\pi$
$884$	0	0
$885$	−2607.79	−0.0990508
$886$	0	0
$887$	37512.9	1.42002	0.710011	−	0.704191i	$-0.248690\pi$
$887$	37512.9	1.42002	0.710011	+	0.704191i	$0.248690\pi$
$888$	0	0
$889$	9418.21	0.355317
$890$	0	0
$891$	26617.8	1.00082
$892$	0	0
$893$	−4219.10	−0.158104
$894$	0	0
$895$	11982.7	0.447528
$896$	0	0
$897$	−36623.6	−1.36324
$898$	0	0
$899$	−63413.0	−2.35255
$900$	0	0
$901$	−9670.11	−0.357556
$902$	0	0
$903$	−11011.7	−0.405812
$904$	0	0
$905$	−3524.99	−0.129475
$906$	0	0
$907$	−36386.8	−1.33209	−0.666043	−	0.745913i	$-0.732013\pi$
$907$	−36386.8	−1.33209	−0.666043	+	0.745913i	$0.732013\pi$
$908$	0	0
$909$	−7506.35	−0.273894
$910$	0	0
$911$	−11154.0	−0.405651	−0.202825	−	0.979215i	$-0.565012\pi$
$911$	−11154.0	−0.405651	−0.202825	+	0.979215i	$0.565012\pi$
$912$	0	0
$913$	−34193.3	−1.23947
$914$	0	0
$915$	−4922.68	−0.177857
$916$	0	0
$917$	3672.08	0.132239
$918$	0	0
$919$	11401.2	0.409239	0.204619	−	0.978842i	$-0.434404\pi$
$919$	11401.2	0.409239	0.204619	+	0.978842i	$0.434404\pi$
$920$	0	0
$921$	−18681.7	−0.668384
$922$	0	0
$923$	−11710.0	−0.417593
$924$	0	0
$925$	18950.8	0.673621
$926$	0	0
$927$	−10391.6	−0.368183
$928$	0	0
$929$	30726.0	1.08513	0.542565	−	0.840014i	$-0.317453\pi$
$929$	30726.0	1.08513	0.542565	+	0.840014i	$0.317453\pi$
$930$	0	0
$931$	−4656.81	−0.163932
$932$	0	0
$933$	−39240.8	−1.37694
$934$	0	0
$935$	4116.97	0.143999
$936$	0	0
$937$	9590.77	0.334383	0.167191	−	0.985924i	$-0.446530\pi$
$937$	9590.77	0.334383	0.167191	+	0.985924i	$0.446530\pi$
$938$	0	0
$939$	−4082.72	−0.141890
$940$	0	0
$941$	27794.8	0.962896	0.481448	−	0.876475i	$-0.340111\pi$
$941$	27794.8	0.962896	0.481448	+	0.876475i	$0.340111\pi$
$942$	0	0
$943$	27058.2	0.934396
$944$	0	0
$945$	3850.86	0.132559
$946$	0	0
$947$	−5053.45	−0.173406	−0.0867028	−	0.996234i	$-0.527633\pi$
$947$	−5053.45	−0.173406	−0.0867028	+	0.996234i	$0.527633\pi$
$948$	0	0
$949$	−46765.7	−1.59966
$950$	0	0
$951$	−12353.5	−0.421229
$952$	0	0
$953$	−53832.9	−1.82982	−0.914910	−	0.403658i	$-0.867739\pi$
$953$	−53832.9	−1.82982	−0.914910	+	0.403658i	$0.867739\pi$
$954$	0	0
$955$	−4422.50	−0.149852
$956$	0	0
$957$	−83153.1	−2.80873
$958$	0	0
$959$	17238.2	0.580449
$960$	0	0
$961$	16999.9	0.570638
$962$	0	0
$963$	2804.98	0.0938623
$964$	0	0
$965$	−14777.5	−0.492957
$966$	0	0
$967$	−7504.78	−0.249574	−0.124787	−	0.992184i	$-0.539825\pi$
$967$	−7504.78	−0.249574	−0.124787	+	0.992184i	$0.539825\pi$
$968$	0	0
$969$	6816.09	0.225969
$970$	0	0
$971$	−40642.1	−1.34322	−0.671611	−	0.740904i	$-0.734397\pi$
$971$	−40642.1	−1.34322	−0.671611	+	0.740904i	$0.734397\pi$
$972$	0	0
$973$	2602.90	0.0857608
$974$	0	0
$975$	−34947.0	−1.14790
$976$	0	0
$977$	21318.3	0.698090	0.349045	−	0.937106i	$-0.386506\pi$
$977$	21318.3	0.698090	0.349045	+	0.937106i	$0.386506\pi$
$978$	0	0
$979$	23327.0	0.761525
$980$	0	0
$981$	4654.91	0.151498
$982$	0	0
$983$	−52415.8	−1.70072	−0.850358	−	0.526205i	$-0.823615\pi$
$983$	−52415.8	−1.70072	−0.850358	+	0.526205i	$0.823615\pi$
$984$	0	0
$985$	8047.66	0.260325
$986$	0	0
$987$	−1311.05	−0.0422808
$988$	0	0
$989$	−43775.7	−1.40747
$990$	0	0
$991$	−20454.7	−0.655664	−0.327832	−	0.944736i	$-0.606318\pi$
$991$	−20454.7	−0.655664	−0.327832	+	0.944736i	$0.606318\pi$
$992$	0	0
$993$	−10421.7	−0.333054
$994$	0	0
$995$	−1913.63	−0.0609710
$996$	0	0
$997$	51753.4	1.64398	0.821989	−	0.569504i	$-0.192865\pi$
$997$	51753.4	1.64398	0.821989	+	0.569504i	$0.192865\pi$
$998$	0	0
$999$	25836.2	0.818239

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	476.4.a.c.1.4	✓	6
4.3	odd	2		1904.4.a.n.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.4.a.c.1.4	✓	6	1.1	even	1	trivial
1904.4.a.n.1.3		6	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:	\( N \)	\(=\)	\( 476 = 2^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	476.a (trivial)

\(f(q)\)	\(=\)	\(q+4.21885 q^{3} +3.60198 q^{5} -7.00000 q^{7} -9.20134 q^{9} +O(q^{10})\)	\(q+4.21885 q^{3} +3.60198 q^{5} -7.00000 q^{7} -9.20134 q^{9} -67.2337 q^{11} +73.9432 q^{13} +15.1962 q^{15} -17.0000 q^{17} -95.0370 q^{19} -29.5319 q^{21} -117.400 q^{23} -112.026 q^{25} -152.728 q^{27} +293.155 q^{29} -216.312 q^{31} -283.649 q^{33} -25.2139 q^{35} -169.165 q^{37} +311.955 q^{39} -230.478 q^{41} +372.876 q^{43} -33.1430 q^{45} +44.3943 q^{47} +49.0000 q^{49} -71.7204 q^{51} +568.830 q^{53} -242.175 q^{55} -400.947 q^{57} -171.608 q^{59} -323.941 q^{61} +64.4094 q^{63} +266.342 q^{65} -1035.53 q^{67} -495.294 q^{69} -158.364 q^{71} -632.455 q^{73} -472.619 q^{75} +470.636 q^{77} +613.395 q^{79} -395.899 q^{81} +508.574 q^{83} -61.2337 q^{85} +1236.78 q^{87} -346.953 q^{89} -517.602 q^{91} -912.587 q^{93} -342.322 q^{95} -535.844 q^{97} +618.640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9}+O(q^{10}) \) 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9	\( 6 q + 2 q^{3} + 10 q^{5} - 42 q^{7} + 16 q^{9} - 20 q^{11} + 42 q^{13} - 136 q^{15} - 102 q^{17} - 104 q^{19} - 14 q^{21} - 230 q^{23} + 108 q^{25} - 130 q^{27} - 52 q^{29} - 564 q^{31} - 346 q^{33} - 70 q^{35} - 564 q^{37} - 626 q^{39} - 548 q^{41} - 648 q^{43} - 174 q^{45} - 366 q^{47} + 294 q^{49} - 34 q^{51} - 74 q^{53} - 1460 q^{55} - 316 q^{57} - 558 q^{59} - 620 q^{61} - 112 q^{63} - 1378 q^{65} - 164 q^{67} - 540 q^{69} - 822 q^{71} + 940 q^{73} - 2698 q^{75} + 140 q^{77} - 1838 q^{79} - 3094 q^{81} - 1118 q^{83} - 170 q^{85} - 1354 q^{87} - 1634 q^{89} - 294 q^{91} + 268 q^{93} - 2642 q^{95} + 182 q^{97} + 12 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 + 10 * q^5 - 42 * q^7 + 16 * q^9 - 20 * q^11 + 42 * q^13 - 136 * q^15 - 102 * q^17 - 104 * q^19 - 14 * q^21 - 230 * q^23 + 108 * q^25 - 130 * q^27 - 52 * q^29 - 564 * q^31 - 346 * q^33 - 70 * q^35 - 564 * q^37 - 626 * q^39 - 548 * q^41 - 648 * q^43 - 174 * q^45 - 366 * q^47 + 294 * q^49 - 34 * q^51 - 74 * q^53 - 1460 * q^55 - 316 * q^57 - 558 * q^59 - 620 * q^61 - 112 * q^63 - 1378 * q^65 - 164 * q^67 - 540 * q^69 - 822 * q^71 + 940 * q^73 - 2698 * q^75 + 140 * q^77 - 1838 * q^79 - 3094 * q^81 - 1118 * q^83 - 170 * q^85 - 1354 * q^87 - 1634 * q^89 - 294 * q^91 + 268 * q^93 - 2642 * q^95 + 182 * q^97 + 12 * q^99

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