LMFDB - Embedded newform 775.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [775,2,Mod(1,775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(775, 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("775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.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:	$6.18840615665$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 155)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} -1.93185 q^{3} -2.73205 q^{6} +2.44949 q^{7} -2.82843 q^{8} +0.732051 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} -1.93185 q^{3} -2.73205 q^{6} +2.44949 q^{7} -2.82843 q^{8} +0.732051 q^{9} -1.26795 q^{11} +2.44949 q^{13} +3.46410 q^{14} -4.00000 q^{16} -4.38134 q^{17} +1.03528 q^{18} -7.19615 q^{19} -4.73205 q^{21} -1.79315 q^{22} -1.41421 q^{23} +5.46410 q^{24} +3.46410 q^{26} +4.38134 q^{27} -8.19615 q^{29} +1.00000 q^{31} +2.44949 q^{33} -6.19615 q^{34} -3.34607 q^{37} -10.1769 q^{38} -4.73205 q^{39} -12.4641 q^{41} -6.69213 q^{42} +0.240237 q^{43} -2.00000 q^{46} -5.93426 q^{47} +7.72741 q^{48} -1.00000 q^{49} +8.46410 q^{51} +8.62398 q^{53} +6.19615 q^{54} -6.92820 q^{56} +13.9019 q^{57} -11.5911 q^{58} +3.92820 q^{59} +6.39230 q^{61} +1.41421 q^{62} +1.79315 q^{63} +8.00000 q^{64} +3.46410 q^{66} -2.44949 q^{67} +2.73205 q^{69} -4.26795 q^{71} -2.07055 q^{72} +4.00240 q^{73} -4.73205 q^{74} -3.10583 q^{77} -6.69213 q^{78} +6.19615 q^{79} -10.6603 q^{81} -17.6269 q^{82} +7.48717 q^{83} +0.339746 q^{86} +15.8338 q^{87} +3.58630 q^{88} -3.80385 q^{89} +6.00000 q^{91} -1.93185 q^{93} -8.39230 q^{94} +3.10583 q^{97} -1.41421 q^{98} -0.928203 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^6 - 4 * q^9	$ 4 q - 4 q^{6} - 4 q^{9} - 12 q^{11} - 16 q^{16} - 8 q^{19} - 12 q^{21} + 8 q^{24} - 12 q^{29} + 4 q^{31} - 4 q^{34} - 12 q^{39} - 36 q^{41} - 8 q^{46} - 4 q^{49} + 20 q^{51} + 4 q^{54} - 12 q^{59} - 16 q^{61} + 32 q^{64} + 4 q^{69} - 24 q^{71} - 12 q^{74} + 4 q^{79} - 8 q^{81} + 36 q^{86} - 36 q^{89} + 24 q^{91} + 8 q^{94} + 24 q^{99}+O(q^{100}) $ 4 * q - 4 * q^6 - 4 * q^9 - 12 * q^11 - 16 * q^16 - 8 * q^19 - 12 * q^21 + 8 * q^24 - 12 * q^29 + 4 * q^31 - 4 * q^34 - 12 * q^39 - 36 * q^41 - 8 * q^46 - 4 * q^49 + 20 * q^51 + 4 * q^54 - 12 * q^59 - 16 * q^61 + 32 * q^64 + 4 * q^69 - 24 * q^71 - 12 * q^74 + 4 * q^79 - 8 * q^81 + 36 * q^86 - 36 * q^89 + 24 * q^91 + 8 * q^94 + 24 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	−1.93185	−1.11536	−0.557678	−	0.830058i	$-0.688307\pi$
$3$	−1.93185	−1.11536	−0.557678	+	0.830058i	$0.688307\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	−2.73205	−1.11536
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	−2.82843	−1.00000
$9$	0.732051	0.244017
$10$	0	0
$11$	−1.26795	−0.382301	−0.191151	−	0.981561i	$-0.561222\pi$
$11$	−1.26795	−0.382301	−0.191151	+	0.981561i	$0.561222\pi$
$12$	0	0
$13$	2.44949	0.679366	0.339683	−	0.940540i	$-0.389680\pi$
$13$	2.44949	0.679366	0.339683	+	0.940540i	$0.389680\pi$
$14$	3.46410	0.925820
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−4.38134	−1.06263	−0.531316	−	0.847174i	$-0.678302\pi$
$17$	−4.38134	−1.06263	−0.531316	+	0.847174i	$0.678302\pi$
$18$	1.03528	0.244017
$19$	−7.19615	−1.65091	−0.825455	−	0.564467i	$-0.809082\pi$
$19$	−7.19615	−1.65091	−0.825455	+	0.564467i	$0.809082\pi$
$20$	0	0
$21$	−4.73205	−1.03262
$22$	−1.79315	−0.382301
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	5.46410	1.11536
$25$	0	0
$26$	3.46410	0.679366
$27$	4.38134	0.843190
$28$	0	0
$29$	−8.19615	−1.52199	−0.760994	−	0.648759i	$-0.775288\pi$
$29$	−8.19615	−1.52199	−0.760994	+	0.648759i	$0.775288\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	2.44949	0.426401
$34$	−6.19615	−1.06263
$35$	0	0
$36$	0	0
$37$	−3.34607	−0.550090	−0.275045	−	0.961431i	$-0.588693\pi$
$37$	−3.34607	−0.550090	−0.275045	+	0.961431i	$0.588693\pi$
$38$	−10.1769	−1.65091
$39$	−4.73205	−0.757735
$40$	0	0
$41$	−12.4641	−1.94657	−0.973283	−	0.229610i	$-0.926255\pi$
$41$	−12.4641	−1.94657	−0.973283	+	0.229610i	$0.926255\pi$
$42$	−6.69213	−1.03262
$43$	0.240237	0.0366357	0.0183179	−	0.999832i	$-0.494169\pi$
$43$	0.240237	0.0366357	0.0183179	+	0.999832i	$0.494169\pi$
$44$	0	0
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−5.93426	−0.865600	−0.432800	−	0.901490i	$-0.642474\pi$
$47$	−5.93426	−0.865600	−0.432800	+	0.901490i	$0.642474\pi$
$48$	7.72741	1.11536
$49$	−1.00000	−0.142857
$50$	0	0
$51$	8.46410	1.18521
$52$	0	0
$53$	8.62398	1.18460	0.592298	−	0.805719i	$-0.298221\pi$
$53$	8.62398	1.18460	0.592298	+	0.805719i	$0.298221\pi$
$54$	6.19615	0.843190
$55$	0	0
$56$	−6.92820	−0.925820
$57$	13.9019	1.84135
$58$	−11.5911	−1.52199
$59$	3.92820	0.511409	0.255704	−	0.966755i	$-0.417693\pi$
$59$	3.92820	0.511409	0.255704	+	0.966755i	$0.417693\pi$
$60$	0	0
$61$	6.39230	0.818451	0.409225	−	0.912433i	$-0.365799\pi$
$61$	6.39230	0.818451	0.409225	+	0.912433i	$0.365799\pi$
$62$	1.41421	0.179605
$63$	1.79315	0.225916
$64$	8.00000	1.00000
$65$	0	0
$66$	3.46410	0.426401
$67$	−2.44949	−0.299253	−0.149626	−	0.988743i	$-0.547807\pi$
$67$	−2.44949	−0.299253	−0.149626	+	0.988743i	$0.547807\pi$
$68$	0	0
$69$	2.73205	0.328900
$70$	0	0
$71$	−4.26795	−0.506512	−0.253256	−	0.967399i	$-0.581502\pi$
$71$	−4.26795	−0.506512	−0.253256	+	0.967399i	$0.581502\pi$
$72$	−2.07055	−0.244017
$73$	4.00240	0.468446	0.234223	−	0.972183i	$-0.424745\pi$
$73$	4.00240	0.468446	0.234223	+	0.972183i	$0.424745\pi$
$74$	−4.73205	−0.550090
$75$	0	0
$76$	0	0
$77$	−3.10583	−0.353942
$78$	−6.69213	−0.757735
$79$	6.19615	0.697122	0.348561	−	0.937286i	$-0.386670\pi$
$79$	6.19615	0.697122	0.348561	+	0.937286i	$0.386670\pi$
$80$	0	0
$81$	−10.6603	−1.18447
$82$	−17.6269	−1.94657
$83$	7.48717	0.821824	0.410912	−	0.911675i	$-0.365210\pi$
$83$	7.48717	0.821824	0.410912	+	0.911675i	$0.365210\pi$
$84$	0	0
$85$	0	0
$86$	0.339746	0.0366357
$87$	15.8338	1.69756
$88$	3.58630	0.382301
$89$	−3.80385	−0.403207	−0.201604	−	0.979467i	$-0.564615\pi$
$89$	−3.80385	−0.403207	−0.201604	+	0.979467i	$0.564615\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	−1.93185	−0.200324
$94$	−8.39230	−0.865600
$95$	0	0
$96$	0	0
$97$	3.10583	0.315349	0.157675	−	0.987491i	$-0.449600\pi$
$97$	3.10583	0.315349	0.157675	+	0.987491i	$0.449600\pi$
$98$	−1.41421	−0.142857
$99$	−0.928203	−0.0932879
$100$	0	0
$101$	−11.1962	−1.11406	−0.557029	−	0.830493i	$-0.688059\pi$
$101$	−11.1962	−1.11406	−0.557029	+	0.830493i	$0.688059\pi$
$102$	11.9700	1.18521
$103$	15.1774	1.49547	0.747737	−	0.663995i	$-0.231140\pi$
$103$	15.1774	1.49547	0.747737	+	0.663995i	$0.231140\pi$
$104$	−6.92820	−0.679366
$105$	0	0
$106$	12.1962	1.18460
$107$	8.76268	0.847121	0.423560	−	0.905868i	$-0.360780\pi$
$107$	8.76268	0.847121	0.423560	+	0.905868i	$0.360780\pi$
$108$	0	0
$109$	13.5885	1.30154	0.650769	−	0.759276i	$-0.274447\pi$
$109$	13.5885	1.30154	0.650769	+	0.759276i	$0.274447\pi$
$110$	0	0
$111$	6.46410	0.613545
$112$	−9.79796	−0.925820
$113$	−16.1112	−1.51561	−0.757805	−	0.652481i	$-0.773728\pi$
$113$	−16.1112	−1.51561	−0.757805	+	0.652481i	$0.773728\pi$
$114$	19.6603	1.84135
$115$	0	0
$116$	0	0
$117$	1.79315	0.165777
$118$	5.55532	0.511409
$119$	−10.7321	−0.983805
$120$	0	0
$121$	−9.39230	−0.853846
$122$	9.04008	0.818451
$123$	24.0788	2.17111
$124$	0	0
$125$	0	0
$126$	2.53590	0.225916
$127$	−6.03579	−0.535590	−0.267795	−	0.963476i	$-0.586295\pi$
$127$	−6.03579	−0.535590	−0.267795	+	0.963476i	$0.586295\pi$
$128$	11.3137	1.00000
$129$	−0.464102	−0.0408619
$130$	0	0
$131$	20.6603	1.80509	0.902547	−	0.430591i	$-0.141695\pi$
$131$	20.6603	1.80509	0.902547	+	0.430591i	$0.141695\pi$
$132$	0	0
$133$	−17.6269	−1.52845
$134$	−3.46410	−0.299253
$135$	0	0
$136$	12.3923	1.06263
$137$	19.9377	1.70339	0.851696	−	0.524036i	$-0.175574\pi$
$137$	19.9377	1.70339	0.851696	+	0.524036i	$0.175574\pi$
$138$	3.86370	0.328900
$139$	1.80385	0.153000	0.0765002	−	0.997070i	$-0.475625\pi$
$139$	1.80385	0.153000	0.0765002	+	0.997070i	$0.475625\pi$
$140$	0	0
$141$	11.4641	0.965452
$142$	−6.03579	−0.506512
$143$	−3.10583	−0.259722
$144$	−2.92820	−0.244017
$145$	0	0
$146$	5.66025	0.468446
$147$	1.93185	0.159336
$148$	0	0
$149$	−6.46410	−0.529560	−0.264780	−	0.964309i	$-0.585299\pi$
$149$	−6.46410	−0.529560	−0.264780	+	0.964309i	$0.585299\pi$
$150$	0	0
$151$	−16.5885	−1.34995	−0.674975	−	0.737841i	$-0.735845\pi$
$151$	−16.5885	−1.34995	−0.674975	+	0.737841i	$0.735845\pi$
$152$	20.3538	1.65091
$153$	−3.20736	−0.259300
$154$	−4.39230	−0.353942
$155$	0	0
$156$	0	0
$157$	−10.4543	−0.834344	−0.417172	−	0.908828i	$-0.636979\pi$
$157$	−10.4543	−0.834344	−0.417172	+	0.908828i	$0.636979\pi$
$158$	8.76268	0.697122
$159$	−16.6603	−1.32124
$160$	0	0
$161$	−3.46410	−0.273009
$162$	−15.0759	−1.18447
$163$	−20.0764	−1.57250	−0.786252	−	0.617906i	$-0.787981\pi$
$163$	−20.0764	−1.57250	−0.786252	+	0.617906i	$0.787981\pi$
$164$	0	0
$165$	0	0
$166$	10.5885	0.821824
$167$	−15.9725	−1.23599	−0.617993	−	0.786184i	$-0.712054\pi$
$167$	−15.9725	−1.23599	−0.617993	+	0.786184i	$0.712054\pi$
$168$	13.3843	1.03262
$169$	−7.00000	−0.538462
$170$	0	0
$171$	−5.26795	−0.402850
$172$	0	0
$173$	−7.62587	−0.579784	−0.289892	−	0.957059i	$-0.593619\pi$
$173$	−7.62587	−0.579784	−0.289892	+	0.957059i	$0.593619\pi$
$174$	22.3923	1.69756
$175$	0	0
$176$	5.07180	0.382301
$177$	−7.58871	−0.570402
$178$	−5.37945	−0.403207
$179$	4.73205	0.353690	0.176845	−	0.984239i	$-0.443411\pi$
$179$	4.73205	0.353690	0.176845	+	0.984239i	$0.443411\pi$
$180$	0	0
$181$	24.3923	1.81307	0.906533	−	0.422135i	$-0.138719\pi$
$181$	24.3923	1.81307	0.906533	+	0.422135i	$0.138719\pi$
$182$	8.48528	0.628971
$183$	−12.3490	−0.912863
$184$	4.00000	0.294884
$185$	0	0
$186$	−2.73205	−0.200324
$187$	5.55532	0.406245
$188$	0	0
$189$	10.7321	0.780642
$190$	0	0
$191$	12.9282	0.935452	0.467726	−	0.883874i	$-0.345073\pi$
$191$	12.9282	0.935452	0.467726	+	0.883874i	$0.345073\pi$
$192$	−15.4548	−1.11536
$193$	−3.58630	−0.258148	−0.129074	−	0.991635i	$-0.541200\pi$
$193$	−3.58630	−0.258148	−0.129074	+	0.991635i	$0.541200\pi$
$194$	4.39230	0.315349
$195$	0	0
$196$	0	0
$197$	−12.4505	−0.887063	−0.443531	−	0.896259i	$-0.646275\pi$
$197$	−12.4505	−0.887063	−0.443531	+	0.896259i	$0.646275\pi$
$198$	−1.31268	−0.0932879
$199$	22.5885	1.60125	0.800627	−	0.599164i	$-0.204500\pi$
$199$	22.5885	1.60125	0.800627	+	0.599164i	$0.204500\pi$
$200$	0	0
$201$	4.73205	0.333773
$202$	−15.8338	−1.11406
$203$	−20.0764	−1.40909
$204$	0	0
$205$	0	0
$206$	21.4641	1.49547
$207$	−1.03528	−0.0719567
$208$	−9.79796	−0.679366
$209$	9.12436	0.631145
$210$	0	0
$211$	−2.39230	−0.164693	−0.0823465	−	0.996604i	$-0.526241\pi$
$211$	−2.39230	−0.164693	−0.0823465	+	0.996604i	$0.526241\pi$
$212$	0	0
$213$	8.24504	0.564941
$214$	12.3923	0.847121
$215$	0	0
$216$	−12.3923	−0.843190
$217$	2.44949	0.166282
$218$	19.2170	1.30154
$219$	−7.73205	−0.522484
$220$	0	0
$221$	−10.7321	−0.721916
$222$	9.14162	0.613545
$223$	−17.8671	−1.19647	−0.598236	−	0.801320i	$-0.704131\pi$
$223$	−17.8671	−1.19647	−0.598236	+	0.801320i	$0.704131\pi$
$224$	0	0
$225$	0	0
$226$	−22.7846	−1.51561
$227$	9.89949	0.657053	0.328526	−	0.944495i	$-0.393448\pi$
$227$	9.89949	0.657053	0.328526	+	0.944495i	$0.393448\pi$
$228$	0	0
$229$	−4.19615	−0.277290	−0.138645	−	0.990342i	$-0.544275\pi$
$229$	−4.19615	−0.277290	−0.138645	+	0.990342i	$0.544275\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	23.1822	1.52199
$233$	−15.2789	−1.00096	−0.500479	−	0.865749i	$-0.666843\pi$
$233$	−15.2789	−1.00096	−0.500479	+	0.865749i	$0.666843\pi$
$234$	2.53590	0.165777
$235$	0	0
$236$	0	0
$237$	−11.9700	−0.777538
$238$	−15.1774	−0.983805
$239$	−17.3205	−1.12037	−0.560185	−	0.828367i	$-0.689270\pi$
$239$	−17.3205	−1.12037	−0.560185	+	0.828367i	$0.689270\pi$
$240$	0	0
$241$	−16.5885	−1.06856	−0.534278	−	0.845309i	$-0.679417\pi$
$241$	−16.5885	−1.06856	−0.534278	+	0.845309i	$0.679417\pi$
$242$	−13.2827	−0.853846
$243$	7.45001	0.477918
$244$	0	0
$245$	0	0
$246$	34.0526	2.17111
$247$	−17.6269	−1.12157
$248$	−2.82843	−0.179605
$249$	−14.4641	−0.916625
$250$	0	0
$251$	19.8564	1.25333	0.626663	−	0.779291i	$-0.284420\pi$
$251$	19.8564	1.25333	0.626663	+	0.779291i	$0.284420\pi$
$252$	0	0
$253$	1.79315	0.112734
$254$	−8.53590	−0.535590
$255$	0	0
$256$	0	0
$257$	1.41421	0.0882162	0.0441081	−	0.999027i	$-0.485955\pi$
$257$	1.41421	0.0882162	0.0441081	+	0.999027i	$0.485955\pi$
$258$	−0.656339	−0.0408619
$259$	−8.19615	−0.509284
$260$	0	0
$261$	−6.00000	−0.371391
$262$	29.2180	1.80509
$263$	14.0034	0.863489	0.431744	−	0.901996i	$-0.357898\pi$
$263$	14.0034	0.863489	0.431744	+	0.901996i	$0.357898\pi$
$264$	−6.92820	−0.426401
$265$	0	0
$266$	−24.9282	−1.52845
$267$	7.34847	0.449719
$268$	0	0
$269$	−4.73205	−0.288518	−0.144259	−	0.989540i	$-0.546080\pi$
$269$	−4.73205	−0.288518	−0.144259	+	0.989540i	$0.546080\pi$
$270$	0	0
$271$	−16.5885	−1.00768	−0.503839	−	0.863798i	$-0.668079\pi$
$271$	−16.5885	−1.00768	−0.503839	+	0.863798i	$0.668079\pi$
$272$	17.5254	1.06263
$273$	−11.5911	−0.701526
$274$	28.1962	1.70339
$275$	0	0
$276$	0	0
$277$	−28.9778	−1.74111	−0.870553	−	0.492075i	$-0.836239\pi$
$277$	−28.9778	−1.74111	−0.870553	+	0.492075i	$0.836239\pi$
$278$	2.55103	0.153000
$279$	0.732051	0.0438267
$280$	0	0
$281$	−18.8038	−1.12174	−0.560872	−	0.827903i	$-0.689534\pi$
$281$	−18.8038	−1.12174	−0.560872	+	0.827903i	$0.689534\pi$
$282$	16.2127	0.965452
$283$	3.10583	0.184622	0.0923112	−	0.995730i	$-0.470575\pi$
$283$	3.10583	0.184622	0.0923112	+	0.995730i	$0.470575\pi$
$284$	0	0
$285$	0	0
$286$	−4.39230	−0.259722
$287$	−30.5307	−1.80217
$288$	0	0
$289$	2.19615	0.129185
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	−1.41421	−0.0826192	−0.0413096	−	0.999146i	$-0.513153\pi$
$293$	−1.41421	−0.0826192	−0.0413096	+	0.999146i	$0.513153\pi$
$294$	2.73205	0.159336
$295$	0	0
$296$	9.46410	0.550090
$297$	−5.55532	−0.322352
$298$	−9.14162	−0.529560
$299$	−3.46410	−0.200334
$300$	0	0
$301$	0.588457	0.0339181
$302$	−23.4596	−1.34995
$303$	21.6293	1.24257
$304$	28.7846	1.65091
$305$	0	0
$306$	−4.53590	−0.259300
$307$	19.4201	1.10836	0.554180	−	0.832397i	$-0.313032\pi$
$307$	19.4201	1.10836	0.554180	+	0.832397i	$0.313032\pi$
$308$	0	0
$309$	−29.3205	−1.66799
$310$	0	0
$311$	−10.6077	−0.601507	−0.300754	−	0.953702i	$-0.597238\pi$
$311$	−10.6077	−0.601507	−0.300754	+	0.953702i	$0.597238\pi$
$312$	13.3843	0.757735
$313$	27.6651	1.56372	0.781862	−	0.623452i	$-0.214270\pi$
$313$	27.6651	1.56372	0.781862	+	0.623452i	$0.214270\pi$
$314$	−14.7846	−0.834344
$315$	0	0
$316$	0	0
$317$	−8.20788	−0.461000	−0.230500	−	0.973072i	$-0.574036\pi$
$317$	−8.20788	−0.461000	−0.230500	+	0.973072i	$0.574036\pi$
$318$	−23.5612	−1.32124
$319$	10.3923	0.581857
$320$	0	0
$321$	−16.9282	−0.944840
$322$	−4.89898	−0.273009
$323$	31.5288	1.75431
$324$	0	0
$325$	0	0
$326$	−28.3923	−1.57250
$327$	−26.2509	−1.45168
$328$	35.2538	1.94657
$329$	−14.5359	−0.801390
$330$	0	0
$331$	18.3923	1.01093	0.505466	−	0.862846i	$-0.331321\pi$
$331$	18.3923	1.01093	0.505466	+	0.862846i	$0.331321\pi$
$332$	0	0
$333$	−2.44949	−0.134231
$334$	−22.5885	−1.23599
$335$	0	0
$336$	18.9282	1.03262
$337$	−13.1440	−0.716001	−0.358000	−	0.933721i	$-0.616541\pi$
$337$	−13.1440	−0.716001	−0.358000	+	0.933721i	$0.616541\pi$
$338$	−9.89949	−0.538462
$339$	31.1244	1.69044
$340$	0	0
$341$	−1.26795	−0.0686633
$342$	−7.45001	−0.402850
$343$	−19.5959	−1.05808
$344$	−0.679492	−0.0366357
$345$	0	0
$346$	−10.7846	−0.579784
$347$	30.8081	1.65386	0.826932	−	0.562301i	$-0.190084\pi$
$347$	30.8081	1.65386	0.826932	+	0.562301i	$0.190084\pi$
$348$	0	0
$349$	2.60770	0.139587	0.0697934	−	0.997561i	$-0.477766\pi$
$349$	2.60770	0.139587	0.0697934	+	0.997561i	$0.477766\pi$
$350$	0	0
$351$	10.7321	0.572834
$352$	0	0
$353$	35.6327	1.89654	0.948270	−	0.317466i	$-0.102832\pi$
$353$	35.6327	1.89654	0.948270	+	0.317466i	$0.102832\pi$
$354$	−10.7321	−0.570402
$355$	0	0
$356$	0	0
$357$	20.7327	1.09729
$358$	6.69213	0.353690
$359$	−18.9282	−0.998992	−0.499496	−	0.866316i	$-0.666482\pi$
$359$	−18.9282	−0.998992	−0.499496	+	0.866316i	$0.666482\pi$
$360$	0	0
$361$	32.7846	1.72551
$362$	34.4959	1.81307
$363$	18.1445	0.952341
$364$	0	0
$365$	0	0
$366$	−17.4641	−0.912863
$367$	26.5283	1.38477	0.692383	−	0.721531i	$-0.256561\pi$
$367$	26.5283	1.38477	0.692383	+	0.721531i	$0.256561\pi$
$368$	5.65685	0.294884
$369$	−9.12436	−0.474995
$370$	0	0
$371$	21.1244	1.09672
$372$	0	0
$373$	−18.9396	−0.980654	−0.490327	−	0.871538i	$-0.663123\pi$
$373$	−18.9396	−0.980654	−0.490327	+	0.871538i	$0.663123\pi$
$374$	7.85641	0.406245
$375$	0	0
$376$	16.7846	0.865600
$377$	−20.0764	−1.03399
$378$	15.1774	0.780642
$379$	2.60770	0.133948	0.0669742	−	0.997755i	$-0.478665\pi$
$379$	2.60770	0.133948	0.0669742	+	0.997755i	$0.478665\pi$
$380$	0	0
$381$	11.6603	0.597373
$382$	18.2832	0.935452
$383$	−31.5288	−1.61105	−0.805523	−	0.592564i	$-0.798116\pi$
$383$	−31.5288	−1.61105	−0.805523	+	0.592564i	$0.798116\pi$
$384$	−21.8564	−1.11536
$385$	0	0
$386$	−5.07180	−0.258148
$387$	0.175865	0.00893974
$388$	0	0
$389$	−13.8564	−0.702548	−0.351274	−	0.936273i	$-0.614251\pi$
$389$	−13.8564	−0.702548	−0.351274	+	0.936273i	$0.614251\pi$
$390$	0	0
$391$	6.19615	0.313353
$392$	2.82843	0.142857
$393$	−39.9125	−2.01332
$394$	−17.6077	−0.887063
$395$	0	0
$396$	0	0
$397$	1.31268	0.0658814	0.0329407	−	0.999457i	$-0.489513\pi$
$397$	1.31268	0.0658814	0.0329407	+	0.999457i	$0.489513\pi$
$398$	31.9449	1.60125
$399$	34.0526	1.70476
$400$	0	0
$401$	−18.5885	−0.928263	−0.464132	−	0.885766i	$-0.653634\pi$
$401$	−18.5885	−0.928263	−0.464132	+	0.885766i	$0.653634\pi$
$402$	6.69213	0.333773
$403$	2.44949	0.122018
$404$	0	0
$405$	0	0
$406$	−28.3923	−1.40909
$407$	4.24264	0.210300
$408$	−23.9401	−1.18521
$409$	28.5885	1.41361	0.706804	−	0.707409i	$-0.250136\pi$
$409$	28.5885	1.41361	0.706804	+	0.707409i	$0.250136\pi$
$410$	0	0
$411$	−38.5167	−1.89989
$412$	0	0
$413$	9.62209	0.473472
$414$	−1.46410	−0.0719567
$415$	0	0
$416$	0	0
$417$	−3.48477	−0.170650
$418$	12.9038	0.631145
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	−3.38323	−0.164693
$423$	−4.34418	−0.211221
$424$	−24.3923	−1.18460
$425$	0	0
$426$	11.6603	0.564941
$427$	15.6579	0.757738
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	−6.12436	−0.295000	−0.147500	−	0.989062i	$-0.547123\pi$
$431$	−6.12436	−0.295000	−0.147500	+	0.989062i	$0.547123\pi$
$432$	−17.5254	−0.843190
$433$	18.1074	0.870185	0.435092	−	0.900386i	$-0.356716\pi$
$433$	18.1074	0.870185	0.435092	+	0.900386i	$0.356716\pi$
$434$	3.46410	0.166282
$435$	0	0
$436$	0	0
$437$	10.1769	0.486827
$438$	−10.9348	−0.522484
$439$	17.3923	0.830089	0.415045	−	0.909801i	$-0.363766\pi$
$439$	17.3923	0.830089	0.415045	+	0.909801i	$0.363766\pi$
$440$	0	0
$441$	−0.732051	−0.0348596
$442$	−15.1774	−0.721916
$443$	12.1459	0.577070	0.288535	−	0.957469i	$-0.406832\pi$
$443$	12.1459	0.577070	0.288535	+	0.957469i	$0.406832\pi$
$444$	0	0
$445$	0	0
$446$	−25.2679	−1.19647
$447$	12.4877	0.590647
$448$	19.5959	0.925820
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	15.8038	0.744174
$452$	0	0
$453$	32.0464	1.50567
$454$	14.0000	0.657053
$455$	0	0
$456$	−39.3205	−1.84135
$457$	15.8338	0.740672	0.370336	−	0.928898i	$-0.379243\pi$
$457$	15.8338	0.740672	0.370336	+	0.928898i	$0.379243\pi$
$458$	−5.93426	−0.277290
$459$	−19.1962	−0.896000
$460$	0	0
$461$	−38.1962	−1.77897	−0.889486	−	0.456962i	$-0.848937\pi$
$461$	−38.1962	−1.77897	−0.889486	+	0.456962i	$0.848937\pi$
$462$	8.48528	0.394771
$463$	10.4543	0.485852	0.242926	−	0.970045i	$-0.421893\pi$
$463$	10.4543	0.485852	0.242926	+	0.970045i	$0.421893\pi$
$464$	32.7846	1.52199
$465$	0	0
$466$	−21.6077	−1.00096
$467$	−19.7990	−0.916188	−0.458094	−	0.888904i	$-0.651468\pi$
$467$	−19.7990	−0.916188	−0.458094	+	0.888904i	$0.651468\pi$
$468$	0	0
$469$	−6.00000	−0.277054
$470$	0	0
$471$	20.1962	0.930590
$472$	−11.1106	−0.511409
$473$	−0.304608	−0.0140059
$474$	−16.9282	−0.777538
$475$	0	0
$476$	0	0
$477$	6.31319	0.289061
$478$	−24.4949	−1.12037
$479$	24.9282	1.13900	0.569499	−	0.821992i	$-0.307137\pi$
$479$	24.9282	1.13900	0.569499	+	0.821992i	$0.307137\pi$
$480$	0	0
$481$	−8.19615	−0.373712
$482$	−23.4596	−1.06856
$483$	6.69213	0.304502
$484$	0	0
$485$	0	0
$486$	10.5359	0.477918
$487$	21.4534	0.972148	0.486074	−	0.873918i	$-0.338429\pi$
$487$	21.4534	0.972148	0.486074	+	0.873918i	$0.338429\pi$
$488$	−18.0802	−0.818451
$489$	38.7846	1.75390
$490$	0	0
$491$	−34.6410	−1.56333	−0.781664	−	0.623700i	$-0.785629\pi$
$491$	−34.6410	−1.56333	−0.781664	+	0.623700i	$0.785629\pi$
$492$	0	0
$493$	35.9101	1.61731
$494$	−24.9282	−1.12157
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−10.4543	−0.468939
$498$	−20.4553	−0.916625
$499$	13.8038	0.617945	0.308973	−	0.951071i	$-0.400015\pi$
$499$	13.8038	0.617945	0.308973	+	0.951071i	$0.400015\pi$
$500$	0	0
$501$	30.8564	1.37856
$502$	28.0812	1.25333
$503$	−21.4906	−0.958219	−0.479109	−	0.877755i	$-0.659040\pi$
$503$	−21.4906	−0.958219	−0.479109	+	0.877755i	$0.659040\pi$
$504$	−5.07180	−0.225916
$505$	0	0
$506$	2.53590	0.112734
$507$	13.5230	0.600576
$508$	0	0
$509$	−6.92820	−0.307087	−0.153544	−	0.988142i	$-0.549069\pi$
$509$	−6.92820	−0.307087	−0.153544	+	0.988142i	$0.549069\pi$
$510$	0	0
$511$	9.80385	0.433697
$512$	−22.6274	−1.00000
$513$	−31.5288	−1.39203
$514$	2.00000	0.0882162
$515$	0	0
$516$	0	0
$517$	7.52433	0.330920
$518$	−11.5911	−0.509284
$519$	14.7321	0.646665
$520$	0	0
$521$	7.05256	0.308978	0.154489	−	0.987994i	$-0.450627\pi$
$521$	7.05256	0.308978	0.154489	+	0.987994i	$0.450627\pi$
$522$	−8.48528	−0.371391
$523$	−15.4176	−0.674167	−0.337083	−	0.941475i	$-0.609440\pi$
$523$	−15.4176	−0.674167	−0.337083	+	0.941475i	$0.609440\pi$
$524$	0	0
$525$	0	0
$526$	19.8038	0.863489
$527$	−4.38134	−0.190854
$528$	−9.79796	−0.426401
$529$	−21.0000	−0.913043
$530$	0	0
$531$	2.87564	0.124792
$532$	0	0
$533$	−30.5307	−1.32243
$534$	10.3923	0.449719
$535$	0	0
$536$	6.92820	0.299253
$537$	−9.14162	−0.394490
$538$	−6.69213	−0.288518
$539$	1.26795	0.0546144
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	−23.4596	−1.00768
$543$	−47.1223	−2.02221
$544$	0	0
$545$	0	0
$546$	−16.3923	−0.701526
$547$	−25.6317	−1.09593	−0.547966	−	0.836500i	$-0.684598\pi$
$547$	−25.6317	−1.09593	−0.547966	+	0.836500i	$0.684598\pi$
$548$	0	0
$549$	4.67949	0.199716
$550$	0	0
$551$	58.9808	2.51266
$552$	−7.72741	−0.328900
$553$	15.1774	0.645409
$554$	−40.9808	−1.74111
$555$	0	0
$556$	0	0
$557$	24.5964	1.04218	0.521092	−	0.853500i	$-0.325525\pi$
$557$	24.5964	1.04218	0.521092	+	0.853500i	$0.325525\pi$
$558$	1.03528	0.0438267
$559$	0.588457	0.0248891
$560$	0	0
$561$	−10.7321	−0.453108
$562$	−26.5927	−1.12174
$563$	27.9797	1.17920	0.589601	−	0.807695i	$-0.299285\pi$
$563$	27.9797	1.17920	0.589601	+	0.807695i	$0.299285\pi$
$564$	0	0
$565$	0	0
$566$	4.39230	0.184622
$567$	−26.1122	−1.09661
$568$	12.0716	0.506512
$569$	−20.5359	−0.860910	−0.430455	−	0.902612i	$-0.641647\pi$
$569$	−20.5359	−0.860910	−0.430455	+	0.902612i	$0.641647\pi$
$570$	0	0
$571$	1.41154	0.0590712	0.0295356	−	0.999564i	$-0.490597\pi$
$571$	1.41154	0.0590712	0.0295356	+	0.999564i	$0.490597\pi$
$572$	0	0
$573$	−24.9754	−1.04336
$574$	−43.1769	−1.80217
$575$	0	0
$576$	5.85641	0.244017
$577$	−20.7327	−0.863115	−0.431557	−	0.902085i	$-0.642036\pi$
$577$	−20.7327	−0.863115	−0.431557	+	0.902085i	$0.642036\pi$
$578$	3.10583	0.129185
$579$	6.92820	0.287926
$580$	0	0
$581$	18.3397	0.760861
$582$	−8.48528	−0.351726
$583$	−10.9348	−0.452872
$584$	−11.3205	−0.468446
$585$	0	0
$586$	−2.00000	−0.0826192
$587$	1.41421	0.0583708	0.0291854	−	0.999574i	$-0.490709\pi$
$587$	1.41421	0.0583708	0.0291854	+	0.999574i	$0.490709\pi$
$588$	0	0
$589$	−7.19615	−0.296512
$590$	0	0
$591$	24.0526	0.989390
$592$	13.3843	0.550090
$593$	−22.6274	−0.929197	−0.464598	−	0.885522i	$-0.653801\pi$
$593$	−22.6274	−0.929197	−0.464598	+	0.885522i	$0.653801\pi$
$594$	−7.85641	−0.322352
$595$	0	0
$596$	0	0
$597$	−43.6375	−1.78597
$598$	−4.89898	−0.200334
$599$	3.46410	0.141539	0.0707697	−	0.997493i	$-0.477454\pi$
$599$	3.46410	0.141539	0.0707697	+	0.997493i	$0.477454\pi$
$600$	0	0
$601$	−24.1962	−0.986982	−0.493491	−	0.869751i	$-0.664279\pi$
$601$	−24.1962	−0.986982	−0.493491	+	0.869751i	$0.664279\pi$
$602$	0.832204	0.0339181
$603$	−1.79315	−0.0730228
$604$	0	0
$605$	0	0
$606$	30.5885	1.24257
$607$	28.3858	1.15214	0.576072	−	0.817399i	$-0.304585\pi$
$607$	28.3858	1.15214	0.576072	+	0.817399i	$0.304585\pi$
$608$	0	0
$609$	38.7846	1.57163
$610$	0	0
$611$	−14.5359	−0.588060
$612$	0	0
$613$	34.1814	1.38057	0.690286	−	0.723537i	$-0.257485\pi$
$613$	34.1814	1.38057	0.690286	+	0.723537i	$0.257485\pi$
$614$	27.4641	1.10836
$615$	0	0
$616$	8.78461	0.353942
$617$	−10.1769	−0.409706	−0.204853	−	0.978793i	$-0.565672\pi$
$617$	−10.1769	−0.409706	−0.204853	+	0.978793i	$0.565672\pi$
$618$	−41.4655	−1.66799
$619$	14.3923	0.578476	0.289238	−	0.957257i	$-0.406598\pi$
$619$	14.3923	0.578476	0.289238	+	0.957257i	$0.406598\pi$
$620$	0	0
$621$	−6.19615	−0.248643
$622$	−15.0015	−0.601507
$623$	−9.31749	−0.373297
$624$	18.9282	0.757735
$625$	0	0
$626$	39.1244	1.56372
$627$	−17.6269	−0.703951
$628$	0	0
$629$	14.6603	0.584543
$630$	0	0
$631$	12.3923	0.493330	0.246665	−	0.969101i	$-0.420665\pi$
$631$	12.3923	0.493330	0.246665	+	0.969101i	$0.420665\pi$
$632$	−17.5254	−0.697122
$633$	4.62158	0.183691
$634$	−11.6077	−0.461000
$635$	0	0
$636$	0	0
$637$	−2.44949	−0.0970523
$638$	14.6969	0.581857
$639$	−3.12436	−0.123598
$640$	0	0
$641$	2.53590	0.100162	0.0500810	−	0.998745i	$-0.484052\pi$
$641$	2.53590	0.100162	0.0500810	+	0.998745i	$0.484052\pi$
$642$	−23.9401	−0.944840
$643$	−1.55291	−0.0612410	−0.0306205	−	0.999531i	$-0.509748\pi$
$643$	−1.55291	−0.0612410	−0.0306205	+	0.999531i	$0.509748\pi$
$644$	0	0
$645$	0	0
$646$	44.5885	1.75431
$647$	21.9067	0.861242	0.430621	−	0.902533i	$-0.358295\pi$
$647$	21.9067	0.861242	0.430621	+	0.902533i	$0.358295\pi$
$648$	30.1518	1.18447
$649$	−4.98076	−0.195512
$650$	0	0
$651$	−4.73205	−0.185464
$652$	0	0
$653$	9.04008	0.353766	0.176883	−	0.984232i	$-0.443399\pi$
$653$	9.04008	0.353766	0.176883	+	0.984232i	$0.443399\pi$
$654$	−37.1244	−1.45168
$655$	0	0
$656$	49.8564	1.94657
$657$	2.92996	0.114309
$658$	−20.5569	−0.801390
$659$	11.1962	0.436140	0.218070	−	0.975933i	$-0.430024\pi$
$659$	11.1962	0.436140	0.218070	+	0.975933i	$0.430024\pi$
$660$	0	0
$661$	1.19615	0.0465249	0.0232625	−	0.999729i	$-0.492595\pi$
$661$	1.19615	0.0465249	0.0232625	+	0.999729i	$0.492595\pi$
$662$	26.0106	1.01093
$663$	20.7327	0.805193
$664$	−21.1769	−0.821824
$665$	0	0
$666$	−3.46410	−0.134231
$667$	11.5911	0.448810
$668$	0	0
$669$	34.5167	1.33449
$670$	0	0
$671$	−8.10512	−0.312895
$672$	0	0
$673$	22.9420	0.884348	0.442174	−	0.896929i	$-0.354207\pi$
$673$	22.9420	0.884348	0.442174	+	0.896929i	$0.354207\pi$
$674$	−18.5885	−0.716001
$675$	0	0
$676$	0	0
$677$	−19.0783	−0.733238	−0.366619	−	0.930371i	$-0.619485\pi$
$677$	−19.0783	−0.733238	−0.366619	+	0.930371i	$0.619485\pi$
$678$	44.0165	1.69044
$679$	7.60770	0.291957
$680$	0	0
$681$	−19.1244	−0.732847
$682$	−1.79315	−0.0686633
$683$	−9.89949	−0.378794	−0.189397	−	0.981901i	$-0.560653\pi$
$683$	−9.89949	−0.378794	−0.189397	+	0.981901i	$0.560653\pi$
$684$	0	0
$685$	0	0
$686$	−27.7128	−1.05808
$687$	8.10634	0.309276
$688$	−0.960947	−0.0366357
$689$	21.1244	0.804774
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	0	0
$693$	−2.27362	−0.0863678
$694$	43.5692	1.65386
$695$	0	0
$696$	−44.7846	−1.69756
$697$	54.6095	2.06848
$698$	3.68784	0.139587
$699$	29.5167	1.11642
$700$	0	0
$701$	40.6410	1.53499	0.767495	−	0.641055i	$-0.221503\pi$
$701$	40.6410	1.53499	0.767495	+	0.641055i	$0.221503\pi$
$702$	15.1774	0.572834
$703$	24.0788	0.908149
$704$	−10.1436	−0.382301
$705$	0	0
$706$	50.3923	1.89654
$707$	−27.4249	−1.03142
$708$	0	0
$709$	−16.7846	−0.630359	−0.315180	−	0.949032i	$-0.602065\pi$
$709$	−16.7846	−0.630359	−0.315180	+	0.949032i	$0.602065\pi$
$710$	0	0
$711$	4.53590	0.170109
$712$	10.7589	0.403207
$713$	−1.41421	−0.0529627
$714$	29.3205	1.09729
$715$	0	0
$716$	0	0
$717$	33.4607	1.24961
$718$	−26.7685	−0.998992
$719$	−40.6410	−1.51565	−0.757827	−	0.652455i	$-0.773739\pi$
$719$	−40.6410	−1.51565	−0.757827	+	0.652455i	$0.773739\pi$
$720$	0	0
$721$	37.1769	1.38454
$722$	46.3644	1.72551
$723$	32.0464	1.19182
$724$	0	0
$725$	0	0
$726$	25.6603	0.952341
$727$	−39.6723	−1.47136	−0.735682	−	0.677327i	$-0.763138\pi$
$727$	−39.6723	−1.47136	−0.735682	+	0.677327i	$0.763138\pi$
$728$	−16.9706	−0.628971
$729$	17.5885	0.651424
$730$	0	0
$731$	−1.05256	−0.0389303
$732$	0	0
$733$	24.4949	0.904740	0.452370	−	0.891830i	$-0.350579\pi$
$733$	24.4949	0.904740	0.452370	+	0.891830i	$0.350579\pi$
$734$	37.5167	1.38477
$735$	0	0
$736$	0	0
$737$	3.10583	0.114405
$738$	−12.9038	−0.474995
$739$	−22.7846	−0.838145	−0.419073	−	0.907953i	$-0.637645\pi$
$739$	−22.7846	−0.838145	−0.419073	+	0.907953i	$0.637645\pi$
$740$	0	0
$741$	34.0526	1.25095
$742$	29.8744	1.09672
$743$	−28.1184	−1.03156	−0.515781	−	0.856720i	$-0.672498\pi$
$743$	−28.1184	−1.03156	−0.515781	+	0.856720i	$0.672498\pi$
$744$	5.46410	0.200324
$745$	0	0
$746$	−26.7846	−0.980654
$747$	5.48099	0.200539
$748$	0	0
$749$	21.4641	0.784281
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	23.7370	0.865600
$753$	−38.3596	−1.39790
$754$	−28.3923	−1.03399
$755$	0	0
$756$	0	0
$757$	21.4534	0.779739	0.389869	−	0.920870i	$-0.372520\pi$
$757$	21.4534	0.779739	0.389869	+	0.920870i	$0.372520\pi$
$758$	3.68784	0.133948
$759$	−3.46410	−0.125739
$760$	0	0
$761$	−33.7128	−1.22209	−0.611044	−	0.791596i	$-0.709250\pi$
$761$	−33.7128	−1.22209	−0.611044	+	0.791596i	$0.709250\pi$
$762$	16.4901	0.597373
$763$	33.2848	1.20499
$764$	0	0
$765$	0	0
$766$	−44.5885	−1.61105
$767$	9.62209	0.347434
$768$	0	0
$769$	−43.7846	−1.57891	−0.789457	−	0.613806i	$-0.789638\pi$
$769$	−43.7846	−1.57891	−0.789457	+	0.613806i	$0.789638\pi$
$770$	0	0
$771$	−2.73205	−0.0983924
$772$	0	0
$773$	−18.3848	−0.661254	−0.330627	−	0.943761i	$-0.607260\pi$
$773$	−18.3848	−0.661254	−0.330627	+	0.943761i	$0.607260\pi$
$774$	0.248711	0.00893974
$775$	0	0
$776$	−8.78461	−0.315349
$777$	15.8338	0.568033
$778$	−19.5959	−0.702548
$779$	89.6936	3.21361
$780$	0	0
$781$	5.41154	0.193640
$782$	8.76268	0.313353
$783$	−35.9101	−1.28332
$784$	4.00000	0.142857
$785$	0	0
$786$	−56.4449	−2.01332
$787$	16.3142	0.581539	0.290770	−	0.956793i	$-0.406089\pi$
$787$	16.3142	0.581539	0.290770	+	0.956793i	$0.406089\pi$
$788$	0	0
$789$	−27.0526	−0.963097
$790$	0	0
$791$	−39.4641	−1.40318
$792$	2.62536	0.0932879
$793$	15.6579	0.556028
$794$	1.85641	0.0658814
$795$	0	0
$796$	0	0
$797$	30.8081	1.09128	0.545639	−	0.838020i	$-0.316287\pi$
$797$	30.8081	1.09128	0.545639	+	0.838020i	$0.316287\pi$
$798$	48.1576	1.70476
$799$	26.0000	0.919814
$800$	0	0
$801$	−2.78461	−0.0983893
$802$	−26.2880	−0.928263
$803$	−5.07484	−0.179087
$804$	0	0
$805$	0	0
$806$	3.46410	0.122018
$807$	9.14162	0.321800
$808$	31.6675	1.11406
$809$	−23.6603	−0.831850	−0.415925	−	0.909399i	$-0.636542\pi$
$809$	−23.6603	−0.831850	−0.415925	+	0.909399i	$0.636542\pi$
$810$	0	0
$811$	2.21539	0.0777929	0.0388964	−	0.999243i	$-0.487616\pi$
$811$	2.21539	0.0777929	0.0388964	+	0.999243i	$0.487616\pi$
$812$	0	0
$813$	32.0464	1.12392
$814$	6.00000	0.210300
$815$	0	0
$816$	−33.8564	−1.18521
$817$	−1.72878	−0.0604823
$818$	40.4302	1.41361
$819$	4.39230	0.153480
$820$	0	0
$821$	21.3731	0.745925	0.372963	−	0.927846i	$-0.378342\pi$
$821$	21.3731	0.745925	0.372963	+	0.927846i	$0.378342\pi$
$822$	−54.4708	−1.89989
$823$	−44.9874	−1.56816	−0.784081	−	0.620659i	$-0.786865\pi$
$823$	−44.9874	−1.56816	−0.784081	+	0.620659i	$0.786865\pi$
$824$	−42.9282	−1.49547
$825$	0	0
$826$	13.6077	0.473472
$827$	−22.4887	−0.782009	−0.391005	−	0.920389i	$-0.627872\pi$
$827$	−22.4887	−0.782009	−0.391005	+	0.920389i	$0.627872\pi$
$828$	0	0
$829$	−10.7846	−0.374565	−0.187282	−	0.982306i	$-0.559968\pi$
$829$	−10.7846	−0.374565	−0.187282	+	0.982306i	$0.559968\pi$
$830$	0	0
$831$	55.9808	1.94195
$832$	19.5959	0.679366
$833$	4.38134	0.151804
$834$	−4.92820	−0.170650
$835$	0	0
$836$	0	0
$837$	4.38134	0.151441
$838$	−21.2132	−0.732798
$839$	−37.9808	−1.31124	−0.655621	−	0.755090i	$-0.727593\pi$
$839$	−37.9808	−1.31124	−0.655621	+	0.755090i	$0.727593\pi$
$840$	0	0
$841$	38.1769	1.31645
$842$	−26.8701	−0.926003
$843$	36.3262	1.25114
$844$	0	0
$845$	0	0
$846$	−6.14359	−0.211221
$847$	−23.0064	−0.790508
$848$	−34.4959	−1.18460
$849$	−6.00000	−0.205919
$850$	0	0
$851$	4.73205	0.162213
$852$	0	0
$853$	6.21166	0.212683	0.106342	−	0.994330i	$-0.466086\pi$
$853$	6.21166	0.212683	0.106342	+	0.994330i	$0.466086\pi$
$854$	22.1436	0.757738
$855$	0	0
$856$	−24.7846	−0.847121
$857$	33.0817	1.13005	0.565025	−	0.825074i	$-0.308867\pi$
$857$	33.0817	1.13005	0.565025	+	0.825074i	$0.308867\pi$
$858$	8.48528	0.289683
$859$	41.7654	1.42502	0.712508	−	0.701664i	$-0.247559\pi$
$859$	41.7654	1.42502	0.712508	+	0.701664i	$0.247559\pi$
$860$	0	0
$861$	58.9808	2.01006
$862$	−8.66115	−0.295000
$863$	−6.37756	−0.217095	−0.108547	−	0.994091i	$-0.534620\pi$
$863$	−6.37756	−0.217095	−0.108547	+	0.994091i	$0.534620\pi$
$864$	0	0
$865$	0	0
$866$	25.6077	0.870185
$867$	−4.24264	−0.144088
$868$	0	0
$869$	−7.85641	−0.266510
$870$	0	0
$871$	−6.00000	−0.203302
$872$	−38.4340	−1.30154
$873$	2.27362	0.0769505
$874$	14.3923	0.486827
$875$	0	0
$876$	0	0
$877$	−7.82894	−0.264365	−0.132182	−	0.991225i	$-0.542198\pi$
$877$	−7.82894	−0.264365	−0.132182	+	0.991225i	$0.542198\pi$
$878$	24.5964	0.830089
$879$	2.73205	0.0921498
$880$	0	0
$881$	−34.0526	−1.14726	−0.573630	−	0.819115i	$-0.694465\pi$
$881$	−34.0526	−1.14726	−0.573630	+	0.819115i	$0.694465\pi$
$882$	−1.03528	−0.0348596
$883$	8.06918	0.271550	0.135775	−	0.990740i	$-0.456648\pi$
$883$	8.06918	0.271550	0.135775	+	0.990740i	$0.456648\pi$
$884$	0	0
$885$	0	0
$886$	17.1769	0.577070
$887$	24.5964	0.825867	0.412934	−	0.910761i	$-0.364504\pi$
$887$	24.5964	0.825867	0.412934	+	0.910761i	$0.364504\pi$
$888$	−18.2832	−0.613545
$889$	−14.7846	−0.495860
$890$	0	0
$891$	13.5167	0.452825
$892$	0	0
$893$	42.7038	1.42903
$894$	17.6603	0.590647
$895$	0	0
$896$	27.7128	0.925820
$897$	6.69213	0.223444
$898$	−25.4558	−0.849473
$899$	−8.19615	−0.273357
$900$	0	0
$901$	−37.7846	−1.25879
$902$	22.3500	0.744174
$903$	−1.13681	−0.0378307
$904$	45.5692	1.51561
$905$	0	0
$906$	45.3205	1.50567
$907$	3.10583	0.103127	0.0515637	−	0.998670i	$-0.483579\pi$
$907$	3.10583	0.103127	0.0515637	+	0.998670i	$0.483579\pi$
$908$	0	0
$909$	−8.19615	−0.271849
$910$	0	0
$911$	−21.1244	−0.699881	−0.349941	−	0.936772i	$-0.613798\pi$
$911$	−21.1244	−0.699881	−0.349941	+	0.936772i	$0.613798\pi$
$912$	−55.6076	−1.84135
$913$	−9.49335	−0.314184
$914$	22.3923	0.740672
$915$	0	0
$916$	0	0
$917$	50.6071	1.67119
$918$	−27.1475	−0.896000
$919$	0.411543	0.0135755	0.00678777	−	0.999977i	$-0.497839\pi$
$919$	0.411543	0.0135755	0.00678777	+	0.999977i	$0.497839\pi$
$920$	0	0
$921$	−37.5167	−1.23622
$922$	−54.0175	−1.77897
$923$	−10.4543	−0.344107
$924$	0	0
$925$	0	0
$926$	14.7846	0.485852
$927$	11.1106	0.364921
$928$	0	0
$929$	−4.39230	−0.144107	−0.0720534	−	0.997401i	$-0.522955\pi$
$929$	−4.39230	−0.144107	−0.0720534	+	0.997401i	$0.522955\pi$
$930$	0	0
$931$	7.19615	0.235844
$932$	0	0
$933$	20.4925	0.670894
$934$	−28.0000	−0.916188
$935$	0	0
$936$	−5.07180	−0.165777
$937$	28.7375	0.938814	0.469407	−	0.882982i	$-0.344468\pi$
$937$	28.7375	0.938814	0.469407	+	0.882982i	$0.344468\pi$
$938$	−8.48528	−0.277054
$939$	−53.4449	−1.74411
$940$	0	0
$941$	−42.2487	−1.37727	−0.688634	−	0.725109i	$-0.741789\pi$
$941$	−42.2487	−1.37727	−0.688634	+	0.725109i	$0.741789\pi$
$942$	28.5617	0.930590
$943$	17.6269	0.574011
$944$	−15.7128	−0.511409
$945$	0	0
$946$	−0.430781	−0.0140059
$947$	−54.9884	−1.78688	−0.893442	−	0.449179i	$-0.851717\pi$
$947$	−54.9884	−1.78688	−0.893442	+	0.449179i	$0.851717\pi$
$948$	0	0
$949$	9.80385	0.318246
$950$	0	0
$951$	15.8564	0.514179
$952$	30.3548	0.983805
$953$	5.51815	0.178751	0.0893753	−	0.995998i	$-0.471513\pi$
$953$	5.51815	0.178751	0.0893753	+	0.995998i	$0.471513\pi$
$954$	8.92820	0.289061
$955$	0	0
$956$	0	0
$957$	−20.0764	−0.648978
$958$	35.2538	1.13900
$959$	48.8372	1.57703
$960$	0	0
$961$	1.00000	0.0322581
$962$	−11.5911	−0.373712
$963$	6.41473	0.206712
$964$	0	0
$965$	0	0
$966$	9.46410	0.304502
$967$	−43.9149	−1.41221	−0.706105	−	0.708107i	$-0.749549\pi$
$967$	−43.9149	−1.41221	−0.706105	+	0.708107i	$0.749549\pi$
$968$	26.5654	0.853846
$969$	−60.9090	−1.95668
$970$	0	0
$971$	2.32051	0.0744686	0.0372343	−	0.999307i	$-0.488145\pi$
$971$	2.32051	0.0744686	0.0372343	+	0.999307i	$0.488145\pi$
$972$	0	0
$973$	4.41851	0.141651
$974$	30.3397	0.972148
$975$	0	0
$976$	−25.5692	−0.818451
$977$	−1.69161	−0.0541196	−0.0270598	−	0.999634i	$-0.508614\pi$
$977$	−1.69161	−0.0541196	−0.0270598	+	0.999634i	$0.508614\pi$
$978$	54.8497	1.75390
$979$	4.82309	0.154146
$980$	0	0
$981$	9.94744	0.317597
$982$	−48.9898	−1.56333
$983$	0.859411	0.0274109	0.0137055	−	0.999906i	$-0.495637\pi$
$983$	0.859411	0.0274109	0.0137055	+	0.999906i	$0.495637\pi$
$984$	−68.1051	−2.17111
$985$	0	0
$986$	50.7846	1.61731
$987$	28.0812	0.893834
$988$	0	0
$989$	−0.339746	−0.0108033
$990$	0	0
$991$	−3.41154	−0.108371	−0.0541856	−	0.998531i	$-0.517256\pi$
$991$	−3.41154	−0.108371	−0.0541856	+	0.998531i	$0.517256\pi$
$992$	0	0
$993$	−35.5312	−1.12755
$994$	−14.7846	−0.468939
$995$	0	0
$996$	0	0
$997$	7.17260	0.227159	0.113579	−	0.993529i	$-0.463768\pi$
$997$	7.17260	0.227159	0.113579	+	0.993529i	$0.463768\pi$
$998$	19.5216	0.617945
$999$	−14.6603	−0.463830

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.a.f.1.3		4
3.2	odd	2		6975.2.a.bk.1.2		4
5.2	odd	4		155.2.b.a.94.4	yes	4
5.3	odd	4		155.2.b.a.94.1	✓	4
5.4	even	2	inner	775.2.a.f.1.2		4
15.2	even	4		1395.2.c.c.559.2		4
15.8	even	4		1395.2.c.c.559.4		4
15.14	odd	2		6975.2.a.bk.1.3		4
20.3	even	4		2480.2.d.b.1489.4		4
20.7	even	4		2480.2.d.b.1489.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.b.a.94.1	✓	4	5.3	odd	4
155.2.b.a.94.4	yes	4	5.2	odd	4
775.2.a.f.1.2		4	5.4	even	2	inner
775.2.a.f.1.3		4	1.1	even	1	trivial
1395.2.c.c.559.2		4	15.2	even	4
1395.2.c.c.559.4		4	15.8	even	4
2480.2.d.b.1489.1		4	20.7	even	4
2480.2.d.b.1489.4		4	20.3	even	4
6975.2.a.bk.1.2		4	3.2	odd	2
6975.2.a.bk.1.3		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} -1.93185 q^{3} -2.73205 q^{6} +2.44949 q^{7} -2.82843 q^{8} +0.732051 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} -1.93185 q^{3} -2.73205 q^{6} +2.44949 q^{7} -2.82843 q^{8} +0.732051 q^{9} -1.26795 q^{11} +2.44949 q^{13} +3.46410 q^{14} -4.00000 q^{16} -4.38134 q^{17} +1.03528 q^{18} -7.19615 q^{19} -4.73205 q^{21} -1.79315 q^{22} -1.41421 q^{23} +5.46410 q^{24} +3.46410 q^{26} +4.38134 q^{27} -8.19615 q^{29} +1.00000 q^{31} +2.44949 q^{33} -6.19615 q^{34} -3.34607 q^{37} -10.1769 q^{38} -4.73205 q^{39} -12.4641 q^{41} -6.69213 q^{42} +0.240237 q^{43} -2.00000 q^{46} -5.93426 q^{47} +7.72741 q^{48} -1.00000 q^{49} +8.46410 q^{51} +8.62398 q^{53} +6.19615 q^{54} -6.92820 q^{56} +13.9019 q^{57} -11.5911 q^{58} +3.92820 q^{59} +6.39230 q^{61} +1.41421 q^{62} +1.79315 q^{63} +8.00000 q^{64} +3.46410 q^{66} -2.44949 q^{67} +2.73205 q^{69} -4.26795 q^{71} -2.07055 q^{72} +4.00240 q^{73} -4.73205 q^{74} -3.10583 q^{77} -6.69213 q^{78} +6.19615 q^{79} -10.6603 q^{81} -17.6269 q^{82} +7.48717 q^{83} +0.339746 q^{86} +15.8338 q^{87} +3.58630 q^{88} -3.80385 q^{89} +6.00000 q^{91} -1.93185 q^{93} -8.39230 q^{94} +3.10583 q^{97} -1.41421 q^{98} -0.928203 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^6 - 4 * q^9	\( 4 q - 4 q^{6} - 4 q^{9} - 12 q^{11} - 16 q^{16} - 8 q^{19} - 12 q^{21} + 8 q^{24} - 12 q^{29} + 4 q^{31} - 4 q^{34} - 12 q^{39} - 36 q^{41} - 8 q^{46} - 4 q^{49} + 20 q^{51} + 4 q^{54} - 12 q^{59} - 16 q^{61} + 32 q^{64} + 4 q^{69} - 24 q^{71} - 12 q^{74} + 4 q^{79} - 8 q^{81} + 36 q^{86} - 36 q^{89} + 24 q^{91} + 8 q^{94} + 24 q^{99}+O(q^{100}) \) 4 * q - 4 * q^6 - 4 * q^9 - 12 * q^11 - 16 * q^16 - 8 * q^19 - 12 * q^21 + 8 * q^24 - 12 * q^29 + 4 * q^31 - 4 * q^34 - 12 * q^39 - 36 * q^41 - 8 * q^46 - 4 * q^49 + 20 * q^51 + 4 * q^54 - 12 * q^59 - 16 * q^61 + 32 * q^64 + 4 * q^69 - 24 * q^71 - 12 * q^74 + 4 * q^79 - 8 * q^81 + 36 * q^86 - 36 * q^89 + 24 * q^91 + 8 * q^94 + 24 * q^99

Introduction

L-functions

Modular forms

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Database

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