LMFDB - Embedded newform 7105.2.a.o.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7105,2,Mod(1,7105)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7105.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7105 = 5 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7105.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,-2,1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$56.7337106361$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	7105.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.48119 q^{2} -0.806063 q^{3} +0.193937 q^{4} -1.00000 q^{5} +1.19394 q^{6} +2.67513 q^{8} -2.35026 q^{9} +1.48119 q^{10} +4.15633 q^{11} -0.156325 q^{12} -2.96239 q^{13} +0.806063 q^{15} -4.35026 q^{16} -5.50659 q^{17} +3.48119 q^{18} +3.19394 q^{19} -0.193937 q^{20} -6.15633 q^{22} +1.84367 q^{23} -2.15633 q^{24} +1.00000 q^{25} +4.38787 q^{26} +4.31265 q^{27} -1.00000 q^{29} -1.19394 q^{30} +4.80606 q^{31} +1.09332 q^{32} -3.35026 q^{33} +8.15633 q^{34} -0.455802 q^{36} -9.50659 q^{37} -4.73084 q^{38} +2.38787 q^{39} -2.67513 q^{40} +11.2750 q^{41} -0.0303172 q^{43} +0.806063 q^{44} +2.35026 q^{45} -2.73084 q^{46} -4.80606 q^{47} +3.50659 q^{48} -1.48119 q^{50} +4.43866 q^{51} -0.574515 q^{52} -1.35026 q^{53} -6.38787 q^{54} -4.15633 q^{55} -2.57452 q^{57} +1.48119 q^{58} -13.2750 q^{59} +0.156325 q^{60} -8.88717 q^{61} -7.11871 q^{62} +7.08110 q^{64} +2.96239 q^{65} +4.96239 q^{66} +5.84367 q^{67} -1.06793 q^{68} -1.48612 q^{69} -1.27504 q^{71} -6.28726 q^{72} +15.2447 q^{73} +14.0811 q^{74} -0.806063 q^{75} +0.619421 q^{76} -3.53690 q^{78} -4.93207 q^{79} +4.35026 q^{80} +3.57452 q^{81} -16.7005 q^{82} -4.41819 q^{83} +5.50659 q^{85} +0.0449056 q^{86} +0.806063 q^{87} +11.1187 q^{88} +3.61213 q^{89} -3.48119 q^{90} +0.357556 q^{92} -3.87399 q^{93} +7.11871 q^{94} -3.19394 q^{95} -0.881286 q^{96} +1.38058 q^{97} -9.76845 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9} - q^{10} + 2 q^{11} + 10 q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} + 4 q^{17} + 5 q^{18} + 10 q^{19} - q^{20} - 8 q^{22} + 16 q^{23}+ \cdots - 18 q^{99}+O(q^{100}) $ 3 * q + q^2 - 2 * q^3 + q^4 - 3 * q^5 + 4 * q^6 + 3 * q^8 + 3 * q^9 - q^10 + 2 * q^11 + 10 * q^12 + 2 * q^13 + 2 * q^15 - 3 * q^16 + 4 * q^17 + 5 * q^18 + 10 * q^19 - q^20 - 8 * q^22 + 16 * q^23 + 4 * q^24 + 3 * q^25 + 14 * q^26 - 8 * q^27 - 3 * q^29 - 4 * q^30 + 14 * q^31 - 3 * q^32 + 14 * q^34 - 11 * q^36 - 8 * q^37 + 8 * q^38 + 8 * q^39 - 3 * q^40 + 2 * q^41 + 2 * q^43 + 2 * q^44 - 3 * q^45 + 14 * q^46 - 14 * q^47 - 10 * q^48 + q^50 - 16 * q^51 + 10 * q^52 + 6 * q^53 - 20 * q^54 - 2 * q^55 + 4 * q^57 - q^58 - 8 * q^59 - 10 * q^60 + 6 * q^61 - 11 * q^64 - 2 * q^65 + 4 * q^66 + 28 * q^67 - 12 * q^68 - 12 * q^69 + 28 * q^71 - 13 * q^72 + 16 * q^73 + 10 * q^74 - 2 * q^75 + 14 * q^76 + 12 * q^78 - 6 * q^79 + 3 * q^80 - q^81 - 30 * q^82 - 12 * q^83 - 4 * q^85 + 24 * q^86 + 2 * q^87 + 12 * q^88 + 10 * q^89 - 5 * q^90 + 4 * q^92 - 20 * q^93 - 10 * q^95 - 24 * q^96 - 8 * q^97 - 18 * 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 $

$2$	−1.48119	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$2$	−1.48119	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$3$	−0.806063	−0.465381	−0.232690	−	0.972551i	$-0.574753\pi$
$3$	−0.806063	−0.465381	−0.232690	+	0.972551i	$0.574753\pi$
$4$	0.193937	0.0969683
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.19394	0.487423
$7$	0	0
$7$	0	0
$8$	2.67513	0.945802
$9$	−2.35026	−0.783421
$10$	1.48119	0.468395
$11$	4.15633	1.25318	0.626590	−	0.779349i	$-0.284450\pi$
$11$	4.15633	1.25318	0.626590	+	0.779349i	$0.284450\pi$
$12$	−0.156325	−0.0451272
$13$	−2.96239	−0.821619	−0.410809	−	0.911721i	$-0.634754\pi$
$13$	−2.96239	−0.821619	−0.410809	+	0.911721i	$0.634754\pi$
$14$	0	0
$15$	0.806063	0.208125
$16$	−4.35026	−1.08757
$17$	−5.50659	−1.33554	−0.667772	−	0.744366i	$-0.732752\pi$
$17$	−5.50659	−1.33554	−0.667772	+	0.744366i	$0.732752\pi$
$18$	3.48119	0.820525
$19$	3.19394	0.732739	0.366370	−	0.930469i	$-0.380601\pi$
$19$	3.19394	0.732739	0.366370	+	0.930469i	$0.380601\pi$
$20$	−0.193937	−0.0433655
$21$	0	0
$22$	−6.15633	−1.31253
$23$	1.84367	0.384433	0.192216	−	0.981353i	$-0.438432\pi$
$23$	1.84367	0.384433	0.192216	+	0.981353i	$0.438432\pi$
$24$	−2.15633	−0.440158
$25$	1.00000	0.200000
$26$	4.38787	0.860533
$27$	4.31265	0.829970
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	−1.19394	−0.217982
$31$	4.80606	0.863194	0.431597	−	0.902066i	$-0.357950\pi$
$31$	4.80606	0.863194	0.431597	+	0.902066i	$0.357950\pi$
$32$	1.09332	0.193274
$33$	−3.35026	−0.583206
$34$	8.15633	1.39880
$35$	0	0
$36$	−0.455802	−0.0759669
$37$	−9.50659	−1.56287	−0.781437	−	0.623985i	$-0.785513\pi$
$37$	−9.50659	−1.56287	−0.781437	+	0.623985i	$0.785513\pi$
$38$	−4.73084	−0.767444
$39$	2.38787	0.382366
$40$	−2.67513	−0.422975
$41$	11.2750	1.76087	0.880433	−	0.474171i	$-0.157252\pi$
$41$	11.2750	1.76087	0.880433	+	0.474171i	$0.157252\pi$
$42$	0	0
$43$	−0.0303172	−0.00462332	−0.00231166	−	0.999997i	$-0.500736\pi$
$43$	−0.0303172	−0.00462332	−0.00231166	+	0.999997i	$0.500736\pi$
$44$	0.806063	0.121519
$45$	2.35026	0.350356
$46$	−2.73084	−0.402640
$47$	−4.80606	−0.701036	−0.350518	−	0.936556i	$-0.613995\pi$
$47$	−4.80606	−0.701036	−0.350518	+	0.936556i	$0.613995\pi$
$48$	3.50659	0.506132
$49$	0	0
$50$	−1.48119	−0.209473
$51$	4.43866	0.621536
$52$	−0.574515	−0.0796710
$53$	−1.35026	−0.185473	−0.0927364	−	0.995691i	$-0.529561\pi$
$53$	−1.35026	−0.185473	−0.0927364	+	0.995691i	$0.529561\pi$
$54$	−6.38787	−0.869279
$55$	−4.15633	−0.560439
$56$	0	0
$57$	−2.57452	−0.341003
$58$	1.48119	0.194490
$59$	−13.2750	−1.72826	−0.864131	−	0.503266i	$-0.832132\pi$
$59$	−13.2750	−1.72826	−0.864131	+	0.503266i	$0.832132\pi$
$60$	0.156325	0.0201815
$61$	−8.88717	−1.13788	−0.568942	−	0.822377i	$-0.692647\pi$
$61$	−8.88717	−1.13788	−0.568942	+	0.822377i	$0.692647\pi$
$62$	−7.11871	−0.904078
$63$	0	0
$64$	7.08110	0.885138
$65$	2.96239	0.367439
$66$	4.96239	0.610828
$67$	5.84367	0.713919	0.356959	−	0.934120i	$-0.383813\pi$
$67$	5.84367	0.713919	0.356959	+	0.934120i	$0.383813\pi$
$68$	−1.06793	−0.129505
$69$	−1.48612	−0.178908
$70$	0	0
$71$	−1.27504	−0.151319	−0.0756596	−	0.997134i	$-0.524106\pi$
$71$	−1.27504	−0.151319	−0.0756596	+	0.997134i	$0.524106\pi$
$72$	−6.28726	−0.740960
$73$	15.2447	1.78426	0.892130	−	0.451779i	$-0.149210\pi$
$73$	15.2447	1.78426	0.892130	+	0.451779i	$0.149210\pi$
$74$	14.0811	1.63689
$75$	−0.806063	−0.0930762
$76$	0.619421	0.0710525
$77$	0	0
$78$	−3.53690	−0.400476
$79$	−4.93207	−0.554901	−0.277451	−	0.960740i	$-0.589490\pi$
$79$	−4.93207	−0.554901	−0.277451	+	0.960740i	$0.589490\pi$
$80$	4.35026	0.486374
$81$	3.57452	0.397168
$82$	−16.7005	−1.84426
$83$	−4.41819	−0.484959	−0.242480	−	0.970156i	$-0.577961\pi$
$83$	−4.41819	−0.484959	−0.242480	+	0.970156i	$0.577961\pi$
$84$	0	0
$85$	5.50659	0.597273
$86$	0.0449056	0.00484230
$87$	0.806063	0.0864191
$88$	11.1187	1.18526
$89$	3.61213	0.382885	0.191442	−	0.981504i	$-0.438684\pi$
$89$	3.61213	0.382885	0.191442	+	0.981504i	$0.438684\pi$
$90$	−3.48119	−0.366950
$91$	0	0
$92$	0.357556	0.0372778
$93$	−3.87399	−0.401714
$94$	7.11871	0.734239
$95$	−3.19394	−0.327691
$96$	−0.881286	−0.0899459
$97$	1.38058	0.140177	0.0700883	−	0.997541i	$-0.477672\pi$
$97$	1.38058	0.140177	0.0700883	+	0.997541i	$0.477672\pi$
$98$	0	0
$99$	−9.76845	−0.981766
$100$	0.193937	0.0193937
$101$	13.0132	1.29486	0.647430	−	0.762125i	$-0.275844\pi$
$101$	13.0132	1.29486	0.647430	+	0.762125i	$0.275844\pi$
$102$	−6.57452	−0.650974
$103$	−5.31994	−0.524190	−0.262095	−	0.965042i	$-0.584413\pi$
$103$	−5.31994	−0.524190	−0.262095	+	0.965042i	$0.584413\pi$
$104$	−7.92478	−0.777088
$105$	0	0
$106$	2.00000	0.194257
$107$	−13.8192	−1.33596	−0.667978	−	0.744181i	$-0.732840\pi$
$107$	−13.8192	−1.33596	−0.667978	+	0.744181i	$0.732840\pi$
$108$	0.836381	0.0804808
$109$	−1.87399	−0.179496	−0.0897479	−	0.995965i	$-0.528606\pi$
$109$	−1.87399	−0.179496	−0.0897479	+	0.995965i	$0.528606\pi$
$110$	6.15633	0.586983
$111$	7.66291	0.727331
$112$	0	0
$113$	−11.7685	−1.10708	−0.553541	−	0.832822i	$-0.686724\pi$
$113$	−11.7685	−1.10708	−0.553541	+	0.832822i	$0.686724\pi$
$114$	3.81336	0.357154
$115$	−1.84367	−0.171924
$116$	−0.193937	−0.0180066
$117$	6.96239	0.643673
$118$	19.6629	1.81012
$119$	0	0
$120$	2.15633	0.196845
$121$	6.27504	0.570458
$122$	13.1636	1.19178
$123$	−9.08840	−0.819473
$124$	0.932071	0.0837025
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.2677	1.26606	0.633029	−	0.774128i	$-0.281811\pi$
$127$	14.2677	1.26606	0.633029	+	0.774128i	$0.281811\pi$
$128$	−12.6751	−1.12033
$129$	0.0244376	0.00215161
$130$	−4.38787	−0.384842
$131$	−5.89446	−0.515001	−0.257501	−	0.966278i	$-0.582899\pi$
$131$	−5.89446	−0.515001	−0.257501	+	0.966278i	$0.582899\pi$
$132$	−0.649738	−0.0565525
$133$	0	0
$134$	−8.65562	−0.747731
$135$	−4.31265	−0.371174
$136$	−14.7308	−1.26316
$137$	18.2823	1.56197	0.780983	−	0.624553i	$-0.214719\pi$
$137$	18.2823	1.56197	0.780983	+	0.624553i	$0.214719\pi$
$138$	2.20123	0.187381
$139$	11.5369	0.978547	0.489274	−	0.872130i	$-0.337262\pi$
$139$	11.5369	0.978547	0.489274	+	0.872130i	$0.337262\pi$
$140$	0	0
$141$	3.87399	0.326249
$142$	1.88858	0.158486
$143$	−12.3127	−1.02964
$144$	10.2243	0.852021
$145$	1.00000	0.0830455
$146$	−22.5804	−1.86877
$147$	0	0
$148$	−1.84367	−0.151549
$149$	2.77575	0.227398	0.113699	−	0.993515i	$-0.463730\pi$
$149$	2.77575	0.227398	0.113699	+	0.993515i	$0.463730\pi$
$150$	1.19394	0.0974845
$151$	1.79877	0.146382	0.0731909	−	0.997318i	$-0.476682\pi$
$151$	1.79877	0.146382	0.0731909	+	0.997318i	$0.476682\pi$
$152$	8.54420	0.693026
$153$	12.9419	1.04629
$154$	0	0
$155$	−4.80606	−0.386032
$156$	0.463096	0.0370773
$157$	−3.76845	−0.300755	−0.150378	−	0.988629i	$-0.548049\pi$
$157$	−3.76845	−0.300755	−0.150378	+	0.988629i	$0.548049\pi$
$158$	7.30536	0.581183
$159$	1.08840	0.0863155
$160$	−1.09332	−0.0864346
$161$	0	0
$162$	−5.29455	−0.415979
$163$	1.64244	0.128646	0.0643231	−	0.997929i	$-0.479511\pi$
$163$	1.64244	0.128646	0.0643231	+	0.997929i	$0.479511\pi$
$164$	2.18664	0.170748
$165$	3.35026	0.260818
$166$	6.54420	0.507928
$167$	−8.08110	−0.625334	−0.312667	−	0.949863i	$-0.601222\pi$
$167$	−8.08110	−0.625334	−0.312667	+	0.949863i	$0.601222\pi$
$168$	0	0
$169$	−4.22425	−0.324943
$170$	−8.15633	−0.625562
$171$	−7.50659	−0.574043
$172$	−0.00587961	−0.000448316 0
$173$	7.73813	0.588320	0.294160	−	0.955756i	$-0.404960\pi$
$173$	7.73813	0.588320	0.294160	+	0.955756i	$0.404960\pi$
$174$	−1.19394	−0.0905121
$175$	0	0
$176$	−18.0811	−1.36291
$177$	10.7005	0.804301
$178$	−5.35026	−0.401019
$179$	−21.4010	−1.59959	−0.799795	−	0.600274i	$-0.795058\pi$
$179$	−21.4010	−1.59959	−0.799795	+	0.600274i	$0.795058\pi$
$180$	0.455802	0.0339735
$181$	−15.2750	−1.13538	−0.567692	−	0.823241i	$-0.692164\pi$
$181$	−15.2750	−1.13538	−0.567692	+	0.823241i	$0.692164\pi$
$182$	0	0
$183$	7.16362	0.529550
$184$	4.93207	0.363597
$185$	9.50659	0.698938
$186$	5.73813	0.420740
$187$	−22.8872	−1.67368
$188$	−0.932071	−0.0679783
$189$	0	0
$190$	4.73084	0.343211
$191$	3.31994	0.240223	0.120111	−	0.992760i	$-0.461675\pi$
$191$	3.31994	0.240223	0.120111	+	0.992760i	$0.461675\pi$
$192$	−5.70782	−0.411926
$193$	−4.88129	−0.351363	−0.175681	−	0.984447i	$-0.556213\pi$
$193$	−4.88129	−0.351363	−0.175681	+	0.984447i	$0.556213\pi$
$194$	−2.04491	−0.146816
$195$	−2.38787	−0.170999
$196$	0	0
$197$	−24.2374	−1.72685	−0.863423	−	0.504481i	$-0.831684\pi$
$197$	−24.2374	−1.72685	−0.863423	+	0.504481i	$0.831684\pi$
$198$	14.4690	1.02827
$199$	−16.7513	−1.18747	−0.593734	−	0.804661i	$-0.702347\pi$
$199$	−16.7513	−1.18747	−0.593734	+	0.804661i	$0.702347\pi$
$200$	2.67513	0.189160
$201$	−4.71037	−0.332244
$202$	−19.2750	−1.35619
$203$	0	0
$204$	0.860818	0.0602693
$205$	−11.2750	−0.787483
$206$	7.87987	0.549017
$207$	−4.33312	−0.301173
$208$	12.8872	0.893564
$209$	13.2750	0.918254
$210$	0	0
$211$	−25.3054	−1.74209	−0.871046	−	0.491201i	$-0.836558\pi$
$211$	−25.3054	−1.74209	−0.871046	+	0.491201i	$0.836558\pi$
$212$	−0.261865	−0.0179850
$213$	1.02776	0.0704211
$214$	20.4690	1.39923
$215$	0.0303172	0.00206761
$216$	11.5369	0.784987
$217$	0	0
$218$	2.77575	0.187997
$219$	−12.2882	−0.830360
$220$	−0.806063	−0.0543448
$221$	16.3127	1.09731
$222$	−11.3503	−0.761780
$223$	−17.6932	−1.18483	−0.592413	−	0.805634i	$-0.701825\pi$
$223$	−17.6932	−1.18483	−0.592413	+	0.805634i	$0.701825\pi$
$224$	0	0
$225$	−2.35026	−0.156684
$226$	17.4314	1.15952
$227$	−26.8423	−1.78158	−0.890792	−	0.454412i	$-0.849849\pi$
$227$	−26.8423	−1.78158	−0.890792	+	0.454412i	$0.849849\pi$
$228$	−0.499293	−0.0330665
$229$	17.2243	1.13821	0.569105	−	0.822265i	$-0.307290\pi$
$229$	17.2243	1.13821	0.569105	+	0.822265i	$0.307290\pi$
$230$	2.73084	0.180066
$231$	0	0
$232$	−2.67513	−0.175631
$233$	−9.07381	−0.594445	−0.297222	−	0.954808i	$-0.596060\pi$
$233$	−9.07381	−0.594445	−0.297222	+	0.954808i	$0.596060\pi$
$234$	−10.3127	−0.674159
$235$	4.80606	0.313513
$236$	−2.57452	−0.167587
$237$	3.97556	0.258241
$238$	0	0
$239$	20.4993	1.32599	0.662995	−	0.748624i	$-0.269285\pi$
$239$	20.4993	1.32599	0.662995	+	0.748624i	$0.269285\pi$
$240$	−3.50659	−0.226349
$241$	−5.47627	−0.352758	−0.176379	−	0.984322i	$-0.556438\pi$
$241$	−5.47627	−0.352758	−0.176379	+	0.984322i	$0.556438\pi$
$242$	−9.29455	−0.597476
$243$	−15.8192	−1.01480
$244$	−1.72355	−0.110339
$245$	0	0
$246$	13.4617	0.858285
$247$	−9.46168	−0.602032
$248$	12.8568	0.816411
$249$	3.56134	0.225691
$250$	1.48119	0.0936790
$251$	29.6180	1.86947	0.934736	−	0.355343i	$-0.115636\pi$
$251$	29.6180	1.86947	0.934736	+	0.355343i	$0.115636\pi$
$252$	0	0
$253$	7.66291	0.481763
$254$	−21.1333	−1.32602
$255$	−4.43866	−0.277960
$256$	4.61213	0.288258
$257$	−17.6629	−1.10178	−0.550891	−	0.834577i	$-0.685712\pi$
$257$	−17.6629	−1.10178	−0.550891	+	0.834577i	$0.685712\pi$
$258$	−0.0361968	−0.00225351
$259$	0	0
$260$	0.574515	0.0356299
$261$	2.35026	0.145478
$262$	8.73084	0.539393
$263$	27.3561	1.68685	0.843426	−	0.537245i	$-0.180535\pi$
$263$	27.3561	1.68685	0.843426	+	0.537245i	$0.180535\pi$
$264$	−8.96239	−0.551597
$265$	1.35026	0.0829459
$266$	0	0
$267$	−2.91160	−0.178187
$268$	1.13330	0.0692275
$269$	−10.4993	−0.640153	−0.320077	−	0.947392i	$-0.603709\pi$
$269$	−10.4993	−0.640153	−0.320077	+	0.947392i	$0.603709\pi$
$270$	6.38787	0.388754
$271$	9.61801	0.584252	0.292126	−	0.956380i	$-0.405637\pi$
$271$	9.61801	0.584252	0.292126	+	0.956380i	$0.405637\pi$
$272$	23.9551	1.45249
$273$	0	0
$274$	−27.0797	−1.63594
$275$	4.15633	0.250636
$276$	−0.288213	−0.0173484
$277$	13.3503	0.802139	0.401070	−	0.916048i	$-0.368638\pi$
$277$	13.3503	0.802139	0.401070	+	0.916048i	$0.368638\pi$
$278$	−17.0884	−1.02489
$279$	−11.2955	−0.676244
$280$	0	0
$281$	20.4241	1.21840	0.609199	−	0.793017i	$-0.291491\pi$
$281$	20.4241	1.21840	0.609199	+	0.793017i	$0.291491\pi$
$282$	−5.73813	−0.341701
$283$	8.02047	0.476767	0.238384	−	0.971171i	$-0.423382\pi$
$283$	8.02047	0.476767	0.238384	+	0.971171i	$0.423382\pi$
$284$	−0.247277	−0.0146732
$285$	2.57452	0.152501
$286$	18.2374	1.07840
$287$	0	0
$288$	−2.56959	−0.151415
$289$	13.3225	0.783676
$290$	−1.48119	−0.0869787
$291$	−1.11283	−0.0652355
$292$	2.95651	0.173017
$293$	23.3054	1.36151	0.680757	−	0.732510i	$-0.261651\pi$
$293$	23.3054	1.36151	0.680757	+	0.732510i	$0.261651\pi$
$294$	0	0
$295$	13.2750	0.772903
$296$	−25.4314	−1.47817
$297$	17.9248	1.04010
$298$	−4.11142	−0.238168
$299$	−5.46168	−0.315857
$300$	−0.156325	−0.00902544
$301$	0	0
$302$	−2.66433	−0.153315
$303$	−10.4894	−0.602603
$304$	−13.8945	−0.796902
$305$	8.88717	0.508878
$306$	−19.1695	−1.09585
$307$	6.73084	0.384149	0.192075	−	0.981380i	$-0.438478\pi$
$307$	6.73084	0.384149	0.192075	+	0.981380i	$0.438478\pi$
$308$	0	0
$309$	4.28821	0.243948
$310$	7.11871	0.404316
$311$	22.0567	1.25072	0.625359	−	0.780337i	$-0.284952\pi$
$311$	22.0567	1.25072	0.625359	+	0.780337i	$0.284952\pi$
$312$	6.38787	0.361642
$313$	−5.03761	−0.284743	−0.142371	−	0.989813i	$-0.545473\pi$
$313$	−5.03761	−0.284743	−0.142371	+	0.989813i	$0.545473\pi$
$314$	5.58181	0.315000
$315$	0	0
$316$	−0.956509	−0.0538078
$317$	34.2941	1.92615	0.963074	−	0.269237i	$-0.0867714\pi$
$317$	34.2941	1.92615	0.963074	+	0.269237i	$0.0867714\pi$
$318$	−1.61213	−0.0904036
$319$	−4.15633	−0.232710
$320$	−7.08110	−0.395846
$321$	11.1392	0.621729
$322$	0	0
$323$	−17.5877	−0.978605
$324$	0.693229	0.0385127
$325$	−2.96239	−0.164324
$326$	−2.43278	−0.134739
$327$	1.51056	0.0835340
$328$	30.1622	1.66543
$329$	0	0
$330$	−4.96239	−0.273171
$331$	−34.8324	−1.91456	−0.957281	−	0.289159i	$-0.906625\pi$
$331$	−34.8324	−1.91456	−0.957281	+	0.289159i	$0.906625\pi$
$332$	−0.856849	−0.0470257
$333$	22.3430	1.22439
$334$	11.9697	0.654952
$335$	−5.84367	−0.319274
$336$	0	0
$337$	−17.6326	−0.960509	−0.480254	−	0.877129i	$-0.659456\pi$
$337$	−17.6326	−0.960509	−0.480254	+	0.877129i	$0.659456\pi$
$338$	6.25694	0.340333
$339$	9.48612	0.515215
$340$	1.06793	0.0579166
$341$	19.9756	1.08174
$342$	11.1187	0.601231
$343$	0	0
$344$	−0.0811024	−0.00437275
$345$	1.48612	0.0800099
$346$	−11.4617	−0.616184
$347$	−3.11871	−0.167421	−0.0837107	−	0.996490i	$-0.526677\pi$
$347$	−3.11871	−0.167421	−0.0837107	+	0.996490i	$0.526677\pi$
$348$	0.156325	0.00837991
$349$	13.0738	0.699825	0.349912	−	0.936782i	$-0.386211\pi$
$349$	13.0738	0.699825	0.349912	+	0.936782i	$0.386211\pi$
$350$	0	0
$351$	−12.7757	−0.681919
$352$	4.54420	0.242207
$353$	−5.19982	−0.276758	−0.138379	−	0.990379i	$-0.544189\pi$
$353$	−5.19982	−0.276758	−0.138379	+	0.990379i	$0.544189\pi$
$354$	−15.8496	−0.842394
$355$	1.27504	0.0676720
$356$	0.700523	0.0371277
$357$	0	0
$358$	31.6991	1.67535
$359$	30.4182	1.60541	0.802705	−	0.596376i	$-0.203393\pi$
$359$	30.4182	1.60541	0.802705	+	0.596376i	$0.203393\pi$
$360$	6.28726	0.331368
$361$	−8.79877	−0.463093
$362$	22.6253	1.18916
$363$	−5.05808	−0.265480
$364$	0	0
$365$	−15.2447	−0.797945
$366$	−10.6107	−0.554631
$367$	−20.6556	−1.07821	−0.539107	−	0.842237i	$-0.681238\pi$
$367$	−20.6556	−1.07821	−0.539107	+	0.842237i	$0.681238\pi$
$368$	−8.02047	−0.418096
$369$	−26.4993	−1.37950
$370$	−14.0811	−0.732042
$371$	0	0
$372$	−0.751309	−0.0389535
$373$	11.0884	0.574135	0.287068	−	0.957910i	$-0.407320\pi$
$373$	11.0884	0.574135	0.287068	+	0.957910i	$0.407320\pi$
$374$	33.9003	1.75294
$375$	0.806063	0.0416249
$376$	−12.8568	−0.663041
$377$	2.96239	0.152571
$378$	0	0
$379$	10.0811	0.517831	0.258916	−	0.965900i	$-0.416635\pi$
$379$	10.0811	0.517831	0.258916	+	0.965900i	$0.416635\pi$
$380$	−0.619421	−0.0317756
$381$	−11.5007	−0.589199
$382$	−4.91748	−0.251600
$383$	−16.3576	−0.835832	−0.417916	−	0.908486i	$-0.637239\pi$
$383$	−16.3576	−0.835832	−0.417916	+	0.908486i	$0.637239\pi$
$384$	10.2170	0.521382
$385$	0	0
$386$	7.23013	0.368004
$387$	0.0712533	0.00362201
$388$	0.267745	0.0135927
$389$	31.9003	1.61741	0.808706	−	0.588213i	$-0.200169\pi$
$389$	31.9003	1.61741	0.808706	+	0.588213i	$0.200169\pi$
$390$	3.53690	0.179098
$391$	−10.1524	−0.513427
$392$	0	0
$393$	4.75131	0.239672
$394$	35.9003	1.80863
$395$	4.93207	0.248159
$396$	−1.89446	−0.0952002
$397$	−2.98683	−0.149905	−0.0749523	−	0.997187i	$-0.523880\pi$
$397$	−2.98683	−0.149905	−0.0749523	+	0.997187i	$0.523880\pi$
$398$	24.8119	1.24371
$399$	0	0
$400$	−4.35026	−0.217513
$401$	−21.9756	−1.09741	−0.548704	−	0.836017i	$-0.684878\pi$
$401$	−21.9756	−1.09741	−0.548704	+	0.836017i	$0.684878\pi$
$402$	6.97698	0.347980
$403$	−14.2374	−0.709217
$404$	2.52373	0.125560
$405$	−3.57452	−0.177619
$406$	0	0
$407$	−39.5125	−1.95856
$408$	11.8740	0.587850
$409$	22.4387	1.10952	0.554760	−	0.832010i	$-0.312810\pi$
$409$	22.4387	1.10952	0.554760	+	0.832010i	$0.312810\pi$
$410$	16.7005	0.824780
$411$	−14.7367	−0.726909
$412$	−1.03173	−0.0508298
$413$	0	0
$414$	6.41819	0.315437
$415$	4.41819	0.216880
$416$	−3.23884	−0.158797
$417$	−9.29948	−0.455397
$418$	−19.6629	−0.961744
$419$	−10.3634	−0.506287	−0.253143	−	0.967429i	$-0.581464\pi$
$419$	−10.3634	−0.506287	−0.253143	+	0.967429i	$0.581464\pi$
$420$	0	0
$421$	34.0362	1.65882	0.829411	−	0.558638i	$-0.188676\pi$
$421$	34.0362	1.65882	0.829411	+	0.558638i	$0.188676\pi$
$422$	37.4821	1.82460
$423$	11.2955	0.549206
$424$	−3.61213	−0.175420
$425$	−5.50659	−0.267109
$426$	−1.52232	−0.0737564
$427$	0	0
$428$	−2.68006	−0.129545
$429$	9.92478	0.479173
$430$	−0.0449056	−0.00216554
$431$	25.7743	1.24151	0.620753	−	0.784006i	$-0.286827\pi$
$431$	25.7743	1.24151	0.620753	+	0.784006i	$0.286827\pi$
$432$	−18.7612	−0.902647
$433$	−2.18076	−0.104801	−0.0524004	−	0.998626i	$-0.516687\pi$
$433$	−2.18076	−0.104801	−0.0524004	+	0.998626i	$0.516687\pi$
$434$	0	0
$435$	−0.806063	−0.0386478
$436$	−0.363436	−0.0174054
$437$	5.88858	0.281689
$438$	18.2012	0.869688
$439$	35.5125	1.69492	0.847459	−	0.530861i	$-0.178131\pi$
$439$	35.5125	1.69492	0.847459	+	0.530861i	$0.178131\pi$
$440$	−11.1187	−0.530064
$441$	0	0
$442$	−24.1622	−1.14928
$443$	−4.34297	−0.206341	−0.103170	−	0.994664i	$-0.532899\pi$
$443$	−4.34297	−0.206341	−0.103170	+	0.994664i	$0.532899\pi$
$444$	1.48612	0.0705281
$445$	−3.61213	−0.171231
$446$	26.2071	1.24094
$447$	−2.23743	−0.105827
$448$	0	0
$449$	31.3357	1.47882	0.739411	−	0.673254i	$-0.235104\pi$
$449$	31.3357	1.47882	0.739411	+	0.673254i	$0.235104\pi$
$450$	3.48119	0.164105
$451$	46.8627	2.20668
$452$	−2.28233	−0.107352
$453$	−1.44992	−0.0681233
$454$	39.7586	1.86596
$455$	0	0
$456$	−6.88717	−0.322521
$457$	34.3488	1.60677	0.803386	−	0.595459i	$-0.203030\pi$
$457$	34.3488	1.60677	0.803386	+	0.595459i	$0.203030\pi$
$458$	−25.5125	−1.19212
$459$	−23.7480	−1.10846
$460$	−0.357556	−0.0166711
$461$	−11.8641	−0.552568	−0.276284	−	0.961076i	$-0.589103\pi$
$461$	−11.8641	−0.552568	−0.276284	+	0.961076i	$0.589103\pi$
$462$	0	0
$463$	40.4953	1.88198	0.940989	−	0.338438i	$-0.109899\pi$
$463$	40.4953	1.88198	0.940989	+	0.338438i	$0.109899\pi$
$464$	4.35026	0.201956
$465$	3.87399	0.179652
$466$	13.4401	0.622599
$467$	30.2071	1.39782	0.698909	−	0.715210i	$-0.253669\pi$
$467$	30.2071	1.39782	0.698909	+	0.715210i	$0.253669\pi$
$468$	1.35026	0.0624159
$469$	0	0
$470$	−7.11871	−0.328362
$471$	3.03761	0.139966
$472$	−35.5125	−1.63459
$473$	−0.126008	−0.00579385
$474$	−5.88858	−0.270471
$475$	3.19394	0.146548
$476$	0	0
$477$	3.17347	0.145303
$478$	−30.3634	−1.38879
$479$	−0.0547547	−0.00250181	−0.00125090	−	0.999999i	$-0.500398\pi$
$479$	−0.0547547	−0.00250181	−0.00125090	+	0.999999i	$0.500398\pi$
$480$	0.881286	0.0402250
$481$	28.1622	1.28409
$482$	8.11142	0.369465
$483$	0	0
$484$	1.21696	0.0553163
$485$	−1.38058	−0.0626889
$486$	23.4314	1.06287
$487$	−0.881286	−0.0399349	−0.0199674	−	0.999801i	$-0.506356\pi$
$487$	−0.881286	−0.0399349	−0.0199674	+	0.999801i	$0.506356\pi$
$488$	−23.7743	−1.07621
$489$	−1.32391	−0.0598695
$490$	0	0
$491$	41.0698	1.85346	0.926728	−	0.375733i	$-0.122609\pi$
$491$	41.0698	1.85346	0.926728	+	0.375733i	$0.122609\pi$
$492$	−1.76257	−0.0794629
$493$	5.50659	0.248004
$494$	14.0146	0.630546
$495$	9.76845	0.439059
$496$	−20.9076	−0.938780
$497$	0	0
$498$	−5.27504	−0.236380
$499$	12.3733	0.553904	0.276952	−	0.960884i	$-0.410676\pi$
$499$	12.3733	0.553904	0.276952	+	0.960884i	$0.410676\pi$
$500$	−0.193937	−0.00867311
$501$	6.51388	0.291019
$502$	−43.8700	−1.95801
$503$	2.26774	0.101114	0.0505569	−	0.998721i	$-0.483900\pi$
$503$	2.26774	0.101114	0.0505569	+	0.998721i	$0.483900\pi$
$504$	0	0
$505$	−13.0132	−0.579079
$506$	−11.3503	−0.504581
$507$	3.40502	0.151222
$508$	2.76704	0.122767
$509$	10.9018	0.483212	0.241606	−	0.970374i	$-0.422326\pi$
$509$	10.9018	0.483212	0.241606	+	0.970374i	$0.422326\pi$
$510$	6.57452	0.291124
$511$	0	0
$512$	18.5188	0.818423
$513$	13.7743	0.608152
$514$	26.1622	1.15397
$515$	5.31994	0.234425
$516$	0.00473934	0.000208638 0
$517$	−19.9756	−0.878524
$518$	0	0
$519$	−6.23743	−0.273793
$520$	7.92478	0.347524
$521$	4.72496	0.207004	0.103502	−	0.994629i	$-0.466995\pi$
$521$	4.72496	0.207004	0.103502	+	0.994629i	$0.466995\pi$
$522$	−3.48119	−0.152368
$523$	1.06793	0.0466973	0.0233486	−	0.999727i	$-0.492567\pi$
$523$	1.06793	0.0466973	0.0233486	+	0.999727i	$0.492567\pi$
$524$	−1.14315	−0.0499388
$525$	0	0
$526$	−40.5198	−1.76675
$527$	−26.4650	−1.15283
$528$	14.5745	0.634274
$529$	−19.6009	−0.852211
$530$	−2.00000	−0.0868744
$531$	31.1998	1.35396
$532$	0	0
$533$	−33.4010	−1.44676
$534$	4.31265	0.186627
$535$	13.8192	0.597458
$536$	15.6326	0.675225
$537$	17.2506	0.744418
$538$	15.5515	0.670472
$539$	0	0
$540$	−0.836381	−0.0359921
$541$	−7.46168	−0.320803	−0.160401	−	0.987052i	$-0.551279\pi$
$541$	−7.46168	−0.320803	−0.160401	+	0.987052i	$0.551279\pi$
$542$	−14.2461	−0.611924
$543$	12.3127	0.528386
$544$	−6.02047	−0.258125
$545$	1.87399	0.0802730
$546$	0	0
$547$	−38.9683	−1.66616	−0.833081	−	0.553150i	$-0.813425\pi$
$547$	−38.9683	−1.66616	−0.833081	+	0.553150i	$0.813425\pi$
$548$	3.54561	0.151461
$549$	20.8872	0.891443
$550$	−6.15633	−0.262507
$551$	−3.19394	−0.136066
$552$	−3.97556	−0.169211
$553$	0	0
$554$	−19.7743	−0.840131
$555$	−7.66291	−0.325273
$556$	2.23743	0.0948881
$557$	−22.9986	−0.974481	−0.487241	−	0.873268i	$-0.661997\pi$
$557$	−22.9986	−0.974481	−0.487241	+	0.873268i	$0.661997\pi$
$558$	16.7308	0.708273
$559$	0.0898112	0.00379861
$560$	0	0
$561$	18.4485	0.778897
$562$	−30.2520	−1.27610
$563$	−11.6688	−0.491781	−0.245890	−	0.969298i	$-0.579080\pi$
$563$	−11.6688	−0.491781	−0.245890	+	0.969298i	$0.579080\pi$
$564$	0.751309	0.0316358
$565$	11.7685	0.495102
$566$	−11.8799	−0.499348
$567$	0	0
$568$	−3.41090	−0.143118
$569$	11.3357	0.475216	0.237608	−	0.971361i	$-0.423637\pi$
$569$	11.3357	0.475216	0.237608	+	0.971361i	$0.423637\pi$
$570$	−3.81336	−0.159724
$571$	27.1754	1.13725	0.568627	−	0.822595i	$-0.307475\pi$
$571$	27.1754	1.13725	0.568627	+	0.822595i	$0.307475\pi$
$572$	−2.38787	−0.0998420
$573$	−2.67609	−0.111795
$574$	0	0
$575$	1.84367	0.0768866
$576$	−16.6424	−0.693435
$577$	−22.5950	−0.940641	−0.470321	−	0.882496i	$-0.655862\pi$
$577$	−22.5950	−0.940641	−0.470321	+	0.882496i	$0.655862\pi$
$578$	−19.7332	−0.820793
$579$	3.93463	0.163517
$580$	0.193937	0.00805278
$581$	0	0
$582$	1.64832	0.0683252
$583$	−5.61213	−0.232431
$584$	40.7816	1.68756
$585$	−6.96239	−0.287859
$586$	−34.5198	−1.42600
$587$	−9.31994	−0.384675	−0.192338	−	0.981329i	$-0.561607\pi$
$587$	−9.31994	−0.384675	−0.192338	+	0.981329i	$0.561607\pi$
$588$	0	0
$589$	15.3503	0.632497
$590$	−19.6629	−0.809509
$591$	19.5369	0.803641
$592$	41.3561	1.69973
$593$	15.1246	0.621093	0.310546	−	0.950558i	$-0.399488\pi$
$593$	15.1246	0.621093	0.310546	+	0.950558i	$0.399488\pi$
$594$	−26.5501	−1.08936
$595$	0	0
$596$	0.538319	0.0220504
$597$	13.5026	0.552625
$598$	8.08981	0.330817
$599$	4.09569	0.167345	0.0836727	−	0.996493i	$-0.473335\pi$
$599$	4.09569	0.167345	0.0836727	+	0.996493i	$0.473335\pi$
$600$	−2.15633	−0.0880316
$601$	−22.2276	−0.906682	−0.453341	−	0.891337i	$-0.649768\pi$
$601$	−22.2276	−0.906682	−0.453341	+	0.891337i	$0.649768\pi$
$602$	0	0
$603$	−13.7342	−0.559298
$604$	0.348847	0.0141944
$605$	−6.27504	−0.255117
$606$	15.5369	0.631144
$607$	48.2941	1.96020	0.980098	−	0.198512i	$-0.0636110\pi$
$607$	48.2941	1.96020	0.980098	+	0.198512i	$0.0636110\pi$
$608$	3.49200	0.141619
$609$	0	0
$610$	−13.1636	−0.532979
$611$	14.2374	0.575985
$612$	2.50991	0.101457
$613$	−9.74798	−0.393717	−0.196859	−	0.980432i	$-0.563074\pi$
$613$	−9.74798	−0.393717	−0.196859	+	0.980432i	$0.563074\pi$
$614$	−9.96968	−0.402344
$615$	9.08840	0.366480
$616$	0	0
$617$	18.2170	0.733387	0.366694	−	0.930342i	$-0.380490\pi$
$617$	18.2170	0.733387	0.366694	+	0.930342i	$0.380490\pi$
$618$	−6.35168	−0.255502
$619$	−25.0943	−1.00862	−0.504312	−	0.863521i	$-0.668254\pi$
$619$	−25.0943	−1.00862	−0.504312	+	0.863521i	$0.668254\pi$
$620$	−0.932071	−0.0374329
$621$	7.95112	0.319068
$622$	−32.6702	−1.30996
$623$	0	0
$624$	−10.3879	−0.415848
$625$	1.00000	0.0400000
$626$	7.46168	0.298229
$627$	−10.7005	−0.427338
$628$	−0.730841	−0.0291637
$629$	52.3488	2.08729
$630$	0	0
$631$	21.4617	0.854376	0.427188	−	0.904163i	$-0.359504\pi$
$631$	21.4617	0.854376	0.427188	+	0.904163i	$0.359504\pi$
$632$	−13.1939	−0.524827
$633$	20.3977	0.810737
$634$	−50.7962	−2.01738
$635$	−14.2677	−0.566198
$636$	0.211080	0.00836986
$637$	0	0
$638$	6.15633	0.243731
$639$	2.99668	0.118547
$640$	12.6751	0.501029
$641$	3.17347	0.125344	0.0626722	−	0.998034i	$-0.480038\pi$
$641$	3.17347	0.125344	0.0626722	+	0.998034i	$0.480038\pi$
$642$	−16.4993	−0.651175
$643$	2.74069	0.108082	0.0540411	−	0.998539i	$-0.482790\pi$
$643$	2.74069	0.108082	0.0540411	+	0.998539i	$0.482790\pi$
$644$	0	0
$645$	−0.0244376	−0.000962228 0
$646$	26.0508	1.02495
$647$	−6.34297	−0.249368	−0.124684	−	0.992197i	$-0.539792\pi$
$647$	−6.34297	−0.249368	−0.124684	+	0.992197i	$0.539792\pi$
$648$	9.56230	0.375642
$649$	−55.1754	−2.16582
$650$	4.38787	0.172107
$651$	0	0
$652$	0.318530	0.0124746
$653$	4.08110	0.159706	0.0798529	−	0.996807i	$-0.474555\pi$
$653$	4.08110	0.159706	0.0798529	+	0.996807i	$0.474555\pi$
$654$	−2.23743	−0.0874903
$655$	5.89446	0.230316
$656$	−49.0494	−1.91506
$657$	−35.8291	−1.39783
$658$	0	0
$659$	9.58181	0.373254	0.186627	−	0.982431i	$-0.440244\pi$
$659$	9.58181	0.373254	0.186627	+	0.982431i	$0.440244\pi$
$660$	0.649738	0.0252910
$661$	27.5271	1.07068	0.535339	−	0.844637i	$-0.320184\pi$
$661$	27.5271	1.07068	0.535339	+	0.844637i	$0.320184\pi$
$662$	51.5936	2.00524
$663$	−13.1490	−0.510666
$664$	−11.8192	−0.458675
$665$	0	0
$666$	−33.0943	−1.28238
$667$	−1.84367	−0.0713874
$668$	−1.56722	−0.0606376
$669$	14.2619	0.551396
$670$	8.65562	0.334396
$671$	−36.9380	−1.42597
$672$	0	0
$673$	3.13727	0.120933	0.0604665	−	0.998170i	$-0.480741\pi$
$673$	3.13727	0.120933	0.0604665	+	0.998170i	$0.480741\pi$
$674$	26.1173	1.00600
$675$	4.31265	0.165994
$676$	−0.819237	−0.0315091
$677$	46.2579	1.77784	0.888918	−	0.458067i	$-0.151458\pi$
$677$	46.2579	1.77784	0.888918	+	0.458067i	$0.151458\pi$
$678$	−14.0508	−0.539617
$679$	0	0
$680$	14.7308	0.564902
$681$	21.6366	0.829115
$682$	−29.5877	−1.13297
$683$	−9.01905	−0.345104	−0.172552	−	0.985000i	$-0.555201\pi$
$683$	−9.01905	−0.345104	−0.172552	+	0.985000i	$0.555201\pi$
$684$	−1.45580	−0.0556640
$685$	−18.2823	−0.698532
$686$	0	0
$687$	−13.8838	−0.529702
$688$	0.131888	0.00502817
$689$	4.00000	0.152388
$690$	−2.20123	−0.0837994
$691$	50.0625	1.90447	0.952234	−	0.305368i	$-0.0987794\pi$
$691$	50.0625	1.90447	0.952234	+	0.305368i	$0.0987794\pi$
$692$	1.50071	0.0570483
$693$	0	0
$694$	4.61942	0.175351
$695$	−11.5369	−0.437620
$696$	2.15633	0.0817353
$697$	−62.0870	−2.35171
$698$	−19.3649	−0.732970
$699$	7.31406	0.276643
$700$	0	0
$701$	45.3014	1.71101	0.855505	−	0.517795i	$-0.173247\pi$
$701$	45.3014	1.71101	0.855505	+	0.517795i	$0.173247\pi$
$702$	18.9234	0.714216
$703$	−30.3634	−1.14518
$704$	29.4314	1.10924
$705$	−3.87399	−0.145903
$706$	7.70194	0.289866
$707$	0	0
$708$	2.07522	0.0779916
$709$	−3.27504	−0.122997	−0.0614983	−	0.998107i	$-0.519588\pi$
$709$	−3.27504	−0.122997	−0.0614983	+	0.998107i	$0.519588\pi$
$710$	−1.88858	−0.0708772
$711$	11.5917	0.434721
$712$	9.66291	0.362133
$713$	8.86082	0.331840
$714$	0	0
$715$	12.3127	0.460467
$716$	−4.15045	−0.155109
$717$	−16.5237	−0.617090
$718$	−45.0553	−1.68145
$719$	−27.7235	−1.03391	−0.516957	−	0.856011i	$-0.672935\pi$
$719$	−27.7235	−1.03391	−0.516957	+	0.856011i	$0.672935\pi$
$720$	−10.2243	−0.381035
$721$	0	0
$722$	13.0327	0.485026
$723$	4.41422	0.164167
$724$	−2.96239	−0.110096
$725$	−1.00000	−0.0371391
$726$	7.49200	0.278054
$727$	26.8930	0.997408	0.498704	−	0.866772i	$-0.333810\pi$
$727$	26.8930	0.997408	0.498704	+	0.866772i	$0.333810\pi$
$728$	0	0
$729$	2.02776	0.0751023
$730$	22.5804	0.835738
$731$	0.166944	0.00617465
$732$	1.38929	0.0513496
$733$	−3.17935	−0.117432	−0.0587160	−	0.998275i	$-0.518701\pi$
$733$	−3.17935	−0.117432	−0.0587160	+	0.998275i	$0.518701\pi$
$734$	30.5950	1.12928
$735$	0	0
$736$	2.01573	0.0743007
$737$	24.2882	0.894668
$738$	39.2506	1.44483
$739$	−29.7440	−1.09415	−0.547076	−	0.837083i	$-0.684259\pi$
$739$	−29.7440	−1.09415	−0.547076	+	0.837083i	$0.684259\pi$
$740$	1.84367	0.0677748
$741$	7.62672	0.280174
$742$	0	0
$743$	−4.34297	−0.159328	−0.0796640	−	0.996822i	$-0.525385\pi$
$743$	−4.34297	−0.159328	−0.0796640	+	0.996822i	$0.525385\pi$
$744$	−10.3634	−0.379942
$745$	−2.77575	−0.101695
$746$	−16.4241	−0.601328
$747$	10.3839	0.379927
$748$	−4.43866	−0.162293
$749$	0	0
$750$	−1.19394	−0.0435964
$751$	22.5804	0.823970	0.411985	−	0.911191i	$-0.364836\pi$
$751$	22.5804	0.823970	0.411985	+	0.911191i	$0.364836\pi$
$752$	20.9076	0.762423
$753$	−23.8740	−0.870017
$754$	−4.38787	−0.159797
$755$	−1.79877	−0.0654639
$756$	0	0
$757$	9.88461	0.359262	0.179631	−	0.983734i	$-0.442510\pi$
$757$	9.88461	0.359262	0.179631	+	0.983734i	$0.442510\pi$
$758$	−14.9321	−0.542357
$759$	−6.17679	−0.224203
$760$	−8.54420	−0.309931
$761$	−13.6991	−0.496592	−0.248296	−	0.968684i	$-0.579871\pi$
$761$	−13.6991	−0.496592	−0.248296	+	0.968684i	$0.579871\pi$
$762$	17.0348	0.617105
$763$	0	0
$764$	0.643859	0.0232940
$765$	−12.9419	−0.467916
$766$	24.2287	0.875419
$767$	39.3258	1.41997
$768$	−3.71767	−0.134150
$769$	−25.0132	−0.901998	−0.450999	−	0.892524i	$-0.648932\pi$
$769$	−25.0132	−0.901998	−0.450999	+	0.892524i	$0.648932\pi$
$770$	0	0
$771$	14.2374	0.512748
$772$	−0.946660	−0.0340710
$773$	35.9062	1.29146	0.645728	−	0.763567i	$-0.276554\pi$
$773$	35.9062	1.29146	0.645728	+	0.763567i	$0.276554\pi$
$774$	−0.105540	−0.00379356
$775$	4.80606	0.172639
$776$	3.69323	0.132579
$777$	0	0
$778$	−47.2506	−1.69402
$779$	36.0118	1.29026
$780$	−0.463096	−0.0165815
$781$	−5.29948	−0.189630
$782$	15.0376	0.537744
$783$	−4.31265	−0.154122
$784$	0	0
$785$	3.76845	0.134502
$786$	−7.03761	−0.251023
$787$	−50.3839	−1.79599	−0.897996	−	0.440003i	$-0.854977\pi$
$787$	−50.3839	−1.79599	−0.897996	+	0.440003i	$0.854977\pi$
$788$	−4.70052	−0.167449
$789$	−22.0508	−0.785029
$790$	−7.30536	−0.259913
$791$	0	0
$792$	−26.1319	−0.928556
$793$	26.3272	0.934908
$794$	4.42407	0.157004
$795$	−1.08840	−0.0386014
$796$	−3.24869	−0.115147
$797$	5.69323	0.201665	0.100832	−	0.994903i	$-0.467849\pi$
$797$	5.69323	0.201665	0.100832	+	0.994903i	$0.467849\pi$
$798$	0	0
$799$	26.4650	0.936265
$800$	1.09332	0.0386547
$801$	−8.48944	−0.299960
$802$	32.5501	1.14938
$803$	63.3620	2.23600
$804$	−0.913513	−0.0322171
$805$	0	0
$806$	21.0884	0.742807
$807$	8.46310	0.297915
$808$	34.8119	1.22468
$809$	−7.76257	−0.272918	−0.136459	−	0.990646i	$-0.543572\pi$
$809$	−7.76257	−0.272918	−0.136459	+	0.990646i	$0.543572\pi$
$810$	5.29455	0.186032
$811$	26.4894	0.930170	0.465085	−	0.885266i	$-0.346024\pi$
$811$	26.4894	0.930170	0.465085	+	0.885266i	$0.346024\pi$
$812$	0	0
$813$	−7.75272	−0.271900
$814$	58.5256	2.05132
$815$	−1.64244	−0.0575323
$816$	−19.3093	−0.675962
$817$	−0.0968311	−0.00338769
$818$	−33.2360	−1.16207
$819$	0	0
$820$	−2.18664	−0.0763609
$821$	25.4763	0.889128	0.444564	−	0.895747i	$-0.353359\pi$
$821$	25.4763	0.889128	0.444564	+	0.895747i	$0.353359\pi$
$822$	21.8279	0.761337
$823$	−9.22028	−0.321399	−0.160699	−	0.987003i	$-0.551375\pi$
$823$	−9.22028	−0.321399	−0.160699	+	0.987003i	$0.551375\pi$
$824$	−14.2315	−0.495779
$825$	−3.35026	−0.116641
$826$	0	0
$827$	24.5343	0.853143	0.426571	−	0.904454i	$-0.359721\pi$
$827$	24.5343	0.853143	0.426571	+	0.904454i	$0.359721\pi$
$828$	−0.840350	−0.0292042
$829$	−0.201231	−0.00698903	−0.00349452	−	0.999994i	$-0.501112\pi$
$829$	−0.201231	−0.00698903	−0.00349452	+	0.999994i	$0.501112\pi$
$830$	−6.54420	−0.227152
$831$	−10.7612	−0.373300
$832$	−20.9770	−0.727246
$833$	0	0
$834$	13.7743	0.476966
$835$	8.08110	0.279658
$836$	2.57452	0.0890415
$837$	20.7269	0.716425
$838$	15.3503	0.530266
$839$	−1.45580	−0.0502599	−0.0251299	−	0.999684i	$-0.508000\pi$
$839$	−1.45580	−0.0502599	−0.0251299	+	0.999684i	$0.508000\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−50.4142	−1.73739
$843$	−16.4631	−0.567019
$844$	−4.90763	−0.168928
$845$	4.22425	0.145319
$846$	−16.7308	−0.575218
$847$	0	0
$848$	5.87399	0.201714
$849$	−6.46501	−0.221878
$850$	8.15633	0.279760
$851$	−17.5271	−0.600820
$852$	0.199321	0.00682861
$853$	−43.1793	−1.47843	−0.739216	−	0.673468i	$-0.764804\pi$
$853$	−43.1793	−1.47843	−0.739216	+	0.673468i	$0.764804\pi$
$854$	0	0
$855$	7.50659	0.256720
$856$	−36.9683	−1.26355
$857$	20.9887	0.716962	0.358481	−	0.933537i	$-0.383295\pi$
$857$	20.9887	0.716962	0.358481	+	0.933537i	$0.383295\pi$
$858$	−14.7005	−0.501868
$859$	49.4069	1.68574	0.842871	−	0.538115i	$-0.180863\pi$
$859$	49.4069	1.68574	0.842871	+	0.538115i	$0.180863\pi$
$860$	0.00587961	0.000200493 0
$861$	0	0
$862$	−38.1768	−1.30031
$863$	56.6820	1.92948	0.964738	−	0.263211i	$-0.0847816\pi$
$863$	56.6820	1.92948	0.964738	+	0.263211i	$0.0847816\pi$
$864$	4.71511	0.160411
$865$	−7.73813	−0.263104
$866$	3.23013	0.109764
$867$	−10.7388	−0.364708
$868$	0	0
$869$	−20.4993	−0.695391
$870$	1.19394	0.0404782
$871$	−17.3112	−0.586569
$872$	−5.01317	−0.169767
$873$	−3.24472	−0.109817
$874$	−8.72213	−0.295031
$875$	0	0
$876$	−2.38313	−0.0805186
$877$	−13.1998	−0.445726	−0.222863	−	0.974850i	$-0.571540\pi$
$877$	−13.1998	−0.445726	−0.222863	+	0.974850i	$0.571540\pi$
$878$	−52.6009	−1.77519
$879$	−18.7856	−0.633622
$880$	18.0811	0.609514
$881$	−6.37802	−0.214881	−0.107441	−	0.994212i	$-0.534266\pi$
$881$	−6.37802	−0.214881	−0.107441	+	0.994212i	$0.534266\pi$
$882$	0	0
$883$	48.6213	1.63624	0.818119	−	0.575049i	$-0.195017\pi$
$883$	48.6213	1.63624	0.818119	+	0.575049i	$0.195017\pi$
$884$	3.16362	0.106404
$885$	−10.7005	−0.359694
$886$	6.43278	0.216113
$887$	15.0317	0.504716	0.252358	−	0.967634i	$-0.418794\pi$
$887$	15.0317	0.504716	0.252358	+	0.967634i	$0.418794\pi$
$888$	20.4993	0.687911
$889$	0	0
$890$	5.35026	0.179341
$891$	14.8568	0.497723
$892$	−3.43136	−0.114891
$893$	−15.3503	−0.513677
$894$	3.31406	0.110839
$895$	21.4010	0.715358
$896$	0	0
$897$	4.40246	0.146994
$898$	−46.4142	−1.54886
$899$	−4.80606	−0.160291
$900$	−0.455802	−0.0151934
$901$	7.43533	0.247707
$902$	−69.4128	−2.31119
$903$	0	0
$904$	−31.4821	−1.04708
$905$	15.2750	0.507759
$906$	2.14762	0.0713498
$907$	−0.342968	−0.0113880	−0.00569402	−	0.999984i	$-0.501812\pi$
$907$	−0.342968	−0.0113880	−0.00569402	+	0.999984i	$0.501812\pi$
$908$	−5.20570	−0.172757
$909$	−30.5844	−1.01442
$910$	0	0
$911$	20.9076	0.692701	0.346350	−	0.938105i	$-0.387421\pi$
$911$	20.9076	0.692701	0.346350	+	0.938105i	$0.387421\pi$
$912$	11.1998	0.370863
$913$	−18.3634	−0.607741
$914$	−50.8773	−1.68287
$915$	−7.16362	−0.236822
$916$	3.34041	0.110370
$917$	0	0
$918$	35.1754	1.16096
$919$	−1.90034	−0.0626864	−0.0313432	−	0.999509i	$-0.509978\pi$
$919$	−1.90034	−0.0626864	−0.0313432	+	0.999509i	$0.509978\pi$
$920$	−4.93207	−0.162606
$921$	−5.42548	−0.178776
$922$	17.5731	0.578739
$923$	3.77716	0.124327
$924$	0	0
$925$	−9.50659	−0.312575
$926$	−59.9814	−1.97111
$927$	12.5033	0.410661
$928$	−1.09332	−0.0358900
$929$	−39.3522	−1.29110	−0.645551	−	0.763717i	$-0.723372\pi$
$929$	−39.3522	−1.29110	−0.645551	+	0.763717i	$0.723372\pi$
$930$	−5.73813	−0.188161
$931$	0	0
$932$	−1.75974	−0.0576423
$933$	−17.7791	−0.582061
$934$	−44.7426	−1.46402
$935$	22.8872	0.748490
$936$	18.6253	0.608787
$937$	6.37802	0.208361	0.104180	−	0.994558i	$-0.466778\pi$
$937$	6.37802	0.208361	0.104180	+	0.994558i	$0.466778\pi$
$938$	0	0
$939$	4.06063	0.132514
$940$	0.932071	0.0304008
$941$	26.6253	0.867960	0.433980	−	0.900923i	$-0.357109\pi$
$941$	26.6253	0.867960	0.433980	+	0.900923i	$0.357109\pi$
$942$	−4.49929	−0.146595
$943$	20.7875	0.676934
$944$	57.7499	1.87960
$945$	0	0
$946$	0.186642	0.00606827
$947$	−12.2823	−0.399122	−0.199561	−	0.979885i	$-0.563952\pi$
$947$	−12.2823	−0.399122	−0.199561	+	0.979885i	$0.563952\pi$
$948$	0.771007	0.0250411
$949$	−45.1608	−1.46598
$950$	−4.73084	−0.153489
$951$	−27.6432	−0.896393
$952$	0	0
$953$	0.821792	0.0266205	0.0133102	−	0.999911i	$-0.495763\pi$
$953$	0.821792	0.0266205	0.0133102	+	0.999911i	$0.495763\pi$
$954$	−4.70052	−0.152185
$955$	−3.31994	−0.107431
$956$	3.97556	0.128579
$957$	3.35026	0.108299
$958$	0.0811024	0.00262030
$959$	0	0
$960$	5.70782	0.184219
$961$	−7.90175	−0.254895
$962$	−41.7137	−1.34490
$963$	32.4788	1.04662
$964$	−1.06205	−0.0342063
$965$	4.88129	0.157134
$966$	0	0
$967$	−37.4314	−1.20371	−0.601856	−	0.798605i	$-0.705572\pi$
$967$	−37.4314	−1.20371	−0.601856	+	0.798605i	$0.705572\pi$
$968$	16.7866	0.539540
$969$	14.1768	0.455424
$970$	2.04491	0.0656580
$971$	8.71625	0.279718	0.139859	−	0.990171i	$-0.455335\pi$
$971$	8.71625	0.279718	0.139859	+	0.990171i	$0.455335\pi$
$972$	−3.06793	−0.0984039
$973$	0	0
$974$	1.30536	0.0418263
$975$	2.38787	0.0764731
$976$	38.6615	1.23752
$977$	−33.7645	−1.08022	−0.540111	−	0.841594i	$-0.681618\pi$
$977$	−33.7645	−1.08022	−0.540111	+	0.841594i	$0.681618\pi$
$978$	1.96097	0.0627050
$979$	15.0132	0.479823
$980$	0	0
$981$	4.40437	0.140621
$982$	−60.8324	−1.94124
$983$	−43.6082	−1.39088	−0.695442	−	0.718582i	$-0.744791\pi$
$983$	−43.6082	−1.39088	−0.695442	+	0.718582i	$0.744791\pi$
$984$	−24.3127	−0.775059
$985$	24.2374	0.772269
$986$	−8.15633	−0.259750
$987$	0	0
$988$	−1.83497	−0.0583780
$989$	−0.0558950	−0.00177736
$990$	−14.4690	−0.459854
$991$	−52.9741	−1.68278	−0.841390	−	0.540429i	$-0.818262\pi$
$991$	−52.9741	−1.68278	−0.841390	+	0.540429i	$0.818262\pi$
$992$	5.25457	0.166833
$993$	28.0771	0.891001
$994$	0	0
$995$	16.7513	0.531052
$996$	0.690674	0.0218849
$997$	−13.6326	−0.431749	−0.215874	−	0.976421i	$-0.569260\pi$
$997$	−13.6326	−0.431749	−0.215874	+	0.976421i	$0.569260\pi$
$998$	−18.3272	−0.580139
$999$	−40.9986	−1.29714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.o.1.1		3
7.6	odd	2		145.2.a.c.1.1	✓	3
21.20	even	2		1305.2.a.p.1.3		3
28.27	even	2		2320.2.a.n.1.2		3
35.13	even	4		725.2.b.e.349.5		6
35.27	even	4		725.2.b.e.349.2		6
35.34	odd	2		725.2.a.e.1.3		3
56.13	odd	2		9280.2.a.bj.1.2		3
56.27	even	2		9280.2.a.br.1.2		3
105.104	even	2		6525.2.a.be.1.1		3
203.202	odd	2		4205.2.a.f.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.c.1.1	✓	3	7.6	odd	2
725.2.a.e.1.3		3	35.34	odd	2
725.2.b.e.349.2		6	35.27	even	4
725.2.b.e.349.5		6	35.13	even	4
1305.2.a.p.1.3		3	21.20	even	2
2320.2.a.n.1.2		3	28.27	even	2
4205.2.a.f.1.3		3	203.202	odd	2
6525.2.a.be.1.1		3	105.104	even	2
7105.2.a.o.1.1		3	1.1	even	1	trivial
9280.2.a.bj.1.2		3	56.13	odd	2
9280.2.a.br.1.2		3	56.27	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7105.a (trivial)

\(f(q)\)	\(=\)	\(q-1.48119 q^{2} -0.806063 q^{3} +0.193937 q^{4} -1.00000 q^{5} +1.19394 q^{6} +2.67513 q^{8} -2.35026 q^{9} +1.48119 q^{10} +4.15633 q^{11} -0.156325 q^{12} -2.96239 q^{13} +0.806063 q^{15} -4.35026 q^{16} -5.50659 q^{17} +3.48119 q^{18} +3.19394 q^{19} -0.193937 q^{20} -6.15633 q^{22} +1.84367 q^{23} -2.15633 q^{24} +1.00000 q^{25} +4.38787 q^{26} +4.31265 q^{27} -1.00000 q^{29} -1.19394 q^{30} +4.80606 q^{31} +1.09332 q^{32} -3.35026 q^{33} +8.15633 q^{34} -0.455802 q^{36} -9.50659 q^{37} -4.73084 q^{38} +2.38787 q^{39} -2.67513 q^{40} +11.2750 q^{41} -0.0303172 q^{43} +0.806063 q^{44} +2.35026 q^{45} -2.73084 q^{46} -4.80606 q^{47} +3.50659 q^{48} -1.48119 q^{50} +4.43866 q^{51} -0.574515 q^{52} -1.35026 q^{53} -6.38787 q^{54} -4.15633 q^{55} -2.57452 q^{57} +1.48119 q^{58} -13.2750 q^{59} +0.156325 q^{60} -8.88717 q^{61} -7.11871 q^{62} +7.08110 q^{64} +2.96239 q^{65} +4.96239 q^{66} +5.84367 q^{67} -1.06793 q^{68} -1.48612 q^{69} -1.27504 q^{71} -6.28726 q^{72} +15.2447 q^{73} +14.0811 q^{74} -0.806063 q^{75} +0.619421 q^{76} -3.53690 q^{78} -4.93207 q^{79} +4.35026 q^{80} +3.57452 q^{81} -16.7005 q^{82} -4.41819 q^{83} +5.50659 q^{85} +0.0449056 q^{86} +0.806063 q^{87} +11.1187 q^{88} +3.61213 q^{89} -3.48119 q^{90} +0.357556 q^{92} -3.87399 q^{93} +7.11871 q^{94} -3.19394 q^{95} -0.881286 q^{96} +1.38058 q^{97} -9.76845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9} - q^{10} + 2 q^{11} + 10 q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} + 4 q^{17} + 5 q^{18} + 10 q^{19} - q^{20} - 8 q^{22} + 16 q^{23}+ \cdots - 18 q^{99}+O(q^{100}) \) 3 * q + q^2 - 2 * q^3 + q^4 - 3 * q^5 + 4 * q^6 + 3 * q^8 + 3 * q^9 - q^10 + 2 * q^11 + 10 * q^12 + 2 * q^13 + 2 * q^15 - 3 * q^16 + 4 * q^17 + 5 * q^18 + 10 * q^19 - q^20 - 8 * q^22 + 16 * q^23 + 4 * q^24 + 3 * q^25 + 14 * q^26 - 8 * q^27 - 3 * q^29 - 4 * q^30 + 14 * q^31 - 3 * q^32 + 14 * q^34 - 11 * q^36 - 8 * q^37 + 8 * q^38 + 8 * q^39 - 3 * q^40 + 2 * q^41 + 2 * q^43 + 2 * q^44 - 3 * q^45 + 14 * q^46 - 14 * q^47 - 10 * q^48 + q^50 - 16 * q^51 + 10 * q^52 + 6 * q^53 - 20 * q^54 - 2 * q^55 + 4 * q^57 - q^58 - 8 * q^59 - 10 * q^60 + 6 * q^61 - 11 * q^64 - 2 * q^65 + 4 * q^66 + 28 * q^67 - 12 * q^68 - 12 * q^69 + 28 * q^71 - 13 * q^72 + 16 * q^73 + 10 * q^74 - 2 * q^75 + 14 * q^76 + 12 * q^78 - 6 * q^79 + 3 * q^80 - q^81 - 30 * q^82 - 12 * q^83 - 4 * q^85 + 24 * q^86 + 2 * q^87 + 12 * q^88 + 10 * q^89 - 5 * q^90 + 4 * q^92 - 20 * q^93 - 10 * q^95 - 24 * q^96 - 8 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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