LMFDB - Embedded newform 4925.2.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4925,2,Mod(1,4925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.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:	$39.3263229955$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 6 x^{13} - x^{12} + 66 x^{11} - 83 x^{10} - 218 x^{9} + 446 x^{8} + 178 x^{7} - 782 x^{6} + \cdots - 1 $ x^14 - 6x^13 - x^12 + 66x^11 - 83x^10 - 218x^9 + 446x^8 + 178x^7 - 782x^6 + 205x^5 + 408x^4 - 216x^3 - 4x^2 + 11x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 985)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.236045$ of defining polynomial
Character	$\chi$	$=$	4925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.236045 q^{2} +2.87112 q^{3} -1.94428 q^{4} -0.677712 q^{6} -1.02221 q^{7} +0.931027 q^{8} +5.24331 q^{9} +0.916844 q^{11} -5.58226 q^{12} -4.99719 q^{13} +0.241286 q^{14} +3.66880 q^{16} -0.345734 q^{17} -1.23766 q^{18} -5.30778 q^{19} -2.93487 q^{21} -0.216416 q^{22} +7.21308 q^{23} +2.67309 q^{24} +1.17956 q^{26} +6.44081 q^{27} +1.98746 q^{28} +4.17022 q^{29} +9.99162 q^{31} -2.72806 q^{32} +2.63237 q^{33} +0.0816088 q^{34} -10.1945 q^{36} -2.64110 q^{37} +1.25287 q^{38} -14.3475 q^{39} -9.80072 q^{41} +0.692761 q^{42} +12.7545 q^{43} -1.78260 q^{44} -1.70261 q^{46} +8.43174 q^{47} +10.5336 q^{48} -5.95510 q^{49} -0.992644 q^{51} +9.71595 q^{52} +3.66822 q^{53} -1.52032 q^{54} -0.951701 q^{56} -15.2393 q^{57} -0.984359 q^{58} +4.19018 q^{59} +2.70353 q^{61} -2.35847 q^{62} -5.35974 q^{63} -6.69366 q^{64} -0.621356 q^{66} -9.72899 q^{67} +0.672205 q^{68} +20.7096 q^{69} +0.435154 q^{71} +4.88167 q^{72} +6.14822 q^{73} +0.623419 q^{74} +10.3198 q^{76} -0.937203 q^{77} +3.38665 q^{78} -4.36466 q^{79} +2.76239 q^{81} +2.31341 q^{82} +9.26666 q^{83} +5.70622 q^{84} -3.01064 q^{86} +11.9732 q^{87} +0.853607 q^{88} +14.2366 q^{89} +5.10815 q^{91} -14.0243 q^{92} +28.6871 q^{93} -1.99027 q^{94} -7.83257 q^{96} +11.0151 q^{97} +1.40567 q^{98} +4.80730 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 6 q^{2} + 7 q^{3} + 10 q^{4} + 5 q^{6} + 4 q^{7} + 12 q^{8} + 13 q^{9} - 5 q^{11} + 10 q^{12} + 15 q^{13} - 15 q^{14} + 18 q^{16} + 18 q^{17} + 29 q^{18} + 20 q^{19} - 5 q^{21} - 12 q^{22} + 42 q^{23}+ \cdots + 32 q^{99}+O(q^{100}) $ 14 * q + 6 * q^2 + 7 * q^3 + 10 * q^4 + 5 * q^6 + 4 * q^7 + 12 * q^8 + 13 * q^9 - 5 * q^11 + 10 * q^12 + 15 * q^13 - 15 * q^14 + 18 * q^16 + 18 * q^17 + 29 * q^18 + 20 * q^19 - 5 * q^21 - 12 * q^22 + 42 * q^23 + 7 * q^24 - 13 * q^26 + 19 * q^27 + 3 * q^28 - 2 * q^29 + 40 * q^32 + 10 * q^33 + 13 * q^34 + q^36 + 15 * q^37 + 3 * q^38 - 19 * q^39 + 3 * q^41 + 9 * q^42 + 15 * q^43 - 29 * q^44 - 8 * q^46 + 40 * q^47 + 28 * q^48 + 12 * q^49 - 13 * q^51 + 17 * q^52 + 39 * q^53 + 28 * q^54 - 41 * q^56 + 17 * q^57 + 10 * q^58 + 19 * q^59 - 18 * q^61 - 26 * q^62 + 28 * q^63 + 36 * q^64 - 8 * q^66 + 22 * q^67 + 57 * q^68 + 12 * q^69 - 50 * q^71 + 37 * q^72 + 10 * q^73 - 3 * q^74 - 3 * q^76 + 36 * q^77 - 10 * q^78 + q^79 - 42 * q^81 + 40 * q^82 + 12 * q^83 + 35 * q^84 + 5 * q^86 + 35 * q^87 - 13 * q^88 + 7 * q^89 + 16 * q^91 + 4 * q^92 + 37 * q^93 + 27 * q^94 + 16 * q^96 - 2 * q^97 - 13 * q^98 + 32 * q^99

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 $

$2$	−0.236045	−0.166909	−0.0834544	−	0.996512i	$-0.526595\pi$
$2$	−0.236045	−0.166909	−0.0834544	+	0.996512i	$0.526595\pi$
$3$	2.87112	1.65764	0.828820	−	0.559515i	$-0.189013\pi$
$3$	2.87112	1.65764	0.828820	+	0.559515i	$0.189013\pi$
$4$	−1.94428	−0.972141
$5$	0	0
$5$	0	0
$6$	−0.677712	−0.276675
$7$	−1.02221	−0.386357	−0.193179	−	0.981164i	$-0.561880\pi$
$7$	−1.02221	−0.386357	−0.193179	+	0.981164i	$0.561880\pi$
$8$	0.931027	0.329168
$9$	5.24331	1.74777
$10$	0	0
$11$	0.916844	0.276439	0.138219	−	0.990402i	$-0.455862\pi$
$11$	0.916844	0.276439	0.138219	+	0.990402i	$0.455862\pi$
$12$	−5.58226	−1.61146
$13$	−4.99719	−1.38597	−0.692985	−	0.720952i	$-0.743705\pi$
$13$	−4.99719	−1.38597	−0.692985	+	0.720952i	$0.743705\pi$
$14$	0.241286	0.0644864
$15$	0	0
$16$	3.66880	0.917200
$17$	−0.345734	−0.0838529	−0.0419264	−	0.999121i	$-0.513350\pi$
$17$	−0.345734	−0.0838529	−0.0419264	+	0.999121i	$0.513350\pi$
$18$	−1.23766	−0.291718
$19$	−5.30778	−1.21769	−0.608845	−	0.793290i	$-0.708367\pi$
$19$	−5.30778	−1.21769	−0.608845	+	0.793290i	$0.708367\pi$
$20$	0	0
$21$	−2.93487	−0.640441
$22$	−0.216416	−0.0461401
$23$	7.21308	1.50403	0.752015	−	0.659146i	$-0.229082\pi$
$23$	7.21308	1.50403	0.752015	+	0.659146i	$0.229082\pi$
$24$	2.67309	0.545642
$25$	0	0
$26$	1.17956	0.231331
$27$	6.44081	1.23953
$28$	1.98746	0.375594
$29$	4.17022	0.774391	0.387195	−	0.921998i	$-0.373444\pi$
$29$	4.17022	0.774391	0.387195	+	0.921998i	$0.373444\pi$
$30$	0	0
$31$	9.99162	1.79455	0.897274	−	0.441474i	$-0.145544\pi$
$31$	9.99162	1.79455	0.897274	+	0.441474i	$0.145544\pi$
$32$	−2.72806	−0.482257
$33$	2.63237	0.458236
$34$	0.0816088	0.0139958
$35$	0	0
$36$	−10.1945	−1.69908
$37$	−2.64110	−0.434195	−0.217097	−	0.976150i	$-0.569659\pi$
$37$	−2.64110	−0.434195	−0.217097	+	0.976150i	$0.569659\pi$
$38$	1.25287	0.203243
$39$	−14.3475	−2.29744
$40$	0	0
$41$	−9.80072	−1.53061	−0.765307	−	0.643665i	$-0.777413\pi$
$41$	−9.80072	−1.53061	−0.765307	+	0.643665i	$0.777413\pi$
$42$	0.692761	0.106895
$43$	12.7545	1.94505	0.972524	−	0.232804i	$-0.0747899\pi$
$43$	12.7545	1.94505	0.972524	+	0.232804i	$0.0747899\pi$
$44$	−1.78260	−0.268738
$45$	0	0
$46$	−1.70261	−0.251036
$47$	8.43174	1.22990	0.614948	−	0.788568i	$-0.289177\pi$
$47$	8.43174	1.22990	0.614948	+	0.788568i	$0.289177\pi$
$48$	10.5336	1.52039
$49$	−5.95510	−0.850728
$50$	0	0
$51$	−0.992644	−0.138998
$52$	9.71595	1.34736
$53$	3.66822	0.503869	0.251935	−	0.967744i	$-0.418933\pi$
$53$	3.66822	0.503869	0.251935	+	0.967744i	$0.418933\pi$
$54$	−1.52032	−0.206889
$55$	0	0
$56$	−0.951701	−0.127176
$57$	−15.2393	−2.01849
$58$	−0.984359	−0.129253
$59$	4.19018	0.545515	0.272757	−	0.962083i	$-0.412064\pi$
$59$	4.19018	0.545515	0.272757	+	0.962083i	$0.412064\pi$
$60$	0	0
$61$	2.70353	0.346152	0.173076	−	0.984908i	$-0.444629\pi$
$61$	2.70353	0.346152	0.173076	+	0.984908i	$0.444629\pi$
$62$	−2.35847	−0.299526
$63$	−5.35974	−0.675264
$64$	−6.69366	−0.836708
$65$	0	0
$66$	−0.621356	−0.0764837
$67$	−9.72899	−1.18859	−0.594293	−	0.804249i	$-0.702568\pi$
$67$	−9.72899	−1.18859	−0.594293	+	0.804249i	$0.702568\pi$
$68$	0.672205	0.0815169
$69$	20.7096	2.49314
$70$	0	0
$71$	0.435154	0.0516432	0.0258216	−	0.999667i	$-0.491780\pi$
$71$	0.435154	0.0516432	0.0258216	+	0.999667i	$0.491780\pi$
$72$	4.88167	0.575310
$73$	6.14822	0.719595	0.359797	−	0.933030i	$-0.382846\pi$
$73$	6.14822	0.719595	0.359797	+	0.933030i	$0.382846\pi$
$74$	0.623419	0.0724709
$75$	0	0
$76$	10.3198	1.18377
$77$	−0.937203	−0.106804
$78$	3.38665	0.383463
$79$	−4.36466	−0.491063	−0.245531	−	0.969389i	$-0.578962\pi$
$79$	−4.36466	−0.491063	−0.245531	+	0.969389i	$0.578962\pi$
$80$	0	0
$81$	2.76239	0.306932
$82$	2.31341	0.255473
$83$	9.26666	1.01715	0.508574	−	0.861018i	$-0.330173\pi$
$83$	9.26666	1.01715	0.508574	+	0.861018i	$0.330173\pi$
$84$	5.70622	0.622600
$85$	0	0
$86$	−3.01064	−0.324646
$87$	11.9732	1.28366
$88$	0.853607	0.0909948
$89$	14.2366	1.50908	0.754538	−	0.656256i	$-0.227861\pi$
$89$	14.2366	1.50908	0.754538	+	0.656256i	$0.227861\pi$
$90$	0	0
$91$	5.10815	0.535480
$92$	−14.0243	−1.46213
$93$	28.6871	2.97471
$94$	−1.99027	−0.205281
$95$	0	0
$96$	−7.83257	−0.799408
$97$	11.0151	1.11842	0.559209	−	0.829027i	$-0.311105\pi$
$97$	11.0151	1.11842	0.559209	+	0.829027i	$0.311105\pi$
$98$	1.40567	0.141994
$99$	4.80730	0.483152
$100$	0	0
$101$	11.5252	1.14680	0.573400	−	0.819276i	$-0.305624\pi$
$101$	11.5252	1.14680	0.573400	+	0.819276i	$0.305624\pi$
$102$	0.234308	0.0232000
$103$	−4.12363	−0.406313	−0.203157	−	0.979146i	$-0.565120\pi$
$103$	−4.12363	−0.406313	−0.203157	+	0.979146i	$0.565120\pi$
$104$	−4.65252	−0.456217
$105$	0	0
$106$	−0.865864	−0.0841002
$107$	10.2697	0.992805	0.496402	−	0.868092i	$-0.334654\pi$
$107$	10.2697	0.992805	0.496402	+	0.868092i	$0.334654\pi$
$108$	−12.5228	−1.20500
$109$	19.6212	1.87937	0.939685	−	0.342041i	$-0.111118\pi$
$109$	19.6212	1.87937	0.939685	+	0.342041i	$0.111118\pi$
$110$	0	0
$111$	−7.58292	−0.719739
$112$	−3.75027	−0.354367
$113$	14.9685	1.40812	0.704058	−	0.710142i	$-0.251369\pi$
$113$	14.9685	1.40812	0.704058	+	0.710142i	$0.251369\pi$
$114$	3.59715	0.336904
$115$	0	0
$116$	−8.10809	−0.752817
$117$	−26.2018	−2.42236
$118$	−0.989070	−0.0910512
$119$	0.353411	0.0323972
$120$	0	0
$121$	−10.1594	−0.923582
$122$	−0.638154	−0.0577758
$123$	−28.1390	−2.53721
$124$	−19.4265	−1.74455
$125$	0	0
$126$	1.26514	0.112708
$127$	−5.21953	−0.463159	−0.231579	−	0.972816i	$-0.574389\pi$
$127$	−5.21953	−0.463159	−0.231579	+	0.972816i	$0.574389\pi$
$128$	7.03611	0.621910
$129$	36.6197	3.22419
$130$	0	0
$131$	−1.76360	−0.154087	−0.0770433	−	0.997028i	$-0.524548\pi$
$131$	−1.76360	−0.154087	−0.0770433	+	0.997028i	$0.524548\pi$
$132$	−5.11807	−0.445470
$133$	5.42564	0.470463
$134$	2.29648	0.198385
$135$	0	0
$136$	−0.321888	−0.0276017
$137$	14.9252	1.27514	0.637571	−	0.770392i	$-0.279939\pi$
$137$	14.9252	1.27514	0.637571	+	0.770392i	$0.279939\pi$
$138$	−4.88839	−0.416127
$139$	−10.6367	−0.902195	−0.451097	−	0.892475i	$-0.648967\pi$
$139$	−10.6367	−0.902195	−0.451097	+	0.892475i	$0.648967\pi$
$140$	0	0
$141$	24.2085	2.03873
$142$	−0.102716	−0.00861971
$143$	−4.58164	−0.383136
$144$	19.2367	1.60306
$145$	0	0
$146$	−1.45125	−0.120107
$147$	−17.0978	−1.41020
$148$	5.13505	0.422099
$149$	−7.93007	−0.649656	−0.324828	−	0.945773i	$-0.605306\pi$
$149$	−7.93007	−0.649656	−0.324828	+	0.945773i	$0.605306\pi$
$150$	0	0
$151$	−23.5423	−1.91584	−0.957921	−	0.287033i	$-0.907331\pi$
$151$	−23.5423	−1.91584	−0.957921	+	0.287033i	$0.907331\pi$
$152$	−4.94169	−0.400824
$153$	−1.81279	−0.146556
$154$	0.221222	0.0178266
$155$	0	0
$156$	27.8956	2.23344
$157$	−6.13158	−0.489353	−0.244677	−	0.969605i	$-0.578682\pi$
$157$	−6.13158	−0.489353	−0.244677	+	0.969605i	$0.578682\pi$
$158$	1.03025	0.0819627
$159$	10.5319	0.835234
$160$	0	0
$161$	−7.37325	−0.581093
$162$	−0.652047	−0.0512297
$163$	7.33018	0.574144	0.287072	−	0.957909i	$-0.407318\pi$
$163$	7.33018	0.574144	0.287072	+	0.957909i	$0.407318\pi$
$164$	19.0554	1.48797
$165$	0	0
$166$	−2.18735	−0.169771
$167$	10.1434	0.784921	0.392461	−	0.919769i	$-0.371624\pi$
$167$	10.1434	0.784921	0.392461	+	0.919769i	$0.371624\pi$
$168$	−2.73244	−0.210813
$169$	11.9719	0.920915
$170$	0	0
$171$	−27.8304	−2.12824
$172$	−24.7984	−1.89086
$173$	−2.03837	−0.154975	−0.0774873	−	0.996993i	$-0.524690\pi$
$173$	−2.03837	−0.154975	−0.0774873	+	0.996993i	$0.524690\pi$
$174$	−2.82621	−0.214254
$175$	0	0
$176$	3.36372	0.253550
$177$	12.0305	0.904267
$178$	−3.36047	−0.251878
$179$	0.681587	0.0509442	0.0254721	−	0.999676i	$-0.491891\pi$
$179$	0.681587	0.0509442	0.0254721	+	0.999676i	$0.491891\pi$
$180$	0	0
$181$	3.58428	0.266417	0.133209	−	0.991088i	$-0.457472\pi$
$181$	3.58428	0.266417	0.133209	+	0.991088i	$0.457472\pi$
$182$	−1.20575	−0.0893763
$183$	7.76216	0.573795
$184$	6.71557	0.495078
$185$	0	0
$186$	−6.77144	−0.496506
$187$	−0.316985	−0.0231802
$188$	−16.3937	−1.19563
$189$	−6.58383	−0.478903
$190$	0	0
$191$	15.4485	1.11782	0.558908	−	0.829229i	$-0.311220\pi$
$191$	15.4485	1.11782	0.558908	+	0.829229i	$0.311220\pi$
$192$	−19.2183	−1.38696
$193$	4.87672	0.351034	0.175517	−	0.984476i	$-0.443840\pi$
$193$	4.87672	0.351034	0.175517	+	0.984476i	$0.443840\pi$
$194$	−2.60006	−0.186674
$195$	0	0
$196$	11.5784	0.827028
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	−1.13474	−0.0806423
$199$	−20.0659	−1.42244	−0.711218	−	0.702971i	$-0.751856\pi$
$199$	−20.0659	−1.42244	−0.711218	+	0.702971i	$0.751856\pi$
$200$	0	0
$201$	−27.9331	−1.97025
$202$	−2.72046	−0.191411
$203$	−4.26282	−0.299192
$204$	1.92998	0.135126
$205$	0	0
$206$	0.973361	0.0678173
$207$	37.8204	2.62870
$208$	−18.3337	−1.27121
$209$	−4.86641	−0.336617
$210$	0	0
$211$	−4.29538	−0.295706	−0.147853	−	0.989009i	$-0.547236\pi$
$211$	−4.29538	−0.295706	−0.147853	+	0.989009i	$0.547236\pi$
$212$	−7.13206	−0.489832
$213$	1.24938	0.0856059
$214$	−2.42410	−0.165708
$215$	0	0
$216$	5.99657	0.408015
$217$	−10.2135	−0.693337
$218$	−4.63148	−0.313683
$219$	17.6523	1.19283
$220$	0	0
$221$	1.72770	0.116218
$222$	1.78991	0.120131
$223$	−13.6428	−0.913589	−0.456795	−	0.889572i	$-0.651003\pi$
$223$	−13.6428	−0.913589	−0.456795	+	0.889572i	$0.651003\pi$
$224$	2.78863	0.186323
$225$	0	0
$226$	−3.53323	−0.235027
$227$	8.32757	0.552720	0.276360	−	0.961054i	$-0.410872\pi$
$227$	8.32757	0.552720	0.276360	+	0.961054i	$0.410872\pi$
$228$	29.6294	1.96226
$229$	−27.7853	−1.83611	−0.918053	−	0.396458i	$-0.870239\pi$
$229$	−27.7853	−1.83611	−0.918053	+	0.396458i	$0.870239\pi$
$230$	0	0
$231$	−2.69082	−0.177043
$232$	3.88259	0.254904
$233$	15.7592	1.03242	0.516211	−	0.856462i	$-0.327342\pi$
$233$	15.7592	1.03242	0.516211	+	0.856462i	$0.327342\pi$
$234$	6.18480	0.404313
$235$	0	0
$236$	−8.14689	−0.530318
$237$	−12.5314	−0.814005
$238$	−0.0834209	−0.00540737
$239$	−26.6777	−1.72564	−0.862818	−	0.505515i	$-0.831303\pi$
$239$	−26.6777	−1.72564	−0.862818	+	0.505515i	$0.831303\pi$
$240$	0	0
$241$	7.59582	0.489290	0.244645	−	0.969613i	$-0.421329\pi$
$241$	7.59582	0.489290	0.244645	+	0.969613i	$0.421329\pi$
$242$	2.39807	0.154154
$243$	−11.3913	−0.730752
$244$	−5.25643	−0.336509
$245$	0	0
$246$	6.64206	0.423482
$247$	26.5240	1.68768
$248$	9.30247	0.590707
$249$	26.6057	1.68607
$250$	0	0
$251$	−10.2993	−0.650088	−0.325044	−	0.945699i	$-0.605379\pi$
$251$	−10.2993	−0.650088	−0.325044	+	0.945699i	$0.605379\pi$
$252$	10.4209	0.656452
$253$	6.61327	0.415773
$254$	1.23204	0.0773053
$255$	0	0
$256$	11.7265	0.732905
$257$	16.8749	1.05263	0.526315	−	0.850290i	$-0.323573\pi$
$257$	16.8749	1.05263	0.526315	+	0.850290i	$0.323573\pi$
$258$	−8.64390	−0.538146
$259$	2.69975	0.167754
$260$	0	0
$261$	21.8658	1.35346
$262$	0.416289	0.0257184
$263$	−8.57005	−0.528452	−0.264226	−	0.964461i	$-0.585116\pi$
$263$	−8.57005	−0.528452	−0.264226	+	0.964461i	$0.585116\pi$
$264$	2.45080	0.150837
$265$	0	0
$266$	−1.28069	−0.0785244
$267$	40.8749	2.50151
$268$	18.9159	1.15547
$269$	26.3927	1.60919	0.804596	−	0.593822i	$-0.202382\pi$
$269$	26.3927	1.60919	0.804596	+	0.593822i	$0.202382\pi$
$270$	0	0
$271$	27.5722	1.67489	0.837447	−	0.546518i	$-0.184047\pi$
$271$	27.5722	1.67489	0.837447	+	0.546518i	$0.184047\pi$
$272$	−1.26843	−0.0769099
$273$	14.6661	0.887633
$274$	−3.52300	−0.212832
$275$	0	0
$276$	−40.2653	−2.42369
$277$	−9.89475	−0.594518	−0.297259	−	0.954797i	$-0.596072\pi$
$277$	−9.89475	−0.594518	−0.297259	+	0.954797i	$0.596072\pi$
$278$	2.51074	0.150584
$279$	52.3892	3.13646
$280$	0	0
$281$	27.3790	1.63330	0.816648	−	0.577135i	$-0.195830\pi$
$281$	27.3790	1.63330	0.816648	+	0.577135i	$0.195830\pi$
$282$	−5.71429	−0.340281
$283$	7.11007	0.422650	0.211325	−	0.977416i	$-0.432222\pi$
$283$	7.11007	0.422650	0.211325	+	0.977416i	$0.432222\pi$
$284$	−0.846062	−0.0502045
$285$	0	0
$286$	1.08147	0.0639488
$287$	10.0183	0.591364
$288$	−14.3040	−0.842874
$289$	−16.8805	−0.992969
$290$	0	0
$291$	31.6257	1.85393
$292$	−11.9539	−0.699548
$293$	−15.7053	−0.917512	−0.458756	−	0.888562i	$-0.651705\pi$
$293$	−15.7053	−0.917512	−0.458756	+	0.888562i	$0.651705\pi$
$294$	4.03584	0.235375
$295$	0	0
$296$	−2.45894	−0.142923
$297$	5.90522	0.342656
$298$	1.87185	0.108433
$299$	−36.0451	−2.08454
$300$	0	0
$301$	−13.0377	−0.751483
$302$	5.55702	0.319771
$303$	33.0902	1.90098
$304$	−19.4732	−1.11686
$305$	0	0
$306$	0.427900	0.0244614
$307$	8.97401	0.512174	0.256087	−	0.966654i	$-0.417567\pi$
$307$	8.97401	0.512174	0.256087	+	0.966654i	$0.417567\pi$
$308$	1.82219	0.103829
$309$	−11.8394	−0.673521
$310$	0	0
$311$	−9.76960	−0.553983	−0.276992	−	0.960872i	$-0.589337\pi$
$311$	−9.76960	−0.553983	−0.276992	+	0.960872i	$0.589337\pi$
$312$	−13.3579	−0.756243
$313$	0.449986	0.0254347	0.0127174	−	0.999919i	$-0.495952\pi$
$313$	0.449986	0.0254347	0.0127174	+	0.999919i	$0.495952\pi$
$314$	1.44733	0.0816773
$315$	0	0
$316$	8.48613	0.477382
$317$	27.2882	1.53266	0.766330	−	0.642447i	$-0.222081\pi$
$317$	27.2882	1.53266	0.766330	+	0.642447i	$0.222081\pi$
$318$	−2.48600	−0.139408
$319$	3.82344	0.214072
$320$	0	0
$321$	29.4854	1.64571
$322$	1.74042	0.0969896
$323$	1.83508	0.102107
$324$	−5.37087	−0.298381
$325$	0	0
$326$	−1.73025	−0.0958297
$327$	56.3348	3.11532
$328$	−9.12473	−0.503829
$329$	−8.61897	−0.475179
$330$	0	0
$331$	4.69448	0.258032	0.129016	−	0.991643i	$-0.458818\pi$
$331$	4.69448	0.258032	0.129016	+	0.991643i	$0.458818\pi$
$332$	−18.0170	−0.988812
$333$	−13.8481	−0.758873
$334$	−2.39430	−0.131010
$335$	0	0
$336$	−10.7675	−0.587413
$337$	1.18729	0.0646760	0.0323380	−	0.999477i	$-0.489705\pi$
$337$	1.18729	0.0646760	0.0323380	+	0.999477i	$0.489705\pi$
$338$	−2.82590	−0.153709
$339$	42.9763	2.33415
$340$	0	0
$341$	9.16076	0.496083
$342$	6.56921	0.355222
$343$	13.2428	0.715042
$344$	11.8748	0.640247
$345$	0	0
$346$	0.481147	0.0258666
$347$	−2.21480	−0.118896	−0.0594482	−	0.998231i	$-0.518934\pi$
$347$	−2.21480	−0.118896	−0.0594482	+	0.998231i	$0.518934\pi$
$348$	−23.2793	−1.24790
$349$	22.6828	1.21418	0.607092	−	0.794632i	$-0.292336\pi$
$349$	22.6828	1.21418	0.607092	+	0.794632i	$0.292336\pi$
$350$	0	0
$351$	−32.1860	−1.71796
$352$	−2.50120	−0.133315
$353$	12.2691	0.653016	0.326508	−	0.945194i	$-0.394128\pi$
$353$	12.2691	0.653016	0.326508	+	0.945194i	$0.394128\pi$
$354$	−2.83973	−0.150930
$355$	0	0
$356$	−27.6800	−1.46704
$357$	1.01469	0.0537029
$358$	−0.160885	−0.00850304
$359$	−18.9950	−1.00252	−0.501258	−	0.865298i	$-0.667129\pi$
$359$	−18.9950	−1.00252	−0.501258	+	0.865298i	$0.667129\pi$
$360$	0	0
$361$	9.17256	0.482767
$362$	−0.846050	−0.0444674
$363$	−29.1688	−1.53097
$364$	−9.93169	−0.520562
$365$	0	0
$366$	−1.83222	−0.0957715
$367$	−35.4991	−1.85304	−0.926519	−	0.376249i	$-0.877214\pi$
$367$	−35.4991	−1.85304	−0.926519	+	0.376249i	$0.877214\pi$
$368$	26.4633	1.37950
$369$	−51.3882	−2.67516
$370$	0	0
$371$	−3.74968	−0.194673
$372$	−55.7759	−2.89184
$373$	−22.1497	−1.14687	−0.573434	−	0.819252i	$-0.694389\pi$
$373$	−22.1497	−1.14687	−0.573434	+	0.819252i	$0.694389\pi$
$374$	0.0748225	0.00386898
$375$	0	0
$376$	7.85018	0.404842
$377$	−20.8394	−1.07328
$378$	1.55408	0.0799332
$379$	−0.0471576	−0.00242232	−0.00121116	−	0.999999i	$-0.500386\pi$
$379$	−0.0471576	−0.00242232	−0.00121116	+	0.999999i	$0.500386\pi$
$380$	0	0
$381$	−14.9859	−0.767750
$382$	−3.64654	−0.186573
$383$	27.8084	1.42094	0.710470	−	0.703727i	$-0.248482\pi$
$383$	27.8084	1.42094	0.710470	+	0.703727i	$0.248482\pi$
$384$	20.2015	1.03090
$385$	0	0
$386$	−1.15112	−0.0585907
$387$	66.8760	3.39950
$388$	−21.4165	−1.08726
$389$	−5.91034	−0.299666	−0.149833	−	0.988711i	$-0.547874\pi$
$389$	−5.91034	−0.299666	−0.149833	+	0.988711i	$0.547874\pi$
$390$	0	0
$391$	−2.49381	−0.126117
$392$	−5.54436	−0.280032
$393$	−5.06351	−0.255420
$394$	−0.236045	−0.0118918
$395$	0	0
$396$	−9.34675	−0.469692
$397$	10.7928	0.541673	0.270836	−	0.962625i	$-0.412700\pi$
$397$	10.7928	0.541673	0.270836	+	0.962625i	$0.412700\pi$
$398$	4.73646	0.237417
$399$	15.5777	0.779858
$400$	0	0
$401$	−25.3129	−1.26407	−0.632034	−	0.774941i	$-0.717780\pi$
$401$	−25.3129	−1.26407	−0.632034	+	0.774941i	$0.717780\pi$
$402$	6.59345	0.328851
$403$	−49.9300	−2.48719
$404$	−22.4082	−1.11485
$405$	0	0
$406$	1.00622	0.0499377
$407$	−2.42148	−0.120028
$408$	−0.924178	−0.0457536
$409$	11.0910	0.548415	0.274207	−	0.961671i	$-0.411585\pi$
$409$	11.0910	0.548415	0.274207	+	0.961671i	$0.411585\pi$
$410$	0	0
$411$	42.8519	2.11373
$412$	8.01750	0.394994
$413$	−4.28322	−0.210764
$414$	−8.92731	−0.438753
$415$	0	0
$416$	13.6326	0.668394
$417$	−30.5393	−1.49551
$418$	1.14869	0.0561843
$419$	−29.8386	−1.45771	−0.728856	−	0.684667i	$-0.759947\pi$
$419$	−29.8386	−1.45771	−0.728856	+	0.684667i	$0.759947\pi$
$420$	0	0
$421$	−31.3685	−1.52881	−0.764403	−	0.644739i	$-0.776966\pi$
$421$	−31.3685	−1.52881	−0.764403	+	0.644739i	$0.776966\pi$
$422$	1.01390	0.0493560
$423$	44.2103	2.14958
$424$	3.41521	0.165857
$425$	0	0
$426$	−0.294909	−0.0142884
$427$	−2.76356	−0.133738
$428$	−19.9671	−0.965147
$429$	−13.1544	−0.635102
$430$	0	0
$431$	−28.4126	−1.36859	−0.684293	−	0.729207i	$-0.739889\pi$
$431$	−28.4126	−1.36859	−0.684293	+	0.729207i	$0.739889\pi$
$432$	23.6301	1.13690
$433$	−29.6797	−1.42631	−0.713157	−	0.701005i	$-0.752735\pi$
$433$	−29.6797	−1.42631	−0.713157	+	0.701005i	$0.752735\pi$
$434$	2.41084	0.115724
$435$	0	0
$436$	−38.1492	−1.82701
$437$	−38.2855	−1.83144
$438$	−4.16672	−0.199094
$439$	19.0177	0.907664	0.453832	−	0.891087i	$-0.350057\pi$
$439$	19.0177	0.907664	0.453832	+	0.891087i	$0.350057\pi$
$440$	0	0
$441$	−31.2244	−1.48688
$442$	−0.407814	−0.0193977
$443$	12.5365	0.595628	0.297814	−	0.954624i	$-0.403742\pi$
$443$	12.5365	0.595628	0.297814	+	0.954624i	$0.403742\pi$
$444$	14.7433	0.699688
$445$	0	0
$446$	3.22031	0.152486
$447$	−22.7681	−1.07690
$448$	6.84230	0.323268
$449$	17.9879	0.848900	0.424450	−	0.905451i	$-0.360467\pi$
$449$	17.9879	0.848900	0.424450	+	0.905451i	$0.360467\pi$
$450$	0	0
$451$	−8.98573	−0.423122
$452$	−29.1030	−1.36889
$453$	−67.5926	−3.17578
$454$	−1.96568	−0.0922539
$455$	0	0
$456$	−14.1882	−0.664422
$457$	27.3085	1.27744	0.638718	−	0.769441i	$-0.279465\pi$
$457$	27.3085	1.27744	0.638718	+	0.769441i	$0.279465\pi$
$458$	6.55858	0.306462
$459$	−2.22681	−0.103939
$460$	0	0
$461$	−8.85519	−0.412427	−0.206214	−	0.978507i	$-0.566114\pi$
$461$	−8.85519	−0.412427	−0.206214	+	0.978507i	$0.566114\pi$
$462$	0.635154	0.0295500
$463$	17.7027	0.822714	0.411357	−	0.911474i	$-0.365055\pi$
$463$	17.7027	0.822714	0.411357	+	0.911474i	$0.365055\pi$
$464$	15.2997	0.710271
$465$	0	0
$466$	−3.71988	−0.172320
$467$	12.5779	0.582035	0.291018	−	0.956718i	$-0.406006\pi$
$467$	12.5779	0.582035	0.291018	+	0.956718i	$0.406006\pi$
$468$	50.9437	2.35488
$469$	9.94502	0.459219
$470$	0	0
$471$	−17.6045	−0.811171
$472$	3.90117	0.179566
$473$	11.6939	0.537687
$474$	2.95798	0.135865
$475$	0	0
$476$	−0.687132	−0.0314946
$477$	19.2336	0.880648
$478$	6.29713	0.288024
$479$	21.5408	0.984224	0.492112	−	0.870532i	$-0.336225\pi$
$479$	21.5408	0.984224	0.492112	+	0.870532i	$0.336225\pi$
$480$	0	0
$481$	13.1981	0.601781
$482$	−1.79295	−0.0816668
$483$	−21.1695	−0.963243
$484$	19.7527	0.897852
$485$	0	0
$486$	2.68886	0.121969
$487$	−4.30185	−0.194935	−0.0974677	−	0.995239i	$-0.531074\pi$
$487$	−4.30185	−0.194935	−0.0974677	+	0.995239i	$0.531074\pi$
$488$	2.51706	0.113942
$489$	21.0458	0.951724
$490$	0	0
$491$	−7.55139	−0.340789	−0.170395	−	0.985376i	$-0.554504\pi$
$491$	−7.55139	−0.340789	−0.170395	+	0.985376i	$0.554504\pi$
$492$	54.7102	2.46653
$493$	−1.44179	−0.0649349
$494$	−6.26085	−0.281689
$495$	0	0
$496$	36.6573	1.64596
$497$	−0.444816	−0.0199527
$498$	−6.28013	−0.281419
$499$	37.8523	1.69450	0.847250	−	0.531194i	$-0.178256\pi$
$499$	37.8523	1.69450	0.847250	+	0.531194i	$0.178256\pi$
$500$	0	0
$501$	29.1229	1.30112
$502$	2.43110	0.108505
$503$	−20.1705	−0.899360	−0.449680	−	0.893190i	$-0.648462\pi$
$503$	−20.1705	−0.899360	−0.449680	+	0.893190i	$0.648462\pi$
$504$	−4.99007	−0.222275
$505$	0	0
$506$	−1.56103	−0.0693961
$507$	34.3727	1.52654
$508$	10.1483	0.450256
$509$	19.5293	0.865623	0.432811	−	0.901484i	$-0.357522\pi$
$509$	19.5293	0.865623	0.432811	+	0.901484i	$0.357522\pi$
$510$	0	0
$511$	−6.28474	−0.278021
$512$	−16.8402	−0.744239
$513$	−34.1864	−1.50937
$514$	−3.98324	−0.175693
$515$	0	0
$516$	−71.1991	−3.13437
$517$	7.73060	0.339991
$518$	−0.637262	−0.0279997
$519$	−5.85241	−0.256892
$520$	0	0
$521$	19.9256	0.872958	0.436479	−	0.899714i	$-0.356225\pi$
$521$	19.9256	0.872958	0.436479	+	0.899714i	$0.356225\pi$
$522$	−5.16130	−0.225904
$523$	−0.645222	−0.0282136	−0.0141068	−	0.999900i	$-0.504490\pi$
$523$	−0.645222	−0.0282136	−0.0141068	+	0.999900i	$0.504490\pi$
$524$	3.42894	0.149794
$525$	0	0
$526$	2.02291	0.0882033
$527$	−3.45445	−0.150478
$528$	9.65763	0.420295
$529$	29.0285	1.26211
$530$	0	0
$531$	21.9704	0.953435
$532$	−10.5490	−0.457357
$533$	48.9760	2.12139
$534$	−9.64831	−0.417523
$535$	0	0
$536$	−9.05795	−0.391244
$537$	1.95692	0.0844472
$538$	−6.22986	−0.268588
$539$	−5.45990	−0.235174
$540$	0	0
$541$	0.776828	0.0333984	0.0166992	−	0.999861i	$-0.494684\pi$
$541$	0.776828	0.0333984	0.0166992	+	0.999861i	$0.494684\pi$
$542$	−6.50828	−0.279555
$543$	10.2909	0.441624
$544$	0.943182	0.0404386
$545$	0	0
$546$	−3.46186	−0.148154
$547$	−8.27780	−0.353933	−0.176967	−	0.984217i	$-0.556628\pi$
$547$	−8.27780	−0.353933	−0.176967	+	0.984217i	$0.556628\pi$
$548$	−29.0187	−1.23962
$549$	14.1755	0.604994
$550$	0	0
$551$	−22.1346	−0.942967
$552$	19.2812	0.820662
$553$	4.46158	0.189726
$554$	2.33560	0.0992303
$555$	0	0
$556$	20.6808	0.877061
$557$	11.5911	0.491130	0.245565	−	0.969380i	$-0.421026\pi$
$557$	11.5911	0.491130	0.245565	+	0.969380i	$0.421026\pi$
$558$	−12.3662	−0.523503
$559$	−63.7368	−2.69578
$560$	0	0
$561$	−0.910100	−0.0384244
$562$	−6.46268	−0.272612
$563$	−13.9955	−0.589840	−0.294920	−	0.955522i	$-0.595293\pi$
$563$	−13.9955	−0.589840	−0.294920	+	0.955522i	$0.595293\pi$
$564$	−47.0682	−1.98193
$565$	0	0
$566$	−1.67829	−0.0705440
$567$	−2.82373	−0.118585
$568$	0.405140	0.0169993
$569$	−40.7482	−1.70825	−0.854127	−	0.520065i	$-0.825908\pi$
$569$	−40.7482	−1.70825	−0.854127	+	0.520065i	$0.825908\pi$
$570$	0	0
$571$	−17.0553	−0.713742	−0.356871	−	0.934154i	$-0.616156\pi$
$571$	−17.0553	−0.713742	−0.356871	+	0.934154i	$0.616156\pi$
$572$	8.90801	0.372463
$573$	44.3545	1.85294
$574$	−2.36478	−0.0987039
$575$	0	0
$576$	−35.0970	−1.46237
$577$	−23.5705	−0.981255	−0.490627	−	0.871369i	$-0.663232\pi$
$577$	−23.5705	−0.981255	−0.490627	+	0.871369i	$0.663232\pi$
$578$	3.98455	0.165735
$579$	14.0016	0.581888
$580$	0	0
$581$	−9.47243	−0.392983
$582$	−7.46509	−0.309438
$583$	3.36319	0.139289
$584$	5.72416	0.236867
$585$	0	0
$586$	3.70715	0.153141
$587$	−0.0140745	−0.000580917 0	−0.000290459	−	1.00000i	$-0.500092\pi$
$587$	−0.0140745	−0.000580917 0	−0.000290459		1.00000i	$0.500092\pi$
$588$	33.2429	1.37091
$589$	−53.0334	−2.18520
$590$	0	0
$591$	2.87112	0.118102
$592$	−9.68969	−0.398244
$593$	7.67621	0.315224	0.157612	−	0.987501i	$-0.449620\pi$
$593$	7.67621	0.315224	0.157612	+	0.987501i	$0.449620\pi$
$594$	−1.39390	−0.0571923
$595$	0	0
$596$	15.4183	0.631558
$597$	−57.6117	−2.35789
$598$	8.50826	0.347928
$599$	−9.42711	−0.385181	−0.192591	−	0.981279i	$-0.561689\pi$
$599$	−9.42711	−0.385181	−0.192591	+	0.981279i	$0.561689\pi$
$600$	0	0
$601$	3.49753	0.142667	0.0713337	−	0.997453i	$-0.477274\pi$
$601$	3.49753	0.142667	0.0713337	+	0.997453i	$0.477274\pi$
$602$	3.07749	0.125429
$603$	−51.0121	−2.07737
$604$	45.7728	1.86247
$605$	0	0
$606$	−7.81076	−0.317290
$607$	−11.1635	−0.453112	−0.226556	−	0.973998i	$-0.572747\pi$
$607$	−11.1635	−0.453112	−0.226556	+	0.973998i	$0.572747\pi$
$608$	14.4799	0.587239
$609$	−12.2391	−0.495952
$610$	0	0
$611$	−42.1350	−1.70460
$612$	3.52458	0.142473
$613$	−44.5755	−1.80039	−0.900193	−	0.435490i	$-0.856575\pi$
$613$	−44.5755	−1.80039	−0.900193	+	0.435490i	$0.856575\pi$
$614$	−2.11827	−0.0854864
$615$	0	0
$616$	−0.872561	−0.0351565
$617$	−25.3603	−1.02097	−0.510484	−	0.859887i	$-0.670534\pi$
$617$	−25.3603	−1.02097	−0.510484	+	0.859887i	$0.670534\pi$
$618$	2.79463	0.112417
$619$	17.2831	0.694667	0.347333	−	0.937742i	$-0.387087\pi$
$619$	17.2831	0.694667	0.347333	+	0.937742i	$0.387087\pi$
$620$	0	0
$621$	46.4581	1.86430
$622$	2.30606	0.0924647
$623$	−14.5527	−0.583043
$624$	−52.6382	−2.10721
$625$	0	0
$626$	−0.106217	−0.00424528
$627$	−13.9720	−0.557989
$628$	11.9215	0.475720
$629$	0.913120	0.0364085
$630$	0	0
$631$	13.6827	0.544700	0.272350	−	0.962198i	$-0.412199\pi$
$631$	13.6827	0.544700	0.272350	+	0.962198i	$0.412199\pi$
$632$	−4.06362	−0.161642
$633$	−12.3325	−0.490175
$634$	−6.44124	−0.255814
$635$	0	0
$636$	−20.4770	−0.811965
$637$	29.7587	1.17908
$638$	−0.902504	−0.0357305
$639$	2.28165	0.0902606
$640$	0	0
$641$	4.80495	0.189784	0.0948921	−	0.995488i	$-0.469749\pi$
$641$	4.80495	0.189784	0.0948921	+	0.995488i	$0.469749\pi$
$642$	−6.95987	−0.274684
$643$	11.5218	0.454376	0.227188	−	0.973851i	$-0.427047\pi$
$643$	11.5218	0.454376	0.227188	+	0.973851i	$0.427047\pi$
$644$	14.3357	0.564905
$645$	0	0
$646$	−0.433162	−0.0170425
$647$	19.8337	0.779742	0.389871	−	0.920870i	$-0.372520\pi$
$647$	19.8337	0.779742	0.389871	+	0.920870i	$0.372520\pi$
$648$	2.57186	0.101032
$649$	3.84174	0.150802
$650$	0	0
$651$	−29.3241	−1.14930
$652$	−14.2519	−0.558149
$653$	−37.8833	−1.48249	−0.741244	−	0.671235i	$-0.765764\pi$
$653$	−37.8833	−1.48249	−0.741244	+	0.671235i	$0.765764\pi$
$654$	−13.2975	−0.519974
$655$	0	0
$656$	−35.9569	−1.40388
$657$	32.2370	1.25769
$658$	2.03446	0.0793116
$659$	−18.2539	−0.711072	−0.355536	−	0.934663i	$-0.615702\pi$
$659$	−18.2539	−0.711072	−0.355536	+	0.934663i	$0.615702\pi$
$660$	0	0
$661$	39.5364	1.53779	0.768895	−	0.639375i	$-0.220807\pi$
$661$	39.5364	1.53779	0.768895	+	0.639375i	$0.220807\pi$
$662$	−1.10811	−0.0430678
$663$	4.96043	0.192647
$664$	8.62751	0.334813
$665$	0	0
$666$	3.26878	0.126663
$667$	30.0801	1.16471
$668$	−19.7217	−0.763055
$669$	−39.1701	−1.51440
$670$	0	0
$671$	2.47872	0.0956898
$672$	8.00649	0.308857
$673$	−14.7965	−0.570364	−0.285182	−	0.958473i	$-0.592054\pi$
$673$	−14.7965	−0.570364	−0.285182	+	0.958473i	$0.592054\pi$
$674$	−0.280254	−0.0107950
$675$	0	0
$676$	−23.2767	−0.895259
$677$	−37.1563	−1.42803	−0.714017	−	0.700128i	$-0.753126\pi$
$677$	−37.1563	−1.42803	−0.714017	+	0.700128i	$0.753126\pi$
$678$	−10.1443	−0.389590
$679$	−11.2597	−0.432109
$680$	0	0
$681$	23.9094	0.916212
$682$	−2.16235	−0.0828006
$683$	29.5420	1.13039	0.565196	−	0.824957i	$-0.308801\pi$
$683$	29.5420	1.13039	0.565196	+	0.824957i	$0.308801\pi$
$684$	54.1101	2.06895
$685$	0	0
$686$	−3.12589	−0.119347
$687$	−79.7749	−3.04360
$688$	46.7938	1.78400
$689$	−18.3308	−0.698348
$690$	0	0
$691$	24.4783	0.931198	0.465599	−	0.884996i	$-0.345839\pi$
$691$	24.4783	0.931198	0.465599	+	0.884996i	$0.345839\pi$
$692$	3.96317	0.150657
$693$	−4.91405	−0.186669
$694$	0.522791	0.0198449
$695$	0	0
$696$	11.1474	0.422540
$697$	3.38844	0.128346
$698$	−5.35416	−0.202658
$699$	45.2466	1.71138
$700$	0	0
$701$	21.2361	0.802077	0.401039	−	0.916061i	$-0.368649\pi$
$701$	21.2361	0.802077	0.401039	+	0.916061i	$0.368649\pi$
$702$	7.59732	0.286742
$703$	14.0184	0.528714
$704$	−6.13704	−0.231299
$705$	0	0
$706$	−2.89605	−0.108994
$707$	−11.7811	−0.443074
$708$	−23.3907	−0.879076
$709$	−10.4005	−0.390599	−0.195300	−	0.980744i	$-0.562568\pi$
$709$	−10.4005	−0.390599	−0.195300	+	0.980744i	$0.562568\pi$
$710$	0	0
$711$	−22.8853	−0.858265
$712$	13.2547	0.496739
$713$	72.0703	2.69905
$714$	−0.239511	−0.00896348
$715$	0	0
$716$	−1.32520	−0.0495250
$717$	−76.5948	−2.86048
$718$	4.48366	0.167329
$719$	−35.5394	−1.32540	−0.662698	−	0.748887i	$-0.730589\pi$
$719$	−35.5394	−1.32540	−0.662698	+	0.748887i	$0.730589\pi$
$720$	0	0
$721$	4.21520	0.156982
$722$	−2.16514	−0.0805780
$723$	21.8085	0.811067
$724$	−6.96885	−0.258995
$725$	0	0
$726$	6.88514	0.255532
$727$	20.0589	0.743944	0.371972	−	0.928244i	$-0.378682\pi$
$727$	20.0589	0.743944	0.371972	+	0.928244i	$0.378682\pi$
$728$	4.75583	0.176263
$729$	−40.9929	−1.51826
$730$	0	0
$731$	−4.40968	−0.163098
$732$	−15.0918	−0.557810
$733$	7.54137	0.278547	0.139273	−	0.990254i	$-0.455523\pi$
$733$	7.54137	0.278547	0.139273	+	0.990254i	$0.455523\pi$
$734$	8.37937	0.309288
$735$	0	0
$736$	−19.6777	−0.725329
$737$	−8.91997	−0.328571
$738$	12.1299	0.446508
$739$	−2.97975	−0.109612	−0.0548059	−	0.998497i	$-0.517454\pi$
$739$	−2.97975	−0.109612	−0.0548059	+	0.998497i	$0.517454\pi$
$740$	0	0
$741$	76.1535	2.79757
$742$	0.885091	0.0324927
$743$	−50.0156	−1.83490	−0.917448	−	0.397856i	$-0.869754\pi$
$743$	−50.0156	−1.83490	−0.917448	+	0.397856i	$0.869754\pi$
$744$	26.7085	0.979180
$745$	0	0
$746$	5.22832	0.191422
$747$	48.5880	1.77774
$748$	0.616308	0.0225344
$749$	−10.4977	−0.383577
$750$	0	0
$751$	25.8694	0.943989	0.471995	−	0.881601i	$-0.343534\pi$
$751$	25.8694	0.943989	0.471995	+	0.881601i	$0.343534\pi$
$752$	30.9344	1.12806
$753$	−29.5706	−1.07761
$754$	4.91903	0.179140
$755$	0	0
$756$	12.8008	0.465562
$757$	52.5795	1.91103	0.955517	−	0.294936i	$-0.0952983\pi$
$757$	52.5795	1.91103	0.955517	+	0.294936i	$0.0952983\pi$
$758$	0.0111313	0.000404307 0
$759$	18.9875	0.689201
$760$	0	0
$761$	−47.2393	−1.71242	−0.856211	−	0.516626i	$-0.827188\pi$
$761$	−47.2393	−1.71242	−0.856211	+	0.516626i	$0.827188\pi$
$762$	3.53734	0.128144
$763$	−20.0569	−0.726109
$764$	−30.0363	−1.08668
$765$	0	0
$766$	−6.56402	−0.237168
$767$	−20.9391	−0.756068
$768$	33.6681	1.21489
$769$	19.1183	0.689425	0.344712	−	0.938708i	$-0.387976\pi$
$769$	19.1183	0.689425	0.344712	+	0.938708i	$0.387976\pi$
$770$	0	0
$771$	48.4499	1.74488
$772$	−9.48172	−0.341255
$773$	35.0337	1.26007	0.630037	−	0.776565i	$-0.283040\pi$
$773$	35.0337	1.26007	0.630037	+	0.776565i	$0.283040\pi$
$774$	−15.7857	−0.567406
$775$	0	0
$776$	10.2554	0.368147
$777$	7.75130	0.278076
$778$	1.39510	0.0500169
$779$	52.0201	1.86381
$780$	0	0
$781$	0.398968	0.0142762
$782$	0.588650	0.0210501
$783$	26.8596	0.959884
$784$	−21.8481	−0.780288
$785$	0	0
$786$	1.19521	0.0426319
$787$	−31.2968	−1.11561	−0.557806	−	0.829971i	$-0.688357\pi$
$787$	−31.2968	−1.11561	−0.557806	+	0.829971i	$0.688357\pi$
$788$	−1.94428	−0.0692622
$789$	−24.6056	−0.875983
$790$	0	0
$791$	−15.3009	−0.544036
$792$	4.47573	0.159038
$793$	−13.5101	−0.479756
$794$	−2.54757	−0.0904099
$795$	0	0
$796$	39.0139	1.38281
$797$	−31.3216	−1.10947	−0.554734	−	0.832027i	$-0.687180\pi$
$797$	−31.3216	−1.10947	−0.554734	+	0.832027i	$0.687180\pi$
$798$	−3.67702	−0.130165
$799$	−2.91514	−0.103130
$800$	0	0
$801$	74.6469	2.63752
$802$	5.97498	0.210984
$803$	5.63696	0.198924
$804$	54.3098	1.91536
$805$	0	0
$806$	11.7857	0.415134
$807$	75.7766	2.66746
$808$	10.7303	0.377489
$809$	4.76429	0.167504	0.0837518	−	0.996487i	$-0.473310\pi$
$809$	4.76429	0.167504	0.0837518	+	0.996487i	$0.473310\pi$
$810$	0	0
$811$	−0.713408	−0.0250511	−0.0125256	−	0.999922i	$-0.503987\pi$
$811$	−0.713408	−0.0250511	−0.0125256	+	0.999922i	$0.503987\pi$
$812$	8.28813	0.290856
$813$	79.1631	2.77637
$814$	0.571578	0.0200338
$815$	0	0
$816$	−3.64181	−0.127489
$817$	−67.6983	−2.36846
$818$	−2.61797	−0.0915352
$819$	26.7836	0.935896
$820$	0	0
$821$	1.05247	0.0367315	0.0183658	−	0.999831i	$-0.494154\pi$
$821$	1.05247	0.0367315	0.0183658	+	0.999831i	$0.494154\pi$
$822$	−10.1150	−0.352800
$823$	−36.9290	−1.28726	−0.643631	−	0.765336i	$-0.722573\pi$
$823$	−36.9290	−1.28726	−0.643631	+	0.765336i	$0.722573\pi$
$824$	−3.83921	−0.133745
$825$	0	0
$826$	1.01103	0.0351783
$827$	10.4375	0.362948	0.181474	−	0.983396i	$-0.441913\pi$
$827$	10.4375	0.362948	0.181474	+	0.983396i	$0.441913\pi$
$828$	−73.5336	−2.55547
$829$	−17.5141	−0.608291	−0.304145	−	0.952626i	$-0.598371\pi$
$829$	−17.5141	−0.608291	−0.304145	+	0.952626i	$0.598371\pi$
$830$	0	0
$831$	−28.4090	−0.985497
$832$	33.4495	1.15965
$833$	2.05888	0.0713360
$834$	7.20863	0.249614
$835$	0	0
$836$	9.46168	0.327239
$837$	64.3541	2.22440
$838$	7.04325	0.243305
$839$	23.0283	0.795026	0.397513	−	0.917597i	$-0.369873\pi$
$839$	23.0283	0.795026	0.397513	+	0.917597i	$0.369873\pi$
$840$	0	0
$841$	−11.6093	−0.400319
$842$	7.40436	0.255171
$843$	78.6084	2.70742
$844$	8.35144	0.287468
$845$	0	0
$846$	−10.4356	−0.358783
$847$	10.3850	0.356832
$848$	13.4580	0.462149
$849$	20.4138	0.700601
$850$	0	0
$851$	−19.0505	−0.653042
$852$	−2.42914	−0.0832211
$853$	−41.3312	−1.41515	−0.707577	−	0.706637i	$-0.750212\pi$
$853$	−41.3312	−1.41515	−0.707577	+	0.706637i	$0.750212\pi$
$854$	0.652325	0.0223221
$855$	0	0
$856$	9.56133	0.326799
$857$	14.1899	0.484718	0.242359	−	0.970187i	$-0.422079\pi$
$857$	14.1899	0.484718	0.242359	+	0.970187i	$0.422079\pi$
$858$	3.10503	0.106004
$859$	−15.4345	−0.526618	−0.263309	−	0.964712i	$-0.584814\pi$
$859$	−15.4345	−0.526618	−0.263309	+	0.964712i	$0.584814\pi$
$860$	0	0
$861$	28.7638	0.980269
$862$	6.70664	0.228429
$863$	3.73755	0.127228	0.0636138	−	0.997975i	$-0.479737\pi$
$863$	3.73755	0.127228	0.0636138	+	0.997975i	$0.479737\pi$
$864$	−17.5709	−0.597774
$865$	0	0
$866$	7.00573	0.238064
$867$	−48.4658	−1.64598
$868$	19.8579	0.674021
$869$	−4.00171	−0.135749
$870$	0	0
$871$	48.6176	1.64734
$872$	18.2679	0.618628
$873$	57.7558	1.95474
$874$	9.03708	0.305684
$875$	0	0
$876$	−34.3210	−1.15960
$877$	37.4630	1.26504	0.632518	−	0.774546i	$-0.282022\pi$
$877$	37.4630	1.26504	0.632518	+	0.774546i	$0.282022\pi$
$878$	−4.48902	−0.151497
$879$	−45.0917	−1.52091
$880$	0	0
$881$	−21.1306	−0.711907	−0.355953	−	0.934504i	$-0.615844\pi$
$881$	−21.1306	−0.711907	−0.355953	+	0.934504i	$0.615844\pi$
$882$	7.37036	0.248173
$883$	21.3153	0.717317	0.358658	−	0.933469i	$-0.383234\pi$
$883$	21.3153	0.717317	0.358658	+	0.933469i	$0.383234\pi$
$884$	−3.35914	−0.112980
$885$	0	0
$886$	−2.95918	−0.0994156
$887$	−28.5267	−0.957834	−0.478917	−	0.877860i	$-0.658971\pi$
$887$	−28.5267	−0.957834	−0.478917	+	0.877860i	$0.658971\pi$
$888$	−7.05990	−0.236915
$889$	5.33544	0.178945
$890$	0	0
$891$	2.53268	0.0848480
$892$	26.5255	0.888138
$893$	−44.7539	−1.49763
$894$	5.37430	0.179743
$895$	0	0
$896$	−7.19235	−0.240280
$897$	−103.490	−3.45542
$898$	−4.24594	−0.141689
$899$	41.6673	1.38968
$900$	0	0
$901$	−1.26823	−0.0422509
$902$	2.12103	0.0706227
$903$	−37.4329	−1.24569
$904$	13.9361	0.463507
$905$	0	0
$906$	15.9549	0.530065
$907$	7.24122	0.240441	0.120220	−	0.992747i	$-0.461640\pi$
$907$	7.24122	0.240441	0.120220	+	0.992747i	$0.461640\pi$
$908$	−16.1912	−0.537322
$909$	60.4302	2.00434
$910$	0	0
$911$	−30.4699	−1.00951	−0.504756	−	0.863262i	$-0.668418\pi$
$911$	−30.4699	−1.00951	−0.504756	+	0.863262i	$0.668418\pi$
$912$	−55.9098	−1.85136
$913$	8.49609	0.281179
$914$	−6.44602	−0.213215
$915$	0	0
$916$	54.0225	1.78495
$917$	1.80276	0.0595325
$918$	0.525627	0.0173483
$919$	−56.0316	−1.84831	−0.924157	−	0.382013i	$-0.875231\pi$
$919$	−56.0316	−1.84831	−0.924157	+	0.382013i	$0.875231\pi$
$920$	0	0
$921$	25.7654	0.849000
$922$	2.09022	0.0688378
$923$	−2.17454	−0.0715760
$924$	5.23171	0.172111
$925$	0	0
$926$	−4.17863	−0.137318
$927$	−21.6215	−0.710142
$928$	−11.3766	−0.373455
$929$	−17.8294	−0.584963	−0.292481	−	0.956271i	$-0.594481\pi$
$929$	−17.8294	−0.584963	−0.292481	+	0.956271i	$0.594481\pi$
$930$	0	0
$931$	31.6084	1.03592
$932$	−30.6404	−1.00366
$933$	−28.0497	−0.918305
$934$	−2.96894	−0.0971468
$935$	0	0
$936$	−24.3946	−0.797363
$937$	−10.3366	−0.337681	−0.168841	−	0.985643i	$-0.554002\pi$
$937$	−10.3366	−0.337681	−0.168841	+	0.985643i	$0.554002\pi$
$938$	−2.34747	−0.0766476
$939$	1.29196	0.0421616
$940$	0	0
$941$	4.39565	0.143294	0.0716470	−	0.997430i	$-0.477174\pi$
$941$	4.39565	0.143294	0.0716470	+	0.997430i	$0.477174\pi$
$942$	4.15544	0.135392
$943$	−70.6933	−2.30209
$944$	15.3729	0.500347
$945$	0	0
$946$	−2.76029	−0.0897447
$947$	13.1530	0.427415	0.213708	−	0.976898i	$-0.431446\pi$
$947$	13.1530	0.427415	0.213708	+	0.976898i	$0.431446\pi$
$948$	24.3647	0.791328
$949$	−30.7238	−0.997337
$950$	0	0
$951$	78.3477	2.54060
$952$	0.329036	0.0106641
$953$	−52.7707	−1.70941	−0.854706	−	0.519113i	$-0.826262\pi$
$953$	−52.7707	−1.70941	−0.854706	+	0.519113i	$0.826262\pi$
$954$	−4.54000	−0.146988
$955$	0	0
$956$	51.8690	1.67756
$957$	10.9776	0.354854
$958$	−5.08459	−0.164276
$959$	−15.2566	−0.492661
$960$	0	0
$961$	68.8325	2.22040
$962$	−3.11534	−0.100443
$963$	53.8470	1.73520
$964$	−14.7684	−0.475659
$965$	0	0
$966$	4.99694	0.160774
$967$	−40.7983	−1.31198	−0.655992	−	0.754768i	$-0.727750\pi$
$967$	−40.7983	−1.31198	−0.655992	+	0.754768i	$0.727750\pi$
$968$	−9.45867	−0.304013
$969$	5.26874	0.169256
$970$	0	0
$971$	59.3602	1.90496	0.952479	−	0.304603i	$-0.0985239\pi$
$971$	59.3602	1.90496	0.952479	+	0.304603i	$0.0985239\pi$
$972$	22.1479	0.710394
$973$	10.8729	0.348569
$974$	1.01543	0.0325364
$975$	0	0
$976$	9.91872	0.317491
$977$	21.8738	0.699806	0.349903	−	0.936786i	$-0.386215\pi$
$977$	21.8738	0.699806	0.349903	+	0.936786i	$0.386215\pi$
$978$	−4.96775	−0.158851
$979$	13.0527	0.417167
$980$	0	0
$981$	102.880	3.28471
$982$	1.78247	0.0568808
$983$	32.5706	1.03884	0.519421	−	0.854519i	$-0.326148\pi$
$983$	32.5706	1.03884	0.519421	+	0.854519i	$0.326148\pi$
$984$	−26.1982	−0.835167
$985$	0	0
$986$	0.340327	0.0108382
$987$	−24.7461	−0.787677
$988$	−51.5701	−1.64066
$989$	91.9994	2.92541
$990$	0	0
$991$	−0.493203	−0.0156671	−0.00783356	−	0.999969i	$-0.502494\pi$
$991$	−0.493203	−0.0156671	−0.00783356	+	0.999969i	$0.502494\pi$
$992$	−27.2577	−0.865433
$993$	13.4784	0.427724
$994$	0.104997	0.00333029
$995$	0	0
$996$	−51.7290	−1.63909
$997$	−19.0675	−0.603872	−0.301936	−	0.953328i	$-0.597633\pi$
$997$	−19.0675	−0.603872	−0.301936	+	0.953328i	$0.597633\pi$
$998$	−8.93483	−0.282827
$999$	−17.0109	−0.538200

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.k.1.5		14
5.4	even	2		985.2.a.f.1.10	✓	14
15.14	odd	2		8865.2.a.v.1.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
985.2.a.f.1.10	✓	14	5.4	even	2
4925.2.a.k.1.5		14	1.1	even	1	trivial
8865.2.a.v.1.5		14	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:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q-0.236045 q^{2} +2.87112 q^{3} -1.94428 q^{4} -0.677712 q^{6} -1.02221 q^{7} +0.931027 q^{8} +5.24331 q^{9} +0.916844 q^{11} -5.58226 q^{12} -4.99719 q^{13} +0.241286 q^{14} +3.66880 q^{16} -0.345734 q^{17} -1.23766 q^{18} -5.30778 q^{19} -2.93487 q^{21} -0.216416 q^{22} +7.21308 q^{23} +2.67309 q^{24} +1.17956 q^{26} +6.44081 q^{27} +1.98746 q^{28} +4.17022 q^{29} +9.99162 q^{31} -2.72806 q^{32} +2.63237 q^{33} +0.0816088 q^{34} -10.1945 q^{36} -2.64110 q^{37} +1.25287 q^{38} -14.3475 q^{39} -9.80072 q^{41} +0.692761 q^{42} +12.7545 q^{43} -1.78260 q^{44} -1.70261 q^{46} +8.43174 q^{47} +10.5336 q^{48} -5.95510 q^{49} -0.992644 q^{51} +9.71595 q^{52} +3.66822 q^{53} -1.52032 q^{54} -0.951701 q^{56} -15.2393 q^{57} -0.984359 q^{58} +4.19018 q^{59} +2.70353 q^{61} -2.35847 q^{62} -5.35974 q^{63} -6.69366 q^{64} -0.621356 q^{66} -9.72899 q^{67} +0.672205 q^{68} +20.7096 q^{69} +0.435154 q^{71} +4.88167 q^{72} +6.14822 q^{73} +0.623419 q^{74} +10.3198 q^{76} -0.937203 q^{77} +3.38665 q^{78} -4.36466 q^{79} +2.76239 q^{81} +2.31341 q^{82} +9.26666 q^{83} +5.70622 q^{84} -3.01064 q^{86} +11.9732 q^{87} +0.853607 q^{88} +14.2366 q^{89} +5.10815 q^{91} -14.0243 q^{92} +28.6871 q^{93} -1.99027 q^{94} -7.83257 q^{96} +11.0151 q^{97} +1.40567 q^{98} +4.80730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 6 q^{2} + 7 q^{3} + 10 q^{4} + 5 q^{6} + 4 q^{7} + 12 q^{8} + 13 q^{9} - 5 q^{11} + 10 q^{12} + 15 q^{13} - 15 q^{14} + 18 q^{16} + 18 q^{17} + 29 q^{18} + 20 q^{19} - 5 q^{21} - 12 q^{22} + 42 q^{23}+ \cdots + 32 q^{99}+O(q^{100}) \) 14 * q + 6 * q^2 + 7 * q^3 + 10 * q^4 + 5 * q^6 + 4 * q^7 + 12 * q^8 + 13 * q^9 - 5 * q^11 + 10 * q^12 + 15 * q^13 - 15 * q^14 + 18 * q^16 + 18 * q^17 + 29 * q^18 + 20 * q^19 - 5 * q^21 - 12 * q^22 + 42 * q^23 + 7 * q^24 - 13 * q^26 + 19 * q^27 + 3 * q^28 - 2 * q^29 + 40 * q^32 + 10 * q^33 + 13 * q^34 + q^36 + 15 * q^37 + 3 * q^38 - 19 * q^39 + 3 * q^41 + 9 * q^42 + 15 * q^43 - 29 * q^44 - 8 * q^46 + 40 * q^47 + 28 * q^48 + 12 * q^49 - 13 * q^51 + 17 * q^52 + 39 * q^53 + 28 * q^54 - 41 * q^56 + 17 * q^57 + 10 * q^58 + 19 * q^59 - 18 * q^61 - 26 * q^62 + 28 * q^63 + 36 * q^64 - 8 * q^66 + 22 * q^67 + 57 * q^68 + 12 * q^69 - 50 * q^71 + 37 * q^72 + 10 * q^73 - 3 * q^74 - 3 * q^76 + 36 * q^77 - 10 * q^78 + q^79 - 42 * q^81 + 40 * q^82 + 12 * q^83 + 35 * q^84 + 5 * q^86 + 35 * q^87 - 13 * q^88 + 7 * q^89 + 16 * q^91 + 4 * q^92 + 37 * q^93 + 27 * q^94 + 16 * q^96 - 2 * q^97 - 13 * q^98 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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