LMFDB - Embedded newform 4675.2.a.be.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4675,2,Mod(1,4675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4675, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4675 = 5^{2} \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4675.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:	$37.3300629449$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 935)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.825785$ of defining polynomial
Character	$\chi$	$=$	4675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14386 q^{2} +0.825785 q^{3} +2.59615 q^{4} -1.77037 q^{6} +2.42194 q^{7} -1.27807 q^{8} -2.31808 q^{9} +O(q^{10})$	\(q-2.14386 q^{2} +0.825785 q^{3} +2.59615 q^{4} -1.77037 q^{6} +2.42194 q^{7} -1.27807 q^{8} -2.31808 q^{9} -1.00000 q^{11} +2.14386 q^{12} +2.72193 q^{13} -5.19231 q^{14} -2.45229 q^{16} +1.00000 q^{17} +4.96965 q^{18} -3.54074 q^{19} +2.00000 q^{21} +2.14386 q^{22} -3.87423 q^{23} -1.05541 q^{24} -5.83544 q^{26} -4.39159 q^{27} +6.28773 q^{28} -0.0884500 q^{29} +0.556150 q^{31} +7.81353 q^{32} -0.825785 q^{33} -2.14386 q^{34} -6.01809 q^{36} -4.66122 q^{37} +7.59087 q^{38} +2.24772 q^{39} +6.78846 q^{41} -4.28773 q^{42} -5.00268 q^{43} -2.59615 q^{44} +8.30582 q^{46} +11.6142 q^{47} -2.02506 q^{48} -1.13421 q^{49} +0.825785 q^{51} +7.06654 q^{52} +12.1704 q^{53} +9.41497 q^{54} -3.09542 q^{56} -2.92389 q^{57} +0.189625 q^{58} -8.06775 q^{59} +4.01112 q^{61} -1.19231 q^{62} -5.61425 q^{63} -11.8466 q^{64} +1.77037 q^{66} -6.36124 q^{67} +2.59615 q^{68} -3.19928 q^{69} +6.57546 q^{71} +2.96268 q^{72} +14.3612 q^{73} +9.99303 q^{74} -9.19231 q^{76} -2.42194 q^{77} -4.81882 q^{78} -3.06339 q^{79} +3.32773 q^{81} -14.5535 q^{82} +1.67663 q^{83} +5.19231 q^{84} +10.7251 q^{86} -0.0730406 q^{87} +1.27807 q^{88} +4.84917 q^{89} +6.59234 q^{91} -10.0581 q^{92} +0.459260 q^{93} -24.8994 q^{94} +6.45229 q^{96} +1.82410 q^{97} +2.43159 q^{98} +2.31808 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 2 q^{4} - q^{6} - q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) $ 4 * q + q^3 + 2 * q^4 - q^6 - q^7 - 3 * q^8 - 3 * q^9	$ 4 q + q^{3} + 2 q^{4} - q^{6} - q^{7} - 3 q^{8} - 3 q^{9} - 4 q^{11} + 13 q^{13} - 4 q^{14} - 10 q^{16} + 4 q^{17} + 9 q^{18} - 2 q^{19} + 8 q^{21} - 5 q^{23} - 8 q^{24} - 6 q^{26} + 4 q^{27} + 8 q^{28} + 12 q^{29} - 2 q^{31} - q^{32} - q^{33} - 5 q^{36} + q^{37} + 4 q^{38} - 4 q^{39} + 2 q^{41} + 7 q^{43} - 2 q^{44} - 3 q^{46} + 19 q^{47} - q^{48} - 11 q^{49} + q^{51} + q^{52} + 17 q^{53} + 15 q^{54} - 12 q^{56} + 18 q^{57} + 11 q^{58} + 6 q^{59} - 15 q^{61} + 12 q^{62} + 5 q^{63} + q^{64} + q^{66} + 7 q^{67} + 2 q^{68} - 8 q^{69} - 8 q^{71} - 11 q^{72} + 25 q^{73} + 28 q^{74} - 20 q^{76} + q^{77} - 5 q^{78} - 7 q^{79} - 8 q^{81} - 9 q^{82} - 5 q^{83} + 4 q^{84} + 23 q^{86} + 20 q^{87} + 3 q^{88} + 16 q^{89} - 16 q^{91} - 17 q^{92} + 14 q^{93} - 10 q^{94} + 26 q^{96} + 11 q^{97} - 16 q^{98} + 3 q^{99}+O(q^{100}) $ 4 * q + q^3 + 2 * q^4 - q^6 - q^7 - 3 * q^8 - 3 * q^9 - 4 * q^11 + 13 * q^13 - 4 * q^14 - 10 * q^16 + 4 * q^17 + 9 * q^18 - 2 * q^19 + 8 * q^21 - 5 * q^23 - 8 * q^24 - 6 * q^26 + 4 * q^27 + 8 * q^28 + 12 * q^29 - 2 * q^31 - q^32 - q^33 - 5 * q^36 + q^37 + 4 * q^38 - 4 * q^39 + 2 * q^41 + 7 * q^43 - 2 * q^44 - 3 * q^46 + 19 * q^47 - q^48 - 11 * q^49 + q^51 + q^52 + 17 * q^53 + 15 * q^54 - 12 * q^56 + 18 * q^57 + 11 * q^58 + 6 * q^59 - 15 * q^61 + 12 * q^62 + 5 * q^63 + q^64 + q^66 + 7 * q^67 + 2 * q^68 - 8 * q^69 - 8 * q^71 - 11 * q^72 + 25 * q^73 + 28 * q^74 - 20 * q^76 + q^77 - 5 * q^78 - 7 * q^79 - 8 * q^81 - 9 * q^82 - 5 * q^83 + 4 * q^84 + 23 * q^86 + 20 * q^87 + 3 * q^88 + 16 * q^89 - 16 * q^91 - 17 * q^92 + 14 * q^93 - 10 * q^94 + 26 * q^96 + 11 * q^97 - 16 * q^98 + 3 * 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 $

$2$	−2.14386	−1.51594	−0.757971	−	0.652289i	$-0.773809\pi$
$2$	−2.14386	−1.51594	−0.757971	+	0.652289i	$0.773809\pi$
$3$	0.825785	0.476767	0.238383	−	0.971171i	$-0.423382\pi$
$3$	0.825785	0.476767	0.238383	+	0.971171i	$0.423382\pi$
$4$	2.59615	1.29808
$5$	0	0
$5$	0	0
$6$	−1.77037	−0.722751
$7$	2.42194	0.915407	0.457703	−	0.889105i	$-0.348672\pi$
$7$	2.42194	0.915407	0.457703	+	0.889105i	$0.348672\pi$
$8$	−1.27807	−0.451868
$9$	−2.31808	−0.772693
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.14386	0.618880
$13$	2.72193	0.754926	0.377463	−	0.926025i	$-0.376796\pi$
$13$	2.72193	0.754926	0.377463	+	0.926025i	$0.376796\pi$
$14$	−5.19231	−1.38770
$15$	0	0
$16$	−2.45229	−0.613073
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	4.96965	1.17136
$19$	−3.54074	−0.812302	−0.406151	−	0.913806i	$-0.633129\pi$
$19$	−3.54074	−0.812302	−0.406151	+	0.913806i	$0.633129\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	2.14386	0.457073
$23$	−3.87423	−0.807833	−0.403916	−	0.914796i	$-0.632351\pi$
$23$	−3.87423	−0.807833	−0.403916	+	0.914796i	$0.632351\pi$
$24$	−1.05541	−0.215436
$25$	0	0
$26$	−5.83544	−1.14442
$27$	−4.39159	−0.845162
$28$	6.28773	1.18827
$29$	−0.0884500	−0.0164248	−0.00821238	−	0.999966i	$-0.502614\pi$
$29$	−0.0884500	−0.0164248	−0.00821238	+	0.999966i	$0.502614\pi$
$30$	0	0
$31$	0.556150	0.0998874	0.0499437	−	0.998752i	$-0.484096\pi$
$31$	0.556150	0.0998874	0.0499437	+	0.998752i	$0.484096\pi$
$32$	7.81353	1.38125
$33$	−0.825785	−0.143751
$34$	−2.14386	−0.367670
$35$	0	0
$36$	−6.01809	−1.00302
$37$	−4.66122	−0.766300	−0.383150	−	0.923686i	$-0.625161\pi$
$37$	−4.66122	−0.766300	−0.383150	+	0.923686i	$0.625161\pi$
$38$	7.59087	1.23140
$39$	2.24772	0.359924
$40$	0	0
$41$	6.78846	1.06018	0.530090	−	0.847941i	$-0.322158\pi$
$41$	6.78846	1.06018	0.530090	+	0.847941i	$0.322158\pi$
$42$	−4.28773	−0.661611
$43$	−5.00268	−0.762902	−0.381451	−	0.924389i	$-0.624576\pi$
$43$	−5.00268	−0.762902	−0.381451	+	0.924389i	$0.624576\pi$
$44$	−2.59615	−0.391385
$45$	0	0
$46$	8.30582	1.22463
$47$	11.6142	1.69411	0.847056	−	0.531503i	$-0.178373\pi$
$47$	11.6142	1.69411	0.847056	+	0.531503i	$0.178373\pi$
$48$	−2.02506	−0.292293
$49$	−1.13421	−0.162030
$50$	0	0
$51$	0.825785	0.115633
$52$	7.06654	0.979953
$53$	12.1704	1.67173	0.835866	−	0.548933i	$-0.184966\pi$
$53$	12.1704	1.67173	0.835866	+	0.548933i	$0.184966\pi$
$54$	9.41497	1.28122
$55$	0	0
$56$	−3.09542	−0.413643
$57$	−2.92389	−0.387279
$58$	0.189625	0.0248990
$59$	−8.06775	−1.05033	−0.525166	−	0.851000i	$-0.675997\pi$
$59$	−8.06775	−1.05033	−0.525166	+	0.851000i	$0.675997\pi$
$60$	0	0
$61$	4.01112	0.513572	0.256786	−	0.966468i	$-0.417336\pi$
$61$	4.01112	0.513572	0.256786	+	0.966468i	$0.417336\pi$
$62$	−1.19231	−0.151423
$63$	−5.61425	−0.707329
$64$	−11.8466	−1.48082
$65$	0	0
$66$	1.77037	0.217917
$67$	−6.36124	−0.777149	−0.388574	−	0.921417i	$-0.627032\pi$
$67$	−6.36124	−0.777149	−0.388574	+	0.921417i	$0.627032\pi$
$68$	2.59615	0.314830
$69$	−3.19928	−0.385148
$70$	0	0
$71$	6.57546	0.780363	0.390182	−	0.920738i	$-0.372412\pi$
$71$	6.57546	0.780363	0.390182	+	0.920738i	$0.372412\pi$
$72$	2.96268	0.349155
$73$	14.3612	1.68086	0.840428	−	0.541923i	$-0.182304\pi$
$73$	14.3612	1.68086	0.840428	+	0.541923i	$0.182304\pi$
$74$	9.99303	1.16167
$75$	0	0
$76$	−9.19231	−1.05443
$77$	−2.42194	−0.276006
$78$	−4.81882	−0.545623
$79$	−3.06339	−0.344658	−0.172329	−	0.985039i	$-0.555129\pi$
$79$	−3.06339	−0.344658	−0.172329	+	0.985039i	$0.555129\pi$
$80$	0	0
$81$	3.32773	0.369748
$82$	−14.5535	−1.60717
$83$	1.67663	0.184034	0.0920172	−	0.995757i	$-0.470669\pi$
$83$	1.67663	0.184034	0.0920172	+	0.995757i	$0.470669\pi$
$84$	5.19231	0.566527
$85$	0	0
$86$	10.7251	1.15651
$87$	−0.0730406	−0.00783078
$88$	1.27807	0.136243
$89$	4.84917	0.514011	0.257005	−	0.966410i	$-0.417264\pi$
$89$	4.84917	0.514011	0.257005	+	0.966410i	$0.417264\pi$
$90$	0	0
$91$	6.59234	0.691065
$92$	−10.0581	−1.04863
$93$	0.459260	0.0476230
$94$	−24.8994	−2.56817
$95$	0	0
$96$	6.45229	0.658534
$97$	1.82410	0.185210	0.0926048	−	0.995703i	$-0.470481\pi$
$97$	1.82410	0.185210	0.0926048	+	0.995703i	$0.470481\pi$
$98$	2.43159	0.245628
$99$	2.31808	0.232976
$100$	0	0
$101$	9.27232	0.922630	0.461315	−	0.887236i	$-0.347378\pi$
$101$	9.27232	0.922630	0.461315	+	0.887236i	$0.347378\pi$
$102$	−1.77037	−0.175293
$103$	−2.26435	−0.223113	−0.111556	−	0.993758i	$-0.535584\pi$
$103$	−2.26435	−0.223113	−0.111556	+	0.993758i	$0.535584\pi$
$104$	−3.47882	−0.341127
$105$	0	0
$106$	−26.0917	−2.53425
$107$	3.19881	0.309241	0.154620	−	0.987974i	$-0.450585\pi$
$107$	3.19881	0.309241	0.154620	+	0.987974i	$0.450585\pi$
$108$	−11.4012	−1.09709
$109$	8.13542	0.779232	0.389616	−	0.920977i	$-0.372608\pi$
$109$	8.13542	0.779232	0.389616	+	0.920977i	$0.372608\pi$
$110$	0	0
$111$	−3.84917	−0.365347
$112$	−5.93930	−0.561211
$113$	20.6835	1.94574	0.972869	−	0.231358i	$-0.0743169\pi$
$113$	20.6835	1.94574	0.972869	+	0.231358i	$0.0743169\pi$
$114$	6.26842	0.587091
$115$	0	0
$116$	−0.229630	−0.0213206
$117$	−6.30964	−0.583326
$118$	17.2962	1.59224
$119$	2.42194	0.222019
$120$	0	0
$121$	1.00000	0.0909091
$122$	−8.59931	−0.778545
$123$	5.60581	0.505459
$124$	1.44385	0.129662
$125$	0	0
$126$	12.0362	1.07227
$127$	−6.89501	−0.611833	−0.305917	−	0.952058i	$-0.598963\pi$
$127$	−6.89501	−0.611833	−0.305917	+	0.952058i	$0.598963\pi$
$128$	9.77037	0.863587
$129$	−4.13114	−0.363727
$130$	0	0
$131$	2.25322	0.196865	0.0984325	−	0.995144i	$-0.468617\pi$
$131$	2.25322	0.196865	0.0984325	+	0.995144i	$0.468617\pi$
$132$	−2.14386	−0.186599
$133$	−8.57546	−0.743586
$134$	13.6376	1.17811
$135$	0	0
$136$	−1.27807	−0.109594
$137$	3.49805	0.298859	0.149429	−	0.988772i	$-0.452256\pi$
$137$	3.49805	0.298859	0.149429	+	0.988772i	$0.452256\pi$
$138$	6.85882	0.583862
$139$	2.22534	0.188751	0.0943756	−	0.995537i	$-0.469915\pi$
$139$	2.22534	0.188751	0.0943756	+	0.995537i	$0.469915\pi$
$140$	0	0
$141$	9.59087	0.807697
$142$	−14.0969	−1.18298
$143$	−2.72193	−0.227619
$144$	5.68460	0.473717
$145$	0	0
$146$	−30.7885	−2.54808
$147$	−0.936613	−0.0772506
$148$	−12.1013	−0.994717
$149$	−5.13161	−0.420398	−0.210199	−	0.977659i	$-0.567411\pi$
$149$	−5.13161	−0.420398	−0.210199	+	0.977659i	$0.567411\pi$
$150$	0	0
$151$	−5.76240	−0.468937	−0.234469	−	0.972124i	$-0.575335\pi$
$151$	−5.76240	−0.468937	−0.234469	+	0.972124i	$0.575335\pi$
$152$	4.52533	0.367053
$153$	−2.31808	−0.187406
$154$	5.19231	0.418408
$155$	0	0
$156$	5.83544	0.467209
$157$	18.9367	1.51131	0.755656	−	0.654968i	$-0.227318\pi$
$157$	18.9367	1.51131	0.755656	+	0.654968i	$0.227318\pi$
$158$	6.56749	0.522481
$159$	10.0501	0.797027
$160$	0	0
$161$	−9.38315	−0.739496
$162$	−7.13421	−0.560517
$163$	17.2936	1.35454	0.677268	−	0.735736i	$-0.263164\pi$
$163$	17.2936	1.35454	0.677268	+	0.735736i	$0.263164\pi$
$164$	17.6239	1.37620
$165$	0	0
$166$	−3.59447	−0.278985
$167$	−6.03325	−0.466867	−0.233433	−	0.972373i	$-0.574996\pi$
$167$	−6.03325	−0.466867	−0.233433	+	0.972373i	$0.574996\pi$
$168$	−2.55615	−0.197211
$169$	−5.59112	−0.430086
$170$	0	0
$171$	8.20772	0.627660
$172$	−12.9877	−0.990306
$173$	21.1882	1.61091	0.805456	−	0.592656i	$-0.201921\pi$
$173$	21.1882	1.61091	0.805456	+	0.592656i	$0.201921\pi$
$174$	0.156589	0.0118710
$175$	0	0
$176$	2.45229	0.184848
$177$	−6.66223	−0.500764
$178$	−10.3960	−0.779210
$179$	20.3324	1.51971	0.759856	−	0.650092i	$-0.225269\pi$
$179$	20.3324	1.51971	0.759856	+	0.650092i	$0.225269\pi$
$180$	0	0
$181$	−19.9148	−1.48025	−0.740127	−	0.672467i	$-0.765235\pi$
$181$	−19.9148	−1.48025	−0.740127	+	0.672467i	$0.765235\pi$
$182$	−14.1331	−1.04761
$183$	3.31232	0.244854
$184$	4.95156	0.365033
$185$	0	0
$186$	−0.984591	−0.0721937
$187$	−1.00000	−0.0731272
$188$	30.1524	2.19909
$189$	−10.6362	−0.773667
$190$	0	0
$191$	15.3682	1.11200	0.556002	−	0.831181i	$-0.312335\pi$
$191$	15.3682	1.11200	0.556002	+	0.831181i	$0.312335\pi$
$192$	−9.78271	−0.706006
$193$	−1.90605	−0.137201	−0.0686003	−	0.997644i	$-0.521853\pi$
$193$	−1.90605	−0.137201	−0.0686003	+	0.997644i	$0.521853\pi$
$194$	−3.91063	−0.280767
$195$	0	0
$196$	−2.94459	−0.210328
$197$	−24.5784	−1.75114	−0.875569	−	0.483093i	$-0.839513\pi$
$197$	−24.5784	−1.75114	−0.875569	+	0.483093i	$0.839513\pi$
$198$	−4.96965	−0.353178
$199$	−18.8940	−1.33936	−0.669680	−	0.742650i	$-0.733569\pi$
$199$	−18.8940	−1.33936	−0.669680	+	0.742650i	$0.733569\pi$
$200$	0	0
$201$	−5.25301	−0.370519
$202$	−19.8786	−1.39865
$203$	−0.214221	−0.0150353
$204$	2.14386	0.150101
$205$	0	0
$206$	4.85445	0.338226
$207$	8.98077	0.624207
$208$	−6.67495	−0.462825
$209$	3.54074	0.244918
$210$	0	0
$211$	17.4885	1.20396	0.601978	−	0.798512i	$-0.294379\pi$
$211$	17.4885	1.20396	0.601978	+	0.798512i	$0.294379\pi$
$212$	31.5962	2.17004
$213$	5.42991	0.372051
$214$	−6.85782	−0.468791
$215$	0	0
$216$	5.61278	0.381901
$217$	1.34696	0.0914376
$218$	−17.4412	−1.18127
$219$	11.8593	0.801376
$220$	0	0
$221$	2.72193	0.183097
$222$	8.25209	0.553844
$223$	−18.7587	−1.25617	−0.628086	−	0.778144i	$-0.716162\pi$
$223$	−18.7587	−1.25617	−0.628086	+	0.778144i	$0.716162\pi$
$224$	18.9239	1.26441
$225$	0	0
$226$	−44.3426	−2.94962
$227$	−13.9458	−0.925615	−0.462808	−	0.886459i	$-0.653158\pi$
$227$	−13.9458	−0.925615	−0.462808	+	0.886459i	$0.653158\pi$
$228$	−7.59087	−0.502717
$229$	6.42609	0.424648	0.212324	−	0.977199i	$-0.431897\pi$
$229$	6.42609	0.424648	0.212324	+	0.977199i	$0.431897\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0.113046	0.00742181
$233$	−15.8582	−1.03890	−0.519451	−	0.854500i	$-0.673864\pi$
$233$	−15.8582	−1.03890	−0.519451	+	0.854500i	$0.673864\pi$
$234$	13.5270	0.884288
$235$	0	0
$236$	−20.9451	−1.36341
$237$	−2.52970	−0.164321
$238$	−5.19231	−0.336567
$239$	−9.13404	−0.590832	−0.295416	−	0.955369i	$-0.595458\pi$
$239$	−9.13404	−0.590832	−0.295416	+	0.955369i	$0.595458\pi$
$240$	0	0
$241$	−8.31011	−0.535301	−0.267651	−	0.963516i	$-0.586247\pi$
$241$	−8.31011	−0.535301	−0.267651	+	0.963516i	$0.586247\pi$
$242$	−2.14386	−0.137813
$243$	15.9228	1.02145
$244$	10.4135	0.666656
$245$	0	0
$246$	−12.0181	−0.766246
$247$	−9.63763	−0.613228
$248$	−0.710801	−0.0451359
$249$	1.38454	0.0877415
$250$	0	0
$251$	4.62101	0.291675	0.145838	−	0.989309i	$-0.453412\pi$
$251$	4.62101	0.291675	0.145838	+	0.989309i	$0.453412\pi$
$252$	−14.5755	−0.918168
$253$	3.87423	0.243571
$254$	14.7820	0.927503
$255$	0	0
$256$	2.74678	0.171674
$257$	14.6297	0.912573	0.456286	−	0.889833i	$-0.349179\pi$
$257$	14.6297	0.912573	0.456286	+	0.889833i	$0.349179\pi$
$258$	8.85660	0.551388
$259$	−11.2892	−0.701477
$260$	0	0
$261$	0.205034	0.0126913
$262$	−4.83060	−0.298436
$263$	4.30582	0.265508	0.132754	−	0.991149i	$-0.457618\pi$
$263$	4.30582	0.265508	0.132754	+	0.991149i	$0.457618\pi$
$264$	1.05541	0.0649563
$265$	0	0
$266$	18.3846	1.12723
$267$	4.00437	0.245063
$268$	−16.5148	−1.00880
$269$	16.9601	1.03407	0.517037	−	0.855963i	$-0.327035\pi$
$269$	16.9601	1.03407	0.517037	+	0.855963i	$0.327035\pi$
$270$	0	0
$271$	1.42697	0.0866823	0.0433411	−	0.999060i	$-0.486200\pi$
$271$	1.42697	0.0866823	0.0433411	+	0.999060i	$0.486200\pi$
$272$	−2.45229	−0.148692
$273$	5.44385	0.329477
$274$	−7.49935	−0.453052
$275$	0	0
$276$	−8.30582	−0.499952
$277$	31.4958	1.89240	0.946199	−	0.323585i	$-0.104888\pi$
$277$	31.4958	1.89240	0.946199	+	0.323585i	$0.104888\pi$
$278$	−4.77084	−0.286136
$279$	−1.28920	−0.0771823
$280$	0	0
$281$	−23.8348	−1.42186	−0.710932	−	0.703261i	$-0.751727\pi$
$281$	−23.8348	−1.42186	−0.710932	+	0.703261i	$0.751727\pi$
$282$	−20.5615	−1.22442
$283$	15.0902	0.897020	0.448510	−	0.893778i	$-0.351955\pi$
$283$	15.0902	0.897020	0.448510	+	0.893778i	$0.351955\pi$
$284$	17.0709	1.01297
$285$	0	0
$286$	5.83544	0.345057
$287$	16.4412	0.970496
$288$	−18.1124	−1.06728
$289$	1.00000	0.0588235
$290$	0	0
$291$	1.50632	0.0883018
$292$	37.2840	2.18188
$293$	17.4813	1.02127	0.510633	−	0.859799i	$-0.329411\pi$
$293$	17.4813	1.02127	0.510633	+	0.859799i	$0.329411\pi$
$294$	2.00797	0.117107
$295$	0	0
$296$	5.95739	0.346266
$297$	4.39159	0.254826
$298$	11.0015	0.637298
$299$	−10.5454	−0.609854
$300$	0	0
$301$	−12.1162	−0.698366
$302$	12.3538	0.710881
$303$	7.65694	0.439880
$304$	8.68292	0.498000
$305$	0	0
$306$	4.96965	0.284096
$307$	23.8989	1.36398	0.681991	−	0.731360i	$-0.261114\pi$
$307$	23.8989	1.36398	0.681991	+	0.731360i	$0.261114\pi$
$308$	−6.28773	−0.358277
$309$	−1.86986	−0.106373
$310$	0	0
$311$	−12.9378	−0.733637	−0.366818	−	0.930293i	$-0.619553\pi$
$311$	−12.9378	−0.733637	−0.366818	+	0.930293i	$0.619553\pi$
$312$	−2.87276	−0.162638
$313$	−4.42202	−0.249947	−0.124974	−	0.992160i	$-0.539885\pi$
$313$	−4.42202	−0.249947	−0.124974	+	0.992160i	$0.539885\pi$
$314$	−40.5977	−2.29106
$315$	0	0
$316$	−7.95303	−0.447393
$317$	−4.23639	−0.237939	−0.118970	−	0.992898i	$-0.537959\pi$
$317$	−4.23639	−0.237939	−0.118970	+	0.992898i	$0.537959\pi$
$318$	−21.5461	−1.20825
$319$	0.0884500	0.00495225
$320$	0	0
$321$	2.64153	0.147436
$322$	20.1162	1.12103
$323$	−3.54074	−0.197012
$324$	8.63931	0.479962
$325$	0	0
$326$	−37.0751	−2.05340
$327$	6.71811	0.371512
$328$	−8.67616	−0.479061
$329$	28.1290	1.55080
$330$	0	0
$331$	11.9266	0.655544	0.327772	−	0.944757i	$-0.393702\pi$
$331$	11.9266	0.655544	0.327772	+	0.944757i	$0.393702\pi$
$332$	4.35280	0.238891
$333$	10.8051	0.592115
$334$	12.9345	0.707742
$335$	0	0
$336$	−4.90458	−0.267567
$337$	−2.04269	−0.111272	−0.0556362	−	0.998451i	$-0.517719\pi$
$337$	−2.04269	−0.111272	−0.0556362	+	0.998451i	$0.517719\pi$
$338$	11.9866	0.651986
$339$	17.0801	0.927663
$340$	0	0
$341$	−0.556150	−0.0301172
$342$	−17.5962	−0.951495
$343$	−19.7006	−1.06373
$344$	6.39381	0.344731
$345$	0	0
$346$	−45.4247	−2.44205
$347$	−29.8606	−1.60300	−0.801500	−	0.597995i	$-0.795964\pi$
$347$	−29.8606	−1.60300	−0.801500	+	0.597995i	$0.795964\pi$
$348$	−0.189625	−0.0101650
$349$	−34.9216	−1.86931	−0.934656	−	0.355554i	$-0.884292\pi$
$349$	−34.9216	−1.86931	−0.934656	+	0.355554i	$0.884292\pi$
$350$	0	0
$351$	−11.9536	−0.638035
$352$	−7.81353	−0.416462
$353$	16.3549	0.870485	0.435243	−	0.900313i	$-0.356663\pi$
$353$	16.3549	0.870485	0.435243	+	0.900313i	$0.356663\pi$
$354$	14.2829	0.759128
$355$	0	0
$356$	12.5892	0.667225
$357$	2.00000	0.105851
$358$	−43.5898	−2.30379
$359$	−12.9432	−0.683116	−0.341558	−	0.939861i	$-0.610955\pi$
$359$	−12.9432	−0.683116	−0.341558	+	0.939861i	$0.610955\pi$
$360$	0	0
$361$	−6.46316	−0.340166
$362$	42.6946	2.24398
$363$	0.825785	0.0433424
$364$	17.1147	0.897055
$365$	0	0
$366$	−7.10117	−0.371184
$367$	−0.743625	−0.0388169	−0.0194085	−	0.999812i	$-0.506178\pi$
$367$	−0.743625	−0.0388169	−0.0194085	+	0.999812i	$0.506178\pi$
$368$	9.50074	0.495260
$369$	−15.7362	−0.819194
$370$	0	0
$371$	29.4760	1.53032
$372$	1.19231	0.0618184
$373$	−20.5316	−1.06309	−0.531544	−	0.847031i	$-0.678388\pi$
$373$	−20.5316	−1.06309	−0.531544	+	0.847031i	$0.678388\pi$
$374$	2.14386	0.110857
$375$	0	0
$376$	−14.8439	−0.765515
$377$	−0.240754	−0.0123995
$378$	22.8025	1.17283
$379$	5.45015	0.279956	0.139978	−	0.990155i	$-0.455297\pi$
$379$	5.45015	0.279956	0.139978	+	0.990155i	$0.455297\pi$
$380$	0	0
$381$	−5.69379	−0.291702
$382$	−32.9474	−1.68573
$383$	5.54410	0.283290	0.141645	−	0.989917i	$-0.454761\pi$
$383$	5.54410	0.283290	0.141645	+	0.989917i	$0.454761\pi$
$384$	8.06822	0.411730
$385$	0	0
$386$	4.08631	0.207988
$387$	11.5966	0.589489
$388$	4.73565	0.240416
$389$	3.41925	0.173363	0.0866815	−	0.996236i	$-0.472374\pi$
$389$	3.41925	0.173363	0.0866815	+	0.996236i	$0.472374\pi$
$390$	0	0
$391$	−3.87423	−0.195928
$392$	1.44961	0.0732161
$393$	1.86068	0.0938587
$394$	52.6928	2.65462
$395$	0	0
$396$	6.01809	0.302421
$397$	36.7734	1.84560	0.922802	−	0.385275i	$-0.125893\pi$
$397$	36.7734	1.84560	0.922802	+	0.385275i	$0.125893\pi$
$398$	40.5062	2.03039
$399$	−7.08148	−0.354517
$400$	0	0
$401$	15.5003	0.774047	0.387024	−	0.922070i	$-0.373503\pi$
$401$	15.5003	0.774047	0.387024	+	0.922070i	$0.373503\pi$
$402$	11.2617	0.561685
$403$	1.51380	0.0754076
$404$	24.0724	1.19765
$405$	0	0
$406$	0.459260	0.0227927
$407$	4.66122	0.231048
$408$	−1.05541	−0.0522508
$409$	1.89548	0.0937252	0.0468626	−	0.998901i	$-0.485078\pi$
$409$	1.89548	0.0937252	0.0468626	+	0.998901i	$0.485078\pi$
$410$	0	0
$411$	2.88864	0.142486
$412$	−5.87860	−0.289618
$413$	−19.5396	−0.961481
$414$	−19.2536	−0.946261
$415$	0	0
$416$	21.2678	1.04274
$417$	1.83766	0.0899904
$418$	−7.59087	−0.371281
$419$	−15.5214	−0.758272	−0.379136	−	0.925341i	$-0.623779\pi$
$419$	−15.5214	−0.758272	−0.379136	+	0.925341i	$0.623779\pi$
$420$	0	0
$421$	19.5377	0.952208	0.476104	−	0.879389i	$-0.342049\pi$
$421$	19.5377	0.952208	0.476104	+	0.879389i	$0.342049\pi$
$422$	−37.4929	−1.82513
$423$	−26.9228	−1.30903
$424$	−15.5547	−0.755402
$425$	0	0
$426$	−11.6410	−0.564008
$427$	9.71470	0.470127
$428$	8.30461	0.401418
$429$	−2.24772	−0.108521
$430$	0	0
$431$	8.26917	0.398312	0.199156	−	0.979968i	$-0.436180\pi$
$431$	8.26917	0.398312	0.199156	+	0.979968i	$0.436180\pi$
$432$	10.7694	0.518145
$433$	21.7882	1.04707	0.523537	−	0.852003i	$-0.324612\pi$
$433$	21.7882	1.04707	0.523537	+	0.852003i	$0.324612\pi$
$434$	−2.88770	−0.138614
$435$	0	0
$436$	21.1208	1.01150
$437$	13.7176	0.656204
$438$	−25.4247	−1.21484
$439$	−7.72171	−0.368537	−0.184269	−	0.982876i	$-0.558992\pi$
$439$	−7.72171	−0.368537	−0.184269	+	0.982876i	$0.558992\pi$
$440$	0	0
$441$	2.62919	0.125200
$442$	−5.83544	−0.277564
$443$	13.8752	0.659232	0.329616	−	0.944115i	$-0.393081\pi$
$443$	13.8752	0.659232	0.329616	+	0.944115i	$0.393081\pi$
$444$	−9.99303	−0.474248
$445$	0	0
$446$	40.2160	1.90428
$447$	−4.23760	−0.200432
$448$	−28.6917	−1.35555
$449$	−21.9924	−1.03788	−0.518942	−	0.854809i	$-0.673674\pi$
$449$	−21.9924	−1.03788	−0.518942	+	0.854809i	$0.673674\pi$
$450$	0	0
$451$	−6.78846	−0.319656
$452$	53.6975	2.52572
$453$	−4.75850	−0.223574
$454$	29.8979	1.40318
$455$	0	0
$456$	3.73695	0.174999
$457$	29.6283	1.38595	0.692976	−	0.720961i	$-0.256299\pi$
$457$	29.6283	1.38595	0.692976	+	0.720961i	$0.256299\pi$
$458$	−13.7767	−0.643742
$459$	−4.39159	−0.204982
$460$	0	0
$461$	39.3471	1.83258	0.916289	−	0.400517i	$-0.131170\pi$
$461$	39.3471	1.83258	0.916289	+	0.400517i	$0.131170\pi$
$462$	4.28773	0.199483
$463$	−31.2934	−1.45433	−0.727165	−	0.686463i	$-0.759162\pi$
$463$	−31.2934	−1.45433	−0.727165	+	0.686463i	$0.759162\pi$
$464$	0.216905	0.0100696
$465$	0	0
$466$	33.9977	1.57491
$467$	4.59683	0.212716	0.106358	−	0.994328i	$-0.466081\pi$
$467$	4.59683	0.212716	0.106358	+	0.994328i	$0.466081\pi$
$468$	−16.3808	−0.757203
$469$	−15.4065	−0.711407
$470$	0	0
$471$	15.6376	0.720544
$472$	10.3112	0.474611
$473$	5.00268	0.230024
$474$	5.42333	0.249102
$475$	0	0
$476$	6.28773	0.288198
$477$	−28.2120	−1.29174
$478$	19.5821	0.895666
$479$	31.8072	1.45331	0.726654	−	0.687004i	$-0.241074\pi$
$479$	31.8072	1.45331	0.726654	+	0.687004i	$0.241074\pi$
$480$	0	0
$481$	−12.6875	−0.578500
$482$	17.8157	0.811485
$483$	−7.74846	−0.352567
$484$	2.59615	0.118007
$485$	0	0
$486$	−34.1362	−1.54845
$487$	−9.26171	−0.419688	−0.209844	−	0.977735i	$-0.567296\pi$
$487$	−9.26171	−0.419688	−0.209844	+	0.977735i	$0.567296\pi$
$488$	−5.12652	−0.232067
$489$	14.2808	0.645798
$490$	0	0
$491$	−14.1086	−0.636711	−0.318355	−	0.947971i	$-0.603130\pi$
$491$	−14.1086	−0.636711	−0.318355	+	0.947971i	$0.603130\pi$
$492$	14.5535	0.656125
$493$	−0.0884500	−0.00398359
$494$	20.6618	0.929617
$495$	0	0
$496$	−1.36384	−0.0612382
$497$	15.9254	0.714350
$498$	−2.96826	−0.133011
$499$	42.6031	1.90718	0.953588	−	0.301115i	$-0.0973588\pi$
$499$	42.6031	1.90718	0.953588	+	0.301115i	$0.0973588\pi$
$500$	0	0
$501$	−4.98216	−0.222587
$502$	−9.90681	−0.442163
$503$	13.3766	0.596435	0.298218	−	0.954498i	$-0.403608\pi$
$503$	13.3766	0.596435	0.298218	+	0.954498i	$0.403608\pi$
$504$	7.17543	0.319619
$505$	0	0
$506$	−8.30582	−0.369239
$507$	−4.61706	−0.205051
$508$	−17.9005	−0.794207
$509$	36.1131	1.60069	0.800343	−	0.599542i	$-0.204651\pi$
$509$	36.1131	1.60069	0.800343	+	0.599542i	$0.204651\pi$
$510$	0	0
$511$	34.7820	1.53867
$512$	−25.4295	−1.12383
$513$	15.5495	0.686526
$514$	−31.3640	−1.38341
$515$	0	0
$516$	−10.7251	−0.472145
$517$	−11.6142	−0.510794
$518$	24.2025	1.06340
$519$	17.4969	0.768029
$520$	0	0
$521$	−4.23333	−0.185466	−0.0927328	−	0.995691i	$-0.529560\pi$
$521$	−4.23333	−0.185466	−0.0927328	+	0.995691i	$0.529560\pi$
$522$	−0.439565	−0.0192393
$523$	17.9765	0.786057	0.393028	−	0.919526i	$-0.371427\pi$
$523$	17.9765	0.786057	0.393028	+	0.919526i	$0.371427\pi$
$524$	5.84972	0.255546
$525$	0	0
$526$	−9.23110	−0.402495
$527$	0.556150	0.0242263
$528$	2.02506	0.0881296
$529$	−7.99035	−0.347406
$530$	0	0
$531$	18.7017	0.811584
$532$	−22.2632	−0.965233
$533$	18.4777	0.800358
$534$	−8.58482	−0.371501
$535$	0	0
$536$	8.13014	0.351168
$537$	16.7901	0.724548
$538$	−36.3601	−1.56760
$539$	1.13421	0.0488539
$540$	0	0
$541$	−4.97709	−0.213982	−0.106991	−	0.994260i	$-0.534122\pi$
$541$	−4.97709	−0.213982	−0.106991	+	0.994260i	$0.534122\pi$
$542$	−3.05923	−0.131405
$543$	−16.4453	−0.705736
$544$	7.81353	0.335002
$545$	0	0
$546$	−11.6709	−0.499467
$547$	1.47373	0.0630123	0.0315062	−	0.999504i	$-0.489970\pi$
$547$	1.47373	0.0630123	0.0315062	+	0.999504i	$0.489970\pi$
$548$	9.08148	0.387942
$549$	−9.29811	−0.396834
$550$	0	0
$551$	0.313178	0.0133419
$552$	4.08892	0.174036
$553$	−7.41934	−0.315502
$554$	−67.5227	−2.86876
$555$	0	0
$556$	5.77734	0.245014
$557$	−41.2360	−1.74723	−0.873613	−	0.486621i	$-0.838229\pi$
$557$	−41.2360	−1.74723	−0.873613	+	0.486621i	$0.838229\pi$
$558$	2.76387	0.117004
$559$	−13.6169	−0.575935
$560$	0	0
$561$	−0.825785	−0.0348647
$562$	51.0985	2.15546
$563$	7.77834	0.327818	0.163909	−	0.986475i	$-0.447590\pi$
$563$	7.77834	0.327818	0.163909	+	0.986475i	$0.447590\pi$
$564$	24.8994	1.04845
$565$	0	0
$566$	−32.3514	−1.35983
$567$	8.05957	0.338470
$568$	−8.40393	−0.352621
$569$	−9.12918	−0.382715	−0.191358	−	0.981520i	$-0.561289\pi$
$569$	−9.12918	−0.382715	−0.191358	+	0.981520i	$0.561289\pi$
$570$	0	0
$571$	23.1432	0.968513	0.484256	−	0.874926i	$-0.339090\pi$
$571$	23.1432	0.968513	0.484256	+	0.874926i	$0.339090\pi$
$572$	−7.06654	−0.295467
$573$	12.6908	0.530167
$574$	−35.2478	−1.47121
$575$	0	0
$576$	27.4613	1.14422
$577$	16.9443	0.705402	0.352701	−	0.935736i	$-0.385263\pi$
$577$	16.9443	0.705402	0.352701	+	0.935736i	$0.385263\pi$
$578$	−2.14386	−0.0891730
$579$	−1.57399	−0.0654127
$580$	0	0
$581$	4.06070	0.168466
$582$	−3.22934	−0.133860
$583$	−12.1704	−0.504046
$584$	−18.3547	−0.759524
$585$	0	0
$586$	−37.4774	−1.54818
$587$	15.0791	0.622379	0.311190	−	0.950348i	$-0.399273\pi$
$587$	15.0791	0.622379	0.311190	+	0.950348i	$0.399273\pi$
$588$	−2.43159	−0.100277
$589$	−1.96918	−0.0811387
$590$	0	0
$591$	−20.2965	−0.834885
$592$	11.4307	0.469798
$593$	−20.4762	−0.840856	−0.420428	−	0.907326i	$-0.638120\pi$
$593$	−20.4762	−0.840856	−0.420428	+	0.907326i	$0.638120\pi$
$594$	−9.41497	−0.386301
$595$	0	0
$596$	−13.3224	−0.545709
$597$	−15.6024	−0.638563
$598$	22.6078	0.924503
$599$	−5.75551	−0.235164	−0.117582	−	0.993063i	$-0.537514\pi$
$599$	−5.75551	−0.235164	−0.117582	+	0.993063i	$0.537514\pi$
$600$	0	0
$601$	13.2830	0.541824	0.270912	−	0.962604i	$-0.412675\pi$
$601$	13.2830	0.541824	0.270912	+	0.962604i	$0.412675\pi$
$602$	25.9755	1.05868
$603$	14.7459	0.600498
$604$	−14.9601	−0.608717
$605$	0	0
$606$	−16.4154	−0.666832
$607$	30.0531	1.21982	0.609908	−	0.792472i	$-0.291206\pi$
$607$	30.0531	1.21982	0.609908	+	0.792472i	$0.291206\pi$
$608$	−27.6657	−1.12199
$609$	−0.176900	−0.00716835
$610$	0	0
$611$	31.6131	1.27893
$612$	−6.01809	−0.243267
$613$	21.5037	0.868527	0.434263	−	0.900786i	$-0.357009\pi$
$613$	21.5037	0.868527	0.434263	+	0.900786i	$0.357009\pi$
$614$	−51.2360	−2.06772
$615$	0	0
$616$	3.09542	0.124718
$617$	19.5696	0.787843	0.393922	−	0.919144i	$-0.371118\pi$
$617$	19.5696	0.787843	0.393922	+	0.919144i	$0.371118\pi$
$618$	4.00873	0.161255
$619$	18.1524	0.729606	0.364803	−	0.931085i	$-0.381136\pi$
$619$	18.1524	0.729606	0.364803	+	0.931085i	$0.381136\pi$
$620$	0	0
$621$	17.0140	0.682749
$622$	27.7369	1.11215
$623$	11.7444	0.470529
$624$	−5.51207	−0.220659
$625$	0	0
$626$	9.48021	0.378906
$627$	2.92389	0.116769
$628$	49.1626	1.96180
$629$	−4.66122	−0.185855
$630$	0	0
$631$	24.4553	0.973551	0.486775	−	0.873527i	$-0.338173\pi$
$631$	24.4553	0.973551	0.486775	+	0.873527i	$0.338173\pi$
$632$	3.91524	0.155740
$633$	14.4417	0.574007
$634$	9.08224	0.360702
$635$	0	0
$636$	26.0917	1.03460
$637$	−3.08724	−0.122321
$638$	−0.189625	−0.00750732
$639$	−15.2424	−0.602981
$640$	0	0
$641$	1.49398	0.0590086	0.0295043	−	0.999565i	$-0.490607\pi$
$641$	1.49398	0.0590086	0.0295043	+	0.999565i	$0.490607\pi$
$642$	−5.66308	−0.223504
$643$	1.44146	0.0568456	0.0284228	−	0.999596i	$-0.490952\pi$
$643$	1.44146	0.0568456	0.0284228	+	0.999596i	$0.490952\pi$
$644$	−24.3601	−0.959923
$645$	0	0
$646$	7.59087	0.298659
$647$	37.9112	1.49044	0.745222	−	0.666816i	$-0.232344\pi$
$647$	37.9112	1.49044	0.745222	+	0.666816i	$0.232344\pi$
$648$	−4.25309	−0.167077
$649$	8.06775	0.316687
$650$	0	0
$651$	1.11230	0.0435944
$652$	44.8968	1.75829
$653$	18.2442	0.713950	0.356975	−	0.934114i	$-0.383808\pi$
$653$	18.2442	0.713950	0.356975	+	0.934114i	$0.383808\pi$
$654$	−14.4027	−0.563191
$655$	0	0
$656$	−16.6473	−0.649967
$657$	−33.2905	−1.29879
$658$	−60.3048	−2.35092
$659$	27.2970	1.06334	0.531670	−	0.846952i	$-0.321565\pi$
$659$	27.2970	1.06334	0.531670	+	0.846952i	$0.321565\pi$
$660$	0	0
$661$	−0.782371	−0.0304307	−0.0152154	−	0.999884i	$-0.504843\pi$
$661$	−0.782371	−0.0304307	−0.0152154	+	0.999884i	$0.504843\pi$
$662$	−25.5690	−0.993766
$663$	2.24772	0.0872944
$664$	−2.14286	−0.0831592
$665$	0	0
$666$	−23.1646	−0.897612
$667$	0.342676	0.0132685
$668$	−15.6632	−0.606029
$669$	−15.4906	−0.598902
$670$	0	0
$671$	−4.01112	−0.154848
$672$	15.6271	0.602827
$673$	2.62093	0.101029	0.0505146	−	0.998723i	$-0.483914\pi$
$673$	2.62093	0.101029	0.0505146	+	0.998723i	$0.483914\pi$
$674$	4.37925	0.168682
$675$	0	0
$676$	−14.5154	−0.558285
$677$	−26.4302	−1.01580	−0.507898	−	0.861417i	$-0.669578\pi$
$677$	−26.4302	−1.01580	−0.507898	+	0.861417i	$0.669578\pi$
$678$	−36.6174	−1.40628
$679$	4.41787	0.169542
$680$	0	0
$681$	−11.5162	−0.441303
$682$	1.19231	0.0456559
$683$	22.3904	0.856744	0.428372	−	0.903603i	$-0.359087\pi$
$683$	22.3904	0.856744	0.428372	+	0.903603i	$0.359087\pi$
$684$	21.3085	0.814751
$685$	0	0
$686$	42.2353	1.61255
$687$	5.30657	0.202458
$688$	12.2680	0.467714
$689$	33.1269	1.26203
$690$	0	0
$691$	5.27753	0.200767	0.100383	−	0.994949i	$-0.467993\pi$
$691$	5.27753	0.200767	0.100383	+	0.994949i	$0.467993\pi$
$692$	55.0079	2.09109
$693$	5.61425	0.213268
$694$	64.0170	2.43005
$695$	0	0
$696$	0.0933514	0.00353848
$697$	6.78846	0.257131
$698$	74.8672	2.83377
$699$	−13.0954	−0.495314
$700$	0	0
$701$	11.8627	0.448046	0.224023	−	0.974584i	$-0.428081\pi$
$701$	11.8627	0.448046	0.224023	+	0.974584i	$0.428081\pi$
$702$	25.6268	0.967223
$703$	16.5042	0.622467
$704$	11.8466	0.446484
$705$	0	0
$706$	−35.0628	−1.31960
$707$	22.4570	0.844582
$708$	−17.2962	−0.650030
$709$	−30.8079	−1.15701	−0.578507	−	0.815678i	$-0.696364\pi$
$709$	−30.8079	−1.15701	−0.578507	+	0.815678i	$0.696364\pi$
$710$	0	0
$711$	7.10117	0.266315
$712$	−6.19760	−0.232265
$713$	−2.15465	−0.0806923
$714$	−4.28773	−0.160464
$715$	0	0
$716$	52.7859	1.97270
$717$	−7.54275	−0.281689
$718$	27.7485	1.03556
$719$	51.5110	1.92104	0.960518	−	0.278216i	$-0.0897432\pi$
$719$	51.5110	1.92104	0.960518	+	0.278216i	$0.0897432\pi$
$720$	0	0
$721$	−5.48411	−0.204239
$722$	13.8561	0.515672
$723$	−6.86236	−0.255214
$724$	−51.7019	−1.92148
$725$	0	0
$726$	−1.77037	−0.0657046
$727$	−15.4297	−0.572256	−0.286128	−	0.958191i	$-0.592368\pi$
$727$	−15.4297	−0.572256	−0.286128	+	0.958191i	$0.592368\pi$
$728$	−8.42550	−0.312270
$729$	3.16556	0.117243
$730$	0	0
$731$	−5.00268	−0.185031
$732$	8.59931	0.317840
$733$	−36.6268	−1.35284	−0.676421	−	0.736515i	$-0.736470\pi$
$733$	−36.6268	−1.35284	−0.676421	+	0.736515i	$0.736470\pi$
$734$	1.59423	0.0588441
$735$	0	0
$736$	−30.2714	−1.11582
$737$	6.36124	0.234319
$738$	33.7363	1.24185
$739$	−35.9818	−1.32361	−0.661806	−	0.749675i	$-0.730210\pi$
$739$	−35.9818	−1.32361	−0.661806	+	0.749675i	$0.730210\pi$
$740$	0	0
$741$	−7.95861	−0.292367
$742$	−63.1925	−2.31987
$743$	−41.3011	−1.51519	−0.757595	−	0.652725i	$-0.773626\pi$
$743$	−41.3011	−1.51519	−0.757595	+	0.652725i	$0.773626\pi$
$744$	−0.586968	−0.0215193
$745$	0	0
$746$	44.0170	1.61158
$747$	−3.88657	−0.142202
$748$	−2.59615	−0.0949248
$749$	7.74733	0.283081
$750$	0	0
$751$	−34.2248	−1.24888	−0.624440	−	0.781073i	$-0.714673\pi$
$751$	−34.2248	−1.24888	−0.624440	+	0.781073i	$0.714673\pi$
$752$	−28.4815	−1.03861
$753$	3.81596	0.139061
$754$	0.516145	0.0187969
$755$	0	0
$756$	−27.6131	−1.00428
$757$	44.5732	1.62004	0.810020	−	0.586402i	$-0.199456\pi$
$757$	44.5732	1.62004	0.810020	+	0.586402i	$0.199456\pi$
$758$	−11.6844	−0.424396
$759$	3.19928	0.116126
$760$	0	0
$761$	−24.4878	−0.887683	−0.443841	−	0.896105i	$-0.646385\pi$
$761$	−24.4878	−0.887683	−0.443841	+	0.896105i	$0.646385\pi$
$762$	12.2067	0.442203
$763$	19.7035	0.713315
$764$	39.8982	1.44347
$765$	0	0
$766$	−11.8858	−0.429452
$767$	−21.9598	−0.792923
$768$	2.26825	0.0818483
$769$	46.2850	1.66908	0.834539	−	0.550949i	$-0.185734\pi$
$769$	46.2850	1.66908	0.834539	+	0.550949i	$0.185734\pi$
$770$	0	0
$771$	12.0809	0.435085
$772$	−4.94840	−0.178097
$773$	−1.40297	−0.0504613	−0.0252306	−	0.999682i	$-0.508032\pi$
$773$	−1.40297	−0.0504613	−0.0252306	+	0.999682i	$0.508032\pi$
$774$	−24.8616	−0.893631
$775$	0	0
$776$	−2.33134	−0.0836902
$777$	−9.32245	−0.334441
$778$	−7.33042	−0.262808
$779$	−24.0362	−0.861186
$780$	0	0
$781$	−6.57546	−0.235288
$782$	8.30582	0.297016
$783$	0.388436	0.0138816
$784$	2.78141	0.0993362
$785$	0	0
$786$	−3.98904	−0.142284
$787$	16.8890	0.602028	0.301014	−	0.953620i	$-0.402675\pi$
$787$	16.8890	0.602028	0.301014	+	0.953620i	$0.402675\pi$
$788$	−63.8093	−2.27311
$789$	3.55568	0.126586
$790$	0	0
$791$	50.0941	1.78114
$792$	−2.96268	−0.105274
$793$	10.9180	0.387709
$794$	−78.8372	−2.79783
$795$	0	0
$796$	−49.0518	−1.73859
$797$	−32.1780	−1.13980	−0.569902	−	0.821713i	$-0.693019\pi$
$797$	−32.1780	−1.13980	−0.569902	+	0.821713i	$0.693019\pi$
$798$	15.1817	0.537428
$799$	11.6142	0.410883
$800$	0	0
$801$	−11.2408	−0.397173
$802$	−33.2305	−1.17341
$803$	−14.3612	−0.506797
$804$	−13.6376	−0.480962
$805$	0	0
$806$	−3.24538	−0.114314
$807$	14.0054	0.493012
$808$	−11.8507	−0.416907
$809$	26.1011	0.917667	0.458833	−	0.888522i	$-0.348267\pi$
$809$	26.1011	0.917667	0.458833	+	0.888522i	$0.348267\pi$
$810$	0	0
$811$	14.5976	0.512592	0.256296	−	0.966598i	$-0.417498\pi$
$811$	14.5976	0.512592	0.256296	+	0.966598i	$0.417498\pi$
$812$	−0.556150	−0.0195170
$813$	1.17837	0.0413273
$814$	−9.99303	−0.350256
$815$	0	0
$816$	−2.02506	−0.0708914
$817$	17.7132	0.619707
$818$	−4.06364	−0.142082
$819$	−15.2816	−0.533981
$820$	0	0
$821$	−16.6929	−0.582585	−0.291293	−	0.956634i	$-0.594085\pi$
$821$	−16.6929	−0.582585	−0.291293	+	0.956634i	$0.594085\pi$
$822$	−6.19284	−0.216000
$823$	−15.0521	−0.524683	−0.262341	−	0.964975i	$-0.584495\pi$
$823$	−15.0521	−0.524683	−0.262341	+	0.964975i	$0.584495\pi$
$824$	2.89401	0.100817
$825$	0	0
$826$	41.8903	1.45755
$827$	−34.2099	−1.18960	−0.594798	−	0.803875i	$-0.702768\pi$
$827$	−34.2099	−1.18960	−0.594798	+	0.803875i	$0.702768\pi$
$828$	23.3155	0.810269
$829$	−27.3346	−0.949370	−0.474685	−	0.880156i	$-0.657438\pi$
$829$	−27.3346	−0.949370	−0.474685	+	0.880156i	$0.657438\pi$
$830$	0	0
$831$	26.0087	0.902233
$832$	−32.2455	−1.11791
$833$	−1.13421	−0.0392981
$834$	−3.93968	−0.136420
$835$	0	0
$836$	9.19231	0.317923
$837$	−2.44238	−0.0844210
$838$	33.2758	1.14950
$839$	17.5495	0.605875	0.302938	−	0.953010i	$-0.402033\pi$
$839$	17.5495	0.605875	0.302938	+	0.953010i	$0.402033\pi$
$840$	0	0
$841$	−28.9922	−0.999730
$842$	−41.8861	−1.44349
$843$	−19.6824	−0.677897
$844$	45.4028	1.56283
$845$	0	0
$846$	57.7187	1.98441
$847$	2.42194	0.0832188
$848$	−29.8453	−1.02489
$849$	12.4613	0.427670
$850$	0	0
$851$	18.0586	0.619042
$852$	14.0969	0.482951
$853$	36.7989	1.25997	0.629986	−	0.776607i	$-0.283061\pi$
$853$	36.7989	1.25997	0.629986	+	0.776607i	$0.283061\pi$
$854$	−20.8270	−0.712685
$855$	0	0
$856$	−4.08832	−0.139736
$857$	32.8522	1.12221	0.561105	−	0.827745i	$-0.310377\pi$
$857$	32.8522	1.12221	0.561105	+	0.827745i	$0.310377\pi$
$858$	4.81882	0.164512
$859$	−42.8934	−1.46350	−0.731752	−	0.681571i	$-0.761297\pi$
$859$	−42.8934	−1.46350	−0.731752	+	0.681571i	$0.761297\pi$
$860$	0	0
$861$	13.5769	0.462700
$862$	−17.7280	−0.603817
$863$	8.49788	0.289271	0.144636	−	0.989485i	$-0.453799\pi$
$863$	8.49788	0.289271	0.144636	+	0.989485i	$0.453799\pi$
$864$	−34.3138	−1.16738
$865$	0	0
$866$	−46.7110	−1.58730
$867$	0.825785	0.0280451
$868$	3.49692	0.118693
$869$	3.06339	0.103918
$870$	0	0
$871$	−17.3148	−0.586690
$872$	−10.3977	−0.352110
$873$	−4.22842	−0.143110
$874$	−29.4088	−0.994766
$875$	0	0
$876$	30.7885	1.04025
$877$	−58.2913	−1.96836	−0.984178	−	0.177183i	$-0.943302\pi$
$877$	−58.2913	−1.96836	−0.984178	+	0.177183i	$0.943302\pi$
$878$	16.5543	0.558681
$879$	14.4357	0.486906
$880$	0	0
$881$	14.6053	0.492066	0.246033	−	0.969261i	$-0.420873\pi$
$881$	14.6053	0.492066	0.246033	+	0.969261i	$0.420873\pi$
$882$	−5.63663	−0.189795
$883$	−53.1585	−1.78893	−0.894463	−	0.447141i	$-0.852442\pi$
$883$	−53.1585	−1.78893	−0.894463	+	0.447141i	$0.852442\pi$
$884$	7.06654	0.237673
$885$	0	0
$886$	−29.7466	−0.999357
$887$	−33.5187	−1.12545	−0.562723	−	0.826645i	$-0.690246\pi$
$887$	−33.5187	−1.12545	−0.562723	+	0.826645i	$0.690246\pi$
$888$	4.91952	0.165088
$889$	−16.6993	−0.560076
$890$	0	0
$891$	−3.32773	−0.111483
$892$	−48.7004	−1.63061
$893$	−41.1230	−1.37613
$894$	9.08484	0.303843
$895$	0	0
$896$	23.6632	0.790533
$897$	−8.70820	−0.290758
$898$	47.1486	1.57337
$899$	−0.0491914	−0.00164063
$900$	0	0
$901$	12.1704	0.405455
$902$	14.5535	0.484580
$903$	−10.0054	−0.332958
$904$	−26.4350	−0.879216
$905$	0	0
$906$	10.2016	0.338925
$907$	5.70664	0.189486	0.0947430	−	0.995502i	$-0.469797\pi$
$907$	5.70664	0.189486	0.0947430	+	0.995502i	$0.469797\pi$
$908$	−36.2055	−1.20152
$909$	−21.4940	−0.712910
$910$	0	0
$911$	39.9586	1.32389	0.661944	−	0.749553i	$-0.269732\pi$
$911$	39.9586	1.32389	0.661944	+	0.749553i	$0.269732\pi$
$912$	7.17022	0.237430
$913$	−1.67663	−0.0554884
$914$	−63.5190	−2.10102
$915$	0	0
$916$	16.6831	0.551226
$917$	5.45717	0.180212
$918$	9.41497	0.310740
$919$	3.28289	0.108293	0.0541464	−	0.998533i	$-0.482756\pi$
$919$	3.28289	0.108293	0.0541464	+	0.998533i	$0.482756\pi$
$920$	0	0
$921$	19.7353	0.650302
$922$	−84.3549	−2.77808
$923$	17.8979	0.589117
$924$	−5.19231	−0.170814
$925$	0	0
$926$	67.0889	2.20468
$927$	5.24894	0.172398
$928$	−0.691106	−0.0226867
$929$	28.2911	0.928201	0.464100	−	0.885783i	$-0.346378\pi$
$929$	28.2911	0.928201	0.464100	+	0.885783i	$0.346378\pi$
$930$	0	0
$931$	4.01594	0.131617
$932$	−41.1702	−1.34858
$933$	−10.6839	−0.349774
$934$	−9.85499	−0.322465
$935$	0	0
$936$	8.06419	0.263586
$937$	−22.3515	−0.730191	−0.365095	−	0.930970i	$-0.618964\pi$
$937$	−22.3515	−0.730191	−0.365095	+	0.930970i	$0.618964\pi$
$938$	33.0295	1.07845
$939$	−3.65164	−0.119167
$940$	0	0
$941$	58.8901	1.91976	0.959881	−	0.280407i	$-0.0904695\pi$
$941$	58.8901	1.91976	0.959881	+	0.280407i	$0.0904695\pi$
$942$	−33.5250	−1.09230
$943$	−26.3001	−0.856448
$944$	19.7845	0.643930
$945$	0	0
$946$	−10.7251	−0.348702
$947$	−41.7043	−1.35521	−0.677603	−	0.735428i	$-0.736981\pi$
$947$	−41.7043	−1.35521	−0.677603	+	0.735428i	$0.736981\pi$
$948$	−6.56749	−0.213302
$949$	39.0902	1.26892
$950$	0	0
$951$	−3.49834	−0.113442
$952$	−3.09542	−0.100323
$953$	−37.2955	−1.20812	−0.604061	−	0.796938i	$-0.706451\pi$
$953$	−37.2955	−1.20812	−0.604061	+	0.796938i	$0.706451\pi$
$954$	60.4826	1.95820
$955$	0	0
$956$	−23.7134	−0.766945
$957$	0.0730406	0.00236107
$958$	−68.1903	−2.20313
$959$	8.47207	0.273577
$960$	0	0
$961$	−30.6907	−0.990023
$962$	27.2003	0.876972
$963$	−7.41510	−0.238948
$964$	−21.5743	−0.694862
$965$	0	0
$966$	16.6116	0.534471
$967$	8.93047	0.287185	0.143592	−	0.989637i	$-0.454135\pi$
$967$	8.93047	0.287185	0.143592	+	0.989637i	$0.454135\pi$
$968$	−1.27807	−0.0410789
$969$	−2.92389	−0.0939288
$970$	0	0
$971$	5.23856	0.168113	0.0840567	−	0.996461i	$-0.473212\pi$
$971$	5.23856	0.168113	0.0840567	+	0.996461i	$0.473212\pi$
$972$	41.3379	1.32591
$973$	5.38965	0.172784
$974$	19.8558	0.636222
$975$	0	0
$976$	−9.83644	−0.314857
$977$	56.9893	1.82325	0.911624	−	0.411026i	$-0.134829\pi$
$977$	56.9893	1.82325	0.911624	+	0.411026i	$0.134829\pi$
$978$	−30.6160	−0.978992
$979$	−4.84917	−0.154980
$980$	0	0
$981$	−18.8586	−0.602108
$982$	30.2468	0.965216
$983$	−2.10559	−0.0671578	−0.0335789	−	0.999436i	$-0.510690\pi$
$983$	−2.10559	−0.0671578	−0.0335789	+	0.999436i	$0.510690\pi$
$984$	−7.16464	−0.228400
$985$	0	0
$986$	0.189625	0.00603888
$987$	23.2285	0.739371
$988$	−25.0208	−0.796017
$989$	19.3815	0.616297
$990$	0	0
$991$	4.34937	0.138162	0.0690811	−	0.997611i	$-0.477993\pi$
$991$	4.34937	0.138162	0.0690811	+	0.997611i	$0.477993\pi$
$992$	4.34549	0.137969
$993$	9.84878	0.312542
$994$	−34.1418	−1.08291
$995$	0	0
$996$	3.59447	0.113895
$997$	−17.8042	−0.563863	−0.281932	−	0.959435i	$-0.590975\pi$
$997$	−17.8042	−0.563863	−0.281932	+	0.959435i	$0.590975\pi$
$998$	−91.3352	−2.89117
$999$	20.4702	0.647648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4675.2.a.be.1.1		4
5.4	even	2		935.2.a.g.1.4	✓	4
15.14	odd	2		8415.2.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
935.2.a.g.1.4	✓	4	5.4	even	2
4675.2.a.be.1.1		4	1.1	even	1	trivial
8415.2.a.ba.1.1		4	15.14	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:	\( N \)	\(=\)	\( 4675 = 5^{2} \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4675.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14386 q^{2} +0.825785 q^{3} +2.59615 q^{4} -1.77037 q^{6} +2.42194 q^{7} -1.27807 q^{8} -2.31808 q^{9} +O(q^{10})\)	\(q-2.14386 q^{2} +0.825785 q^{3} +2.59615 q^{4} -1.77037 q^{6} +2.42194 q^{7} -1.27807 q^{8} -2.31808 q^{9} -1.00000 q^{11} +2.14386 q^{12} +2.72193 q^{13} -5.19231 q^{14} -2.45229 q^{16} +1.00000 q^{17} +4.96965 q^{18} -3.54074 q^{19} +2.00000 q^{21} +2.14386 q^{22} -3.87423 q^{23} -1.05541 q^{24} -5.83544 q^{26} -4.39159 q^{27} +6.28773 q^{28} -0.0884500 q^{29} +0.556150 q^{31} +7.81353 q^{32} -0.825785 q^{33} -2.14386 q^{34} -6.01809 q^{36} -4.66122 q^{37} +7.59087 q^{38} +2.24772 q^{39} +6.78846 q^{41} -4.28773 q^{42} -5.00268 q^{43} -2.59615 q^{44} +8.30582 q^{46} +11.6142 q^{47} -2.02506 q^{48} -1.13421 q^{49} +0.825785 q^{51} +7.06654 q^{52} +12.1704 q^{53} +9.41497 q^{54} -3.09542 q^{56} -2.92389 q^{57} +0.189625 q^{58} -8.06775 q^{59} +4.01112 q^{61} -1.19231 q^{62} -5.61425 q^{63} -11.8466 q^{64} +1.77037 q^{66} -6.36124 q^{67} +2.59615 q^{68} -3.19928 q^{69} +6.57546 q^{71} +2.96268 q^{72} +14.3612 q^{73} +9.99303 q^{74} -9.19231 q^{76} -2.42194 q^{77} -4.81882 q^{78} -3.06339 q^{79} +3.32773 q^{81} -14.5535 q^{82} +1.67663 q^{83} +5.19231 q^{84} +10.7251 q^{86} -0.0730406 q^{87} +1.27807 q^{88} +4.84917 q^{89} +6.59234 q^{91} -10.0581 q^{92} +0.459260 q^{93} -24.8994 q^{94} +6.45229 q^{96} +1.82410 q^{97} +2.43159 q^{98} +2.31808 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 2 q^{4} - q^{6} - q^{7} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) 4 * q + q^3 + 2 * q^4 - q^6 - q^7 - 3 * q^8 - 3 * q^9	\( 4 q + q^{3} + 2 q^{4} - q^{6} - q^{7} - 3 q^{8} - 3 q^{9} - 4 q^{11} + 13 q^{13} - 4 q^{14} - 10 q^{16} + 4 q^{17} + 9 q^{18} - 2 q^{19} + 8 q^{21} - 5 q^{23} - 8 q^{24} - 6 q^{26} + 4 q^{27} + 8 q^{28} + 12 q^{29} - 2 q^{31} - q^{32} - q^{33} - 5 q^{36} + q^{37} + 4 q^{38} - 4 q^{39} + 2 q^{41} + 7 q^{43} - 2 q^{44} - 3 q^{46} + 19 q^{47} - q^{48} - 11 q^{49} + q^{51} + q^{52} + 17 q^{53} + 15 q^{54} - 12 q^{56} + 18 q^{57} + 11 q^{58} + 6 q^{59} - 15 q^{61} + 12 q^{62} + 5 q^{63} + q^{64} + q^{66} + 7 q^{67} + 2 q^{68} - 8 q^{69} - 8 q^{71} - 11 q^{72} + 25 q^{73} + 28 q^{74} - 20 q^{76} + q^{77} - 5 q^{78} - 7 q^{79} - 8 q^{81} - 9 q^{82} - 5 q^{83} + 4 q^{84} + 23 q^{86} + 20 q^{87} + 3 q^{88} + 16 q^{89} - 16 q^{91} - 17 q^{92} + 14 q^{93} - 10 q^{94} + 26 q^{96} + 11 q^{97} - 16 q^{98} + 3 q^{99}+O(q^{100}) \) 4 * q + q^3 + 2 * q^4 - q^6 - q^7 - 3 * q^8 - 3 * q^9 - 4 * q^11 + 13 * q^13 - 4 * q^14 - 10 * q^16 + 4 * q^17 + 9 * q^18 - 2 * q^19 + 8 * q^21 - 5 * q^23 - 8 * q^24 - 6 * q^26 + 4 * q^27 + 8 * q^28 + 12 * q^29 - 2 * q^31 - q^32 - q^33 - 5 * q^36 + q^37 + 4 * q^38 - 4 * q^39 + 2 * q^41 + 7 * q^43 - 2 * q^44 - 3 * q^46 + 19 * q^47 - q^48 - 11 * q^49 + q^51 + q^52 + 17 * q^53 + 15 * q^54 - 12 * q^56 + 18 * q^57 + 11 * q^58 + 6 * q^59 - 15 * q^61 + 12 * q^62 + 5 * q^63 + q^64 + q^66 + 7 * q^67 + 2 * q^68 - 8 * q^69 - 8 * q^71 - 11 * q^72 + 25 * q^73 + 28 * q^74 - 20 * q^76 + q^77 - 5 * q^78 - 7 * q^79 - 8 * q^81 - 9 * q^82 - 5 * q^83 + 4 * q^84 + 23 * q^86 + 20 * q^87 + 3 * q^88 + 16 * q^89 - 16 * q^91 - 17 * q^92 + 14 * q^93 - 10 * q^94 + 26 * q^96 + 11 * q^97 - 16 * q^98 + 3 * q^99

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