LMFDB - Embedded newform 3025.2.a.bn.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3025,2,Mod(1,3025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3025, 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("3025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3025 = 5^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3025.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:	$24.1547466114$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 $ x^8 - 3x^7 - 7x^6 + 18x^5 + 16x^4 - 30x^3 - 12x^2 + 12*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.56247$ of defining polynomial
Character	$\chi$	$=$	3025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.56247 q^{2} +1.79260 q^{3} +4.56626 q^{4} +4.59347 q^{6} +3.80088 q^{7} +6.57595 q^{8} +0.213399 q^{9} +8.18545 q^{12} -4.49758 q^{13} +9.73964 q^{14} +7.71818 q^{16} +5.01207 q^{17} +0.546828 q^{18} -3.46053 q^{19} +6.81344 q^{21} +0.105727 q^{23} +11.7880 q^{24} -11.5249 q^{26} -4.99525 q^{27} +17.3558 q^{28} -2.35008 q^{29} +4.74526 q^{31} +6.62570 q^{32} +12.8433 q^{34} +0.974434 q^{36} -5.60088 q^{37} -8.86750 q^{38} -8.06233 q^{39} +0.916922 q^{41} +17.4592 q^{42} -4.46162 q^{43} +0.270922 q^{46} +1.25360 q^{47} +13.8356 q^{48} +7.44667 q^{49} +8.98461 q^{51} -20.5371 q^{52} -5.49861 q^{53} -12.8002 q^{54} +24.9944 q^{56} -6.20333 q^{57} -6.02200 q^{58} -1.18305 q^{59} -7.32605 q^{61} +12.1596 q^{62} +0.811103 q^{63} +1.54180 q^{64} +7.84414 q^{67} +22.8864 q^{68} +0.189526 q^{69} -2.18022 q^{71} +1.40330 q^{72} +8.95263 q^{73} -14.3521 q^{74} -15.8017 q^{76} -20.6595 q^{78} +14.3825 q^{79} -9.59466 q^{81} +2.34959 q^{82} -4.71142 q^{83} +31.1119 q^{84} -11.4328 q^{86} -4.21274 q^{87} -0.172993 q^{89} -17.0947 q^{91} +0.482777 q^{92} +8.50634 q^{93} +3.21232 q^{94} +11.8772 q^{96} -4.46043 q^{97} +19.0819 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + q^{3} + 9 q^{4} + q^{6} + 8 q^{7} + 12 q^{8} + 9 q^{9} + 3 q^{12} + 9 q^{13} + 9 q^{14} + 23 q^{16} + 19 q^{17} + 22 q^{18} + q^{19} + 5 q^{21} + 2 q^{23} + q^{24} - 2 q^{26} - 2 q^{27}+ \cdots + 26 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + q^3 + 9 * q^4 + q^6 + 8 * q^7 + 12 * q^8 + 9 * q^9 + 3 * q^12 + 9 * q^13 + 9 * q^14 + 23 * q^16 + 19 * q^17 + 22 * q^18 + q^19 + 5 * q^21 + 2 * q^23 + q^24 - 2 * q^26 - 2 * q^27 + 9 * q^28 + 7 * q^29 - 5 * q^31 + 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 - q^39 + 41 * q^42 + 14 * q^43 - 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 - 25 * q^51 - 7 * q^52 + 11 * q^53 + 30 * q^54 - 10 * q^56 - 2 * q^57 + 27 * q^58 + 17 * q^59 - 2 * q^61 + 25 * q^62 + 41 * q^63 + 30 * q^64 + 7 * q^67 + 66 * q^68 + 17 * q^71 - 19 * q^72 + 34 * q^73 - 6 * q^74 - 31 * q^76 - 17 * q^78 + 23 * q^79 - 4 * q^81 - 17 * q^82 + 41 * q^83 + 83 * q^84 + q^86 + 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 + 50 * q^94 - 61 * q^96 + 2 * q^97 + 26 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	2.56247	1.81194	0.905970	−	0.423342i	$-0.139143\pi$
$2$	2.56247	1.81194	0.905970	+	0.423342i	$0.139143\pi$
$3$	1.79260	1.03496	0.517478	−	0.855697i	$-0.326871\pi$
$3$	1.79260	1.03496	0.517478	+	0.855697i	$0.326871\pi$
$4$	4.56626	2.28313
$5$	0	0
$5$	0	0
$6$	4.59347	1.87528
$7$	3.80088	1.43660	0.718298	−	0.695735i	$-0.244921\pi$
$7$	3.80088	1.43660	0.718298	+	0.695735i	$0.244921\pi$
$8$	6.57595	2.32495
$9$	0.213399	0.0711330
$10$	0	0
$11$	0	0
$11$	0	0
$12$	8.18545	2.36294
$13$	−4.49758	−1.24740	−0.623702	−	0.781663i	$-0.714372\pi$
$13$	−4.49758	−1.24740	−0.623702	+	0.781663i	$0.714372\pi$
$14$	9.73964	2.60303
$15$	0	0
$16$	7.71818	1.92955
$17$	5.01207	1.21560	0.607802	−	0.794088i	$-0.292051\pi$
$17$	5.01207	1.21560	0.607802	+	0.794088i	$0.292051\pi$
$18$	0.546828	0.128889
$19$	−3.46053	−0.793900	−0.396950	−	0.917840i	$-0.629931\pi$
$19$	−3.46053	−0.793900	−0.396950	+	0.917840i	$0.629931\pi$
$20$	0	0
$21$	6.81344	1.48681
$22$	0	0
$23$	0.105727	0.0220456	0.0110228	−	0.999939i	$-0.496491\pi$
$23$	0.105727	0.0220456	0.0110228	+	0.999939i	$0.496491\pi$
$24$	11.7880	2.40622
$25$	0	0
$26$	−11.5249	−2.26022
$27$	−4.99525	−0.961336
$28$	17.3558	3.27993
$29$	−2.35008	−0.436398	−0.218199	−	0.975904i	$-0.570018\pi$
$29$	−2.35008	−0.436398	−0.218199	+	0.975904i	$0.570018\pi$
$30$	0	0
$31$	4.74526	0.852274	0.426137	−	0.904659i	$-0.359874\pi$
$31$	4.74526	0.852274	0.426137	+	0.904659i	$0.359874\pi$
$32$	6.62570	1.17127
$33$	0	0
$34$	12.8433	2.20260
$35$	0	0
$36$	0.974434	0.162406
$37$	−5.60088	−0.920779	−0.460390	−	0.887717i	$-0.652290\pi$
$37$	−5.60088	−0.920779	−0.460390	+	0.887717i	$0.652290\pi$
$38$	−8.86750	−1.43850
$39$	−8.06233	−1.29101
$40$	0	0
$41$	0.916922	0.143199	0.0715996	−	0.997433i	$-0.477190\pi$
$41$	0.916922	0.143199	0.0715996	+	0.997433i	$0.477190\pi$
$42$	17.4592	2.69402
$43$	−4.46162	−0.680390	−0.340195	−	0.940355i	$-0.610493\pi$
$43$	−4.46162	−0.680390	−0.340195	+	0.940355i	$0.610493\pi$
$44$	0	0
$45$	0	0
$46$	0.270922	0.0399453
$47$	1.25360	0.182857	0.0914283	−	0.995812i	$-0.470857\pi$
$47$	1.25360	0.182857	0.0914283	+	0.995812i	$0.470857\pi$
$48$	13.8356	1.99699
$49$	7.44667	1.06381
$50$	0	0
$51$	8.98461	1.25810
$52$	−20.5371	−2.84798
$53$	−5.49861	−0.755292	−0.377646	−	0.925950i	$-0.623266\pi$
$53$	−5.49861	−0.755292	−0.377646	+	0.925950i	$0.623266\pi$
$54$	−12.8002	−1.74188
$55$	0	0
$56$	24.9944	3.34002
$57$	−6.20333	−0.821651
$58$	−6.02200	−0.790728
$59$	−1.18305	−0.154020	−0.0770099	−	0.997030i	$-0.524537\pi$
$59$	−1.18305	−0.154020	−0.0770099	+	0.997030i	$0.524537\pi$
$60$	0	0
$61$	−7.32605	−0.938004	−0.469002	−	0.883197i	$-0.655386\pi$
$61$	−7.32605	−0.938004	−0.469002	+	0.883197i	$0.655386\pi$
$62$	12.1596	1.54427
$63$	0.811103	0.102189
$64$	1.54180	0.192725
$65$	0	0
$66$	0	0
$67$	7.84414	0.958315	0.479157	−	0.877729i	$-0.340942\pi$
$67$	7.84414	0.958315	0.479157	+	0.877729i	$0.340942\pi$
$68$	22.8864	2.77538
$69$	0.189526	0.0228162
$70$	0	0
$71$	−2.18022	−0.258745	−0.129372	−	0.991596i	$-0.541296\pi$
$71$	−2.18022	−0.258745	−0.129372	+	0.991596i	$0.541296\pi$
$72$	1.40330	0.165381
$73$	8.95263	1.04783	0.523913	−	0.851772i	$-0.324472\pi$
$73$	8.95263	1.04783	0.523913	+	0.851772i	$0.324472\pi$
$74$	−14.3521	−1.66840
$75$	0	0
$76$	−15.8017	−1.81257
$77$	0	0
$78$	−20.6595	−2.33923
$79$	14.3825	1.61815	0.809077	−	0.587702i	$-0.199967\pi$
$79$	14.3825	1.61815	0.809077	+	0.587702i	$0.199967\pi$
$80$	0	0
$81$	−9.59466	−1.06607
$82$	2.34959	0.259468
$83$	−4.71142	−0.517146	−0.258573	−	0.965992i	$-0.583252\pi$
$83$	−4.71142	−0.517146	−0.258573	+	0.965992i	$0.583252\pi$
$84$	31.1119	3.39459
$85$	0	0
$86$	−11.4328	−1.23283
$87$	−4.21274	−0.451653
$88$	0	0
$89$	−0.172993	−0.0183372	−0.00916862	−	0.999958i	$-0.502919\pi$
$89$	−0.172993	−0.0183372	−0.00916862	+	0.999958i	$0.502919\pi$
$90$	0	0
$91$	−17.0947	−1.79202
$92$	0.482777	0.0503330
$93$	8.50634	0.882066
$94$	3.21232	0.331325
$95$	0	0
$96$	11.8772	1.21221
$97$	−4.46043	−0.452888	−0.226444	−	0.974024i	$-0.572710\pi$
$97$	−4.46043	−0.452888	−0.226444	+	0.974024i	$0.572710\pi$
$98$	19.0819	1.92756
$99$	0	0
$100$	0	0
$101$	8.80519	0.876149	0.438074	−	0.898939i	$-0.355661\pi$
$101$	8.80519	0.876149	0.438074	+	0.898939i	$0.355661\pi$
$102$	23.0228	2.27960
$103$	−16.4740	−1.62323	−0.811615	−	0.584193i	$-0.801411\pi$
$103$	−16.4740	−1.62323	−0.811615	+	0.584193i	$0.801411\pi$
$104$	−29.5759	−2.90015
$105$	0	0
$106$	−14.0900	−1.36854
$107$	17.1724	1.66012	0.830060	−	0.557674i	$-0.188306\pi$
$107$	17.1724	1.66012	0.830060	+	0.557674i	$0.188306\pi$
$108$	−22.8096	−2.19485
$109$	17.9060	1.71508	0.857542	−	0.514414i	$-0.171991\pi$
$109$	17.9060	1.71508	0.857542	+	0.514414i	$0.171991\pi$
$110$	0	0
$111$	−10.0401	−0.952965
$112$	29.3359	2.77198
$113$	−14.3161	−1.34675	−0.673374	−	0.739302i	$-0.735156\pi$
$113$	−14.3161	−1.34675	−0.673374	+	0.739302i	$0.735156\pi$
$114$	−15.8958	−1.48878
$115$	0	0
$116$	−10.7311	−0.996353
$117$	−0.959778	−0.0887315
$118$	−3.03153	−0.279075
$119$	19.0503	1.74633
$120$	0	0
$121$	0	0
$122$	−18.7728	−1.69961
$123$	1.64367	0.148205
$124$	21.6681	1.94585
$125$	0	0
$126$	2.07843	0.185161
$127$	−5.57000	−0.494257	−0.247129	−	0.968983i	$-0.579487\pi$
$127$	−5.57000	−0.494257	−0.247129	+	0.968983i	$0.579487\pi$
$128$	−9.30058	−0.822063
$129$	−7.99788	−0.704174
$130$	0	0
$131$	−13.8367	−1.20892	−0.604458	−	0.796637i	$-0.706610\pi$
$131$	−13.8367	−1.20892	−0.604458	+	0.796637i	$0.706610\pi$
$132$	0	0
$133$	−13.1530	−1.14051
$134$	20.1004	1.73641
$135$	0	0
$136$	32.9591	2.82622
$137$	−9.94447	−0.849614	−0.424807	−	0.905284i	$-0.639658\pi$
$137$	−9.94447	−0.849614	−0.424807	+	0.905284i	$0.639658\pi$
$138$	0.485654	0.0413417
$139$	2.03061	0.172234	0.0861170	−	0.996285i	$-0.472554\pi$
$139$	2.03061	0.172234	0.0861170	+	0.996285i	$0.472554\pi$
$140$	0	0
$141$	2.24720	0.189248
$142$	−5.58676	−0.468830
$143$	0	0
$144$	1.64705	0.137254
$145$	0	0
$146$	22.9408	1.89860
$147$	13.3489	1.10100
$148$	−25.5751	−2.10226
$149$	10.6201	0.870033	0.435017	−	0.900422i	$-0.356742\pi$
$149$	10.6201	0.870033	0.435017	+	0.900422i	$0.356742\pi$
$150$	0	0
$151$	−3.43448	−0.279494	−0.139747	−	0.990187i	$-0.544629\pi$
$151$	−3.43448	−0.279494	−0.139747	+	0.990187i	$0.544629\pi$
$152$	−22.7563	−1.84578
$153$	1.06957	0.0864696
$154$	0	0
$155$	0	0
$156$	−36.8147	−2.94753
$157$	−0.298585	−0.0238296	−0.0119148	−	0.999929i	$-0.503793\pi$
$157$	−0.298585	−0.0238296	−0.0119148	+	0.999929i	$0.503793\pi$
$158$	36.8547	2.93200
$159$	−9.85678	−0.781693
$160$	0	0
$161$	0.401856	0.0316707
$162$	−24.5860	−1.93166
$163$	14.2597	1.11691	0.558454	−	0.829536i	$-0.311395\pi$
$163$	14.2597	1.11691	0.558454	+	0.829536i	$0.311395\pi$
$164$	4.18690	0.326942
$165$	0	0
$166$	−12.0729	−0.937037
$167$	17.9977	1.39270	0.696352	−	0.717700i	$-0.254805\pi$
$167$	17.9977	1.39270	0.696352	+	0.717700i	$0.254805\pi$
$168$	44.8049	3.45677
$169$	7.22819	0.556014
$170$	0	0
$171$	−0.738473	−0.0564724
$172$	−20.3729	−1.55342
$173$	−8.75353	−0.665519	−0.332759	−	0.943012i	$-0.607980\pi$
$173$	−8.75353	−0.665519	−0.332759	+	0.943012i	$0.607980\pi$
$174$	−10.7950	−0.818368
$175$	0	0
$176$	0	0
$177$	−2.12073	−0.159404
$178$	−0.443290	−0.0332260
$179$	−22.8999	−1.71162	−0.855811	−	0.517289i	$-0.826941\pi$
$179$	−22.8999	−1.71162	−0.855811	+	0.517289i	$0.826941\pi$
$180$	0	0
$181$	17.2705	1.28370	0.641852	−	0.766828i	$-0.278166\pi$
$181$	17.2705	1.28370	0.641852	+	0.766828i	$0.278166\pi$
$182$	−43.8048	−3.24702
$183$	−13.1326	−0.970793
$184$	0.695256	0.0512550
$185$	0	0
$186$	21.7972	1.59825
$187$	0	0
$188$	5.72427	0.417485
$189$	−18.9863	−1.38105
$190$	0	0
$191$	−18.2017	−1.31703	−0.658513	−	0.752569i	$-0.728814\pi$
$191$	−18.2017	−1.31703	−0.658513	+	0.752569i	$0.728814\pi$
$192$	2.76383	0.199462
$193$	−11.0361	−0.794397	−0.397199	−	0.917733i	$-0.630018\pi$
$193$	−11.0361	−0.794397	−0.397199	+	0.917733i	$0.630018\pi$
$194$	−11.4297	−0.820607
$195$	0	0
$196$	34.0034	2.42882
$197$	−15.7213	−1.12009	−0.560047	−	0.828461i	$-0.689217\pi$
$197$	−15.7213	−1.12009	−0.560047	+	0.828461i	$0.689217\pi$
$198$	0	0
$199$	4.44523	0.315114	0.157557	−	0.987510i	$-0.449638\pi$
$199$	4.44523	0.315114	0.157557	+	0.987510i	$0.449638\pi$
$200$	0	0
$201$	14.0614	0.991813
$202$	22.5630	1.58753
$203$	−8.93236	−0.626929
$204$	41.0260	2.87240
$205$	0	0
$206$	−42.2141	−2.94119
$207$	0.0225620	0.00156817
$208$	−34.7131	−2.40692
$209$	0	0
$210$	0	0
$211$	2.31213	0.159173	0.0795866	−	0.996828i	$-0.474640\pi$
$211$	2.31213	0.159173	0.0795866	+	0.996828i	$0.474640\pi$
$212$	−25.1080	−1.72443
$213$	−3.90826	−0.267790
$214$	44.0038	3.00804
$215$	0	0
$216$	−32.8485	−2.23506
$217$	18.0362	1.22437
$218$	45.8836	3.10763
$219$	16.0484	1.08445
$220$	0	0
$221$	−22.5422	−1.51635
$222$	−25.7275	−1.72672
$223$	10.2973	0.689555	0.344777	−	0.938684i	$-0.387954\pi$
$223$	10.2973	0.689555	0.344777	+	0.938684i	$0.387954\pi$
$224$	25.1835	1.68264
$225$	0	0
$226$	−36.6847	−2.44023
$227$	−7.17833	−0.476442	−0.238221	−	0.971211i	$-0.576564\pi$
$227$	−7.17833	−0.476442	−0.238221	+	0.971211i	$0.576564\pi$
$228$	−28.3260	−1.87593
$229$	2.05937	0.136087	0.0680435	−	0.997682i	$-0.478324\pi$
$229$	2.05937	0.136087	0.0680435	+	0.997682i	$0.478324\pi$
$230$	0	0
$231$	0	0
$232$	−15.4540	−1.01461
$233$	17.5865	1.15213	0.576065	−	0.817404i	$-0.304587\pi$
$233$	17.5865	1.15213	0.576065	+	0.817404i	$0.304587\pi$
$234$	−2.45940	−0.160776
$235$	0	0
$236$	−5.40210	−0.351647
$237$	25.7820	1.67472
$238$	48.8157	3.16425
$239$	18.6889	1.20888	0.604442	−	0.796649i	$-0.293396\pi$
$239$	18.6889	1.20888	0.604442	+	0.796649i	$0.293396\pi$
$240$	0	0
$241$	30.1916	1.94481	0.972406	−	0.233296i	$-0.0749510\pi$
$241$	30.1916	1.94481	0.972406	+	0.233296i	$0.0749510\pi$
$242$	0	0
$243$	−2.21359	−0.142002
$244$	−33.4526	−2.14158
$245$	0	0
$246$	4.21186	0.268538
$247$	15.5640	0.990313
$248$	31.2046	1.98150
$249$	−8.44568	−0.535223
$250$	0	0
$251$	−25.5114	−1.61026	−0.805132	−	0.593096i	$-0.797906\pi$
$251$	−25.5114	−1.61026	−0.805132	+	0.593096i	$0.797906\pi$
$252$	3.70370	0.233311
$253$	0	0
$254$	−14.2730	−0.895565
$255$	0	0
$256$	−26.9161	−1.68225
$257$	23.9417	1.49344	0.746721	−	0.665137i	$-0.231627\pi$
$257$	23.9417	1.49344	0.746721	+	0.665137i	$0.231627\pi$
$258$	−20.4943	−1.27592
$259$	−21.2883	−1.32279
$260$	0	0
$261$	−0.501504	−0.0310423
$262$	−35.4561	−2.19048
$263$	26.7696	1.65069	0.825343	−	0.564631i	$-0.190982\pi$
$263$	26.7696	1.65069	0.825343	+	0.564631i	$0.190982\pi$
$264$	0	0
$265$	0	0
$266$	−33.7043	−2.06654
$267$	−0.310107	−0.0189782
$268$	35.8184	2.18796
$269$	−5.96585	−0.363744	−0.181872	−	0.983322i	$-0.558216\pi$
$269$	−5.96585	−0.363744	−0.181872	+	0.983322i	$0.558216\pi$
$270$	0	0
$271$	−7.49365	−0.455207	−0.227603	−	0.973754i	$-0.573089\pi$
$271$	−7.49365	−0.455207	−0.227603	+	0.973754i	$0.573089\pi$
$272$	38.6840	2.34556
$273$	−30.6439	−1.85466
$274$	−25.4824	−1.53945
$275$	0	0
$276$	0.865424	0.0520924
$277$	1.44255	0.0866742	0.0433371	−	0.999061i	$-0.486201\pi$
$277$	1.44255	0.0866742	0.0433371	+	0.999061i	$0.486201\pi$
$278$	5.20338	0.312078
$279$	1.01263	0.0606248
$280$	0	0
$281$	−0.239273	−0.0142738	−0.00713691	−	0.999975i	$-0.502272\pi$
$281$	−0.239273	−0.0142738	−0.00713691	+	0.999975i	$0.502272\pi$
$282$	5.75839	0.342907
$283$	5.27885	0.313795	0.156898	−	0.987615i	$-0.449851\pi$
$283$	5.27885	0.313795	0.156898	+	0.987615i	$0.449851\pi$
$284$	−9.95546	−0.590748
$285$	0	0
$286$	0	0
$287$	3.48511	0.205720
$288$	1.41392	0.0833159
$289$	8.12082	0.477695
$290$	0	0
$291$	−7.99575	−0.468719
$292$	40.8800	2.39232
$293$	14.9959	0.876069	0.438035	−	0.898958i	$-0.355675\pi$
$293$	14.9959	0.876069	0.438035	+	0.898958i	$0.355675\pi$
$294$	34.2061	1.99494
$295$	0	0
$296$	−36.8311	−2.14077
$297$	0	0
$298$	27.2137	1.57645
$299$	−0.475515	−0.0274998
$300$	0	0
$301$	−16.9581	−0.977447
$302$	−8.80076	−0.506427
$303$	15.7841	0.906775
$304$	−26.7090	−1.53187
$305$	0	0
$306$	2.74074	0.156678
$307$	−2.11034	−0.120443	−0.0602217	−	0.998185i	$-0.519181\pi$
$307$	−2.11034	−0.120443	−0.0602217	+	0.998185i	$0.519181\pi$
$308$	0	0
$309$	−29.5312	−1.67997
$310$	0	0
$311$	−16.6612	−0.944768	−0.472384	−	0.881393i	$-0.656606\pi$
$311$	−16.6612	−0.944768	−0.472384	+	0.881393i	$0.656606\pi$
$312$	−53.0175	−3.00153
$313$	29.0799	1.64369	0.821845	−	0.569710i	$-0.192945\pi$
$313$	29.0799	1.64369	0.821845	+	0.569710i	$0.192945\pi$
$314$	−0.765114	−0.0431779
$315$	0	0
$316$	65.6740	3.69445
$317$	18.3165	1.02876	0.514379	−	0.857563i	$-0.328023\pi$
$317$	18.3165	1.02876	0.514379	+	0.857563i	$0.328023\pi$
$318$	−25.2577	−1.41638
$319$	0	0
$320$	0	0
$321$	30.7832	1.71815
$322$	1.02974	0.0573854
$323$	−17.3444	−0.965068
$324$	−43.8117	−2.43398
$325$	0	0
$326$	36.5401	2.02377
$327$	32.0982	1.77504
$328$	6.02964	0.332931
$329$	4.76479	0.262691
$330$	0	0
$331$	21.5599	1.18504	0.592520	−	0.805556i	$-0.298133\pi$
$331$	21.5599	1.18504	0.592520	+	0.805556i	$0.298133\pi$
$332$	−21.5136	−1.18071
$333$	−1.19522	−0.0654977
$334$	46.1186	2.52350
$335$	0	0
$336$	52.5873	2.86887
$337$	−14.0614	−0.765973	−0.382986	−	0.923754i	$-0.625104\pi$
$337$	−14.0614	−0.765973	−0.382986	+	0.923754i	$0.625104\pi$
$338$	18.5220	1.00746
$339$	−25.6630	−1.39382
$340$	0	0
$341$	0	0
$342$	−1.89232	−0.102325
$343$	1.69775	0.0916699
$344$	−29.3394	−1.58187
$345$	0	0
$346$	−22.4307	−1.20588
$347$	28.6497	1.53800	0.768999	−	0.639250i	$-0.220755\pi$
$347$	28.6497	1.53800	0.768999	+	0.639250i	$0.220755\pi$
$348$	−19.2364	−1.03118
$349$	−17.9222	−0.959353	−0.479676	−	0.877446i	$-0.659246\pi$
$349$	−17.9222	−0.959353	−0.479676	+	0.877446i	$0.659246\pi$
$350$	0	0
$351$	22.4665	1.19917
$352$	0	0
$353$	24.7757	1.31868	0.659339	−	0.751845i	$-0.270836\pi$
$353$	24.7757	1.31868	0.659339	+	0.751845i	$0.270836\pi$
$354$	−5.43430	−0.288830
$355$	0	0
$356$	−0.789931	−0.0418663
$357$	34.1494	1.80738
$358$	−58.6804	−3.10136
$359$	−10.4523	−0.551651	−0.275826	−	0.961208i	$-0.588951\pi$
$359$	−10.4523	−0.551651	−0.275826	+	0.961208i	$0.588951\pi$
$360$	0	0
$361$	−7.02474	−0.369723
$362$	44.2551	2.32600
$363$	0	0
$364$	−78.0589	−4.09140
$365$	0	0
$366$	−33.6520	−1.75902
$367$	−15.1951	−0.793176	−0.396588	−	0.917997i	$-0.629806\pi$
$367$	−15.1951	−0.793176	−0.396588	+	0.917997i	$0.629806\pi$
$368$	0.816020	0.0425380
$369$	0.195670	0.0101862
$370$	0	0
$371$	−20.8995	−1.08505
$372$	38.8421	2.01387
$373$	−21.1774	−1.09653	−0.548263	−	0.836306i	$-0.684711\pi$
$373$	−21.1774	−1.09653	−0.548263	+	0.836306i	$0.684711\pi$
$374$	0	0
$375$	0	0
$376$	8.24363	0.425133
$377$	10.5697	0.544365
$378$	−48.6519	−2.50238
$379$	−35.1644	−1.80627	−0.903137	−	0.429351i	$-0.858742\pi$
$379$	−35.1644	−1.80627	−0.903137	+	0.429351i	$0.858742\pi$
$380$	0	0
$381$	−9.98475	−0.511534
$382$	−46.6412	−2.38637
$383$	−1.98986	−0.101677	−0.0508385	−	0.998707i	$-0.516189\pi$
$383$	−1.98986	−0.101677	−0.0508385	+	0.998707i	$0.516189\pi$
$384$	−16.6722	−0.850799
$385$	0	0
$386$	−28.2797	−1.43940
$387$	−0.952104	−0.0483982
$388$	−20.3675	−1.03400
$389$	−6.05849	−0.307178	−0.153589	−	0.988135i	$-0.549083\pi$
$389$	−6.05849	−0.307178	−0.153589	+	0.988135i	$0.549083\pi$
$390$	0	0
$391$	0.529911	0.0267988
$392$	48.9690	2.47331
$393$	−24.8036	−1.25118
$394$	−40.2853	−2.02954
$395$	0	0
$396$	0	0
$397$	35.0663	1.75993	0.879964	−	0.475040i	$-0.157566\pi$
$397$	35.0663	1.75993	0.879964	+	0.475040i	$0.157566\pi$
$398$	11.3908	0.570968
$399$	−23.5781	−1.18038
$400$	0	0
$401$	16.4747	0.822708	0.411354	−	0.911476i	$-0.365056\pi$
$401$	16.4747	0.822708	0.411354	+	0.911476i	$0.365056\pi$
$402$	36.0319	1.79711
$403$	−21.3422	−1.06313
$404$	40.2067	2.00036
$405$	0	0
$406$	−22.8889	−1.13596
$407$	0	0
$408$	59.0824	2.92501
$409$	4.31034	0.213133	0.106566	−	0.994306i	$-0.466014\pi$
$409$	4.31034	0.213133	0.106566	+	0.994306i	$0.466014\pi$
$410$	0	0
$411$	−17.8264	−0.879312
$412$	−75.2244	−3.70604
$413$	−4.49662	−0.221264
$414$	0.0578146	0.00284143
$415$	0	0
$416$	−29.7996	−1.46104
$417$	3.64006	0.178255
$418$	0	0
$419$	−18.7114	−0.914110	−0.457055	−	0.889438i	$-0.651096\pi$
$419$	−18.7114	−0.914110	−0.457055	+	0.889438i	$0.651096\pi$
$420$	0	0
$421$	−30.5078	−1.48686	−0.743429	−	0.668815i	$-0.766802\pi$
$421$	−30.5078	−1.48686	−0.743429	+	0.668815i	$0.766802\pi$
$422$	5.92475	0.288412
$423$	0.267517	0.0130071
$424$	−36.1586	−1.75602
$425$	0	0
$426$	−10.0148	−0.485219
$427$	−27.8454	−1.34753
$428$	78.4136	3.79027
$429$	0	0
$430$	0	0
$431$	8.31340	0.400442	0.200221	−	0.979751i	$-0.435834\pi$
$431$	8.31340	0.400442	0.200221	+	0.979751i	$0.435834\pi$
$432$	−38.5542	−1.85494
$433$	−27.4497	−1.31915	−0.659574	−	0.751640i	$-0.729263\pi$
$433$	−27.4497	−1.31915	−0.659574	+	0.751640i	$0.729263\pi$
$434$	46.2171	2.21849
$435$	0	0
$436$	81.7634	3.91576
$437$	−0.365872	−0.0175020
$438$	41.1237	1.96496
$439$	−23.1527	−1.10502	−0.552510	−	0.833506i	$-0.686330\pi$
$439$	−23.1527	−1.10502	−0.552510	+	0.833506i	$0.686330\pi$
$440$	0	0
$441$	1.58911	0.0756720
$442$	−57.7636	−2.74753
$443$	−20.1835	−0.958947	−0.479474	−	0.877556i	$-0.659172\pi$
$443$	−20.1835	−0.958947	−0.479474	+	0.877556i	$0.659172\pi$
$444$	−45.8457	−2.17574
$445$	0	0
$446$	26.3864	1.24943
$447$	19.0376	0.900446
$448$	5.86020	0.276868
$449$	14.0341	0.662312	0.331156	−	0.943576i	$-0.392561\pi$
$449$	14.0341	0.662312	0.331156	+	0.943576i	$0.392561\pi$
$450$	0	0
$451$	0	0
$452$	−65.3711	−3.07480
$453$	−6.15664	−0.289264
$454$	−18.3943	−0.863285
$455$	0	0
$456$	−40.7928	−1.91030
$457$	−15.6236	−0.730843	−0.365421	−	0.930842i	$-0.619075\pi$
$457$	−15.6236	−0.730843	−0.365421	+	0.930842i	$0.619075\pi$
$458$	5.27707	0.246581
$459$	−25.0365	−1.16860
$460$	0	0
$461$	29.6082	1.37899	0.689496	−	0.724289i	$-0.257832\pi$
$461$	29.6082	1.37899	0.689496	+	0.724289i	$0.257832\pi$
$462$	0	0
$463$	1.31629	0.0611730	0.0305865	−	0.999532i	$-0.490262\pi$
$463$	1.31629	0.0611730	0.0305865	+	0.999532i	$0.490262\pi$
$464$	−18.1383	−0.842050
$465$	0	0
$466$	45.0649	2.08759
$467$	12.0995	0.559898	0.279949	−	0.960015i	$-0.409682\pi$
$467$	12.0995	0.559898	0.279949	+	0.960015i	$0.409682\pi$
$468$	−4.38259	−0.202585
$469$	29.8146	1.37671
$470$	0	0
$471$	−0.535242	−0.0246626
$472$	−7.77967	−0.358088
$473$	0	0
$474$	66.0655	3.03449
$475$	0	0
$476$	86.9883	3.98710
$477$	−1.17340	−0.0537261
$478$	47.8897	2.19043
$479$	3.05400	0.139541	0.0697703	−	0.997563i	$-0.477773\pi$
$479$	3.05400	0.139541	0.0697703	+	0.997563i	$0.477773\pi$
$480$	0	0
$481$	25.1904	1.14858
$482$	77.3651	3.52388
$483$	0.720365	0.0327777
$484$	0	0
$485$	0	0
$486$	−5.67227	−0.257299
$487$	−24.2398	−1.09841	−0.549205	−	0.835688i	$-0.685069\pi$
$487$	−24.2398	−1.09841	−0.549205	+	0.835688i	$0.685069\pi$
$488$	−48.1758	−2.18081
$489$	25.5619	1.15595
$490$	0	0
$491$	−32.6796	−1.47481	−0.737404	−	0.675452i	$-0.763949\pi$
$491$	−32.6796	−1.47481	−0.737404	+	0.675452i	$0.763949\pi$
$492$	7.50542	0.338370
$493$	−11.7787	−0.530488
$494$	39.8823	1.79439
$495$	0	0
$496$	36.6248	1.64450
$497$	−8.28676	−0.371712
$498$	−21.6418	−0.969792
$499$	−30.0059	−1.34325	−0.671625	−	0.740891i	$-0.734403\pi$
$499$	−30.0059	−1.34325	−0.671625	+	0.740891i	$0.734403\pi$
$500$	0	0
$501$	32.2626	1.44139
$502$	−65.3721	−2.91770
$503$	−13.5516	−0.604236	−0.302118	−	0.953271i	$-0.597694\pi$
$503$	−13.5516	−0.604236	−0.302118	+	0.953271i	$0.597694\pi$
$504$	5.33378	0.237585
$505$	0	0
$506$	0	0
$507$	12.9572	0.575450
$508$	−25.4340	−1.12845
$509$	38.0843	1.68806	0.844028	−	0.536299i	$-0.180178\pi$
$509$	38.0843	1.68806	0.844028	+	0.536299i	$0.180178\pi$
$510$	0	0
$511$	34.0278	1.50530
$512$	−50.3705	−2.22608
$513$	17.2862	0.763204
$514$	61.3499	2.70603
$515$	0	0
$516$	−36.5203	−1.60772
$517$	0	0
$518$	−54.5505	−2.39681
$519$	−15.6915	−0.688782
$520$	0	0
$521$	2.20872	0.0967658	0.0483829	−	0.998829i	$-0.484593\pi$
$521$	2.20872	0.0967658	0.0483829	+	0.998829i	$0.484593\pi$
$522$	−1.28509	−0.0562468
$523$	2.40261	0.105059	0.0525294	−	0.998619i	$-0.483272\pi$
$523$	2.40261	0.105059	0.0525294	+	0.998619i	$0.483272\pi$
$524$	−63.1818	−2.76011
$525$	0	0
$526$	68.5964	2.99095
$527$	23.7836	1.03603
$528$	0	0
$529$	−22.9888	−0.999514
$530$	0	0
$531$	−0.252461	−0.0109559
$532$	−60.0602	−2.60394
$533$	−4.12393	−0.178627
$534$	−0.794639	−0.0343874
$535$	0	0
$536$	51.5827	2.22804
$537$	−41.0503	−1.77145
$538$	−15.2873	−0.659083
$539$	0	0
$540$	0	0
$541$	−23.5101	−1.01078	−0.505388	−	0.862892i	$-0.668651\pi$
$541$	−23.5101	−1.01078	−0.505388	+	0.862892i	$0.668651\pi$
$542$	−19.2023	−0.824807
$543$	30.9590	1.32858
$544$	33.2085	1.42380
$545$	0	0
$546$	−78.5242	−3.36053
$547$	1.41425	0.0604688	0.0302344	−	0.999543i	$-0.490375\pi$
$547$	1.41425	0.0604688	0.0302344	+	0.999543i	$0.490375\pi$
$548$	−45.4090	−1.93978
$549$	−1.56337	−0.0667230
$550$	0	0
$551$	8.13251	0.346457
$552$	1.24631	0.0530466
$553$	54.6660	2.32464
$554$	3.69648	0.157049
$555$	0	0
$556$	9.27228	0.393232
$557$	4.96462	0.210358	0.105179	−	0.994453i	$-0.466458\pi$
$557$	4.96462	0.210358	0.105179	+	0.994453i	$0.466458\pi$
$558$	2.59484	0.109849
$559$	20.0665	0.848721
$560$	0	0
$561$	0	0
$562$	−0.613129	−0.0258633
$563$	16.6024	0.699707	0.349853	−	0.936804i	$-0.386231\pi$
$563$	16.6024	0.699707	0.349853	+	0.936804i	$0.386231\pi$
$564$	10.2613	0.432078
$565$	0	0
$566$	13.5269	0.568578
$567$	−36.4681	−1.53152
$568$	−14.3371	−0.601569
$569$	−27.0868	−1.13554	−0.567768	−	0.823189i	$-0.692193\pi$
$569$	−27.0868	−1.13554	−0.567768	+	0.823189i	$0.692193\pi$
$570$	0	0
$571$	−25.9648	−1.08659	−0.543296	−	0.839541i	$-0.682824\pi$
$571$	−25.9648	−1.08659	−0.543296	+	0.839541i	$0.682824\pi$
$572$	0	0
$573$	−32.6282	−1.36306
$574$	8.93049	0.372751
$575$	0	0
$576$	0.329019	0.0137091
$577$	−0.746507	−0.0310775	−0.0155388	−	0.999879i	$-0.504946\pi$
$577$	−0.746507	−0.0310775	−0.0155388	+	0.999879i	$0.504946\pi$
$578$	20.8094	0.865555
$579$	−19.7833	−0.822166
$580$	0	0
$581$	−17.9075	−0.742930
$582$	−20.4889	−0.849291
$583$	0	0
$584$	58.8721	2.43614
$585$	0	0
$586$	38.4265	1.58738
$587$	17.9002	0.738819	0.369409	−	0.929267i	$-0.379560\pi$
$587$	17.9002	0.738819	0.369409	+	0.929267i	$0.379560\pi$
$588$	60.9544	2.51372
$589$	−16.4211	−0.676620
$590$	0	0
$591$	−28.1819	−1.15925
$592$	−43.2286	−1.77668
$593$	11.1992	0.459895	0.229948	−	0.973203i	$-0.426145\pi$
$593$	11.1992	0.459895	0.229948	+	0.973203i	$0.426145\pi$
$594$	0	0
$595$	0	0
$596$	48.4941	1.98640
$597$	7.96851	0.326129
$598$	−1.21849	−0.0498279
$599$	−1.81590	−0.0741958	−0.0370979	−	0.999312i	$-0.511811\pi$
$599$	−1.81590	−0.0741958	−0.0370979	+	0.999312i	$0.511811\pi$
$600$	0	0
$601$	46.8125	1.90952	0.954761	−	0.297373i	$-0.0961106\pi$
$601$	46.8125	1.90952	0.954761	+	0.297373i	$0.0961106\pi$
$602$	−43.4545	−1.77107
$603$	1.67393	0.0681678
$604$	−15.6827	−0.638121
$605$	0	0
$606$	40.4464	1.64302
$607$	32.8999	1.33537	0.667683	−	0.744446i	$-0.267286\pi$
$607$	32.8999	1.33537	0.667683	+	0.744446i	$0.267286\pi$
$608$	−22.9284	−0.929870
$609$	−16.0121	−0.648843
$610$	0	0
$611$	−5.63817	−0.228096
$612$	4.88393	0.197421
$613$	14.8860	0.601240	0.300620	−	0.953744i	$-0.402806\pi$
$613$	14.8860	0.601240	0.300620	+	0.953744i	$0.402806\pi$
$614$	−5.40768	−0.218236
$615$	0	0
$616$	0	0
$617$	37.4741	1.50865	0.754324	−	0.656502i	$-0.227965\pi$
$617$	37.4741	1.50865	0.754324	+	0.656502i	$0.227965\pi$
$618$	−75.6728	−3.04401
$619$	−12.3920	−0.498077	−0.249039	−	0.968494i	$-0.580115\pi$
$619$	−12.3920	−0.498077	−0.249039	+	0.968494i	$0.580115\pi$
$620$	0	0
$621$	−0.528133	−0.0211932
$622$	−42.6937	−1.71186
$623$	−0.657526	−0.0263432
$624$	−62.2265	−2.49106
$625$	0	0
$626$	74.5163	2.97827
$627$	0	0
$628$	−1.36341	−0.0544061
$629$	−28.0720	−1.11930
$630$	0	0
$631$	1.54279	0.0614174	0.0307087	−	0.999528i	$-0.490224\pi$
$631$	1.54279	0.0614174	0.0307087	+	0.999528i	$0.490224\pi$
$632$	94.5785	3.76213
$633$	4.14471	0.164737
$634$	46.9355	1.86405
$635$	0	0
$636$	−45.0086	−1.78471
$637$	−33.4920	−1.32700
$638$	0	0
$639$	−0.465257	−0.0184053
$640$	0	0
$641$	31.2172	1.23300	0.616502	−	0.787353i	$-0.288549\pi$
$641$	31.2172	1.23300	0.616502	+	0.787353i	$0.288549\pi$
$642$	78.8810	3.11319
$643$	20.3183	0.801275	0.400638	−	0.916237i	$-0.368789\pi$
$643$	20.3183	0.801275	0.400638	+	0.916237i	$0.368789\pi$
$644$	1.83498	0.0723082
$645$	0	0
$646$	−44.4445	−1.74865
$647$	−22.1687	−0.871541	−0.435771	−	0.900058i	$-0.643524\pi$
$647$	−22.1687	−0.871541	−0.435771	+	0.900058i	$0.643524\pi$
$648$	−63.0940	−2.47857
$649$	0	0
$650$	0	0
$651$	32.3316	1.26717
$652$	65.1135	2.55004
$653$	−11.8719	−0.464583	−0.232292	−	0.972646i	$-0.574622\pi$
$653$	−11.8719	−0.464583	−0.232292	+	0.972646i	$0.574622\pi$
$654$	82.2507	3.21626
$655$	0	0
$656$	7.07697	0.276309
$657$	1.91048	0.0745349
$658$	12.2096	0.475981
$659$	13.3368	0.519527	0.259764	−	0.965672i	$-0.416355\pi$
$659$	13.3368	0.519527	0.259764	+	0.965672i	$0.416355\pi$
$660$	0	0
$661$	−44.8817	−1.74570	−0.872848	−	0.487992i	$-0.837730\pi$
$661$	−44.8817	−1.74570	−0.872848	+	0.487992i	$0.837730\pi$
$662$	55.2467	2.14722
$663$	−40.4090	−1.56935
$664$	−30.9821	−1.20234
$665$	0	0
$666$	−3.06272	−0.118678
$667$	−0.248467	−0.00962067
$668$	82.1821	3.17972
$669$	18.4588	0.713659
$670$	0	0
$671$	0	0
$672$	45.1438	1.74146
$673$	5.88475	0.226840	0.113420	−	0.993547i	$-0.463819\pi$
$673$	5.88475	0.226840	0.113420	+	0.993547i	$0.463819\pi$
$674$	−36.0319	−1.38790
$675$	0	0
$676$	33.0057	1.26945
$677$	29.6008	1.13765	0.568825	−	0.822458i	$-0.307398\pi$
$677$	29.6008	1.13765	0.568825	+	0.822458i	$0.307398\pi$
$678$	−65.7608	−2.52553
$679$	−16.9536	−0.650618
$680$	0	0
$681$	−12.8678	−0.493097
$682$	0	0
$683$	13.1257	0.502239	0.251120	−	0.967956i	$-0.419201\pi$
$683$	13.1257	0.502239	0.251120	+	0.967956i	$0.419201\pi$
$684$	−3.37206	−0.128934
$685$	0	0
$686$	4.35043	0.166100
$687$	3.69162	0.140844
$688$	−34.4356	−1.31284
$689$	24.7304	0.942153
$690$	0	0
$691$	−7.04276	−0.267919	−0.133960	−	0.990987i	$-0.542769\pi$
$691$	−7.04276	−0.267919	−0.133960	+	0.990987i	$0.542769\pi$
$692$	−39.9709	−1.51946
$693$	0	0
$694$	73.4141	2.78676
$695$	0	0
$696$	−27.7028	−1.05007
$697$	4.59568	0.174074
$698$	−45.9251	−1.73829
$699$	31.5255	1.19240
$700$	0	0
$701$	44.3634	1.67558	0.837791	−	0.545992i	$-0.183847\pi$
$701$	44.3634	1.67558	0.837791	+	0.545992i	$0.183847\pi$
$702$	57.5698	2.17283
$703$	19.3820	0.731006
$704$	0	0
$705$	0	0
$706$	63.4871	2.38937
$707$	33.4674	1.25867
$708$	−9.68378	−0.363939
$709$	−19.5161	−0.732943	−0.366471	−	0.930429i	$-0.619434\pi$
$709$	−19.5161	−0.732943	−0.366471	+	0.930429i	$0.619434\pi$
$710$	0	0
$711$	3.06920	0.115104
$712$	−1.13760	−0.0426332
$713$	0.501703	0.0187889
$714$	87.5068	3.27486
$715$	0	0
$716$	−104.567	−3.90785
$717$	33.5016	1.25114
$718$	−26.7837	−0.999560
$719$	47.4024	1.76781	0.883906	−	0.467665i	$-0.154905\pi$
$719$	47.4024	1.76781	0.883906	+	0.467665i	$0.154905\pi$
$720$	0	0
$721$	−62.6156	−2.33193
$722$	−18.0007	−0.669916
$723$	54.1213	2.01279
$724$	78.8614	2.93086
$725$	0	0
$726$	0	0
$727$	−37.2483	−1.38146	−0.690731	−	0.723112i	$-0.742711\pi$
$727$	−37.2483	−1.38146	−0.690731	+	0.723112i	$0.742711\pi$
$728$	−112.414	−4.16635
$729$	24.8159	0.919107
$730$	0	0
$731$	−22.3619	−0.827086
$732$	−59.9670	−2.21644
$733$	−11.6209	−0.429226	−0.214613	−	0.976699i	$-0.568849\pi$
$733$	−11.6209	−0.429226	−0.214613	+	0.976699i	$0.568849\pi$
$734$	−38.9369	−1.43719
$735$	0	0
$736$	0.700516	0.0258214
$737$	0	0
$738$	0.501399	0.0184568
$739$	37.9801	1.39712	0.698560	−	0.715551i	$-0.253824\pi$
$739$	37.9801	1.39712	0.698560	+	0.715551i	$0.253824\pi$
$740$	0	0
$741$	27.8999	1.02493
$742$	−53.5544	−1.96605
$743$	7.71703	0.283111	0.141555	−	0.989930i	$-0.454790\pi$
$743$	7.71703	0.283111	0.141555	+	0.989930i	$0.454790\pi$
$744$	55.9373	2.05076
$745$	0	0
$746$	−54.2665	−1.98684
$747$	−1.00541	−0.0367861
$748$	0	0
$749$	65.2703	2.38492
$750$	0	0
$751$	16.0280	0.584870	0.292435	−	0.956285i	$-0.405534\pi$
$751$	16.0280	0.584870	0.292435	+	0.956285i	$0.405534\pi$
$752$	9.67552	0.352830
$753$	−45.7316	−1.66655
$754$	27.0844	0.986356
$755$	0	0
$756$	−86.6964	−3.15312
$757$	6.37605	0.231742	0.115871	−	0.993264i	$-0.463034\pi$
$757$	6.37605	0.231742	0.115871	+	0.993264i	$0.463034\pi$
$758$	−90.1078	−3.27286
$759$	0	0
$760$	0	0
$761$	−15.8646	−0.575092	−0.287546	−	0.957767i	$-0.592839\pi$
$761$	−15.8646	−0.575092	−0.287546	+	0.957767i	$0.592839\pi$
$762$	−25.5856	−0.926870
$763$	68.0585	2.46388
$764$	−83.1135	−3.00694
$765$	0	0
$766$	−5.09895	−0.184233
$767$	5.32085	0.192125
$768$	−48.2496	−1.74106
$769$	−51.7503	−1.86616	−0.933082	−	0.359663i	$-0.882892\pi$
$769$	−51.7503	−1.86616	−0.933082	+	0.359663i	$0.882892\pi$
$770$	0	0
$771$	42.9178	1.54565
$772$	−50.3937	−1.81371
$773$	−9.68605	−0.348383	−0.174192	−	0.984712i	$-0.555731\pi$
$773$	−9.68605	−0.348383	−0.174192	+	0.984712i	$0.555731\pi$
$774$	−2.43974	−0.0876946
$775$	0	0
$776$	−29.3316	−1.05294
$777$	−38.1612	−1.36903
$778$	−15.5247	−0.556588
$779$	−3.17304	−0.113686
$780$	0	0
$781$	0	0
$782$	1.35788	0.0485578
$783$	11.7392	0.419526
$784$	57.4748	2.05267
$785$	0	0
$786$	−63.5584	−2.26705
$787$	18.2226	0.649566	0.324783	−	0.945789i	$-0.394709\pi$
$787$	18.2226	0.649566	0.324783	+	0.945789i	$0.394709\pi$
$788$	−71.7874	−2.55732
$789$	47.9871	1.70839
$790$	0	0
$791$	−54.4139	−1.93473
$792$	0	0
$793$	32.9495	1.17007
$794$	89.8564	3.18889
$795$	0	0
$796$	20.2981	0.719446
$797$	20.0132	0.708902	0.354451	−	0.935075i	$-0.384668\pi$
$797$	20.0132	0.708902	0.354451	+	0.935075i	$0.384668\pi$
$798$	−60.4182	−2.13878
$799$	6.28314	0.222281
$800$	0	0
$801$	−0.0369165	−0.00130438
$802$	42.2160	1.49070
$803$	0	0
$804$	64.2079	2.26444
$805$	0	0
$806$	−54.6887	−1.92633
$807$	−10.6944	−0.376459
$808$	57.9025	2.03700
$809$	−31.8062	−1.11825	−0.559123	−	0.829085i	$-0.688862\pi$
$809$	−31.8062	−1.11825	−0.559123	+	0.829085i	$0.688862\pi$
$810$	0	0
$811$	3.65484	0.128339	0.0641694	−	0.997939i	$-0.479560\pi$
$811$	3.65484	0.128339	0.0641694	+	0.997939i	$0.479560\pi$
$812$	−40.7874	−1.43136
$813$	−13.4331	−0.471119
$814$	0	0
$815$	0	0
$816$	69.3448	2.42755
$817$	15.4396	0.540162
$818$	11.0451	0.386184
$819$	−3.64800	−0.127471
$820$	0	0
$821$	13.7678	0.480501	0.240250	−	0.970711i	$-0.422770\pi$
$821$	13.7678	0.480501	0.240250	+	0.970711i	$0.422770\pi$
$822$	−45.6797	−1.59326
$823$	40.5760	1.41439	0.707195	−	0.707018i	$-0.249960\pi$
$823$	40.5760	1.41439	0.707195	+	0.707018i	$0.249960\pi$
$824$	−108.332	−3.77393
$825$	0	0
$826$	−11.5225	−0.400918
$827$	−15.5321	−0.540105	−0.270052	−	0.962846i	$-0.587041\pi$
$827$	−15.5321	−0.540105	−0.270052	+	0.962846i	$0.587041\pi$
$828$	0.103024	0.00358033
$829$	−29.1311	−1.01177	−0.505883	−	0.862602i	$-0.668833\pi$
$829$	−29.1311	−1.01177	−0.505883	+	0.862602i	$0.668833\pi$
$830$	0	0
$831$	2.58590	0.0897040
$832$	−6.93437	−0.240406
$833$	37.3232	1.29317
$834$	9.32755	0.322987
$835$	0	0
$836$	0	0
$837$	−23.7038	−0.819322
$838$	−47.9473	−1.65631
$839$	21.7832	0.752041	0.376020	−	0.926611i	$-0.377292\pi$
$839$	21.7832	0.752041	0.376020	+	0.926611i	$0.377292\pi$
$840$	0	0
$841$	−23.4771	−0.809556
$842$	−78.1753	−2.69410
$843$	−0.428919	−0.0147728
$844$	10.5578	0.363413
$845$	0	0
$846$	0.685505	0.0235681
$847$	0	0
$848$	−42.4392	−1.45737
$849$	9.46285	0.324764
$850$	0	0
$851$	−0.592165	−0.0202991
$852$	−17.8461	−0.611398
$853$	−8.75293	−0.299695	−0.149847	−	0.988709i	$-0.547878\pi$
$853$	−8.75293	−0.299695	−0.149847	+	0.988709i	$0.547878\pi$
$854$	−71.3531	−2.44165
$855$	0	0
$856$	112.925	3.85970
$857$	−14.0273	−0.479163	−0.239582	−	0.970876i	$-0.577010\pi$
$857$	−14.0273	−0.479163	−0.239582	+	0.970876i	$0.577010\pi$
$858$	0	0
$859$	31.8774	1.08764	0.543822	−	0.839201i	$-0.316977\pi$
$859$	31.8774	1.08764	0.543822	+	0.839201i	$0.316977\pi$
$860$	0	0
$861$	6.24739	0.212911
$862$	21.3028	0.725577
$863$	−3.46523	−0.117958	−0.0589789	−	0.998259i	$-0.518784\pi$
$863$	−3.46523	−0.117958	−0.0589789	+	0.998259i	$0.518784\pi$
$864$	−33.0970	−1.12598
$865$	0	0
$866$	−70.3390	−2.39022
$867$	14.5573	0.494393
$868$	82.3577	2.79540
$869$	0	0
$870$	0	0
$871$	−35.2796	−1.19540
$872$	117.749	3.98749
$873$	−0.951851	−0.0322153
$874$	−0.937535	−0.0317126
$875$	0	0
$876$	73.2813	2.47594
$877$	−19.7763	−0.667799	−0.333899	−	0.942609i	$-0.608365\pi$
$877$	−19.7763	−0.667799	−0.333899	+	0.942609i	$0.608365\pi$
$878$	−59.3282	−2.00223
$879$	26.8816	0.906692
$880$	0	0
$881$	57.2097	1.92744	0.963722	−	0.266910i	$-0.0860025\pi$
$881$	57.2097	1.92744	0.963722	+	0.266910i	$0.0860025\pi$
$882$	4.07205	0.137113
$883$	−57.2412	−1.92632	−0.963159	−	0.268931i	$-0.913329\pi$
$883$	−57.2412	−1.92632	−0.963159	+	0.268931i	$0.913329\pi$
$884$	−102.933	−3.46202
$885$	0	0
$886$	−51.7196	−1.73756
$887$	39.7251	1.33384	0.666920	−	0.745129i	$-0.267612\pi$
$887$	39.7251	1.33384	0.666920	+	0.745129i	$0.267612\pi$
$888$	−66.0233	−2.21560
$889$	−21.1709	−0.710049
$890$	0	0
$891$	0	0
$892$	47.0199	1.57434
$893$	−4.33812	−0.145170
$894$	48.7832	1.63155
$895$	0	0
$896$	−35.3504	−1.18097
$897$	−0.852407	−0.0284610
$898$	35.9621	1.20007
$899$	−11.1517	−0.371931
$900$	0	0
$901$	−27.5594	−0.918136
$902$	0	0
$903$	−30.3989	−1.01161
$904$	−94.1422	−3.13112
$905$	0	0
$906$	−15.7762	−0.524129
$907$	−26.6119	−0.883635	−0.441817	−	0.897105i	$-0.645666\pi$
$907$	−26.6119	−0.883635	−0.441817	+	0.897105i	$0.645666\pi$
$908$	−32.7781	−1.08778
$909$	1.87902	0.0623230
$910$	0	0
$911$	7.78778	0.258021	0.129010	−	0.991643i	$-0.458820\pi$
$911$	7.78778	0.258021	0.129010	+	0.991643i	$0.458820\pi$
$912$	−47.8784	−1.58541
$913$	0	0
$914$	−40.0351	−1.32424
$915$	0	0
$916$	9.40361	0.310704
$917$	−52.5915	−1.73673
$918$	−64.1554	−2.11744
$919$	−15.9781	−0.527069	−0.263535	−	0.964650i	$-0.584888\pi$
$919$	−15.9781	−0.527069	−0.263535	+	0.964650i	$0.584888\pi$
$920$	0	0
$921$	−3.78298	−0.124654
$922$	75.8702	2.49865
$923$	9.80572	0.322759
$924$	0	0
$925$	0	0
$926$	3.37295	0.110842
$927$	−3.51553	−0.115465
$928$	−15.5709	−0.511140
$929$	9.92533	0.325639	0.162820	−	0.986656i	$-0.447941\pi$
$929$	9.92533	0.325639	0.162820	+	0.986656i	$0.447941\pi$
$930$	0	0
$931$	−25.7694	−0.844559
$932$	80.3045	2.63046
$933$	−29.8667	−0.977793
$934$	31.0046	1.01450
$935$	0	0
$936$	−6.31145	−0.206296
$937$	−8.91808	−0.291341	−0.145671	−	0.989333i	$-0.546534\pi$
$937$	−8.91808	−0.291341	−0.145671	+	0.989333i	$0.546534\pi$
$938$	76.3991	2.49452
$939$	52.1284	1.70115
$940$	0	0
$941$	37.3215	1.21665	0.608323	−	0.793690i	$-0.291843\pi$
$941$	37.3215	1.21665	0.608323	+	0.793690i	$0.291843\pi$
$942$	−1.37154	−0.0446872
$943$	0.0969435	0.00315691
$944$	−9.13098	−0.297188
$945$	0	0
$946$	0	0
$947$	−1.62118	−0.0526814	−0.0263407	−	0.999653i	$-0.508385\pi$
$947$	−1.62118	−0.0526814	−0.0263407	+	0.999653i	$0.508385\pi$
$948$	117.727	3.82359
$949$	−40.2651	−1.30706
$950$	0	0
$951$	32.8341	1.06472
$952$	125.274	4.06014
$953$	25.3420	0.820907	0.410454	−	0.911881i	$-0.365370\pi$
$953$	25.3420	0.820907	0.410454	+	0.911881i	$0.365370\pi$
$954$	−3.00679	−0.0973486
$955$	0	0
$956$	85.3383	2.76004
$957$	0	0
$958$	7.82578	0.252839
$959$	−37.7977	−1.22055
$960$	0	0
$961$	−8.48248	−0.273628
$962$	64.5496	2.08116
$963$	3.66457	0.118089
$964$	137.863	4.44025
$965$	0	0
$966$	1.84591	0.0593913
$967$	−6.70594	−0.215648	−0.107824	−	0.994170i	$-0.534388\pi$
$967$	−6.70594	−0.215648	−0.107824	+	0.994170i	$0.534388\pi$
$968$	0	0
$969$	−31.0915	−0.998803
$970$	0	0
$971$	5.81959	0.186760	0.0933798	−	0.995631i	$-0.470233\pi$
$971$	5.81959	0.186760	0.0933798	+	0.995631i	$0.470233\pi$
$972$	−10.1078	−0.324209
$973$	7.71810	0.247431
$974$	−62.1137	−1.99025
$975$	0	0
$976$	−56.5438	−1.80992
$977$	5.23201	0.167387	0.0836934	−	0.996492i	$-0.473328\pi$
$977$	5.23201	0.167387	0.0836934	+	0.996492i	$0.473328\pi$
$978$	65.5017	2.09451
$979$	0	0
$980$	0	0
$981$	3.82112	0.121999
$982$	−83.7404	−2.67227
$983$	−14.7913	−0.471768	−0.235884	−	0.971781i	$-0.575799\pi$
$983$	−14.7913	−0.471768	−0.235884	+	0.971781i	$0.575799\pi$
$984$	10.8087	0.344569
$985$	0	0
$986$	−30.1827	−0.961213
$987$	8.54134	0.271874
$988$	71.0692	2.26101
$989$	−0.471714	−0.0149996
$990$	0	0
$991$	24.9189	0.791575	0.395788	−	0.918342i	$-0.370472\pi$
$991$	24.9189	0.791575	0.395788	+	0.918342i	$0.370472\pi$
$992$	31.4407	0.998243
$993$	38.6482	1.22646
$994$	−21.2346	−0.673520
$995$	0	0
$996$	−38.5651	−1.22198
$997$	14.8384	0.469937	0.234969	−	0.972003i	$-0.424501\pi$
$997$	14.8384	0.469937	0.234969	+	0.972003i	$0.424501\pi$
$998$	−76.8893	−2.43389
$999$	27.9778	0.885178

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3025.2.a.bn.1.7		8
5.4	even	2		3025.2.a.bi.1.2		8
11.5	even	5		275.2.h.e.201.1	yes	16
11.9	even	5		275.2.h.e.26.1	yes	16
11.10	odd	2		3025.2.a.bj.1.2		8
55.9	even	10		275.2.h.c.26.4	✓	16
55.27	odd	20		275.2.z.c.124.1		32
55.38	odd	20		275.2.z.c.124.8		32
55.42	odd	20		275.2.z.c.224.8		32
55.49	even	10		275.2.h.c.201.4	yes	16
55.53	odd	20		275.2.z.c.224.1		32
55.54	odd	2		3025.2.a.bm.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.h.c.26.4	✓	16	55.9	even	10
275.2.h.c.201.4	yes	16	55.49	even	10
275.2.h.e.26.1	yes	16	11.9	even	5
275.2.h.e.201.1	yes	16	11.5	even	5
275.2.z.c.124.1		32	55.27	odd	20
275.2.z.c.124.8		32	55.38	odd	20
275.2.z.c.224.1		32	55.53	odd	20
275.2.z.c.224.8		32	55.42	odd	20
3025.2.a.bi.1.2		8	5.4	even	2
3025.2.a.bj.1.2		8	11.10	odd	2
3025.2.a.bm.1.7		8	55.54	odd	2
3025.2.a.bn.1.7		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3025 = 5^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56247 q^{2} +1.79260 q^{3} +4.56626 q^{4} +4.59347 q^{6} +3.80088 q^{7} +6.57595 q^{8} +0.213399 q^{9} +8.18545 q^{12} -4.49758 q^{13} +9.73964 q^{14} +7.71818 q^{16} +5.01207 q^{17} +0.546828 q^{18} -3.46053 q^{19} +6.81344 q^{21} +0.105727 q^{23} +11.7880 q^{24} -11.5249 q^{26} -4.99525 q^{27} +17.3558 q^{28} -2.35008 q^{29} +4.74526 q^{31} +6.62570 q^{32} +12.8433 q^{34} +0.974434 q^{36} -5.60088 q^{37} -8.86750 q^{38} -8.06233 q^{39} +0.916922 q^{41} +17.4592 q^{42} -4.46162 q^{43} +0.270922 q^{46} +1.25360 q^{47} +13.8356 q^{48} +7.44667 q^{49} +8.98461 q^{51} -20.5371 q^{52} -5.49861 q^{53} -12.8002 q^{54} +24.9944 q^{56} -6.20333 q^{57} -6.02200 q^{58} -1.18305 q^{59} -7.32605 q^{61} +12.1596 q^{62} +0.811103 q^{63} +1.54180 q^{64} +7.84414 q^{67} +22.8864 q^{68} +0.189526 q^{69} -2.18022 q^{71} +1.40330 q^{72} +8.95263 q^{73} -14.3521 q^{74} -15.8017 q^{76} -20.6595 q^{78} +14.3825 q^{79} -9.59466 q^{81} +2.34959 q^{82} -4.71142 q^{83} +31.1119 q^{84} -11.4328 q^{86} -4.21274 q^{87} -0.172993 q^{89} -17.0947 q^{91} +0.482777 q^{92} +8.50634 q^{93} +3.21232 q^{94} +11.8772 q^{96} -4.46043 q^{97} +19.0819 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + q^{3} + 9 q^{4} + q^{6} + 8 q^{7} + 12 q^{8} + 9 q^{9} + 3 q^{12} + 9 q^{13} + 9 q^{14} + 23 q^{16} + 19 q^{17} + 22 q^{18} + q^{19} + 5 q^{21} + 2 q^{23} + q^{24} - 2 q^{26} - 2 q^{27}+ \cdots + 26 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + q^3 + 9 * q^4 + q^6 + 8 * q^7 + 12 * q^8 + 9 * q^9 + 3 * q^12 + 9 * q^13 + 9 * q^14 + 23 * q^16 + 19 * q^17 + 22 * q^18 + q^19 + 5 * q^21 + 2 * q^23 + q^24 - 2 * q^26 - 2 * q^27 + 9 * q^28 + 7 * q^29 - 5 * q^31 + 29 * q^32 + 10 * q^34 - 16 * q^36 - 8 * q^37 - 37 * q^38 - q^39 + 41 * q^42 + 14 * q^43 - 20 * q^46 + 11 * q^47 - 27 * q^48 - 12 * q^49 - 25 * q^51 - 7 * q^52 + 11 * q^53 + 30 * q^54 - 10 * q^56 - 2 * q^57 + 27 * q^58 + 17 * q^59 - 2 * q^61 + 25 * q^62 + 41 * q^63 + 30 * q^64 + 7 * q^67 + 66 * q^68 + 17 * q^71 - 19 * q^72 + 34 * q^73 - 6 * q^74 - 31 * q^76 - 17 * q^78 + 23 * q^79 - 4 * q^81 - 17 * q^82 + 41 * q^83 + 83 * q^84 + q^86 + 25 * q^87 - 11 * q^89 - 7 * q^91 + 33 * q^92 - 59 * q^93 + 50 * q^94 - 61 * q^96 + 2 * q^97 + 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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