LMFDB - Embedded newform 3775.2.a.r.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3775,2,Mod(1,3775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3775, 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("3775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3775 = 5^{2} \cdot 151$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3775.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:	$30.1435267630$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{18} - 2 x^{17} - 32 x^{16} + 64 x^{15} + 417 x^{14} - 839 x^{13} - 2829 x^{12} + 5789 x^{11} + \cdots + 576$ x^18 - 2x^17 - 32x^16 + 64x^15 + 417x^14 - 839x^13 - 2829x^12 + 5789x^11 + 10562x^10 - 22471x^9 - 20867x^8 + 48611x^7 + 18052x^6 - 54316x^5 - 788x^4 + 25424x^3 - 5248x^2 - 2240*x + 576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 755)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.47536$ of defining polynomial
Character	$\chi$	$=$	3775.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.47536 q^{2} +2.78489 q^{3} +0.176690 q^{4} +4.10872 q^{6} +0.173538 q^{7} -2.69004 q^{8} +4.75562 q^{9} -3.80721 q^{11} +0.492061 q^{12} +4.71035 q^{13} +0.256031 q^{14} -4.32216 q^{16} +7.22152 q^{17} +7.01626 q^{18} +0.641297 q^{19} +0.483284 q^{21} -5.61701 q^{22} +7.75290 q^{23} -7.49147 q^{24} +6.94946 q^{26} +4.88922 q^{27} +0.0306623 q^{28} +3.13775 q^{29} -4.18639 q^{31} -0.996664 q^{32} -10.6027 q^{33} +10.6544 q^{34} +0.840269 q^{36} +4.45008 q^{37} +0.946145 q^{38} +13.1178 q^{39} +8.48955 q^{41} +0.713018 q^{42} -11.0223 q^{43} -0.672695 q^{44} +11.4383 q^{46} +10.8184 q^{47} -12.0367 q^{48} -6.96988 q^{49} +20.1112 q^{51} +0.832269 q^{52} -12.8581 q^{53} +7.21337 q^{54} -0.466823 q^{56} +1.78594 q^{57} +4.62932 q^{58} -6.72790 q^{59} +9.26506 q^{61} -6.17643 q^{62} +0.825280 q^{63} +7.17388 q^{64} -15.6428 q^{66} +4.78472 q^{67} +1.27597 q^{68} +21.5910 q^{69} -11.7482 q^{71} -12.7928 q^{72} +5.96425 q^{73} +6.56548 q^{74} +0.113311 q^{76} -0.660695 q^{77} +19.3535 q^{78} +12.5255 q^{79} -0.650916 q^{81} +12.5251 q^{82} +2.31172 q^{83} +0.0853912 q^{84} -16.2619 q^{86} +8.73830 q^{87} +10.2416 q^{88} -6.84864 q^{89} +0.817423 q^{91} +1.36986 q^{92} -11.6586 q^{93} +15.9611 q^{94} -2.77560 q^{96} -3.06816 q^{97} -10.2831 q^{98} -18.1057 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$18 q + 2 q^{2} - 2 q^{3} + 32 q^{4} + 10 q^{6} - 4 q^{7} + 32 q^{9} + 11 q^{11} - 12 q^{13} + q^{14} + 60 q^{16} + 25 q^{17} + 9 q^{18} + 8 q^{19} + 20 q^{21} - 29 q^{22} - 12 q^{23} + 15 q^{24} + 5 q^{26}+ \cdots + 11 q^{99}+O(q^{100})$ 18 * q + 2 * q^2 - 2 * q^3 + 32 * q^4 + 10 * q^6 - 4 * q^7 + 32 * q^9 + 11 * q^11 - 12 * q^13 + q^14 + 60 * q^16 + 25 * q^17 + 9 * q^18 + 8 * q^19 + 20 * q^21 - 29 * q^22 - 12 * q^23 + 15 * q^24 + 5 * q^26 - 5 * q^27 - 15 * q^28 + 24 * q^29 + 11 * q^31 + 17 * q^32 + 33 * q^33 + 17 * q^34 + 90 * q^36 - 45 * q^37 + 28 * q^38 + 38 * q^39 + 12 * q^41 + 9 * q^42 - 19 * q^43 + 6 * q^44 - 5 * q^46 + 16 * q^47 + 3 * q^48 + 62 * q^49 + 7 * q^51 - q^52 + 4 * q^53 - 6 * q^54 + 2 * q^56 - 5 * q^57 - 28 * q^58 - 8 * q^59 + 27 * q^61 + 57 * q^62 - 10 * q^63 + 98 * q^64 - 32 * q^66 - 9 * q^67 + 26 * q^68 + 31 * q^69 + 9 * q^71 + 102 * q^72 - 2 * q^73 - 24 * q^74 - 43 * q^76 - 11 * q^77 - 45 * q^78 + 20 * q^79 + 66 * q^81 + 19 * q^82 + 20 * q^83 - 33 * q^84 + 21 * q^86 + 33 * q^87 - 62 * q^88 + 13 * q^89 + 22 * q^91 + 6 * q^92 - 58 * q^93 + 70 * q^94 - 17 * q^96 + 27 * q^97 + 6 * q^98 + 11 * 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

)

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

$2$	1.47536	1.04324	0.521619	−	0.853179i	$-0.325328\pi$
$2$	1.47536	1.04324	0.521619	+	0.853179i	$0.325328\pi$
$3$	2.78489	1.60786	0.803929	−	0.594725i	$-0.202739\pi$
$3$	2.78489	1.60786	0.803929	+	0.594725i	$0.202739\pi$
$4$	0.176690	0.0883448
$5$	0	0
$5$	0	0
$6$	4.10872	1.67738
$7$	0.173538	0.0655911	0.0327955	−	0.999462i	$-0.489559\pi$
$7$	0.173538	0.0655911	0.0327955	+	0.999462i	$0.489559\pi$
$8$	−2.69004	−0.951073
$9$	4.75562	1.58521
$10$	0	0
$11$	−3.80721	−1.14792	−0.573959	−	0.818884i	$-0.694593\pi$
$11$	−3.80721	−1.14792	−0.573959	+	0.818884i	$0.694593\pi$
$12$	0.492061	0.142046
$13$	4.71035	1.30641	0.653207	−	0.757179i	$-0.273423\pi$
$13$	4.71035	1.30641	0.653207	+	0.757179i	$0.273423\pi$
$14$	0.256031	0.0684271
$15$	0	0
$16$	−4.32216	−1.08054
$17$	7.22152	1.75148	0.875738	−	0.482786i	$-0.160375\pi$
$17$	7.22152	1.75148	0.875738	+	0.482786i	$0.160375\pi$
$18$	7.01626	1.65375
$19$	0.641297	0.147124	0.0735619	−	0.997291i	$-0.476563\pi$
$19$	0.641297	0.147124	0.0735619	+	0.997291i	$0.476563\pi$
$20$	0	0
$21$	0.483284	0.105461
$22$	−5.61701	−1.19755
$23$	7.75290	1.61659	0.808296	−	0.588777i	$-0.200390\pi$
$23$	7.75290	1.61659	0.808296	+	0.588777i	$0.200390\pi$
$24$	−7.49147	−1.52919
$25$	0	0
$26$	6.94946	1.36290
$27$	4.88922	0.940931
$28$	0.0306623	0.00579463
$29$	3.13775	0.582666	0.291333	−	0.956622i	$-0.405901\pi$
$29$	3.13775	0.582666	0.291333	+	0.956622i	$0.405901\pi$
$30$	0	0
$31$	−4.18639	−0.751898	−0.375949	−	0.926640i	$-0.622683\pi$
$31$	−4.18639	−0.751898	−0.375949	+	0.926640i	$0.622683\pi$
$32$	−0.996664	−0.176187
$33$	−10.6027	−1.84569
$34$	10.6544	1.82721
$35$	0	0
$36$	0.840269	0.140045
$37$	4.45008	0.731589	0.365795	−	0.930696i	$-0.380797\pi$
$37$	4.45008	0.731589	0.365795	+	0.930696i	$0.380797\pi$
$38$	0.946145	0.153485
$39$	13.1178	2.10053
$40$	0	0
$41$	8.48955	1.32584	0.662922	−	0.748688i	$-0.269316\pi$
$41$	8.48955	1.32584	0.662922	+	0.748688i	$0.269316\pi$
$42$	0.713018	0.110021
$43$	−11.0223	−1.68089	−0.840446	−	0.541895i	$-0.817707\pi$
$43$	−11.0223	−1.68089	−0.840446	+	0.541895i	$0.817707\pi$
$44$	−0.672695	−0.101413
$45$	0	0
$46$	11.4383	1.68649
$47$	10.8184	1.57803	0.789015	−	0.614374i	$-0.210591\pi$
$47$	10.8184	1.57803	0.789015	+	0.614374i	$0.210591\pi$
$48$	−12.0367	−1.73736
$49$	−6.96988	−0.995698
$50$	0	0
$51$	20.1112	2.81613
$52$	0.832269	0.115415
$53$	−12.8581	−1.76620	−0.883100	−	0.469184i	$-0.844548\pi$
$53$	−12.8581	−1.76620	−0.883100	+	0.469184i	$0.844548\pi$
$54$	7.21337	0.981615
$55$	0	0
$56$	−0.466823	−0.0623819
$57$	1.78594	0.236554
$58$	4.62932	0.607859
$59$	−6.72790	−0.875898	−0.437949	−	0.899000i	$-0.644295\pi$
$59$	−6.72790	−0.875898	−0.437949	+	0.899000i	$0.644295\pi$
$60$	0	0
$61$	9.26506	1.18627	0.593134	−	0.805103i	$-0.297890\pi$
$61$	9.26506	1.18627	0.593134	+	0.805103i	$0.297890\pi$
$62$	−6.17643	−0.784408
$63$	0.825280	0.103975
$64$	7.17388	0.896735
$65$	0	0
$66$	−15.6428	−1.92549
$67$	4.78472	0.584547	0.292273	−	0.956335i	$-0.405588\pi$
$67$	4.78472	0.584547	0.292273	+	0.956335i	$0.405588\pi$
$68$	1.27597	0.154734
$69$	21.5910	2.59925
$70$	0	0
$71$	−11.7482	−1.39426	−0.697128	−	0.716947i	$-0.745539\pi$
$71$	−11.7482	−1.39426	−0.697128	+	0.716947i	$0.745539\pi$
$72$	−12.7928	−1.50765
$73$	5.96425	0.698063	0.349031	−	0.937111i	$-0.386511\pi$
$73$	5.96425	0.698063	0.349031	+	0.937111i	$0.386511\pi$
$74$	6.56548	0.763222
$75$	0	0
$76$	0.113311	0.0129976
$77$	−0.660695	−0.0752932
$78$	19.3535	2.19135
$79$	12.5255	1.40923	0.704614	−	0.709591i	$-0.251120\pi$
$79$	12.5255	1.40923	0.704614	+	0.709591i	$0.251120\pi$
$80$	0	0
$81$	−0.650916	−0.0723240
$82$	12.5251	1.38317
$83$	2.31172	0.253744	0.126872	−	0.991919i	$-0.459506\pi$
$83$	2.31172	0.253744	0.126872	+	0.991919i	$0.459506\pi$
$84$	0.0853912	0.00931694
$85$	0	0
$86$	−16.2619	−1.75357
$87$	8.73830	0.936844
$88$	10.2416	1.09175
$89$	−6.84864	−0.725955	−0.362977	−	0.931798i	$-0.618240\pi$
$89$	−6.84864	−0.725955	−0.362977	+	0.931798i	$0.618240\pi$
$90$	0	0
$91$	0.817423	0.0856892
$92$	1.36986	0.142817
$93$	−11.6586	−1.20894
$94$	15.9611	1.64626
$95$	0	0
$96$	−2.77560	−0.283284
$97$	−3.06816	−0.311525	−0.155762	−	0.987795i	$-0.549783\pi$
$97$	−3.06816	−0.311525	−0.155762	+	0.987795i	$0.549783\pi$
$98$	−10.2831	−1.03875
$99$	−18.1057	−1.81969
$100$	0	0
$101$	5.39034	0.536359	0.268180	−	0.963369i	$-0.413578\pi$
$101$	5.39034	0.536359	0.268180	+	0.963369i	$0.413578\pi$
$102$	29.6712	2.93789
$103$	−5.71787	−0.563399	−0.281699	−	0.959503i	$-0.590898\pi$
$103$	−5.71787	−0.563399	−0.281699	+	0.959503i	$0.590898\pi$
$104$	−12.6710	−1.24250
$105$	0	0
$106$	−18.9704	−1.84257
$107$	−1.49999	−0.145010	−0.0725049	−	0.997368i	$-0.523099\pi$
$107$	−1.49999	−0.145010	−0.0725049	+	0.997368i	$0.523099\pi$
$108$	0.863875	0.0831264
$109$	18.5905	1.78065	0.890323	−	0.455329i	$-0.150478\pi$
$109$	18.5905	1.78065	0.890323	+	0.455329i	$0.150478\pi$
$110$	0	0
$111$	12.3930	1.17629
$112$	−0.750058	−0.0708738
$113$	−15.8434	−1.49042	−0.745212	−	0.666828i	$-0.767652\pi$
$113$	−15.8434	−1.49042	−0.745212	+	0.666828i	$0.767652\pi$
$114$	2.63491	0.246782
$115$	0	0
$116$	0.554408	0.0514755
$117$	22.4006	2.07094
$118$	−9.92608	−0.913770
$119$	1.25321	0.114881
$120$	0	0
$121$	3.49488	0.317716
$122$	13.6693	1.23756
$123$	23.6425	2.13177
$124$	−0.739691	−0.0664262
$125$	0	0
$126$	1.21759	0.108471
$127$	−0.732444	−0.0649939	−0.0324969	−	0.999472i	$-0.510346\pi$
$127$	−0.732444	−0.0649939	−0.0324969	+	0.999472i	$0.510346\pi$
$128$	12.5774	1.11169
$129$	−30.6960	−2.70264
$130$	0	0
$131$	−7.67556	−0.670617	−0.335309	−	0.942108i	$-0.608841\pi$
$131$	−7.67556	−0.670617	−0.335309	+	0.942108i	$0.608841\pi$
$132$	−1.87338	−0.163057
$133$	0.111289	0.00965000
$134$	7.05919	0.609821
$135$	0	0
$136$	−19.4262	−1.66578
$137$	6.21515	0.530996	0.265498	−	0.964111i	$-0.414464\pi$
$137$	6.21515	0.530996	0.265498	+	0.964111i	$0.414464\pi$
$138$	31.8545	2.71163
$139$	18.5423	1.57274	0.786370	−	0.617756i	$-0.211958\pi$
$139$	18.5423	1.57274	0.786370	+	0.617756i	$0.211958\pi$
$140$	0	0
$141$	30.1282	2.53725
$142$	−17.3329	−1.45454
$143$	−17.9333	−1.49966
$144$	−20.5546	−1.71288
$145$	0	0
$146$	8.79942	0.728245
$147$	−19.4104	−1.60094
$148$	0.786284	0.0646321
$149$	0.940244	0.0770278	0.0385139	−	0.999258i	$-0.487738\pi$
$149$	0.940244	0.0770278	0.0385139	+	0.999258i	$0.487738\pi$
$150$	0	0
$151$	1.00000	0.0813788
$151$	1.00000	0.0813788
$152$	−1.72512	−0.139925
$153$	34.3428	2.77645
$154$	−0.974764	−0.0785487
$155$	0	0
$156$	2.31778	0.185571
$157$	−10.4672	−0.835377	−0.417688	−	0.908590i	$-0.637160\pi$
$157$	−10.4672	−0.835377	−0.417688	+	0.908590i	$0.637160\pi$
$158$	18.4796	1.47016
$159$	−35.8085	−2.83980
$160$	0	0
$161$	1.34542	0.106034
$162$	−0.960335	−0.0754511
$163$	0.469671	0.0367875	0.0183937	−	0.999831i	$-0.494145\pi$
$163$	0.469671	0.0367875	0.0183937	+	0.999831i	$0.494145\pi$
$164$	1.50001	0.117131
$165$	0	0
$166$	3.41062	0.264716
$167$	16.2913	1.26066	0.630331	−	0.776326i	$-0.282919\pi$
$167$	16.2913	1.26066	0.630331	+	0.776326i	$0.282919\pi$
$168$	−1.30005	−0.100301
$169$	9.18736	0.706720
$170$	0	0
$171$	3.04977	0.233222
$172$	−1.94753	−0.148498
$173$	−8.43744	−0.641486	−0.320743	−	0.947166i	$-0.603933\pi$
$173$	−8.43744	−0.641486	−0.320743	+	0.947166i	$0.603933\pi$
$174$	12.8921	0.977351
$175$	0	0
$176$	16.4554	1.24037
$177$	−18.7365	−1.40832
$178$	−10.1042	−0.757343
$179$	−24.5628	−1.83591	−0.917955	−	0.396685i	$-0.870160\pi$
$179$	−24.5628	−1.83591	−0.917955	+	0.396685i	$0.870160\pi$
$180$	0	0
$181$	3.72844	0.277133	0.138566	−	0.990353i	$-0.455751\pi$
$181$	3.72844	0.277133	0.138566	+	0.990353i	$0.455751\pi$
$182$	1.20599	0.0893942
$183$	25.8022	1.90735
$184$	−20.8556	−1.53750
$185$	0	0
$186$	−17.2007	−1.26122
$187$	−27.4939	−2.01055
$188$	1.91150	0.139411
$189$	0.848464	0.0617167
$190$	0	0
$191$	−23.5169	−1.70162	−0.850812	−	0.525471i	$-0.823889\pi$
$191$	−23.5169	−1.70162	−0.850812	+	0.525471i	$0.823889\pi$
$192$	19.9785	1.44182
$193$	−23.0958	−1.66248	−0.831238	−	0.555917i	$-0.812367\pi$
$193$	−23.0958	−1.66248	−0.831238	+	0.555917i	$0.812367\pi$
$194$	−4.52665	−0.324994
$195$	0	0
$196$	−1.23151	−0.0879647
$197$	7.82625	0.557597	0.278798	−	0.960350i	$-0.410064\pi$
$197$	7.82625	0.557597	0.278798	+	0.960350i	$0.410064\pi$
$198$	−26.7124	−1.89837
$199$	−17.3065	−1.22683	−0.613414	−	0.789762i	$-0.710204\pi$
$199$	−17.3065	−1.22683	−0.613414	+	0.789762i	$0.710204\pi$
$200$	0	0
$201$	13.3249	0.939868
$202$	7.95270	0.559550
$203$	0.544518	0.0382177
$204$	3.55343	0.248790
$205$	0	0
$206$	−8.43593	−0.587759
$207$	36.8699	2.56263
$208$	−20.3589	−1.41163
$209$	−2.44156	−0.168886
$210$	0	0
$211$	−11.1570	−0.768079	−0.384040	−	0.923317i	$-0.625467\pi$
$211$	−11.1570	−0.768079	−0.384040	+	0.923317i	$0.625467\pi$
$212$	−2.27190	−0.156035
$213$	−32.7175	−2.24177
$214$	−2.21303	−0.151280
$215$	0	0
$216$	−13.1522	−0.894894
$217$	−0.726496	−0.0493178
$218$	27.4277	1.85764
$219$	16.6098	1.12239
$220$	0	0
$221$	34.0159	2.28816
$222$	18.2842	1.22715
$223$	−20.4613	−1.37019	−0.685094	−	0.728454i	$-0.740239\pi$
$223$	−20.4613	−1.37019	−0.685094	+	0.728454i	$0.740239\pi$
$224$	−0.172959	−0.0115563
$225$	0	0
$226$	−23.3748	−1.55487
$227$	−12.0239	−0.798056	−0.399028	−	0.916939i	$-0.630652\pi$
$227$	−12.0239	−0.798056	−0.399028	+	0.916939i	$0.630652\pi$
$228$	0.315558	0.0208983
$229$	−4.13374	−0.273166	−0.136583	−	0.990629i	$-0.543612\pi$
$229$	−4.13374	−0.273166	−0.136583	+	0.990629i	$0.543612\pi$
$230$	0	0
$231$	−1.83996	−0.121061
$232$	−8.44068	−0.554158
$233$	11.9875	0.785324	0.392662	−	0.919683i	$-0.371554\pi$
$233$	11.9875	0.785324	0.392662	+	0.919683i	$0.371554\pi$
$234$	33.0490	2.16048
$235$	0	0
$236$	−1.18875	−0.0773810
$237$	34.8822	2.26584
$238$	1.84893	0.119848
$239$	−2.21028	−0.142971	−0.0714855	−	0.997442i	$-0.522774\pi$
$239$	−2.21028	−0.142971	−0.0714855	+	0.997442i	$0.522774\pi$
$240$	0	0
$241$	−4.24934	−0.273724	−0.136862	−	0.990590i	$-0.543702\pi$
$241$	−4.24934	−0.273724	−0.136862	+	0.990590i	$0.543702\pi$
$242$	5.15621	0.331454
$243$	−16.4804	−1.05722
$244$	1.63704	0.104801
$245$	0	0
$246$	34.8812	2.22394
$247$	3.02073	0.192205
$248$	11.2616	0.715109
$249$	6.43789	0.407985
$250$	0	0
$251$	6.45202	0.407248	0.203624	−	0.979049i	$-0.434728\pi$
$251$	6.45202	0.407248	0.203624	+	0.979049i	$0.434728\pi$
$252$	0.145818	0.00918569
$253$	−29.5169	−1.85571
$254$	−1.08062	−0.0678040
$255$	0	0
$256$	4.20843	0.263027
$257$	−4.36198	−0.272093	−0.136046	−	0.990702i	$-0.543440\pi$
$257$	−4.36198	−0.272093	−0.136046	+	0.990702i	$0.543440\pi$
$258$	−45.2877	−2.81949
$259$	0.772257	0.0479857
$260$	0	0
$261$	14.9220	0.923647
$262$	−11.3242	−0.699613
$263$	−22.5697	−1.39171	−0.695854	−	0.718183i	$-0.744974\pi$
$263$	−22.5697	−1.39171	−0.695854	+	0.718183i	$0.744974\pi$
$264$	28.5216	1.75539
$265$	0	0
$266$	0.164192	0.0100672
$267$	−19.0727	−1.16723
$268$	0.845410	0.0516416
$269$	−16.1107	−0.982285	−0.491142	−	0.871079i	$-0.663421\pi$
$269$	−16.1107	−0.982285	−0.491142	+	0.871079i	$0.663421\pi$
$270$	0	0
$271$	−21.1405	−1.28420	−0.642098	−	0.766623i	$-0.721936\pi$
$271$	−21.1405	−1.28420	−0.642098	+	0.766623i	$0.721936\pi$
$272$	−31.2126	−1.89254
$273$	2.27643	0.137776
$274$	9.16958	0.553955
$275$	0	0
$276$	3.81490	0.229630
$277$	−15.9560	−0.958706	−0.479353	−	0.877622i	$-0.659129\pi$
$277$	−15.9560	−0.958706	−0.479353	+	0.877622i	$0.659129\pi$
$278$	27.3566	1.64074
$279$	−19.9089	−1.19191
$280$	0	0
$281$	15.1372	0.903010	0.451505	−	0.892269i	$-0.350887\pi$
$281$	15.1372	0.903010	0.451505	+	0.892269i	$0.350887\pi$
$282$	44.4499	2.64695
$283$	5.47901	0.325693	0.162847	−	0.986651i	$-0.447932\pi$
$283$	5.47901	0.325693	0.162847	+	0.986651i	$0.447932\pi$
$284$	−2.07579	−0.123175
$285$	0	0
$286$	−26.4581	−1.56450
$287$	1.47326	0.0869636
$288$	−4.73976	−0.279293
$289$	35.1504	2.06767
$290$	0	0
$291$	−8.54451	−0.500888
$292$	1.05382	0.0616702
$293$	6.72375	0.392806	0.196403	−	0.980523i	$-0.437074\pi$
$293$	6.72375	0.392806	0.196403	+	0.980523i	$0.437074\pi$
$294$	−28.6373	−1.67016
$295$	0	0
$296$	−11.9709	−0.695795
$297$	−18.6143	−1.08011
$298$	1.38720	0.0803583
$299$	36.5188	2.11194
$300$	0	0
$301$	−1.91279	−0.110252
$302$	1.47536	0.0848975
$303$	15.0115	0.862390
$304$	−2.77179	−0.158973
$305$	0	0
$306$	50.6681	2.89650
$307$	18.5754	1.06015	0.530076	−	0.847950i	$-0.322164\pi$
$307$	18.5754	1.06015	0.530076	+	0.847950i	$0.322164\pi$
$308$	−0.116738	−0.00665176
$309$	−15.9237	−0.905865
$310$	0	0
$311$	−1.09338	−0.0620002	−0.0310001	−	0.999519i	$-0.509869\pi$
$311$	−1.09338	−0.0620002	−0.0310001	+	0.999519i	$0.509869\pi$
$312$	−35.2874	−1.99776
$313$	14.0269	0.792848	0.396424	−	0.918067i	$-0.370251\pi$
$313$	14.0269	0.792848	0.396424	+	0.918067i	$0.370251\pi$
$314$	−15.4430	−0.871496
$315$	0	0
$316$	2.21312	0.124498
$317$	−8.99835	−0.505398	−0.252699	−	0.967545i	$-0.581318\pi$
$317$	−8.99835	−0.505398	−0.252699	+	0.967545i	$0.581318\pi$
$318$	−52.8305	−2.96259
$319$	−11.9461	−0.668853
$320$	0	0
$321$	−4.17732	−0.233155
$322$	1.98498	0.110619
$323$	4.63114	0.257684
$324$	−0.115010	−0.00638944
$325$	0	0
$326$	0.692934	0.0383781
$327$	51.7725	2.86303
$328$	−22.8372	−1.26097
$329$	1.87741	0.103505
$330$	0	0
$331$	−15.1047	−0.830231	−0.415116	−	0.909769i	$-0.636259\pi$
$331$	−15.1047	−0.830231	−0.415116	+	0.909769i	$0.636259\pi$
$332$	0.408457	0.0224170
$333$	21.1629	1.15972
$334$	24.0356	1.31517
$335$	0	0
$336$	−2.08883	−0.113955
$337$	27.6940	1.50859	0.754294	−	0.656537i	$-0.227979\pi$
$337$	27.6940	1.50859	0.754294	+	0.656537i	$0.227979\pi$
$338$	13.5547	0.737277
$339$	−44.1222	−2.39639
$340$	0	0
$341$	15.9385	0.863117
$342$	4.49951	0.243306
$343$	−2.42430	−0.130900
$344$	29.6506	1.59865
$345$	0	0
$346$	−12.4483	−0.669223
$347$	−21.0757	−1.13140	−0.565700	−	0.824611i	$-0.691394\pi$
$347$	−21.0757	−1.13140	−0.565700	+	0.824611i	$0.691394\pi$
$348$	1.54397	0.0827653
$349$	−7.07610	−0.378775	−0.189388	−	0.981902i	$-0.560650\pi$
$349$	−7.07610	−0.378775	−0.189388	+	0.981902i	$0.560650\pi$
$350$	0	0
$351$	23.0299	1.22925
$352$	3.79451	0.202248
$353$	20.7508	1.10446	0.552228	−	0.833693i	$-0.313778\pi$
$353$	20.7508	1.10446	0.552228	+	0.833693i	$0.313778\pi$
$354$	−27.6431	−1.46921
$355$	0	0
$356$	−1.21008	−0.0641343
$357$	3.49004	0.184713
$358$	−36.2390	−1.91529
$359$	36.8349	1.94407	0.972036	−	0.234831i	$-0.0754537\pi$
$359$	36.8349	1.94407	0.972036	+	0.234831i	$0.0754537\pi$
$360$	0	0
$361$	−18.5887	−0.978355
$362$	5.50079	0.289115
$363$	9.73286	0.510843
$364$	0.144430	0.00757019
$365$	0	0
$366$	38.0675	1.98982
$367$	−14.9639	−0.781108	−0.390554	−	0.920580i	$-0.627717\pi$
$367$	−14.9639	−0.781108	−0.390554	+	0.920580i	$0.627717\pi$
$368$	−33.5093	−1.74679
$369$	40.3731	2.10174
$370$	0	0
$371$	−2.23137	−0.115847
$372$	−2.05996	−0.106804
$373$	12.6009	0.652450	0.326225	−	0.945292i	$-0.394223\pi$
$373$	12.6009	0.652450	0.326225	+	0.945292i	$0.394223\pi$
$374$	−40.5634	−2.09748
$375$	0	0
$376$	−29.1020	−1.50082
$377$	14.7799	0.761204
$378$	1.25179	0.0643852
$379$	8.18303	0.420334	0.210167	−	0.977666i	$-0.432599\pi$
$379$	8.18303	0.420334	0.210167	+	0.977666i	$0.432599\pi$
$380$	0	0
$381$	−2.03978	−0.104501
$382$	−34.6959	−1.77520
$383$	−32.6541	−1.66855	−0.834273	−	0.551352i	$-0.814112\pi$
$383$	−32.6541	−1.66855	−0.834273	+	0.551352i	$0.814112\pi$
$384$	35.0267	1.78745
$385$	0	0
$386$	−34.0747	−1.73436
$387$	−52.4181	−2.66456
$388$	−0.542113	−0.0275216
$389$	−15.6663	−0.794313	−0.397156	−	0.917751i	$-0.630003\pi$
$389$	−15.6663	−0.794313	−0.397156	+	0.917751i	$0.630003\pi$
$390$	0	0
$391$	55.9877	2.83142
$392$	18.7493	0.946981
$393$	−21.3756	−1.07826
$394$	11.5465	0.581706
$395$	0	0
$396$	−3.19908	−0.160760
$397$	−15.2366	−0.764701	−0.382351	−	0.924017i	$-0.624885\pi$
$397$	−15.2366	−0.764701	−0.382351	+	0.924017i	$0.624885\pi$
$398$	−25.5334	−1.27987
$399$	0.309929	0.0155158
$400$	0	0
$401$	29.4129	1.46881	0.734406	−	0.678711i	$-0.237461\pi$
$401$	29.4129	1.46881	0.734406	+	0.678711i	$0.237461\pi$
$402$	19.6591	0.980506
$403$	−19.7193	−0.982290
$404$	0.952418	0.0473845
$405$	0	0
$406$	0.803361	0.0398701
$407$	−16.9424	−0.839805
$408$	−54.0998	−2.67834
$409$	−16.9214	−0.836712	−0.418356	−	0.908283i	$-0.637393\pi$
$409$	−16.9214	−0.836712	−0.418356	+	0.908283i	$0.637393\pi$
$410$	0	0
$411$	17.3085	0.853766
$412$	−1.01029	−0.0497734
$413$	−1.16754	−0.0574511
$414$	54.3964	2.67344
$415$	0	0
$416$	−4.69463	−0.230173
$417$	51.6384	2.52874
$418$	−3.60218	−0.176188
$419$	−10.3963	−0.507893	−0.253947	−	0.967218i	$-0.581729\pi$
$419$	−10.3963	−0.507893	−0.253947	+	0.967218i	$0.581729\pi$
$420$	0	0
$421$	2.99774	0.146101	0.0730504	−	0.997328i	$-0.476727\pi$
$421$	2.99774	0.146101	0.0730504	+	0.997328i	$0.476727\pi$
$422$	−16.4606	−0.801289
$423$	51.4484	2.50151
$424$	34.5889	1.67979
$425$	0	0
$426$	−48.2701	−2.33869
$427$	1.60784	0.0778087
$428$	−0.265033	−0.0128109
$429$	−49.9423	−2.41124
$430$	0	0
$431$	−26.5082	−1.27686	−0.638428	−	0.769682i	$-0.720415\pi$
$431$	−26.5082	−1.27686	−0.638428	+	0.769682i	$0.720415\pi$
$432$	−21.1320	−1.01671
$433$	30.7877	1.47956	0.739782	−	0.672847i	$-0.234929\pi$
$433$	30.7877	1.47956	0.739782	+	0.672847i	$0.234929\pi$
$434$	−1.07184	−0.0514502
$435$	0	0
$436$	3.28475	0.157311
$437$	4.97191	0.237839
$438$	24.5054	1.17091
$439$	−13.5111	−0.644852	−0.322426	−	0.946595i	$-0.604498\pi$
$439$	−13.5111	−0.644852	−0.322426	+	0.946595i	$0.604498\pi$
$440$	0	0
$441$	−33.1461	−1.57839
$442$	50.1857	2.38709
$443$	6.07371	0.288571	0.144285	−	0.989536i	$-0.453912\pi$
$443$	6.07371	0.288571	0.144285	+	0.989536i	$0.453912\pi$
$444$	2.18971	0.103919
$445$	0	0
$446$	−30.1878	−1.42943
$447$	2.61848	0.123850
$448$	1.24494	0.0588178
$449$	12.4983	0.589830	0.294915	−	0.955523i	$-0.404709\pi$
$449$	12.4983	0.589830	0.294915	+	0.955523i	$0.404709\pi$
$450$	0	0
$451$	−32.3215	−1.52196
$452$	−2.79937	−0.131671
$453$	2.78489	0.130846
$454$	−17.7396	−0.832562
$455$	0	0
$456$	−4.80426	−0.224980
$457$	−38.9960	−1.82416	−0.912078	−	0.410017i	$-0.865523\pi$
$457$	−38.9960	−1.82416	−0.912078	+	0.410017i	$0.865523\pi$
$458$	−6.09876	−0.284977
$459$	35.3076	1.64802
$460$	0	0
$461$	−36.4649	−1.69834	−0.849171	−	0.528119i	$-0.822898\pi$
$461$	−36.4649	−1.69834	−0.849171	+	0.528119i	$0.822898\pi$
$462$	−2.71461	−0.126295
$463$	2.82224	0.131161	0.0655803	−	0.997847i	$-0.479110\pi$
$463$	2.82224	0.131161	0.0655803	+	0.997847i	$0.479110\pi$
$464$	−13.5619	−0.629594
$465$	0	0
$466$	17.6858	0.819280
$467$	10.3008	0.476666	0.238333	−	0.971184i	$-0.423399\pi$
$467$	10.3008	0.476666	0.238333	+	0.971184i	$0.423399\pi$
$468$	3.95796	0.182957
$469$	0.830329	0.0383410
$470$	0	0
$471$	−29.1501	−1.34317
$472$	18.0983	0.833043
$473$	41.9644	1.92953
$474$	51.4638	2.36381
$475$	0	0
$476$	0.221429	0.0101492
$477$	−61.1485	−2.79980
$478$	−3.26096	−0.149153
$479$	27.6022	1.26118	0.630588	−	0.776118i	$-0.282814\pi$
$479$	27.6022	1.26118	0.630588	+	0.776118i	$0.282814\pi$
$480$	0	0
$481$	20.9614	0.955759
$482$	−6.26932	−0.285560
$483$	3.74685	0.170488
$484$	0.617509	0.0280686
$485$	0	0
$486$	−24.3145	−1.10293
$487$	−17.7289	−0.803375	−0.401687	−	0.915777i	$-0.631576\pi$
$487$	−17.7289	−0.803375	−0.401687	+	0.915777i	$0.631576\pi$
$488$	−24.9234	−1.12823
$489$	1.30798	0.0591491
$490$	0	0
$491$	1.00695	0.0454429	0.0227214	−	0.999742i	$-0.492767\pi$
$491$	1.00695	0.0454429	0.0227214	+	0.999742i	$0.492767\pi$
$492$	4.17738	0.188331
$493$	22.6594	1.02053
$494$	4.45667	0.200515
$495$	0	0
$496$	18.0942	0.812455
$497$	−2.03876	−0.0914508
$498$	9.49821	0.425625
$499$	−10.8623	−0.486261	−0.243131	−	0.969994i	$-0.578174\pi$
$499$	−10.8623	−0.486261	−0.243131	+	0.969994i	$0.578174\pi$
$500$	0	0
$501$	45.3696	2.02697
$502$	9.51906	0.424857
$503$	−1.80993	−0.0807010	−0.0403505	−	0.999186i	$-0.512847\pi$
$503$	−1.80993	−0.0807010	−0.0403505	+	0.999186i	$0.512847\pi$
$504$	−2.22004	−0.0988883
$505$	0	0
$506$	−43.5481	−1.93595
$507$	25.5858	1.13631
$508$	−0.129415	−0.00574187
$509$	3.56061	0.157821	0.0789106	−	0.996882i	$-0.474856\pi$
$509$	3.56061	0.157821	0.0789106	+	0.996882i	$0.474856\pi$
$510$	0	0
$511$	1.03502	0.0457867
$512$	−18.9458	−0.837295
$513$	3.13545	0.138433
$514$	−6.43549	−0.283857
$515$	0	0
$516$	−5.42367	−0.238764
$517$	−41.1881	−1.81145
$518$	1.13936	0.0500605
$519$	−23.4973	−1.03142
$520$	0	0
$521$	23.4154	1.02585	0.512925	−	0.858434i	$-0.328562\pi$
$521$	23.4154	1.02585	0.512925	+	0.858434i	$0.328562\pi$
$522$	22.0153	0.963583
$523$	7.66919	0.335350	0.167675	−	0.985842i	$-0.446374\pi$
$523$	7.66919	0.335350	0.167675	+	0.985842i	$0.446374\pi$
$524$	−1.35619	−0.0592455
$525$	0	0
$526$	−33.2985	−1.45188
$527$	−30.2321	−1.31693
$528$	45.8265	1.99434
$529$	37.1074	1.61337
$530$	0	0
$531$	−31.9954	−1.38848
$532$	0.0196637	0.000852527 0
$533$	39.9887	1.73210
$534$	−28.1392	−1.21770
$535$	0	0
$536$	−12.8711	−0.555946
$537$	−68.4047	−2.95188
$538$	−23.7690	−1.02476
$539$	26.5358	1.14298
$540$	0	0
$541$	15.5470	0.668419	0.334210	−	0.942499i	$-0.391531\pi$
$541$	15.5470	0.668419	0.334210	+	0.942499i	$0.391531\pi$
$542$	−31.1899	−1.33972
$543$	10.3833	0.445590
$544$	−7.19743	−0.308587
$545$	0	0
$546$	3.35856	0.143733
$547$	8.00310	0.342188	0.171094	−	0.985255i	$-0.445270\pi$
$547$	8.00310	0.342188	0.171094	+	0.985255i	$0.445270\pi$
$548$	1.09815	0.0469107
$549$	44.0611	1.88048
$550$	0	0
$551$	2.01223	0.0857240
$552$	−58.0806	−2.47208
$553$	2.17365	0.0924328
$554$	−23.5409	−1.00016
$555$	0	0
$556$	3.27623	0.138943
$557$	29.6855	1.25781	0.628907	−	0.777481i	$-0.283503\pi$
$557$	29.6855	1.25781	0.628907	+	0.777481i	$0.283503\pi$
$558$	−29.3728	−1.24345
$559$	−51.9191	−2.19594
$560$	0	0
$561$	−76.5675	−3.23268
$562$	22.3328	0.942054
$563$	−6.17526	−0.260256	−0.130128	−	0.991497i	$-0.541539\pi$
$563$	−6.17526	−0.260256	−0.130128	+	0.991497i	$0.541539\pi$
$564$	5.32333	0.224153
$565$	0	0
$566$	8.08352	0.339776
$567$	−0.112958	−0.00474381
$568$	31.6032	1.32604
$569$	−4.48186	−0.187889	−0.0939447	−	0.995577i	$-0.529948\pi$
$569$	−4.48186	−0.187889	−0.0939447	+	0.995577i	$0.529948\pi$
$570$	0	0
$571$	29.7458	1.24482	0.622411	−	0.782691i	$-0.286153\pi$
$571$	29.7458	1.24482	0.622411	+	0.782691i	$0.286153\pi$
$572$	−3.16863	−0.132487
$573$	−65.4920	−2.73597
$574$	2.17358	0.0907237
$575$	0	0
$576$	34.1163	1.42151
$577$	−5.64755	−0.235110	−0.117555	−	0.993066i	$-0.537506\pi$
$577$	−5.64755	−0.235110	−0.117555	+	0.993066i	$0.537506\pi$
$578$	51.8595	2.15707
$579$	−64.3194	−2.67302
$580$	0	0
$581$	0.401170	0.0166434
$582$	−12.6062	−0.522545
$583$	48.9537	2.02745
$584$	−16.0441	−0.663908
$585$	0	0
$586$	9.91996	0.409790
$587$	8.70796	0.359416	0.179708	−	0.983720i	$-0.442485\pi$
$587$	8.70796	0.359416	0.179708	+	0.983720i	$0.442485\pi$
$588$	−3.42961	−0.141435
$589$	−2.68472	−0.110622
$590$	0	0
$591$	21.7953	0.896537
$592$	−19.2340	−0.790512
$593$	1.26578	0.0519793	0.0259897	−	0.999662i	$-0.491726\pi$
$593$	1.26578	0.0519793	0.0259897	+	0.999662i	$0.491726\pi$
$594$	−27.4628	−1.12681
$595$	0	0
$596$	0.166131	0.00680500
$597$	−48.1968	−1.97256
$598$	53.8785	2.20325
$599$	−12.1470	−0.496312	−0.248156	−	0.968720i	$-0.579825\pi$
$599$	−12.1470	−0.496312	−0.248156	+	0.968720i	$0.579825\pi$
$600$	0	0
$601$	−2.95538	−0.120552	−0.0602762	−	0.998182i	$-0.519198\pi$
$601$	−2.95538	−0.120552	−0.0602762	+	0.998182i	$0.519198\pi$
$602$	−2.82206	−0.115019
$603$	22.7543	0.926628
$604$	0.176690	0.00718940
$605$	0	0
$606$	22.1474	0.899677
$607$	9.99565	0.405711	0.202856	−	0.979209i	$-0.434978\pi$
$607$	9.99565	0.405711	0.202856	+	0.979209i	$0.434978\pi$
$608$	−0.639158	−0.0259213
$609$	1.51642	0.0614486
$610$	0	0
$611$	50.9586	2.06156
$612$	6.06802	0.245285
$613$	−2.00209	−0.0808637	−0.0404318	−	0.999182i	$-0.512873\pi$
$613$	−2.00209	−0.0808637	−0.0404318	+	0.999182i	$0.512873\pi$
$614$	27.4053	1.10599
$615$	0	0
$616$	1.77730	0.0716093
$617$	33.6582	1.35503	0.677513	−	0.735511i	$-0.263058\pi$
$617$	33.6582	1.35503	0.677513	+	0.735511i	$0.263058\pi$
$618$	−23.4931	−0.945033
$619$	−8.28261	−0.332906	−0.166453	−	0.986049i	$-0.553231\pi$
$619$	−8.28261	−0.332906	−0.166453	+	0.986049i	$0.553231\pi$
$620$	0	0
$621$	37.9056	1.52110
$622$	−1.61314	−0.0646809
$623$	−1.18850	−0.0476161
$624$	−56.6973	−2.26971
$625$	0	0
$626$	20.6948	0.827129
$627$	−6.79947	−0.271545
$628$	−1.84945	−0.0738012
$629$	32.1364	1.28136
$630$	0	0
$631$	−45.6467	−1.81717	−0.908584	−	0.417702i	$-0.862836\pi$
$631$	−45.6467	−1.81717	−0.908584	+	0.417702i	$0.862836\pi$
$632$	−33.6941	−1.34028
$633$	−31.0710	−1.23496
$634$	−13.2758	−0.527250
$635$	0	0
$636$	−6.32699	−0.250882
$637$	−32.8306	−1.30079
$638$	−17.6248	−0.697773
$639$	−55.8701	−2.21019
$640$	0	0
$641$	29.0348	1.14680	0.573402	−	0.819274i	$-0.305623\pi$
$641$	29.0348	1.14680	0.573402	+	0.819274i	$0.305623\pi$
$642$	−6.16306	−0.243236
$643$	8.92069	0.351798	0.175899	−	0.984408i	$-0.443717\pi$
$643$	8.92069	0.351798	0.175899	+	0.984408i	$0.443717\pi$
$644$	0.237722	0.00936755
$645$	0	0
$646$	6.83261	0.268825
$647$	5.68328	0.223433	0.111716	−	0.993740i	$-0.464365\pi$
$647$	5.68328	0.223433	0.111716	+	0.993740i	$0.464365\pi$
$648$	1.75099	0.0687854
$649$	25.6146	1.00546
$650$	0	0
$651$	−2.02321	−0.0792960
$652$	0.0829860	0.00324998
$653$	8.99659	0.352064	0.176032	−	0.984384i	$-0.443674\pi$
$653$	8.99659	0.352064	0.176032	+	0.984384i	$0.443674\pi$
$654$	76.3832	2.98682
$655$	0	0
$656$	−36.6932	−1.43263
$657$	28.3637	1.10657
$658$	2.76985	0.107980
$659$	26.2820	1.02380	0.511902	−	0.859044i	$-0.328941\pi$
$659$	26.2820	1.02380	0.511902	+	0.859044i	$0.328941\pi$
$660$	0	0
$661$	−45.5396	−1.77129	−0.885643	−	0.464367i	$-0.846282\pi$
$661$	−45.5396	−1.77129	−0.885643	+	0.464367i	$0.846282\pi$
$662$	−22.2849	−0.866128
$663$	94.7305	3.67903
$664$	−6.21862	−0.241329
$665$	0	0
$666$	31.2230	1.20986
$667$	24.3267	0.941933
$668$	2.87851	0.111373
$669$	−56.9825	−2.20307
$670$	0	0
$671$	−35.2741	−1.36174
$672$	−0.481671	−0.0185809
$673$	16.2158	0.625071	0.312536	−	0.949906i	$-0.398822\pi$
$673$	16.2158	0.625071	0.312536	+	0.949906i	$0.398822\pi$
$674$	40.8586	1.57382
$675$	0	0
$676$	1.62331	0.0624350
$677$	19.1783	0.737082	0.368541	−	0.929612i	$-0.379857\pi$
$677$	19.1783	0.737082	0.368541	+	0.929612i	$0.379857\pi$
$678$	−65.0962	−2.50000
$679$	−0.532442	−0.0204332
$680$	0	0
$681$	−33.4853	−1.28316
$682$	23.5150	0.900436
$683$	30.7996	1.17851	0.589257	−	0.807946i	$-0.299421\pi$
$683$	30.7996	1.17851	0.589257	+	0.807946i	$0.299421\pi$
$684$	0.538862	0.0206039
$685$	0	0
$686$	−3.57672	−0.136560
$687$	−11.5120	−0.439211
$688$	47.6403	1.81627
$689$	−60.5663	−2.30739
$690$	0	0
$691$	19.4614	0.740346	0.370173	−	0.928963i	$-0.379298\pi$
$691$	19.4614	0.740346	0.370173	+	0.928963i	$0.379298\pi$
$692$	−1.49081	−0.0566720
$693$	−3.14202	−0.119355
$694$	−31.0942	−1.18032
$695$	0	0
$696$	−23.5064	−0.891007
$697$	61.3075	2.32219
$698$	−10.4398	−0.395153
$699$	33.3838	1.26269
$700$	0	0
$701$	37.6707	1.42280	0.711402	−	0.702786i	$-0.248061\pi$
$701$	37.6707	1.42280	0.711402	+	0.702786i	$0.248061\pi$
$702$	33.9775	1.28240
$703$	2.85383	0.107634
$704$	−27.3125	−1.02938
$705$	0	0
$706$	30.6150	1.15221
$707$	0.935428	0.0351804
$708$	−3.31054	−0.124418
$709$	17.1573	0.644357	0.322179	−	0.946679i	$-0.395585\pi$
$709$	17.1573	0.644357	0.322179	+	0.946679i	$0.395585\pi$
$710$	0	0
$711$	59.5665	2.23392
$712$	18.4231	0.690436
$713$	−32.4566	−1.21551
$714$	5.14907	0.192699
$715$	0	0
$716$	−4.33999	−0.162193
$717$	−6.15539	−0.229877
$718$	54.3448	2.02813
$719$	38.4140	1.43260	0.716300	−	0.697793i	$-0.245834\pi$
$719$	38.4140	1.43260	0.716300	+	0.697793i	$0.245834\pi$
$720$	0	0
$721$	−0.992267	−0.0369539
$722$	−27.4251	−1.02066
$723$	−11.8340	−0.440110
$724$	0.658776	0.0244832
$725$	0	0
$726$	14.3595	0.532930
$727$	15.3354	0.568760	0.284380	−	0.958712i	$-0.408212\pi$
$727$	15.3354	0.568760	0.284380	+	0.958712i	$0.408212\pi$
$728$	−2.19890	−0.0814967
$729$	−43.9434	−1.62753
$730$	0	0
$731$	−79.5981	−2.94404
$732$	4.55898	0.168505
$733$	−35.0005	−1.29277	−0.646386	−	0.763011i	$-0.723720\pi$
$733$	−35.0005	−1.29277	−0.646386	+	0.763011i	$0.723720\pi$
$734$	−22.0771	−0.814881
$735$	0	0
$736$	−7.72704	−0.284822
$737$	−18.2165	−0.671012
$738$	59.5649	2.19261
$739$	17.1231	0.629882	0.314941	−	0.949111i	$-0.398015\pi$
$739$	17.1231	0.629882	0.314941	+	0.949111i	$0.398015\pi$
$740$	0	0
$741$	8.41241	0.309038
$742$	−3.29208	−0.120856
$743$	46.3566	1.70066	0.850330	−	0.526250i	$-0.176402\pi$
$743$	46.3566	1.70066	0.850330	+	0.526250i	$0.176402\pi$
$744$	31.3622	1.14979
$745$	0	0
$746$	18.5909	0.680660
$747$	10.9937	0.402237
$748$	−4.85788	−0.177622
$749$	−0.260305	−0.00951135
$750$	0	0
$751$	22.2542	0.812066	0.406033	−	0.913858i	$-0.366912\pi$
$751$	22.2542	0.812066	0.406033	+	0.913858i	$0.366912\pi$
$752$	−46.7590	−1.70513
$753$	17.9682	0.654797
$754$	21.8057	0.794116
$755$	0	0
$756$	0.149915	0.00545235
$757$	20.9163	0.760214	0.380107	−	0.924942i	$-0.375887\pi$
$757$	20.9163	0.760214	0.380107	+	0.924942i	$0.375887\pi$
$758$	12.0729	0.438508
$759$	−82.2015	−2.98373
$760$	0	0
$761$	−4.33724	−0.157225	−0.0786124	−	0.996905i	$-0.525049\pi$
$761$	−4.33724	−0.157225	−0.0786124	+	0.996905i	$0.525049\pi$
$762$	−3.00941	−0.109019
$763$	3.22615	0.116795
$764$	−4.15519	−0.150330
$765$	0	0
$766$	−48.1766	−1.74069
$767$	−31.6907	−1.14429
$768$	11.7200	0.422910
$769$	−4.64226	−0.167404	−0.0837021	−	0.996491i	$-0.526674\pi$
$769$	−4.64226	−0.167404	−0.0837021	+	0.996491i	$0.526674\pi$
$770$	0	0
$771$	−12.1476	−0.437486
$772$	−4.08080	−0.146871
$773$	5.10251	0.183524	0.0917622	−	0.995781i	$-0.470750\pi$
$773$	5.10251	0.183524	0.0917622	+	0.995781i	$0.470750\pi$
$774$	−77.3357	−2.77977
$775$	0	0
$776$	8.25349	0.296283
$777$	2.15065	0.0771543
$778$	−23.1134	−0.828657
$779$	5.44432	0.195063
$780$	0	0
$781$	44.7280	1.60049
$782$	82.6021	2.95385
$783$	15.3412	0.548249
$784$	30.1250	1.07589
$785$	0	0
$786$	−31.5367	−1.12488
$787$	−18.5582	−0.661527	−0.330763	−	0.943714i	$-0.607306\pi$
$787$	−18.5582	−0.661527	−0.330763	+	0.943714i	$0.607306\pi$
$788$	1.38282	0.0492608
$789$	−62.8542	−2.23767
$790$	0	0
$791$	−2.74943	−0.0977585
$792$	48.7050	1.73066
$793$	43.6416	1.54976
$794$	−22.4794	−0.797765
$795$	0	0
$796$	−3.05788	−0.108384
$797$	−8.18241	−0.289836	−0.144918	−	0.989444i	$-0.546292\pi$
$797$	−8.18241	−0.289836	−0.144918	+	0.989444i	$0.546292\pi$
$798$	0.457256	0.0161867
$799$	78.1256	2.76388
$800$	0	0
$801$	−32.5696	−1.15079
$802$	43.3947	1.53232
$803$	−22.7072	−0.801319
$804$	2.35438	0.0830324
$805$	0	0
$806$	−29.0931	−1.02476
$807$	−44.8665	−1.57937
$808$	−14.5002	−0.510117
$809$	23.0092	0.808962	0.404481	−	0.914546i	$-0.367452\pi$
$809$	23.0092	0.808962	0.404481	+	0.914546i	$0.367452\pi$
$810$	0	0
$811$	−11.8798	−0.417158	−0.208579	−	0.978006i	$-0.566884\pi$
$811$	−11.8798	−0.417158	−0.208579	+	0.978006i	$0.566884\pi$
$812$	0.0962107	0.00337633
$813$	−58.8741	−2.06480
$814$	−24.9962	−0.876116
$815$	0	0
$816$	−86.9237	−3.04294
$817$	−7.06860	−0.247299
$818$	−24.9652	−0.872889
$819$	3.88735	0.135835
$820$	0	0
$821$	17.1711	0.599277	0.299638	−	0.954053i	$-0.403134\pi$
$821$	17.1711	0.599277	0.299638	+	0.954053i	$0.403134\pi$
$822$	25.5363	0.890681
$823$	1.55628	0.0542487	0.0271243	−	0.999632i	$-0.491365\pi$
$823$	1.55628	0.0542487	0.0271243	+	0.999632i	$0.491365\pi$
$824$	15.3813	0.535833
$825$	0	0
$826$	−1.72255	−0.0599351
$827$	21.2420	0.738655	0.369328	−	0.929299i	$-0.379588\pi$
$827$	21.2420	0.738655	0.369328	+	0.929299i	$0.379588\pi$
$828$	6.51452	0.226395
$829$	−29.1520	−1.01249	−0.506245	−	0.862390i	$-0.668967\pi$
$829$	−29.1520	−1.01249	−0.506245	+	0.862390i	$0.668967\pi$
$830$	0	0
$831$	−44.4359	−1.54146
$832$	33.7915	1.17151
$833$	−50.3332	−1.74394
$834$	76.1852	2.63808
$835$	0	0
$836$	−0.431398	−0.0149202
$837$	−20.4682	−0.707484
$838$	−15.3383	−0.529853
$839$	16.9390	0.584799	0.292400	−	0.956296i	$-0.405546\pi$
$839$	16.9390	0.584799	0.292400	+	0.956296i	$0.405546\pi$
$840$	0	0
$841$	−19.1545	−0.660500
$842$	4.42275	0.152418
$843$	42.1555	1.45191
$844$	−1.97133	−0.0678558
$845$	0	0
$846$	75.9049	2.60967
$847$	0.606493	0.0208394
$848$	55.5749	1.90845
$849$	15.2585	0.523669
$850$	0	0
$851$	34.5011	1.18268
$852$	−5.78084	−0.198048
$853$	−54.6344	−1.87065	−0.935323	−	0.353796i	$-0.884891\pi$
$853$	−54.6344	−1.87065	−0.935323	+	0.353796i	$0.884891\pi$
$854$	2.37214	0.0811729
$855$	0	0
$856$	4.03504	0.137915
$857$	−36.5265	−1.24772	−0.623861	−	0.781535i	$-0.714437\pi$
$857$	−36.5265	−1.24772	−0.623861	+	0.781535i	$0.714437\pi$
$858$	−73.6829	−2.51549
$859$	−8.22360	−0.280586	−0.140293	−	0.990110i	$-0.544804\pi$
$859$	−8.22360	−0.280586	−0.140293	+	0.990110i	$0.544804\pi$
$860$	0	0
$861$	4.10286	0.139825
$862$	−39.1092	−1.33206
$863$	−12.1416	−0.413305	−0.206653	−	0.978414i	$-0.566257\pi$
$863$	−12.1416	−0.413305	−0.206653	+	0.978414i	$0.566257\pi$
$864$	−4.87291	−0.165780
$865$	0	0
$866$	45.4230	1.54354
$867$	97.8901	3.32452
$868$	−0.128364	−0.00435697
$869$	−47.6872	−1.61768
$870$	0	0
$871$	22.5377	0.763660
$872$	−50.0092	−1.69353
$873$	−14.5910	−0.493832
$874$	7.33537	0.248122
$875$	0	0
$876$	2.93478	0.0991569
$877$	8.35977	0.282289	0.141145	−	0.989989i	$-0.454922\pi$
$877$	8.35977	0.282289	0.141145	+	0.989989i	$0.454922\pi$
$878$	−19.9338	−0.672734
$879$	18.7249	0.631576
$880$	0	0
$881$	0.277228	0.00934005	0.00467002	−	0.999989i	$-0.498513\pi$
$881$	0.277228	0.00934005	0.00467002	+	0.999989i	$0.498513\pi$
$882$	−48.9025	−1.64663
$883$	−17.3785	−0.584832	−0.292416	−	0.956291i	$-0.594459\pi$
$883$	−17.3785	−0.584832	−0.292416	+	0.956291i	$0.594459\pi$
$884$	6.01025	0.202147
$885$	0	0
$886$	8.96092	0.301048
$887$	−24.3042	−0.816056	−0.408028	−	0.912969i	$-0.633783\pi$
$887$	−24.3042	−0.816056	−0.408028	+	0.912969i	$0.633783\pi$
$888$	−33.3377	−1.11874
$889$	−0.127107	−0.00426302
$890$	0	0
$891$	2.47818	0.0830220
$892$	−3.61529	−0.121049
$893$	6.93783	0.232166
$894$	3.86320	0.129205
$895$	0	0
$896$	2.18265	0.0729173
$897$	101.701	3.39570
$898$	18.4395	0.615333
$899$	−13.1358	−0.438105
$900$	0	0
$901$	−92.8553	−3.09346
$902$	−47.6859	−1.58777
$903$	−5.32692	−0.177269
$904$	42.6194	1.41750
$905$	0	0
$906$	4.10872	0.136503
$907$	−42.0612	−1.39662	−0.698309	−	0.715796i	$-0.746064\pi$
$907$	−42.0612	−1.39662	−0.698309	+	0.715796i	$0.746064\pi$
$908$	−2.12450	−0.0705041
$909$	25.6344	0.850241
$910$	0	0
$911$	13.2981	0.440587	0.220293	−	0.975434i	$-0.429299\pi$
$911$	13.2981	0.440587	0.220293	+	0.975434i	$0.429299\pi$
$912$	−7.71914	−0.255606
$913$	−8.80121	−0.291278
$914$	−57.5332	−1.90303
$915$	0	0
$916$	−0.730390	−0.0241327
$917$	−1.33200	−0.0439865
$918$	52.0915	1.71928
$919$	43.0777	1.42100	0.710501	−	0.703697i	$-0.248469\pi$
$919$	43.0777	1.42100	0.710501	+	0.703697i	$0.248469\pi$
$920$	0	0
$921$	51.7303	1.70457
$922$	−53.7989	−1.77177
$923$	−55.3381	−1.82148
$924$	−0.325103	−0.0106951
$925$	0	0
$926$	4.16382	0.136832
$927$	−27.1921	−0.893104
$928$	−3.12728	−0.102658
$929$	50.9373	1.67120	0.835600	−	0.549339i	$-0.185120\pi$
$929$	50.9373	1.67120	0.835600	+	0.549339i	$0.185120\pi$
$930$	0	0
$931$	−4.46977	−0.146491
$932$	2.11806	0.0693793
$933$	−3.04496	−0.0996875
$934$	15.1974	0.497276
$935$	0	0
$936$	−60.2586	−1.96961
$937$	−41.8394	−1.36683	−0.683417	−	0.730028i	$-0.739507\pi$
$937$	−41.8394	−1.36683	−0.683417	+	0.730028i	$0.739507\pi$
$938$	1.22504	0.0399988
$939$	39.0635	1.27479
$940$	0	0
$941$	−22.5373	−0.734696	−0.367348	−	0.930084i	$-0.619734\pi$
$941$	−22.5373	−0.734696	−0.367348	+	0.930084i	$0.619734\pi$
$942$	−43.0070	−1.40124
$943$	65.8186	2.14335
$944$	29.0791	0.946443
$945$	0	0
$946$	61.9127	2.01295
$947$	16.9670	0.551355	0.275677	−	0.961250i	$-0.411098\pi$
$947$	16.9670	0.551355	0.275677	+	0.961250i	$0.411098\pi$
$948$	6.16331	0.200175
$949$	28.0937	0.911959
$950$	0	0
$951$	−25.0594	−0.812608
$952$	−3.37118	−0.109260
$953$	−45.8873	−1.48644	−0.743218	−	0.669049i	$-0.766702\pi$
$953$	−45.8873	−1.48644	−0.743218	+	0.669049i	$0.766702\pi$
$954$	−90.2160	−2.92085
$955$	0	0
$956$	−0.390533	−0.0126307
$957$	−33.2686	−1.07542
$958$	40.7232	1.31571
$959$	1.07856	0.0348286
$960$	0	0
$961$	−13.4742	−0.434650
$962$	30.9257	0.997084
$963$	−7.13341	−0.229871
$964$	−0.750815	−0.0241821
$965$	0	0
$966$	5.52796	0.177859
$967$	19.8373	0.637926	0.318963	−	0.947767i	$-0.396665\pi$
$967$	19.8373	0.637926	0.318963	+	0.947767i	$0.396665\pi$
$968$	−9.40137	−0.302171
$969$	12.8972	0.414319
$970$	0	0
$971$	60.8748	1.95357	0.976783	−	0.214231i	$-0.0687244\pi$
$971$	60.8748	1.95357	0.976783	+	0.214231i	$0.0687244\pi$
$972$	−2.91191	−0.0933997
$973$	3.21779	0.103158
$974$	−26.1566	−0.838111
$975$	0	0
$976$	−40.0451	−1.28181
$977$	−8.96312	−0.286756	−0.143378	−	0.989668i	$-0.545796\pi$
$977$	−8.96312	−0.286756	−0.143378	+	0.989668i	$0.545796\pi$
$978$	1.92975	0.0617065
$979$	26.0742	0.833337
$980$	0	0
$981$	88.4094	2.82270
$982$	1.48561	0.0474077
$983$	49.3888	1.57526	0.787630	−	0.616149i	$-0.211308\pi$
$983$	49.3888	1.57526	0.787630	+	0.616149i	$0.211308\pi$
$984$	−63.5992	−2.02747
$985$	0	0
$986$	33.4307	1.06465
$987$	5.22837	0.166421
$988$	0.533732	0.0169803
$989$	−85.4551	−2.71732
$990$	0	0
$991$	40.5105	1.28686	0.643430	−	0.765505i	$-0.277511\pi$
$991$	40.5105	1.28686	0.643430	+	0.765505i	$0.277511\pi$
$992$	4.17242	0.132475
$993$	−42.0651	−1.33489
$994$	−3.00790	−0.0954049
$995$	0	0
$996$	1.13751	0.0360433
$997$	27.0457	0.856544	0.428272	−	0.903650i	$-0.359122\pi$
$997$	27.0457	0.856544	0.428272	+	0.903650i	$0.359122\pi$
$998$	−16.0257	−0.507286
$999$	21.7575	0.688375

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3775.2.a.r.1.13		18
5.4	even	2		755.2.a.k.1.6	✓	18
15.14	odd	2		6795.2.a.bi.1.13		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
755.2.a.k.1.6	✓	18	5.4	even	2
3775.2.a.r.1.13		18	1.1	even	1	trivial
6795.2.a.bi.1.13		18	15.14	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Varieties

Fields

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Groups

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3775.2.a.r.1.13

Level	$3775$
Weight	$2$
Character	3775.1
Self dual	yes
Analytic conductor	$30.144$
Analytic rank	$0$
Dimension	$18$
CM	no
Inner twists	$1$