LMFDB - Embedded newform 475.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, 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("475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	475.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:	$3.79289409601$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.44504 q^{2} -2.24698 q^{3} +0.0881460 q^{4} +3.24698 q^{6} -1.35690 q^{7} +2.76271 q^{8} +2.04892 q^{9} +4.85086 q^{11} -0.198062 q^{12} -0.198062 q^{13} +1.96077 q^{14} -4.16852 q^{16} +1.13706 q^{17} -2.96077 q^{18} -1.00000 q^{19} +3.04892 q^{21} -7.00969 q^{22} -2.55496 q^{23} -6.20775 q^{24} +0.286208 q^{26} +2.13706 q^{27} -0.119605 q^{28} -10.2349 q^{29} +2.51573 q^{31} +0.498271 q^{32} -10.8998 q^{33} -1.64310 q^{34} +0.180604 q^{36} +0.137063 q^{37} +1.44504 q^{38} +0.445042 q^{39} -11.7506 q^{41} -4.40581 q^{42} +7.59179 q^{43} +0.427583 q^{44} +3.69202 q^{46} -2.69202 q^{47} +9.36658 q^{48} -5.15883 q^{49} -2.55496 q^{51} -0.0174584 q^{52} -12.8780 q^{53} -3.08815 q^{54} -3.74871 q^{56} +2.24698 q^{57} +14.7899 q^{58} +5.82371 q^{59} -7.58211 q^{61} -3.63533 q^{62} -2.78017 q^{63} +7.61702 q^{64} +15.7506 q^{66} -8.01507 q^{67} +0.100228 q^{68} +5.74094 q^{69} -8.82371 q^{71} +5.66056 q^{72} +11.9705 q^{73} -0.198062 q^{74} -0.0881460 q^{76} -6.58211 q^{77} -0.643104 q^{78} +10.7409 q^{79} -10.9487 q^{81} +16.9801 q^{82} -3.77479 q^{83} +0.268750 q^{84} -10.9705 q^{86} +22.9976 q^{87} +13.4015 q^{88} +9.36658 q^{89} +0.268750 q^{91} -0.225209 q^{92} -5.65279 q^{93} +3.89008 q^{94} -1.11960 q^{96} -0.198062 q^{97} +7.45473 q^{98} +9.93900 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9} + q^{11} - 5 q^{12} - 5 q^{13} - 7 q^{14} + 18 q^{16} - 2 q^{17} + 4 q^{18} - 3 q^{19} + q^{22} - 8 q^{23} - q^{24} + 9 q^{26} + q^{27}+ \cdots + 20 q^{99}+O(q^{100}) $ 3 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^6 - 9 * q^8 - 3 * q^9 + q^11 - 5 * q^12 - 5 * q^13 - 7 * q^14 + 18 * q^16 - 2 * q^17 + 4 * q^18 - 3 * q^19 + q^22 - 8 * q^23 - q^24 + 9 * q^26 + q^27 + 21 * q^28 - 7 * q^29 - 5 * q^31 - 27 * q^32 - 10 * q^33 - 9 * q^34 - 11 * q^36 - 5 * q^37 + 4 * q^38 + q^39 + q^41 - 5 * q^43 - 15 * q^44 + 6 * q^46 - 3 * q^47 + 2 * q^48 - 7 * q^49 - 8 * q^51 - 16 * q^52 - 19 * q^53 - 13 * q^54 - 35 * q^56 + 2 * q^57 + 21 * q^58 + 10 * q^59 - 17 * q^61 + 23 * q^62 - 7 * q^63 + 49 * q^64 + 11 * q^66 + q^67 + 23 * q^68 + 3 * q^69 - 19 * q^71 + 37 * q^72 + q^73 - 5 * q^74 - 4 * q^76 - 14 * q^77 - 6 * q^78 + 18 * q^79 - q^81 - 6 * q^82 - 13 * q^83 - 7 * q^84 + 2 * q^86 + 28 * q^87 + 46 * q^88 + 2 * q^89 - 7 * q^91 + q^92 + q^93 + 11 * q^94 + 18 * q^96 - 5 * q^97 + 20 * 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 $

$2$	−1.44504	−1.02180	−0.510899	−	0.859640i	$-0.670688\pi$
$2$	−1.44504	−1.02180	−0.510899	+	0.859640i	$0.670688\pi$
$3$	−2.24698	−1.29729	−0.648647	−	0.761089i	$-0.724665\pi$
$3$	−2.24698	−1.29729	−0.648647	+	0.761089i	$0.724665\pi$
$4$	0.0881460	0.0440730
$5$	0	0
$5$	0	0
$6$	3.24698	1.32557
$7$	−1.35690	−0.512858	−0.256429	−	0.966563i	$-0.582546\pi$
$7$	−1.35690	−0.512858	−0.256429	+	0.966563i	$0.582546\pi$
$8$	2.76271	0.976765
$9$	2.04892	0.682972
$10$	0	0
$11$	4.85086	1.46259	0.731294	−	0.682062i	$-0.238917\pi$
$11$	4.85086	1.46259	0.731294	+	0.682062i	$0.238917\pi$
$12$	−0.198062	−0.0571757
$13$	−0.198062	−0.0549326	−0.0274663	−	0.999623i	$-0.508744\pi$
$13$	−0.198062	−0.0549326	−0.0274663	+	0.999623i	$0.508744\pi$
$14$	1.96077	0.524038
$15$	0	0
$16$	−4.16852	−1.04213
$17$	1.13706	0.275778	0.137889	−	0.990448i	$-0.455968\pi$
$17$	1.13706	0.275778	0.137889	+	0.990448i	$0.455968\pi$
$18$	−2.96077	−0.697860
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.04892	0.665328
$22$	−7.00969	−1.49447
$23$	−2.55496	−0.532746	−0.266373	−	0.963870i	$-0.585825\pi$
$23$	−2.55496	−0.532746	−0.266373	+	0.963870i	$0.585825\pi$
$24$	−6.20775	−1.26715
$25$	0	0
$26$	0.286208	0.0561301
$27$	2.13706	0.411278
$28$	−0.119605	−0.0226032
$29$	−10.2349	−1.90057	−0.950286	−	0.311377i	$-0.899210\pi$
$29$	−10.2349	−1.90057	−0.950286	+	0.311377i	$0.899210\pi$
$30$	0	0
$31$	2.51573	0.451838	0.225919	−	0.974146i	$-0.427461\pi$
$31$	2.51573	0.451838	0.225919	+	0.974146i	$0.427461\pi$
$32$	0.498271	0.0880827
$33$	−10.8998	−1.89741
$34$	−1.64310	−0.281790
$35$	0	0
$36$	0.180604	0.0301006
$37$	0.137063	0.0225331	0.0112665	−	0.999937i	$-0.496414\pi$
$37$	0.137063	0.0225331	0.0112665	+	0.999937i	$0.496414\pi$
$38$	1.44504	0.234417
$39$	0.445042	0.0712637
$40$	0	0
$41$	−11.7506	−1.83514	−0.917570	−	0.397575i	$-0.869852\pi$
$41$	−11.7506	−1.83514	−0.917570	+	0.397575i	$0.869852\pi$
$42$	−4.40581	−0.679832
$43$	7.59179	1.15774	0.578869	−	0.815421i	$-0.303494\pi$
$43$	7.59179	1.15774	0.578869	+	0.815421i	$0.303494\pi$
$44$	0.427583	0.0644606
$45$	0	0
$46$	3.69202	0.544359
$47$	−2.69202	−0.392672	−0.196336	−	0.980537i	$-0.562904\pi$
$47$	−2.69202	−0.392672	−0.196336	+	0.980537i	$0.562904\pi$
$48$	9.36658	1.35195
$49$	−5.15883	−0.736976
$50$	0	0
$51$	−2.55496	−0.357766
$52$	−0.0174584	−0.00242104
$53$	−12.8780	−1.76893	−0.884465	−	0.466607i	$-0.845476\pi$
$53$	−12.8780	−1.76893	−0.884465	+	0.466607i	$0.845476\pi$
$54$	−3.08815	−0.420243
$55$	0	0
$56$	−3.74871	−0.500942
$57$	2.24698	0.297620
$58$	14.7899	1.94200
$59$	5.82371	0.758182	0.379091	−	0.925359i	$-0.376237\pi$
$59$	5.82371	0.758182	0.379091	+	0.925359i	$0.376237\pi$
$60$	0	0
$61$	−7.58211	−0.970789	−0.485395	−	0.874295i	$-0.661324\pi$
$61$	−7.58211	−0.970789	−0.485395	+	0.874295i	$0.661324\pi$
$62$	−3.63533	−0.461688
$63$	−2.78017	−0.350268
$64$	7.61702	0.952128
$65$	0	0
$66$	15.7506	1.93877
$67$	−8.01507	−0.979196	−0.489598	−	0.871948i	$-0.662856\pi$
$67$	−8.01507	−0.979196	−0.489598	+	0.871948i	$0.662856\pi$
$68$	0.100228	0.0121544
$69$	5.74094	0.691128
$70$	0	0
$71$	−8.82371	−1.04718	−0.523591	−	0.851970i	$-0.675408\pi$
$71$	−8.82371	−1.04718	−0.523591	+	0.851970i	$0.675408\pi$
$72$	5.66056	0.667104
$73$	11.9705	1.40104	0.700518	−	0.713635i	$-0.252952\pi$
$73$	11.9705	1.40104	0.700518	+	0.713635i	$0.252952\pi$
$74$	−0.198062	−0.0230243
$75$	0	0
$76$	−0.0881460	−0.0101110
$77$	−6.58211	−0.750101
$78$	−0.643104	−0.0728172
$79$	10.7409	1.20845	0.604225	−	0.796814i	$-0.293483\pi$
$79$	10.7409	1.20845	0.604225	+	0.796814i	$0.293483\pi$
$80$	0	0
$81$	−10.9487	−1.21652
$82$	16.9801	1.87514
$83$	−3.77479	−0.414337	−0.207169	−	0.978305i	$-0.566425\pi$
$83$	−3.77479	−0.414337	−0.207169	+	0.978305i	$0.566425\pi$
$84$	0.268750	0.0293230
$85$	0	0
$86$	−10.9705	−1.18298
$87$	22.9976	2.46560
$88$	13.4015	1.42860
$89$	9.36658	0.992856	0.496428	−	0.868078i	$-0.334645\pi$
$89$	9.36658	0.992856	0.496428	+	0.868078i	$0.334645\pi$
$90$	0	0
$91$	0.268750	0.0281726
$92$	−0.225209	−0.0234797
$93$	−5.65279	−0.586167
$94$	3.89008	0.401232
$95$	0	0
$96$	−1.11960	−0.114269
$97$	−0.198062	−0.0201102	−0.0100551	−	0.999949i	$-0.503201\pi$
$97$	−0.198062	−0.0201102	−0.0100551	+	0.999949i	$0.503201\pi$
$98$	7.45473	0.753042
$99$	9.93900	0.998907
$100$	0	0
$101$	11.5090	1.14519	0.572595	−	0.819838i	$-0.305937\pi$
$101$	11.5090	1.14519	0.572595	+	0.819838i	$0.305937\pi$
$102$	3.69202	0.365565
$103$	−15.1564	−1.49341	−0.746704	−	0.665156i	$-0.768365\pi$
$103$	−15.1564	−1.49341	−0.746704	+	0.665156i	$0.768365\pi$
$104$	−0.547188	−0.0536562
$105$	0	0
$106$	18.6093	1.80749
$107$	−2.65279	−0.256455	−0.128228	−	0.991745i	$-0.540929\pi$
$107$	−2.65279	−0.256455	−0.128228	+	0.991745i	$0.540929\pi$
$108$	0.188374	0.0181263
$109$	−2.49934	−0.239393	−0.119696	−	0.992811i	$-0.538192\pi$
$109$	−2.49934	−0.239393	−0.119696	+	0.992811i	$0.538192\pi$
$110$	0	0
$111$	−0.307979	−0.0292320
$112$	5.65625	0.534465
$113$	−8.52781	−0.802229	−0.401114	−	0.916028i	$-0.631377\pi$
$113$	−8.52781	−0.802229	−0.401114	+	0.916028i	$0.631377\pi$
$114$	−3.24698	−0.304108
$115$	0	0
$116$	−0.902165	−0.0837639
$117$	−0.405813	−0.0375174
$118$	−8.41550	−0.774710
$119$	−1.54288	−0.141435
$120$	0	0
$121$	12.5308	1.13916
$122$	10.9565	0.991951
$123$	26.4034	2.38072
$124$	0.221751	0.0199139
$125$	0	0
$126$	4.01746	0.357904
$127$	−20.4088	−1.81099	−0.905494	−	0.424359i	$-0.860499\pi$
$127$	−20.4088	−1.81099	−0.905494	+	0.424359i	$0.860499\pi$
$128$	−12.0035	−1.06097
$129$	−17.0586	−1.50193
$130$	0	0
$131$	−13.2131	−1.15444	−0.577218	−	0.816590i	$-0.695862\pi$
$131$	−13.2131	−1.15444	−0.577218	+	0.816590i	$0.695862\pi$
$132$	−0.960771	−0.0836244
$133$	1.35690	0.117658
$134$	11.5821	1.00054
$135$	0	0
$136$	3.14138	0.269371
$137$	6.86054	0.586136	0.293068	−	0.956092i	$-0.405324\pi$
$137$	6.86054	0.586136	0.293068	+	0.956092i	$0.405324\pi$
$138$	−8.29590	−0.706194
$139$	4.28621	0.363551	0.181776	−	0.983340i	$-0.441816\pi$
$139$	4.28621	0.363551	0.181776	+	0.983340i	$0.441816\pi$
$140$	0	0
$141$	6.04892	0.509411
$142$	12.7506	1.07001
$143$	−0.960771	−0.0803437
$144$	−8.54096	−0.711746
$145$	0	0
$146$	−17.2978	−1.43158
$147$	11.5918	0.956075
$148$	0.0120816	0.000993100 0
$149$	−15.3545	−1.25789	−0.628945	−	0.777450i	$-0.716513\pi$
$149$	−15.3545	−1.25789	−0.628945	+	0.777450i	$0.716513\pi$
$150$	0	0
$151$	−10.2295	−0.832467	−0.416233	−	0.909258i	$-0.636650\pi$
$151$	−10.2295	−0.832467	−0.416233	+	0.909258i	$0.636650\pi$
$152$	−2.76271	−0.224085
$153$	2.32975	0.188349
$154$	9.51142	0.766452
$155$	0	0
$156$	0.0392287	0.00314081
$157$	−3.17092	−0.253067	−0.126533	−	0.991962i	$-0.540385\pi$
$157$	−3.17092	−0.253067	−0.126533	+	0.991962i	$0.540385\pi$
$158$	−15.5211	−1.23479
$159$	28.9366	2.29482
$160$	0	0
$161$	3.46681	0.273223
$162$	15.8213	1.24304
$163$	4.63773	0.363255	0.181627	−	0.983367i	$-0.441864\pi$
$163$	4.63773	0.363255	0.181627	+	0.983367i	$0.441864\pi$
$164$	−1.03577	−0.0808801
$165$	0	0
$166$	5.45473	0.423369
$167$	19.6286	1.51891	0.759454	−	0.650560i	$-0.225466\pi$
$167$	19.6286	1.51891	0.759454	+	0.650560i	$0.225466\pi$
$168$	8.42327	0.649870
$169$	−12.9608	−0.996982
$170$	0	0
$171$	−2.04892	−0.156685
$172$	0.669186	0.0510250
$173$	−20.5646	−1.56350	−0.781751	−	0.623591i	$-0.785673\pi$
$173$	−20.5646	−1.56350	−0.781751	+	0.623591i	$0.785673\pi$
$174$	−33.2325	−2.51935
$175$	0	0
$176$	−20.2209	−1.52421
$177$	−13.0858	−0.983585
$178$	−13.5351	−1.01450
$179$	6.92154	0.517340	0.258670	−	0.965966i	$-0.416716\pi$
$179$	6.92154	0.517340	0.258670	+	0.965966i	$0.416716\pi$
$180$	0	0
$181$	−17.6461	−1.31162	−0.655812	−	0.754925i	$-0.727673\pi$
$181$	−17.6461	−1.31162	−0.655812	+	0.754925i	$0.727673\pi$
$182$	−0.388355	−0.0287868
$183$	17.0368	1.25940
$184$	−7.05861	−0.520367
$185$	0	0
$186$	8.16852	0.598945
$187$	5.51573	0.403350
$188$	−0.237291	−0.0173062
$189$	−2.89977	−0.210927
$190$	0	0
$191$	6.92931	0.501387	0.250694	−	0.968066i	$-0.419341\pi$
$191$	6.92931	0.501387	0.250694	+	0.968066i	$0.419341\pi$
$192$	−17.1153	−1.23519
$193$	21.0368	1.51426	0.757132	−	0.653262i	$-0.226600\pi$
$193$	21.0368	1.51426	0.757132	+	0.653262i	$0.226600\pi$
$194$	0.286208	0.0205486
$195$	0	0
$196$	−0.454731	−0.0324808
$197$	−21.5646	−1.53642	−0.768209	−	0.640199i	$-0.778852\pi$
$197$	−21.5646	−1.53642	−0.768209	+	0.640199i	$0.778852\pi$
$198$	−14.3623	−1.02068
$199$	21.9909	1.55889	0.779447	−	0.626468i	$-0.215500\pi$
$199$	21.9909	1.55889	0.779447	+	0.626468i	$0.215500\pi$
$200$	0	0
$201$	18.0097	1.27031
$202$	−16.6310	−1.17015
$203$	13.8877	0.974725
$204$	−0.225209	−0.0157678
$205$	0	0
$206$	21.9017	1.52596
$207$	−5.23490	−0.363851
$208$	0.825627	0.0572469
$209$	−4.85086	−0.335541
$210$	0	0
$211$	−20.6233	−1.41976	−0.709882	−	0.704321i	$-0.751252\pi$
$211$	−20.6233	−1.41976	−0.709882	+	0.704321i	$0.751252\pi$
$212$	−1.13514	−0.0779620
$213$	19.8267	1.35850
$214$	3.83340	0.262046
$215$	0	0
$216$	5.90408	0.401722
$217$	−3.41358	−0.231729
$218$	3.61165	0.244611
$219$	−26.8974	−1.81756
$220$	0	0
$221$	−0.225209	−0.0151492
$222$	0.445042	0.0298693
$223$	−7.97716	−0.534190	−0.267095	−	0.963670i	$-0.586064\pi$
$223$	−7.97716	−0.534190	−0.267095	+	0.963670i	$0.586064\pi$
$224$	−0.676102	−0.0451740
$225$	0	0
$226$	12.3230	0.819717
$227$	−19.7942	−1.31379	−0.656893	−	0.753984i	$-0.728129\pi$
$227$	−19.7942	−1.31379	−0.656893	+	0.753984i	$0.728129\pi$
$228$	0.198062	0.0131170
$229$	−4.03385	−0.266564	−0.133282	−	0.991078i	$-0.542552\pi$
$229$	−4.03385	−0.266564	−0.133282	+	0.991078i	$0.542552\pi$
$230$	0	0
$231$	14.7899	0.973101
$232$	−28.2760	−1.85641
$233$	−26.8159	−1.75677	−0.878385	−	0.477953i	$-0.841379\pi$
$233$	−26.8159	−1.75677	−0.878385	+	0.477953i	$0.841379\pi$
$234$	0.586417	0.0383353
$235$	0	0
$236$	0.513337	0.0334154
$237$	−24.1347	−1.56772
$238$	2.22952	0.144518
$239$	3.36227	0.217487	0.108744	−	0.994070i	$-0.465317\pi$
$239$	3.36227	0.217487	0.108744	+	0.994070i	$0.465317\pi$
$240$	0	0
$241$	27.7506	1.78758	0.893788	−	0.448491i	$-0.148038\pi$
$241$	27.7506	1.78758	0.893788	+	0.448491i	$0.148038\pi$
$242$	−18.1075	−1.16400
$243$	18.1903	1.16691
$244$	−0.668332	−0.0427856
$245$	0	0
$246$	−38.1540	−2.43261
$247$	0.198062	0.0126024
$248$	6.95023	0.441340
$249$	8.48188	0.537517
$250$	0	0
$251$	−5.59419	−0.353102	−0.176551	−	0.984291i	$-0.556494\pi$
$251$	−5.59419	−0.353102	−0.176551	+	0.984291i	$0.556494\pi$
$252$	−0.245061	−0.0154374
$253$	−12.3937	−0.779187
$254$	29.4916	1.85047
$255$	0	0
$256$	2.11146	0.131966
$257$	−10.4668	−0.652902	−0.326451	−	0.945214i	$-0.605853\pi$
$257$	−10.4668	−0.652902	−0.326451	+	0.945214i	$0.605853\pi$
$258$	24.6504	1.53467
$259$	−0.185981	−0.0115563
$260$	0	0
$261$	−20.9705	−1.29804
$262$	19.0935	1.17960
$263$	−15.4795	−0.954506	−0.477253	−	0.878766i	$-0.658367\pi$
$263$	−15.4795	−0.954506	−0.477253	+	0.878766i	$0.658367\pi$
$264$	−30.1129	−1.85332
$265$	0	0
$266$	−1.96077	−0.120223
$267$	−21.0465	−1.28803
$268$	−0.706496	−0.0431561
$269$	9.13036	0.556688	0.278344	−	0.960481i	$-0.410215\pi$
$269$	9.13036	0.556688	0.278344	+	0.960481i	$0.410215\pi$
$270$	0	0
$271$	7.44265	0.452109	0.226054	−	0.974115i	$-0.427417\pi$
$271$	7.44265	0.452109	0.226054	+	0.974115i	$0.427417\pi$
$272$	−4.73987	−0.287397
$273$	−0.603875	−0.0365482
$274$	−9.91377	−0.598913
$275$	0	0
$276$	0.506041	0.0304601
$277$	11.4155	0.685891	0.342946	−	0.939355i	$-0.388575\pi$
$277$	11.4155	0.685891	0.342946	+	0.939355i	$0.388575\pi$
$278$	−6.19375	−0.371476
$279$	5.15452	0.308593
$280$	0	0
$281$	21.5060	1.28294	0.641471	−	0.767147i	$-0.278324\pi$
$281$	21.5060	1.28294	0.641471	+	0.767147i	$0.278324\pi$
$282$	−8.74094	−0.520515
$283$	5.45712	0.324392	0.162196	−	0.986759i	$-0.448142\pi$
$283$	5.45712	0.324392	0.162196	+	0.986759i	$0.448142\pi$
$284$	−0.777775	−0.0461524
$285$	0	0
$286$	1.38835	0.0820951
$287$	15.9444	0.941167
$288$	1.02092	0.0601581
$289$	−15.7071	−0.923946
$290$	0	0
$291$	0.445042	0.0260888
$292$	1.05515	0.0617479
$293$	−7.39075	−0.431772	−0.215886	−	0.976419i	$-0.569264\pi$
$293$	−7.39075	−0.431772	−0.215886	+	0.976419i	$0.569264\pi$
$294$	−16.7506	−0.976916
$295$	0	0
$296$	0.378666	0.0220095
$297$	10.3666	0.601530
$298$	22.1879	1.28531
$299$	0.506041	0.0292651
$300$	0	0
$301$	−10.3013	−0.593756
$302$	14.7821	0.850613
$303$	−25.8605	−1.48565
$304$	4.16852	0.239081
$305$	0	0
$306$	−3.36658	−0.192455
$307$	32.2131	1.83850	0.919250	−	0.393674i	$-0.128796\pi$
$307$	32.2131	1.83850	0.919250	+	0.393674i	$0.128796\pi$
$308$	−0.580186	−0.0330592
$309$	34.0562	1.93739
$310$	0	0
$311$	14.8442	0.841735	0.420867	−	0.907122i	$-0.361726\pi$
$311$	14.8442	0.841735	0.420867	+	0.907122i	$0.361726\pi$
$312$	1.22952	0.0696079
$313$	13.1491	0.743234	0.371617	−	0.928386i	$-0.378804\pi$
$313$	13.1491	0.743234	0.371617	+	0.928386i	$0.378804\pi$
$314$	4.58211	0.258583
$315$	0	0
$316$	0.946771	0.0532600
$317$	5.13467	0.288392	0.144196	−	0.989549i	$-0.453940\pi$
$317$	5.13467	0.288392	0.144196	+	0.989549i	$0.453940\pi$
$318$	−41.8146	−2.34485
$319$	−49.6480	−2.77975
$320$	0	0
$321$	5.96077	0.332698
$322$	−5.00969	−0.279179
$323$	−1.13706	−0.0632679
$324$	−0.965083	−0.0536157
$325$	0	0
$326$	−6.70171	−0.371173
$327$	5.61596	0.310563
$328$	−32.4636	−1.79250
$329$	3.65279	0.201385
$330$	0	0
$331$	2.00969	0.110462	0.0552312	−	0.998474i	$-0.482410\pi$
$331$	2.00969	0.110462	0.0552312	+	0.998474i	$0.482410\pi$
$332$	−0.332733	−0.0182611
$333$	0.280831	0.0153895
$334$	−28.3642	−1.55202
$335$	0	0
$336$	−12.7095	−0.693359
$337$	1.31873	0.0718359	0.0359180	−	0.999355i	$-0.488564\pi$
$337$	1.31873	0.0718359	0.0359180	+	0.999355i	$0.488564\pi$
$338$	18.7289	1.01872
$339$	19.1618	1.04073
$340$	0	0
$341$	12.2034	0.660853
$342$	2.96077	0.160100
$343$	16.4983	0.890823
$344$	20.9739	1.13084
$345$	0	0
$346$	29.7168	1.59758
$347$	−12.0151	−0.645003	−0.322501	−	0.946569i	$-0.604524\pi$
$347$	−12.0151	−0.645003	−0.322501	+	0.946569i	$0.604524\pi$
$348$	2.02715	0.108666
$349$	−10.5579	−0.565154	−0.282577	−	0.959245i	$-0.591189\pi$
$349$	−10.5579	−0.565154	−0.282577	+	0.959245i	$0.591189\pi$
$350$	0	0
$351$	−0.423272	−0.0225926
$352$	2.41704	0.128829
$353$	1.01102	0.0538110	0.0269055	−	0.999638i	$-0.491435\pi$
$353$	1.01102	0.0538110	0.0269055	+	0.999638i	$0.491435\pi$
$354$	18.9095	1.00503
$355$	0	0
$356$	0.825627	0.0437581
$357$	3.46681	0.183483
$358$	−10.0019	−0.528618
$359$	−5.14244	−0.271408	−0.135704	−	0.990749i	$-0.543330\pi$
$359$	−5.14244	−0.271408	−0.135704	+	0.990749i	$0.543330\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	25.4993	1.34022
$363$	−28.1564	−1.47783
$364$	0.0236892	0.00124165
$365$	0	0
$366$	−24.6189	−1.28685
$367$	29.5308	1.54149	0.770747	−	0.637141i	$-0.219883\pi$
$367$	29.5308	1.54149	0.770747	+	0.637141i	$0.219883\pi$
$368$	10.6504	0.555190
$369$	−24.0761	−1.25335
$370$	0	0
$371$	17.4741	0.907210
$372$	−0.498271	−0.0258342
$373$	−8.78986	−0.455121	−0.227561	−	0.973764i	$-0.573075\pi$
$373$	−8.78986	−0.455121	−0.227561	+	0.973764i	$0.573075\pi$
$374$	−7.97046	−0.412143
$375$	0	0
$376$	−7.43727	−0.383548
$377$	2.02715	0.104403
$378$	4.19029	0.215525
$379$	19.1511	0.983724	0.491862	−	0.870673i	$-0.336316\pi$
$379$	19.1511	0.983724	0.491862	+	0.870673i	$0.336316\pi$
$380$	0	0
$381$	45.8582	2.34938
$382$	−10.0131	−0.512317
$383$	6.99894	0.357629	0.178814	−	0.983883i	$-0.442774\pi$
$383$	6.99894	0.357629	0.178814	+	0.983883i	$0.442774\pi$
$384$	26.9715	1.37638
$385$	0	0
$386$	−30.3991	−1.54727
$387$	15.5550	0.790703
$388$	−0.0174584	−0.000886316 0
$389$	8.08575	0.409964	0.204982	−	0.978766i	$-0.434286\pi$
$389$	8.08575	0.409964	0.204982	+	0.978766i	$0.434286\pi$
$390$	0	0
$391$	−2.90515	−0.146920
$392$	−14.2524	−0.719853
$393$	29.6896	1.49764
$394$	31.1618	1.56991
$395$	0	0
$396$	0.876083	0.0440248
$397$	17.7006	0.888370	0.444185	−	0.895935i	$-0.353493\pi$
$397$	17.7006	0.888370	0.444185	+	0.895935i	$0.353493\pi$
$398$	−31.7778	−1.59288
$399$	−3.04892	−0.152637
$400$	0	0
$401$	−15.5418	−0.776121	−0.388061	−	0.921634i	$-0.626855\pi$
$401$	−15.5418	−0.776121	−0.388061	+	0.921634i	$0.626855\pi$
$402$	−26.0248	−1.29800
$403$	−0.498271	−0.0248207
$404$	1.01447	0.0504720
$405$	0	0
$406$	−20.0683	−0.995973
$407$	0.664874	0.0329566
$408$	−7.05861	−0.349453
$409$	13.1661	0.651023	0.325512	−	0.945538i	$-0.394463\pi$
$409$	13.1661	0.651023	0.325512	+	0.945538i	$0.394463\pi$
$410$	0	0
$411$	−15.4155	−0.760391
$412$	−1.33598	−0.0658190
$413$	−7.90217	−0.388840
$414$	7.56465	0.371782
$415$	0	0
$416$	−0.0986887	−0.00483861
$417$	−9.63102	−0.471633
$418$	7.00969	0.342855
$419$	−25.1739	−1.22983	−0.614913	−	0.788595i	$-0.710809\pi$
$419$	−25.1739	−1.22983	−0.614913	+	0.788595i	$0.710809\pi$
$420$	0	0
$421$	26.9420	1.31307	0.656536	−	0.754295i	$-0.272021\pi$
$421$	26.9420	1.31307	0.656536	+	0.754295i	$0.272021\pi$
$422$	29.8015	1.45071
$423$	−5.51573	−0.268184
$424$	−35.5782	−1.72783
$425$	0	0
$426$	−28.6504	−1.38812
$427$	10.2881	0.497877
$428$	−0.233833	−0.0113027
$429$	2.15883	0.104229
$430$	0	0
$431$	18.9487	0.912726	0.456363	−	0.889794i	$-0.349152\pi$
$431$	18.9487	0.912726	0.456363	+	0.889794i	$0.349152\pi$
$432$	−8.90840	−0.428605
$433$	3.68904	0.177284	0.0886419	−	0.996064i	$-0.471747\pi$
$433$	3.68904	0.177284	0.0886419	+	0.996064i	$0.471747\pi$
$434$	4.93277	0.236781
$435$	0	0
$436$	−0.220306	−0.0105508
$437$	2.55496	0.122220
$438$	38.8678	1.85718
$439$	−25.6926	−1.22624	−0.613121	−	0.789989i	$-0.710086\pi$
$439$	−25.6926	−1.22624	−0.613121	+	0.789989i	$0.710086\pi$
$440$	0	0
$441$	−10.5700	−0.503334
$442$	0.325437	0.0154795
$443$	27.3653	1.30016	0.650081	−	0.759865i	$-0.274735\pi$
$443$	27.3653	1.30016	0.650081	+	0.759865i	$0.274735\pi$
$444$	−0.0271471	−0.00128834
$445$	0	0
$446$	11.5273	0.545835
$447$	34.5013	1.63185
$448$	−10.3355	−0.488307
$449$	7.55794	0.356681	0.178341	−	0.983969i	$-0.442927\pi$
$449$	7.55794	0.356681	0.178341	+	0.983969i	$0.442927\pi$
$450$	0	0
$451$	−57.0006	−2.68405
$452$	−0.751692	−0.0353566
$453$	22.9855	1.07995
$454$	28.6034	1.34242
$455$	0	0
$456$	6.20775	0.290705
$457$	7.85623	0.367499	0.183750	−	0.982973i	$-0.441176\pi$
$457$	7.85623	0.367499	0.183750	+	0.982973i	$0.441176\pi$
$458$	5.82908	0.272375
$459$	2.42998	0.113422
$460$	0	0
$461$	3.87907	0.180666	0.0903331	−	0.995912i	$-0.471207\pi$
$461$	3.87907	0.180666	0.0903331	+	0.995912i	$0.471207\pi$
$462$	−21.3720	−0.994314
$463$	13.0954	0.608597	0.304298	−	0.952577i	$-0.401578\pi$
$463$	13.0954	0.608597	0.304298	+	0.952577i	$0.401578\pi$
$464$	42.6644	1.98065
$465$	0	0
$466$	38.7502	1.79507
$467$	−6.44026	−0.298020	−0.149010	−	0.988836i	$-0.547609\pi$
$467$	−6.44026	−0.298020	−0.149010	+	0.988836i	$0.547609\pi$
$468$	−0.0357708	−0.00165351
$469$	10.8756	0.502189
$470$	0	0
$471$	7.12498	0.328302
$472$	16.0892	0.740566
$473$	36.8267	1.69329
$474$	34.8756	1.60189
$475$	0	0
$476$	−0.135998	−0.00623348
$477$	−26.3860	−1.20813
$478$	−4.85862	−0.222228
$479$	5.69096	0.260026	0.130013	−	0.991512i	$-0.458498\pi$
$479$	5.69096	0.260026	0.130013	+	0.991512i	$0.458498\pi$
$480$	0	0
$481$	−0.0271471	−0.00123780
$482$	−40.1008	−1.82654
$483$	−7.78986	−0.354451
$484$	1.10454	0.0502063
$485$	0	0
$486$	−26.2857	−1.19235
$487$	12.9632	0.587417	0.293709	−	0.955895i	$-0.405110\pi$
$487$	12.9632	0.587417	0.293709	+	0.955895i	$0.405110\pi$
$488$	−20.9472	−0.948233
$489$	−10.4209	−0.471248
$490$	0	0
$491$	−19.6045	−0.884737	−0.442369	−	0.896833i	$-0.645862\pi$
$491$	−19.6045	−0.884737	−0.442369	+	0.896833i	$0.645862\pi$
$492$	2.32736	0.104925
$493$	−11.6377	−0.524137
$494$	−0.286208	−0.0128771
$495$	0	0
$496$	−10.4869	−0.470875
$497$	11.9729	0.537056
$498$	−12.2567	−0.549234
$499$	5.49827	0.246136	0.123068	−	0.992398i	$-0.460727\pi$
$499$	5.49827	0.246136	0.123068	+	0.992398i	$0.460727\pi$
$500$	0	0
$501$	−44.1051	−1.97047
$502$	8.08383	0.360799
$503$	35.6939	1.59151	0.795757	−	0.605616i	$-0.207073\pi$
$503$	35.6939	1.59151	0.795757	+	0.605616i	$0.207073\pi$
$504$	−7.68079	−0.342130
$505$	0	0
$506$	17.9095	0.796173
$507$	29.1226	1.29338
$508$	−1.79895	−0.0798157
$509$	−16.4450	−0.728914	−0.364457	−	0.931220i	$-0.618745\pi$
$509$	−16.4450	−0.728914	−0.364457	+	0.931220i	$0.618745\pi$
$510$	0	0
$511$	−16.2427	−0.718533
$512$	20.9558	0.926123
$513$	−2.13706	−0.0943537
$514$	15.1250	0.667134
$515$	0	0
$516$	−1.50365	−0.0661944
$517$	−13.0586	−0.574317
$518$	0.268750	0.0118082
$519$	46.2083	2.02832
$520$	0	0
$521$	−26.5435	−1.16289	−0.581445	−	0.813586i	$-0.697513\pi$
$521$	−26.5435	−1.16289	−0.581445	+	0.813586i	$0.697513\pi$
$522$	30.3032	1.32633
$523$	24.1685	1.05682	0.528408	−	0.848991i	$-0.322789\pi$
$523$	24.1685	1.05682	0.528408	+	0.848991i	$0.322789\pi$
$524$	−1.16468	−0.0508795
$525$	0	0
$526$	22.3685	0.975313
$527$	2.86054	0.124607
$528$	45.4359	1.97735
$529$	−16.4722	−0.716182
$530$	0	0
$531$	11.9323	0.517818
$532$	0.119605	0.00518553
$533$	2.32736	0.100809
$534$	30.4131	1.31610
$535$	0	0
$536$	−22.1433	−0.956445
$537$	−15.5526	−0.671143
$538$	−13.1938	−0.568823
$539$	−25.0248	−1.07789
$540$	0	0
$541$	9.80386	0.421501	0.210750	−	0.977540i	$-0.432409\pi$
$541$	9.80386	0.421501	0.210750	+	0.977540i	$0.432409\pi$
$542$	−10.7549	−0.461964
$543$	39.6504	1.70156
$544$	0.566566	0.0242913
$545$	0	0
$546$	0.872625	0.0373449
$547$	−21.1739	−0.905331	−0.452665	−	0.891681i	$-0.649527\pi$
$547$	−21.1739	−0.905331	−0.452665	+	0.891681i	$0.649527\pi$
$548$	0.604729	0.0258328
$549$	−15.5351	−0.663022
$550$	0	0
$551$	10.2349	0.436021
$552$	15.8605	0.675070
$553$	−14.5743	−0.619764
$554$	−16.4959	−0.700843
$555$	0	0
$556$	0.377812	0.0160228
$557$	−24.4077	−1.03419	−0.517094	−	0.855928i	$-0.672986\pi$
$557$	−24.4077	−1.03419	−0.517094	+	0.855928i	$0.672986\pi$
$558$	−7.44850	−0.315320
$559$	−1.50365	−0.0635975
$560$	0	0
$561$	−12.3937	−0.523264
$562$	−31.0771	−1.31091
$563$	−14.4849	−0.610464	−0.305232	−	0.952278i	$-0.598734\pi$
$563$	−14.4849	−0.610464	−0.305232	+	0.952278i	$0.598734\pi$
$564$	0.533188	0.0224513
$565$	0	0
$566$	−7.88577	−0.331464
$567$	14.8562	0.623903
$568$	−24.3773	−1.02285
$569$	0.811626	0.0340251	0.0170126	−	0.999855i	$-0.494584\pi$
$569$	0.811626	0.0340251	0.0170126	+	0.999855i	$0.494584\pi$
$570$	0	0
$571$	−37.6588	−1.57597	−0.787985	−	0.615694i	$-0.788876\pi$
$571$	−37.6588	−1.57597	−0.787985	+	0.615694i	$0.788876\pi$
$572$	−0.0846882	−0.00354099
$573$	−15.5700	−0.650447
$574$	−23.0403	−0.961683
$575$	0	0
$576$	15.6066	0.650277
$577$	−8.89307	−0.370223	−0.185112	−	0.982718i	$-0.559265\pi$
$577$	−8.89307	−0.370223	−0.185112	+	0.982718i	$0.559265\pi$
$578$	22.6974	0.944087
$579$	−47.2693	−1.96445
$580$	0	0
$581$	5.12200	0.212496
$582$	−0.643104	−0.0266575
$583$	−62.4693	−2.58721
$584$	33.0709	1.36848
$585$	0	0
$586$	10.6799	0.441184
$587$	−20.3327	−0.839222	−0.419611	−	0.907704i	$-0.637833\pi$
$587$	−20.3327	−0.839222	−0.419611	+	0.907704i	$0.637833\pi$
$588$	1.02177	0.0421371
$589$	−2.51573	−0.103659
$590$	0	0
$591$	48.4553	1.99319
$592$	−0.571352	−0.0234824
$593$	−17.0049	−0.698308	−0.349154	−	0.937065i	$-0.613531\pi$
$593$	−17.0049	−0.698308	−0.349154	+	0.937065i	$0.613531\pi$
$594$	−14.9801	−0.614643
$595$	0	0
$596$	−1.35344	−0.0554390
$597$	−49.4131	−2.02234
$598$	−0.731250	−0.0299030
$599$	−12.4004	−0.506668	−0.253334	−	0.967379i	$-0.581527\pi$
$599$	−12.4004	−0.506668	−0.253334	+	0.967379i	$0.581527\pi$
$600$	0	0
$601$	−9.73125	−0.396946	−0.198473	−	0.980106i	$-0.563598\pi$
$601$	−9.73125	−0.396946	−0.198473	+	0.980106i	$0.563598\pi$
$602$	14.8858	0.606699
$603$	−16.4222	−0.668764
$604$	−0.901691	−0.0366893
$605$	0	0
$606$	37.3696	1.51803
$607$	23.4282	0.950920	0.475460	−	0.879737i	$-0.342282\pi$
$607$	23.4282	0.950920	0.475460	+	0.879737i	$0.342282\pi$
$608$	−0.498271	−0.0202076
$609$	−31.2054	−1.26450
$610$	0	0
$611$	0.533188	0.0215705
$612$	0.205358	0.00830111
$613$	−28.1424	−1.13666	−0.568331	−	0.822800i	$-0.692411\pi$
$613$	−28.1424	−1.13666	−0.568331	+	0.822800i	$0.692411\pi$
$614$	−46.5493	−1.87858
$615$	0	0
$616$	−18.1844	−0.732672
$617$	−36.4805	−1.46865	−0.734326	−	0.678797i	$-0.762502\pi$
$617$	−36.4805	−1.46865	−0.734326	+	0.678797i	$0.762502\pi$
$618$	−49.2127	−1.97962
$619$	−38.2097	−1.53578	−0.767888	−	0.640584i	$-0.778692\pi$
$619$	−38.2097	−1.53578	−0.767888	+	0.640584i	$0.778692\pi$
$620$	0	0
$621$	−5.46011	−0.219107
$622$	−21.4504	−0.860083
$623$	−12.7095	−0.509195
$624$	−1.85517	−0.0742661
$625$	0	0
$626$	−19.0011	−0.759435
$627$	10.8998	0.435295
$628$	−0.279503	−0.0111534
$629$	0.155850	0.00621413
$630$	0	0
$631$	−12.5235	−0.498553	−0.249276	−	0.968432i	$-0.580193\pi$
$631$	−12.5235	−0.498553	−0.249276	+	0.968432i	$0.580193\pi$
$632$	29.6741	1.18037
$633$	46.3400	1.84185
$634$	−7.41981	−0.294678
$635$	0	0
$636$	2.55065	0.101140
$637$	1.02177	0.0404840
$638$	71.7434	2.84035
$639$	−18.0790	−0.715196
$640$	0	0
$641$	37.3564	1.47549	0.737745	−	0.675080i	$-0.235891\pi$
$641$	37.3564	1.47549	0.737745	+	0.675080i	$0.235891\pi$
$642$	−8.61356	−0.339950
$643$	14.5483	0.573727	0.286864	−	0.957971i	$-0.407387\pi$
$643$	14.5483	0.573727	0.286864	+	0.957971i	$0.407387\pi$
$644$	0.305586	0.0120418
$645$	0	0
$646$	1.64310	0.0646471
$647$	23.4403	0.921532	0.460766	−	0.887522i	$-0.347575\pi$
$647$	23.4403	0.921532	0.460766	+	0.887522i	$0.347575\pi$
$648$	−30.2480	−1.18826
$649$	28.2500	1.10891
$650$	0	0
$651$	7.67025	0.300621
$652$	0.408797	0.0160097
$653$	−27.1903	−1.06404	−0.532019	−	0.846732i	$-0.678567\pi$
$653$	−27.1903	−1.06404	−0.532019	+	0.846732i	$0.678567\pi$
$654$	−8.11529	−0.317333
$655$	0	0
$656$	48.9828	1.91246
$657$	24.5265	0.956869
$658$	−5.27844	−0.205775
$659$	2.71486	0.105756	0.0528779	−	0.998601i	$-0.483161\pi$
$659$	2.71486	0.105756	0.0528779	+	0.998601i	$0.483161\pi$
$660$	0	0
$661$	−9.41311	−0.366128	−0.183064	−	0.983101i	$-0.558601\pi$
$661$	−9.41311	−0.366128	−0.183064	+	0.983101i	$0.558601\pi$
$662$	−2.90408	−0.112870
$663$	0.506041	0.0196530
$664$	−10.4286	−0.404710
$665$	0	0
$666$	−0.405813	−0.0157249
$667$	26.1497	1.01252
$668$	1.73019	0.0669429
$669$	17.9245	0.693002
$670$	0	0
$671$	−36.7797	−1.41986
$672$	1.51919	0.0586039
$673$	44.8001	1.72692	0.863459	−	0.504419i	$-0.168293\pi$
$673$	44.8001	1.72692	0.863459	+	0.504419i	$0.168293\pi$
$674$	−1.90562	−0.0734019
$675$	0	0
$676$	−1.14244	−0.0439400
$677$	32.9197	1.26521	0.632604	−	0.774475i	$-0.281986\pi$
$677$	32.9197	1.26521	0.632604	+	0.774475i	$0.281986\pi$
$678$	−27.6896	−1.06341
$679$	0.268750	0.0103137
$680$	0	0
$681$	44.4771	1.70437
$682$	−17.6345	−0.675259
$683$	−20.3448	−0.778473	−0.389236	−	0.921138i	$-0.627261\pi$
$683$	−20.3448	−0.778473	−0.389236	+	0.921138i	$0.627261\pi$
$684$	−0.180604	−0.00690556
$685$	0	0
$686$	−23.8407	−0.910242
$687$	9.06398	0.345813
$688$	−31.6466	−1.20651
$689$	2.55065	0.0971719
$690$	0	0
$691$	−46.1473	−1.75553	−0.877764	−	0.479094i	$-0.840965\pi$
$691$	−46.1473	−1.75553	−0.877764	+	0.479094i	$0.840965\pi$
$692$	−1.81269	−0.0689082
$693$	−13.4862	−0.512298
$694$	17.3623	0.659063
$695$	0	0
$696$	63.5357	2.40831
$697$	−13.3612	−0.506092
$698$	15.2567	0.577473
$699$	60.2549	2.27905
$700$	0	0
$701$	13.1933	0.498303	0.249152	−	0.968464i	$-0.419848\pi$
$701$	13.1933	0.498303	0.249152	+	0.968464i	$0.419848\pi$
$702$	0.611645	0.0230851
$703$	−0.137063	−0.00516944
$704$	36.9491	1.39257
$705$	0	0
$706$	−1.46096	−0.0549840
$707$	−15.6165	−0.587321
$708$	−1.15346	−0.0433496
$709$	41.1860	1.54677	0.773386	−	0.633935i	$-0.218562\pi$
$709$	41.1860	1.54677	0.773386	+	0.633935i	$0.218562\pi$
$710$	0	0
$711$	22.0073	0.825338
$712$	25.8771	0.969787
$713$	−6.42758	−0.240715
$714$	−5.00969	−0.187483
$715$	0	0
$716$	0.610106	0.0228007
$717$	−7.55496	−0.282145
$718$	7.43104	0.277324
$719$	−35.6256	−1.32861	−0.664306	−	0.747461i	$-0.731273\pi$
$719$	−35.6256	−1.32861	−0.664306	+	0.747461i	$0.731273\pi$
$720$	0	0
$721$	20.5657	0.765907
$722$	−1.44504	−0.0537789
$723$	−62.3551	−2.31901
$724$	−1.55543	−0.0578072
$725$	0	0
$726$	40.6872	1.51004
$727$	−20.0116	−0.742189	−0.371095	−	0.928595i	$-0.621017\pi$
$727$	−20.0116	−0.742189	−0.371095	+	0.928595i	$0.621017\pi$
$728$	0.742478	0.0275181
$729$	−8.02715	−0.297302
$730$	0	0
$731$	8.63235	0.319279
$732$	1.50173	0.0555055
$733$	18.9952	0.701604	0.350802	−	0.936450i	$-0.385909\pi$
$733$	18.9952	0.701604	0.350802	+	0.936450i	$0.385909\pi$
$734$	−42.6732	−1.57510
$735$	0	0
$736$	−1.27306	−0.0469257
$737$	−38.8799	−1.43216
$738$	34.7909	1.28067
$739$	48.1704	1.77198	0.885989	−	0.463706i	$-0.153481\pi$
$739$	48.1704	1.77198	0.885989	+	0.463706i	$0.153481\pi$
$740$	0	0
$741$	−0.445042	−0.0163490
$742$	−25.2508	−0.926987
$743$	13.2446	0.485897	0.242948	−	0.970039i	$-0.421885\pi$
$743$	13.2446	0.485897	0.242948	+	0.970039i	$0.421885\pi$
$744$	−15.6170	−0.572548
$745$	0	0
$746$	12.7017	0.465043
$747$	−7.73423	−0.282981
$748$	0.486189	0.0177768
$749$	3.59956	0.131525
$750$	0	0
$751$	12.0562	0.439937	0.219969	−	0.975507i	$-0.429404\pi$
$751$	12.0562	0.439937	0.219969	+	0.975507i	$0.429404\pi$
$752$	11.2218	0.409215
$753$	12.5700	0.458077
$754$	−2.92931	−0.106679
$755$	0	0
$756$	−0.255603	−0.00929620
$757$	−15.0054	−0.545380	−0.272690	−	0.962102i	$-0.587913\pi$
$757$	−15.0054	−0.545380	−0.272690	+	0.962102i	$0.587913\pi$
$758$	−27.6741	−1.00517
$759$	27.8485	1.01084
$760$	0	0
$761$	44.3967	1.60938	0.804690	−	0.593695i	$-0.202332\pi$
$761$	44.3967	1.60938	0.804690	+	0.593695i	$0.202332\pi$
$762$	−66.2669	−2.40060
$763$	3.39134	0.122775
$764$	0.610791	0.0220976
$765$	0	0
$766$	−10.1138	−0.365425
$767$	−1.15346	−0.0416489
$768$	−4.74440	−0.171199
$769$	−39.7211	−1.43238	−0.716190	−	0.697906i	$-0.754115\pi$
$769$	−39.7211	−1.43238	−0.716190	+	0.697906i	$0.754115\pi$
$770$	0	0
$771$	23.5187	0.847006
$772$	1.85431	0.0667382
$773$	1.72779	0.0621444	0.0310722	−	0.999517i	$-0.490108\pi$
$773$	1.72779	0.0621444	0.0310722	+	0.999517i	$0.490108\pi$
$774$	−22.4776	−0.807939
$775$	0	0
$776$	−0.547188	−0.0196429
$777$	0.417895	0.0149919
$778$	−11.6843	−0.418901
$779$	11.7506	0.421010
$780$	0	0
$781$	−42.8025	−1.53159
$782$	4.19806	0.150122
$783$	−21.8726	−0.781664
$784$	21.5047	0.768025
$785$	0	0
$786$	−42.9028	−1.53029
$787$	−42.6329	−1.51970	−0.759850	−	0.650098i	$-0.774728\pi$
$787$	−42.6329	−1.51970	−0.759850	+	0.650098i	$0.774728\pi$
$788$	−1.90084	−0.0677145
$789$	34.7821	1.23828
$790$	0	0
$791$	11.5714	0.411430
$792$	27.4586	0.975698
$793$	1.50173	0.0533280
$794$	−25.5782	−0.907735
$795$	0	0
$796$	1.93841	0.0687051
$797$	−38.5864	−1.36680	−0.683401	−	0.730044i	$-0.739500\pi$
$797$	−38.5864	−1.36680	−0.683401	+	0.730044i	$0.739500\pi$
$798$	4.40581	0.155964
$799$	−3.06100	−0.108290
$800$	0	0
$801$	19.1914	0.678093
$802$	22.4586	0.793040
$803$	58.0670	2.04914
$804$	1.58748	0.0559862
$805$	0	0
$806$	0.720023	0.0253617
$807$	−20.5157	−0.722188
$808$	31.7961	1.11858
$809$	−8.38298	−0.294730	−0.147365	−	0.989082i	$-0.547079\pi$
$809$	−8.38298	−0.294730	−0.147365	+	0.989082i	$0.547079\pi$
$810$	0	0
$811$	−0.340765	−0.0119659	−0.00598295	−	0.999982i	$-0.501904\pi$
$811$	−0.340765	−0.0119659	−0.00598295	+	0.999982i	$0.501904\pi$
$812$	1.22414	0.0429590
$813$	−16.7235	−0.586518
$814$	−0.960771	−0.0336750
$815$	0	0
$816$	10.6504	0.372839
$817$	−7.59179	−0.265603
$818$	−19.0256	−0.665215
$819$	0.550646	0.0192411
$820$	0	0
$821$	−33.7506	−1.17791	−0.588953	−	0.808168i	$-0.700460\pi$
$821$	−33.7506	−1.17791	−0.588953	+	0.808168i	$0.700460\pi$
$822$	22.2760	0.776966
$823$	−37.2669	−1.29904	−0.649522	−	0.760343i	$-0.725031\pi$
$823$	−37.2669	−1.29904	−0.649522	+	0.760343i	$0.725031\pi$
$824$	−41.8728	−1.45871
$825$	0	0
$826$	11.4190	0.397316
$827$	21.1691	0.736122	0.368061	−	0.929802i	$-0.380022\pi$
$827$	21.1691	0.736122	0.368061	+	0.929802i	$0.380022\pi$
$828$	−0.461435	−0.0160360
$829$	−31.0374	−1.07797	−0.538987	−	0.842314i	$-0.681193\pi$
$829$	−31.0374	−1.07797	−0.538987	+	0.842314i	$0.681193\pi$
$830$	0	0
$831$	−25.6504	−0.889803
$832$	−1.50864	−0.0523028
$833$	−5.86592	−0.203242
$834$	13.9172	0.481914
$835$	0	0
$836$	−0.427583	−0.0147883
$837$	5.37627	0.185831
$838$	36.3773	1.25663
$839$	33.2403	1.14758	0.573791	−	0.819002i	$-0.305472\pi$
$839$	33.2403	1.14758	0.573791	+	0.819002i	$0.305472\pi$
$840$	0	0
$841$	75.7531	2.61218
$842$	−38.9323	−1.34170
$843$	−48.3236	−1.66435
$844$	−1.81786	−0.0625732
$845$	0	0
$846$	7.97046	0.274030
$847$	−17.0030	−0.584229
$848$	53.6822	1.84346
$849$	−12.2620	−0.420832
$850$	0	0
$851$	−0.350191	−0.0120044
$852$	1.74764	0.0598733
$853$	24.3086	0.832310	0.416155	−	0.909294i	$-0.363377\pi$
$853$	24.3086	0.832310	0.416155	+	0.909294i	$0.363377\pi$
$854$	−14.8668	−0.508731
$855$	0	0
$856$	−7.32889	−0.250496
$857$	57.3889	1.96037	0.980185	−	0.198087i	$-0.0634727\pi$
$857$	57.3889	1.96037	0.980185	+	0.198087i	$0.0634727\pi$
$858$	−3.11960	−0.106502
$859$	−13.8135	−0.471312	−0.235656	−	0.971837i	$-0.575724\pi$
$859$	−13.8135	−0.471312	−0.235656	+	0.971837i	$0.575724\pi$
$860$	0	0
$861$	−35.8267	−1.22097
$862$	−27.3817	−0.932623
$863$	9.01938	0.307023	0.153512	−	0.988147i	$-0.450942\pi$
$863$	9.01938	0.307023	0.153512	+	0.988147i	$0.450942\pi$
$864$	1.06484	0.0362265
$865$	0	0
$866$	−5.33081	−0.181148
$867$	35.2935	1.19863
$868$	−0.300894	−0.0102130
$869$	52.1027	1.76746
$870$	0	0
$871$	1.58748	0.0537898
$872$	−6.90494	−0.233831
$873$	−0.405813	−0.0137347
$874$	−3.69202	−0.124884
$875$	0	0
$876$	−2.37090	−0.0801052
$877$	2.37675	0.0802570	0.0401285	−	0.999195i	$-0.487223\pi$
$877$	2.37675	0.0802570	0.0401285	+	0.999195i	$0.487223\pi$
$878$	37.1269	1.25297
$879$	16.6069	0.560135
$880$	0	0
$881$	−7.99330	−0.269301	−0.134650	−	0.990893i	$-0.542991\pi$
$881$	−7.99330	−0.269301	−0.134650	+	0.990893i	$0.542991\pi$
$882$	15.2741	0.514307
$883$	1.76377	0.0593557	0.0296779	−	0.999560i	$-0.490552\pi$
$883$	1.76377	0.0593557	0.0296779	+	0.999560i	$0.490552\pi$
$884$	−0.0198513	−0.000667672 0
$885$	0	0
$886$	−39.5439	−1.32850
$887$	−45.9197	−1.54183	−0.770917	−	0.636936i	$-0.780202\pi$
$887$	−45.9197	−1.54183	−0.770917	+	0.636936i	$0.780202\pi$
$888$	−0.850855	−0.0285528
$889$	27.6926	0.928780
$890$	0	0
$891$	−53.1105	−1.77927
$892$	−0.703155	−0.0235434
$893$	2.69202	0.0900851
$894$	−49.8558	−1.66743
$895$	0	0
$896$	16.2874	0.544125
$897$	−1.13706	−0.0379654
$898$	−10.9215	−0.364457
$899$	−25.7482	−0.858752
$900$	0	0
$901$	−14.6431	−0.487833
$902$	82.3682	2.74256
$903$	23.1468	0.770276
$904$	−23.5599	−0.783589
$905$	0	0
$906$	−33.2150	−1.10350
$907$	−55.0549	−1.82807	−0.914034	−	0.405638i	$-0.867049\pi$
$907$	−55.0549	−1.82807	−0.914034	+	0.405638i	$0.867049\pi$
$908$	−1.74478	−0.0579024
$909$	23.5810	0.782134
$910$	0	0
$911$	51.8998	1.71952	0.859758	−	0.510702i	$-0.170614\pi$
$911$	51.8998	1.71952	0.859758	+	0.510702i	$0.170614\pi$
$912$	−9.36658	−0.310159
$913$	−18.3110	−0.606004
$914$	−11.3526	−0.375510
$915$	0	0
$916$	−0.355568	−0.0117483
$917$	17.9288	0.592062
$918$	−3.51142	−0.115894
$919$	11.4614	0.378078	0.189039	−	0.981970i	$-0.439463\pi$
$919$	11.4614	0.378078	0.189039	+	0.981970i	$0.439463\pi$
$920$	0	0
$921$	−72.3822	−2.38508
$922$	−5.60541	−0.184604
$923$	1.74764	0.0575244
$924$	1.30367	0.0428875
$925$	0	0
$926$	−18.9235	−0.621864
$927$	−31.0543	−1.01996
$928$	−5.09975	−0.167408
$929$	36.5295	1.19849	0.599246	−	0.800565i	$-0.295467\pi$
$929$	36.5295	1.19849	0.599246	+	0.800565i	$0.295467\pi$
$930$	0	0
$931$	5.15883	0.169074
$932$	−2.36372	−0.0774261
$933$	−33.3545	−1.09198
$934$	9.30644	0.304516
$935$	0	0
$936$	−1.12114	−0.0366457
$937$	48.7845	1.59372	0.796860	−	0.604164i	$-0.206493\pi$
$937$	48.7845	1.59372	0.796860	+	0.604164i	$0.206493\pi$
$938$	−15.7157	−0.513136
$939$	−29.5459	−0.964193
$940$	0	0
$941$	−18.1817	−0.592705	−0.296353	−	0.955079i	$-0.595770\pi$
$941$	−18.1817	−0.592705	−0.296353	+	0.955079i	$0.595770\pi$
$942$	−10.2959	−0.335458
$943$	30.0224	0.977663
$944$	−24.2763	−0.790125
$945$	0	0
$946$	−53.2161	−1.73021
$947$	7.55150	0.245391	0.122695	−	0.992444i	$-0.460846\pi$
$947$	7.55150	0.245391	0.122695	+	0.992444i	$0.460846\pi$
$948$	−2.12737	−0.0690939
$949$	−2.37090	−0.0769626
$950$	0	0
$951$	−11.5375	−0.374129
$952$	−4.26252	−0.138149
$953$	−13.8592	−0.448944	−0.224472	−	0.974481i	$-0.572066\pi$
$953$	−13.8592	−0.448944	−0.224472	+	0.974481i	$0.572066\pi$
$954$	38.1288	1.23447
$955$	0	0
$956$	0.296371	0.00958532
$957$	111.558	3.60616
$958$	−8.22367	−0.265695
$959$	−9.30904	−0.300605
$960$	0	0
$961$	−24.6711	−0.795842
$962$	0.0392287	0.00126478
$963$	−5.43535	−0.175152
$964$	2.44611	0.0787838
$965$	0	0
$966$	11.2567	0.362177
$967$	−4.89977	−0.157566	−0.0787830	−	0.996892i	$-0.525103\pi$
$967$	−4.89977	−0.157566	−0.0787830	+	0.996892i	$0.525103\pi$
$968$	34.6189	1.11269
$969$	2.55496	0.0820771
$970$	0	0
$971$	14.5133	0.465755	0.232878	−	0.972506i	$-0.425186\pi$
$971$	14.5133	0.465755	0.232878	+	0.972506i	$0.425186\pi$
$972$	1.60340	0.0514291
$973$	−5.81594	−0.186450
$974$	−18.7323	−0.600222
$975$	0	0
$976$	31.6062	1.01169
$977$	19.6644	0.629120	0.314560	−	0.949238i	$-0.398143\pi$
$977$	19.6644	0.629120	0.314560	+	0.949238i	$0.398143\pi$
$978$	15.0586	0.481521
$979$	45.4359	1.45214
$980$	0	0
$981$	−5.12093	−0.163499
$982$	28.3293	0.904023
$983$	15.4397	0.492449	0.246224	−	0.969213i	$-0.420810\pi$
$983$	15.4397	0.492449	0.246224	+	0.969213i	$0.420810\pi$
$984$	72.9450	2.32540
$985$	0	0
$986$	16.8170	0.535562
$987$	−8.20775	−0.261256
$988$	0.0174584	0.000555426 0
$989$	−19.3967	−0.616780
$990$	0	0
$991$	11.9377	0.379213	0.189606	−	0.981860i	$-0.439279\pi$
$991$	11.9377	0.379213	0.189606	+	0.981860i	$0.439279\pi$
$992$	1.25352	0.0397991
$993$	−4.51573	−0.143302
$994$	−17.3013	−0.548763
$995$	0	0
$996$	0.747644	0.0236900
$997$	−17.9390	−0.568134	−0.284067	−	0.958804i	$-0.591684\pi$
$997$	−17.9390	−0.568134	−0.284067	+	0.958804i	$0.591684\pi$
$998$	−7.94523	−0.251502
$999$	0.292913	0.00926736

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.2.a.d.1.2	✓	3
3.2	odd	2		4275.2.a.bn.1.2		3
4.3	odd	2		7600.2.a.bw.1.3		3
5.2	odd	4		475.2.b.c.324.2		6
5.3	odd	4		475.2.b.c.324.5		6
5.4	even	2		475.2.a.h.1.2	yes	3
15.14	odd	2		4275.2.a.z.1.2		3
19.18	odd	2		9025.2.a.be.1.2		3
20.19	odd	2		7600.2.a.bn.1.1		3
95.94	odd	2		9025.2.a.w.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.2	✓	3	1.1	even	1	trivial
475.2.a.h.1.2	yes	3	5.4	even	2
475.2.b.c.324.2		6	5.2	odd	4
475.2.b.c.324.5		6	5.3	odd	4
4275.2.a.z.1.2		3	15.14	odd	2
4275.2.a.bn.1.2		3	3.2	odd	2
7600.2.a.bn.1.1		3	20.19	odd	2
7600.2.a.bw.1.3		3	4.3	odd	2
9025.2.a.w.1.2		3	95.94	odd	2
9025.2.a.be.1.2		3	19.18	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q-1.44504 q^{2} -2.24698 q^{3} +0.0881460 q^{4} +3.24698 q^{6} -1.35690 q^{7} +2.76271 q^{8} +2.04892 q^{9} +4.85086 q^{11} -0.198062 q^{12} -0.198062 q^{13} +1.96077 q^{14} -4.16852 q^{16} +1.13706 q^{17} -2.96077 q^{18} -1.00000 q^{19} +3.04892 q^{21} -7.00969 q^{22} -2.55496 q^{23} -6.20775 q^{24} +0.286208 q^{26} +2.13706 q^{27} -0.119605 q^{28} -10.2349 q^{29} +2.51573 q^{31} +0.498271 q^{32} -10.8998 q^{33} -1.64310 q^{34} +0.180604 q^{36} +0.137063 q^{37} +1.44504 q^{38} +0.445042 q^{39} -11.7506 q^{41} -4.40581 q^{42} +7.59179 q^{43} +0.427583 q^{44} +3.69202 q^{46} -2.69202 q^{47} +9.36658 q^{48} -5.15883 q^{49} -2.55496 q^{51} -0.0174584 q^{52} -12.8780 q^{53} -3.08815 q^{54} -3.74871 q^{56} +2.24698 q^{57} +14.7899 q^{58} +5.82371 q^{59} -7.58211 q^{61} -3.63533 q^{62} -2.78017 q^{63} +7.61702 q^{64} +15.7506 q^{66} -8.01507 q^{67} +0.100228 q^{68} +5.74094 q^{69} -8.82371 q^{71} +5.66056 q^{72} +11.9705 q^{73} -0.198062 q^{74} -0.0881460 q^{76} -6.58211 q^{77} -0.643104 q^{78} +10.7409 q^{79} -10.9487 q^{81} +16.9801 q^{82} -3.77479 q^{83} +0.268750 q^{84} -10.9705 q^{86} +22.9976 q^{87} +13.4015 q^{88} +9.36658 q^{89} +0.268750 q^{91} -0.225209 q^{92} -5.65279 q^{93} +3.89008 q^{94} -1.11960 q^{96} -0.198062 q^{97} +7.45473 q^{98} +9.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 5 q^{6} - 9 q^{8} - 3 q^{9} + q^{11} - 5 q^{12} - 5 q^{13} - 7 q^{14} + 18 q^{16} - 2 q^{17} + 4 q^{18} - 3 q^{19} + q^{22} - 8 q^{23} - q^{24} + 9 q^{26} + q^{27}+ \cdots + 20 q^{99}+O(q^{100}) \) 3 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 + 5 * q^6 - 9 * q^8 - 3 * q^9 + q^11 - 5 * q^12 - 5 * q^13 - 7 * q^14 + 18 * q^16 - 2 * q^17 + 4 * q^18 - 3 * q^19 + q^22 - 8 * q^23 - q^24 + 9 * q^26 + q^27 + 21 * q^28 - 7 * q^29 - 5 * q^31 - 27 * q^32 - 10 * q^33 - 9 * q^34 - 11 * q^36 - 5 * q^37 + 4 * q^38 + q^39 + q^41 - 5 * q^43 - 15 * q^44 + 6 * q^46 - 3 * q^47 + 2 * q^48 - 7 * q^49 - 8 * q^51 - 16 * q^52 - 19 * q^53 - 13 * q^54 - 35 * q^56 + 2 * q^57 + 21 * q^58 + 10 * q^59 - 17 * q^61 + 23 * q^62 - 7 * q^63 + 49 * q^64 + 11 * q^66 + q^67 + 23 * q^68 + 3 * q^69 - 19 * q^71 + 37 * q^72 + q^73 - 5 * q^74 - 4 * q^76 - 14 * q^77 - 6 * q^78 + 18 * q^79 - q^81 - 6 * q^82 - 13 * q^83 - 7 * q^84 + 2 * q^86 + 28 * q^87 + 46 * q^88 + 2 * q^89 - 7 * q^91 + q^92 + q^93 + 11 * q^94 + 18 * q^96 - 5 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

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Database

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