LMFDB - Embedded newform 775.2.a.f.1.4

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			1.4
Root			$-0.517638$ 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)

$f(q)$	$=$	$q+1.41421 q^{2} +0.517638 q^{3} +0.732051 q^{6} -2.44949 q^{7} -2.82843 q^{8} -2.73205 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +0.517638 q^{3} +0.732051 q^{6} -2.44949 q^{7} -2.82843 q^{8} -2.73205 q^{9} -4.73205 q^{11} -2.44949 q^{13} -3.46410 q^{14} -4.00000 q^{16} +2.96713 q^{17} -3.86370 q^{18} +3.19615 q^{19} -1.26795 q^{21} -6.69213 q^{22} -1.41421 q^{23} -1.46410 q^{24} -3.46410 q^{26} -2.96713 q^{27} +2.19615 q^{29} +1.00000 q^{31} -2.44949 q^{33} +4.19615 q^{34} -0.896575 q^{37} +4.52004 q^{38} -1.26795 q^{39} -5.53590 q^{41} -1.79315 q^{42} +12.4877 q^{43} -2.00000 q^{46} +8.76268 q^{47} -2.07055 q^{48} -1.00000 q^{49} +1.53590 q^{51} +1.27551 q^{53} -4.19615 q^{54} +6.92820 q^{56} +1.65445 q^{57} +3.10583 q^{58} -9.92820 q^{59} -14.3923 q^{61} +1.41421 q^{62} +6.69213 q^{63} +8.00000 q^{64} -3.46410 q^{66} +2.44949 q^{67} -0.732051 q^{69} -7.73205 q^{71} +7.72741 q^{72} -8.24504 q^{73} -1.26795 q^{74} +11.5911 q^{77} -1.79315 q^{78} -4.19615 q^{79} +6.66025 q^{81} -7.82894 q^{82} -14.5582 q^{83} +17.6603 q^{86} +1.13681 q^{87} +13.3843 q^{88} -14.1962 q^{89} +6.00000 q^{91} +0.517638 q^{93} +12.3923 q^{94} -11.5911 q^{97} -1.41421 q^{98} +12.9282 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$0.333333\pi$
$3$	0.517638	0.298858	0.149429	−	0.988772i	$-0.452256\pi$
$3$	0.517638	0.298858	0.149429	+	0.988772i	$0.452256\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0.732051	0.298858
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	−2.82843	−1.00000
$9$	−2.73205	−0.910684
$10$	0	0
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	−2.44949	−0.679366	−0.339683	−	0.940540i	$-0.610320\pi$
$13$	−2.44949	−0.679366	−0.339683	+	0.940540i	$0.610320\pi$
$14$	−3.46410	−0.925820
$15$	0	0
$16$	−4.00000	−1.00000
$17$	2.96713	0.719634	0.359817	−	0.933023i	$-0.382839\pi$
$17$	2.96713	0.719634	0.359817	+	0.933023i	$0.382839\pi$
$18$	−3.86370	−0.910684
$19$	3.19615	0.733248	0.366624	−	0.930369i	$-0.380514\pi$
$19$	3.19615	0.733248	0.366624	+	0.930369i	$0.380514\pi$
$20$	0	0
$21$	−1.26795	−0.276689
$22$	−6.69213	−1.42677
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	−1.46410	−0.298858
$25$	0	0
$26$	−3.46410	−0.679366
$27$	−2.96713	−0.571024
$28$	0	0
$29$	2.19615	0.407815	0.203908	−	0.978990i	$-0.434636\pi$
$29$	2.19615	0.407815	0.203908	+	0.978990i	$0.434636\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−2.44949	−0.426401
$34$	4.19615	0.719634
$35$	0	0
$36$	0	0
$37$	−0.896575	−0.147396	−0.0736980	−	0.997281i	$-0.523480\pi$
$37$	−0.896575	−0.147396	−0.0736980	+	0.997281i	$0.523480\pi$
$38$	4.52004	0.733248
$39$	−1.26795	−0.203034
$40$	0	0
$41$	−5.53590	−0.864562	−0.432281	−	0.901739i	$-0.642291\pi$
$41$	−5.53590	−0.864562	−0.432281	+	0.901739i	$0.642291\pi$
$42$	−1.79315	−0.276689
$43$	12.4877	1.90435	0.952177	−	0.305547i	$-0.0988392\pi$
$43$	12.4877	1.90435	0.952177	+	0.305547i	$0.0988392\pi$
$44$	0	0
$45$	0	0
$46$	−2.00000	−0.294884
$47$	8.76268	1.27817	0.639084	−	0.769137i	$-0.279313\pi$
$47$	8.76268	1.27817	0.639084	+	0.769137i	$0.279313\pi$
$48$	−2.07055	−0.298858
$49$	−1.00000	−0.142857
$50$	0	0
$51$	1.53590	0.215069
$52$	0	0
$53$	1.27551	0.175205	0.0876026	−	0.996156i	$-0.472079\pi$
$53$	1.27551	0.175205	0.0876026	+	0.996156i	$0.472079\pi$
$54$	−4.19615	−0.571024
$55$	0	0
$56$	6.92820	0.925820
$57$	1.65445	0.219137
$58$	3.10583	0.407815
$59$	−9.92820	−1.29254	−0.646271	−	0.763108i	$-0.723672\pi$
$59$	−9.92820	−1.29254	−0.646271	+	0.763108i	$0.723672\pi$
$60$	0	0
$61$	−14.3923	−1.84275	−0.921373	−	0.388680i	$-0.872931\pi$
$61$	−14.3923	−1.84275	−0.921373	+	0.388680i	$0.872931\pi$
$62$	1.41421	0.179605
$63$	6.69213	0.843129
$64$	8.00000	1.00000
$65$	0	0
$66$	−3.46410	−0.426401
$67$	2.44949	0.299253	0.149626	−	0.988743i	$-0.452193\pi$
$67$	2.44949	0.299253	0.149626	+	0.988743i	$0.452193\pi$
$68$	0	0
$69$	−0.732051	−0.0881286
$70$	0	0
$71$	−7.73205	−0.917626	−0.458813	−	0.888533i	$-0.651725\pi$
$71$	−7.73205	−0.917626	−0.458813	+	0.888533i	$0.651725\pi$
$72$	7.72741	0.910684
$73$	−8.24504	−0.965009	−0.482505	−	0.875893i	$-0.660273\pi$
$73$	−8.24504	−0.965009	−0.482505	+	0.875893i	$0.660273\pi$
$74$	−1.26795	−0.147396
$75$	0	0
$76$	0	0
$77$	11.5911	1.32093
$78$	−1.79315	−0.203034
$79$	−4.19615	−0.472104	−0.236052	−	0.971740i	$-0.575854\pi$
$79$	−4.19615	−0.472104	−0.236052	+	0.971740i	$0.575854\pi$
$80$	0	0
$81$	6.66025	0.740028
$82$	−7.82894	−0.864562
$83$	−14.5582	−1.59797	−0.798987	−	0.601348i	$-0.794630\pi$
$83$	−14.5582	−1.59797	−0.798987	+	0.601348i	$0.794630\pi$
$84$	0	0
$85$	0	0
$86$	17.6603	1.90435
$87$	1.13681	0.121879
$88$	13.3843	1.42677
$89$	−14.1962	−1.50479	−0.752395	−	0.658713i	$-0.771101\pi$
$89$	−14.1962	−1.50479	−0.752395	+	0.658713i	$0.771101\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	0.517638	0.0536766
$94$	12.3923	1.27817
$95$	0	0
$96$	0	0
$97$	−11.5911	−1.17690	−0.588449	−	0.808534i	$-0.700261\pi$
$97$	−11.5911	−1.17690	−0.588449	+	0.808534i	$0.700261\pi$
$98$	−1.41421	−0.142857
$99$	12.9282	1.29933
$100$	0	0
$101$	−0.803848	−0.0799858	−0.0399929	−	0.999200i	$-0.512734\pi$
$101$	−0.803848	−0.0799858	−0.0399929	+	0.999200i	$0.512734\pi$
$102$	2.17209	0.215069
$103$	10.2784	1.01276	0.506382	−	0.862309i	$-0.330983\pi$
$103$	10.2784	1.01276	0.506382	+	0.862309i	$0.330983\pi$
$104$	6.92820	0.679366
$105$	0	0
$106$	1.80385	0.175205
$107$	−5.93426	−0.573686	−0.286843	−	0.957978i	$-0.592606\pi$
$107$	−5.93426	−0.573686	−0.286843	+	0.957978i	$0.592606\pi$
$108$	0	0
$109$	−17.5885	−1.68467	−0.842334	−	0.538955i	$-0.818819\pi$
$109$	−17.5885	−1.68467	−0.842334	+	0.538955i	$0.818819\pi$
$110$	0	0
$111$	−0.464102	−0.0440506
$112$	9.79796	0.925820
$113$	13.2827	1.24953	0.624767	−	0.780811i	$-0.285194\pi$
$113$	13.2827	1.24953	0.624767	+	0.780811i	$0.285194\pi$
$114$	2.33975	0.219137
$115$	0	0
$116$	0	0
$117$	6.69213	0.618688
$118$	−14.0406	−1.29254
$119$	−7.26795	−0.666252
$120$	0	0
$121$	11.3923	1.03566
$122$	−20.3538	−1.84275
$123$	−2.86559	−0.258382
$124$	0	0
$125$	0	0
$126$	9.46410	0.843129
$127$	−10.9348	−0.970304	−0.485152	−	0.874430i	$-0.661236\pi$
$127$	−10.9348	−0.970304	−0.485152	+	0.874430i	$0.661236\pi$
$128$	11.3137	1.00000
$129$	6.46410	0.569132
$130$	0	0
$131$	3.33975	0.291795	0.145897	−	0.989300i	$-0.453393\pi$
$131$	3.33975	0.291795	0.145897	+	0.989300i	$0.453393\pi$
$132$	0	0
$133$	−7.82894	−0.678855
$134$	3.46410	0.299253
$135$	0	0
$136$	−8.39230	−0.719634
$137$	12.5892	1.07557	0.537785	−	0.843082i	$-0.319261\pi$
$137$	12.5892	1.07557	0.537785	+	0.843082i	$0.319261\pi$
$138$	−1.03528	−0.0881286
$139$	12.1962	1.03446	0.517232	−	0.855845i	$-0.326962\pi$
$139$	12.1962	1.03446	0.517232	+	0.855845i	$0.326962\pi$
$140$	0	0
$141$	4.53590	0.381992
$142$	−10.9348	−0.917626
$143$	11.5911	0.969297
$144$	10.9282	0.910684
$145$	0	0
$146$	−11.6603	−0.965009
$147$	−0.517638	−0.0426941
$148$	0	0
$149$	0.464102	0.0380207	0.0190103	−	0.999819i	$-0.493948\pi$
$149$	0.464102	0.0380207	0.0190103	+	0.999819i	$0.493948\pi$
$150$	0	0
$151$	14.5885	1.18719	0.593596	−	0.804763i	$-0.297708\pi$
$151$	14.5885	1.18719	0.593596	+	0.804763i	$0.297708\pi$
$152$	−9.04008	−0.733248
$153$	−8.10634	−0.655359
$154$	16.3923	1.32093
$155$	0	0
$156$	0	0
$157$	18.9396	1.51154	0.755771	−	0.654835i	$-0.227262\pi$
$157$	18.9396	1.51154	0.755771	+	0.654835i	$0.227262\pi$
$158$	−5.93426	−0.472104
$159$	0.660254	0.0523616
$160$	0	0
$161$	3.46410	0.273009
$162$	9.41902	0.740028
$163$	−5.37945	−0.421351	−0.210676	−	0.977556i	$-0.567566\pi$
$163$	−5.37945	−0.421351	−0.210676	+	0.977556i	$0.567566\pi$
$164$	0	0
$165$	0	0
$166$	−20.5885	−1.59797
$167$	6.07296	0.469939	0.234970	−	0.972003i	$-0.424501\pi$
$167$	6.07296	0.469939	0.234970	+	0.972003i	$0.424501\pi$
$168$	3.58630	0.276689
$169$	−7.00000	−0.538462
$170$	0	0
$171$	−8.73205	−0.667757
$172$	0	0
$173$	21.7680	1.65499	0.827495	−	0.561472i	$-0.189765\pi$
$173$	21.7680	1.65499	0.827495	+	0.561472i	$0.189765\pi$
$174$	1.60770	0.121879
$175$	0	0
$176$	18.9282	1.42677
$177$	−5.13922	−0.386287
$178$	−20.0764	−1.50479
$179$	1.26795	0.0947710	0.0473855	−	0.998877i	$-0.484911\pi$
$179$	1.26795	0.0947710	0.0473855	+	0.998877i	$0.484911\pi$
$180$	0	0
$181$	3.60770	0.268158	0.134079	−	0.990971i	$-0.457192\pi$
$181$	3.60770	0.268158	0.134079	+	0.990971i	$0.457192\pi$
$182$	8.48528	0.628971
$183$	−7.45001	−0.550720
$184$	4.00000	0.294884
$185$	0	0
$186$	0.732051	0.0536766
$187$	−14.0406	−1.02675
$188$	0	0
$189$	7.26795	0.528666
$190$	0	0
$191$	−0.928203	−0.0671624	−0.0335812	−	0.999436i	$-0.510691\pi$
$191$	−0.928203	−0.0671624	−0.0335812	+	0.999436i	$0.510691\pi$
$192$	4.14110	0.298858
$193$	−13.3843	−0.963420	−0.481710	−	0.876331i	$-0.659984\pi$
$193$	−13.3843	−0.963420	−0.481710	+	0.876331i	$0.659984\pi$
$194$	−16.3923	−1.17690
$195$	0	0
$196$	0	0
$197$	−27.1475	−1.93418	−0.967088	−	0.254441i	$-0.918108\pi$
$197$	−27.1475	−1.93418	−0.967088	+	0.254441i	$0.918108\pi$
$198$	18.2832	1.29933
$199$	−8.58846	−0.608820	−0.304410	−	0.952541i	$-0.598459\pi$
$199$	−8.58846	−0.608820	−0.304410	+	0.952541i	$0.598459\pi$
$200$	0	0
$201$	1.26795	0.0894342
$202$	−1.13681	−0.0799858
$203$	−5.37945	−0.377564
$204$	0	0
$205$	0	0
$206$	14.5359	1.01276
$207$	3.86370	0.268546
$208$	9.79796	0.679366
$209$	−15.1244	−1.04617
$210$	0	0
$211$	18.3923	1.26618	0.633089	−	0.774079i	$-0.281787\pi$
$211$	18.3923	1.26618	0.633089	+	0.774079i	$0.281787\pi$
$212$	0	0
$213$	−4.00240	−0.274240
$214$	−8.39230	−0.573686
$215$	0	0
$216$	8.39230	0.571024
$217$	−2.44949	−0.166282
$218$	−24.8738	−1.68467
$219$	−4.26795	−0.288401
$220$	0	0
$221$	−7.26795	−0.488895
$222$	−0.656339	−0.0440506
$223$	−20.3166	−1.36050	−0.680251	−	0.732979i	$-0.738129\pi$
$223$	−20.3166	−1.36050	−0.680251	+	0.732979i	$0.738129\pi$
$224$	0	0
$225$	0	0
$226$	18.7846	1.24953
$227$	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
$229$	6.19615	0.409453	0.204727	−	0.978819i	$-0.434369\pi$
$229$	6.19615	0.409453	0.204727	+	0.978819i	$0.434369\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	−6.21166	−0.407815
$233$	−29.9759	−1.96379	−0.981893	−	0.189437i	$-0.939334\pi$
$233$	−29.9759	−1.96379	−0.981893	+	0.189437i	$0.939334\pi$
$234$	9.46410	0.618688
$235$	0	0
$236$	0	0
$237$	−2.17209	−0.141092
$238$	−10.2784	−0.666252
$239$	17.3205	1.12037	0.560185	−	0.828367i	$-0.310730\pi$
$239$	17.3205	1.12037	0.560185	+	0.828367i	$0.310730\pi$
$240$	0	0
$241$	14.5885	0.939725	0.469863	−	0.882740i	$-0.344303\pi$
$241$	14.5885	0.939725	0.469863	+	0.882740i	$0.344303\pi$
$242$	16.1112	1.03566
$243$	12.3490	0.792188
$244$	0	0
$245$	0	0
$246$	−4.05256	−0.258382
$247$	−7.82894	−0.498144
$248$	−2.82843	−0.179605
$249$	−7.53590	−0.477568
$250$	0	0
$251$	−7.85641	−0.495892	−0.247946	−	0.968774i	$-0.579756\pi$
$251$	−7.85641	−0.495892	−0.247946	+	0.968774i	$0.579756\pi$
$252$	0	0
$253$	6.69213	0.420731
$254$	−15.4641	−0.970304
$255$	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	$0.485955\pi$
$258$	9.14162	0.569132
$259$	2.19615	0.136462
$260$	0	0
$261$	−6.00000	−0.371391
$262$	4.72311	0.291795
$263$	21.3519	1.31661	0.658307	−	0.752749i	$-0.271273\pi$
$263$	21.3519	1.31661	0.658307	+	0.752749i	$0.271273\pi$
$264$	6.92820	0.426401
$265$	0	0
$266$	−11.0718	−0.678855
$267$	−7.34847	−0.449719
$268$	0	0
$269$	−1.26795	−0.0773082	−0.0386541	−	0.999253i	$-0.512307\pi$
$269$	−1.26795	−0.0773082	−0.0386541	+	0.999253i	$0.512307\pi$
$270$	0	0
$271$	14.5885	0.886186	0.443093	−	0.896476i	$-0.353881\pi$
$271$	14.5885	0.886186	0.443093	+	0.896476i	$0.353881\pi$
$272$	−11.8685	−0.719634
$273$	3.10583	0.187973
$274$	17.8038	1.07557
$275$	0	0
$276$	0	0
$277$	7.76457	0.466528	0.233264	−	0.972413i	$-0.425059\pi$
$277$	7.76457	0.466528	0.233264	+	0.972413i	$0.425059\pi$
$278$	17.2480	1.03446
$279$	−2.73205	−0.163564
$280$	0	0
$281$	−29.1962	−1.74170	−0.870848	−	0.491552i	$-0.836430\pi$
$281$	−29.1962	−1.74170	−0.870848	+	0.491552i	$0.836430\pi$
$282$	6.41473	0.381992
$283$	−11.5911	−0.689020	−0.344510	−	0.938783i	$-0.611955\pi$
$283$	−11.5911	−0.689020	−0.344510	+	0.938783i	$0.611955\pi$
$284$	0	0
$285$	0	0
$286$	16.3923	0.969297
$287$	13.5601	0.800429
$288$	0	0
$289$	−8.19615	−0.482127
$290$	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	$0.513153\pi$
$294$	−0.732051	−0.0426941
$295$	0	0
$296$	2.53590	0.147396
$297$	14.0406	0.814718
$298$	0.656339	0.0380207
$299$	3.46410	0.200334
$300$	0	0
$301$	−30.5885	−1.76309
$302$	20.6312	1.18719
$303$	−0.416102	−0.0239044
$304$	−12.7846	−0.733248
$305$	0	0
$306$	−11.4641	−0.655359
$307$	14.5211	0.828761	0.414381	−	0.910104i	$-0.363998\pi$
$307$	14.5211	0.828761	0.414381	+	0.910104i	$0.363998\pi$
$308$	0	0
$309$	5.32051	0.302673
$310$	0	0
$311$	−31.3923	−1.78009	−0.890047	−	0.455868i	$-0.849329\pi$
$311$	−31.3923	−1.78009	−0.890047	+	0.455868i	$0.849329\pi$
$312$	3.58630	0.203034
$313$	10.5187	0.594550	0.297275	−	0.954792i	$-0.403922\pi$
$313$	10.5187	0.594550	0.297275	+	0.954792i	$0.403922\pi$
$314$	26.7846	1.51154
$315$	0	0
$316$	0	0
$317$	−22.9048	−1.28646	−0.643231	−	0.765672i	$-0.722407\pi$
$317$	−22.9048	−1.28646	−0.643231	+	0.765672i	$0.722407\pi$
$318$	0.933740	0.0523616
$319$	−10.3923	−0.581857
$320$	0	0
$321$	−3.07180	−0.171451
$322$	4.89898	0.273009
$323$	9.48339	0.527670
$324$	0	0
$325$	0	0
$326$	−7.60770	−0.421351
$327$	−9.10446	−0.503478
$328$	15.6579	0.864562
$329$	−21.4641	−1.18335
$330$	0	0
$331$	−2.39230	−0.131493	−0.0657465	−	0.997836i	$-0.520943\pi$
$331$	−2.39230	−0.131493	−0.0657465	+	0.997836i	$0.520943\pi$
$332$	0	0
$333$	2.44949	0.134231
$334$	8.58846	0.469939
$335$	0	0
$336$	5.07180	0.276689
$337$	8.90138	0.484889	0.242445	−	0.970165i	$-0.422051\pi$
$337$	8.90138	0.484889	0.242445	+	0.970165i	$0.422051\pi$
$338$	−9.89949	−0.538462
$339$	6.87564	0.373434
$340$	0	0
$341$	−4.73205	−0.256255
$342$	−12.3490	−0.667757
$343$	19.5959	1.05808
$344$	−35.3205	−1.90435
$345$	0	0
$346$	30.7846	1.65499
$347$	−27.9797	−1.50203	−0.751014	−	0.660287i	$-0.770435\pi$
$347$	−27.9797	−1.50203	−0.751014	+	0.660287i	$0.770435\pi$
$348$	0	0
$349$	23.3923	1.25216	0.626081	−	0.779758i	$-0.284658\pi$
$349$	23.3923	1.25216	0.626081	+	0.779758i	$0.284658\pi$
$350$	0	0
$351$	7.26795	0.387934
$352$	0	0
$353$	20.9358	1.11430	0.557150	−	0.830412i	$-0.311895\pi$
$353$	20.9358	1.11430	0.557150	+	0.830412i	$0.311895\pi$
$354$	−7.26795	−0.386287
$355$	0	0
$356$	0	0
$357$	−3.76217	−0.199115
$358$	1.79315	0.0947710
$359$	−5.07180	−0.267679	−0.133840	−	0.991003i	$-0.542731\pi$
$359$	−5.07180	−0.267679	−0.133840	+	0.991003i	$0.542731\pi$
$360$	0	0
$361$	−8.78461	−0.462348
$362$	5.10205	0.268158
$363$	5.89709	0.309517
$364$	0	0
$365$	0	0
$366$	−10.5359	−0.550720
$367$	−5.31508	−0.277445	−0.138723	−	0.990331i	$-0.544300\pi$
$367$	−5.31508	−0.277445	−0.138723	+	0.990331i	$0.544300\pi$
$368$	5.65685	0.294884
$369$	15.1244	0.787343
$370$	0	0
$371$	−3.12436	−0.162208
$372$	0	0
$373$	10.4543	0.541303	0.270652	−	0.962677i	$-0.412761\pi$
$373$	10.4543	0.541303	0.270652	+	0.962677i	$0.412761\pi$
$374$	−19.8564	−1.02675
$375$	0	0
$376$	−24.7846	−1.27817
$377$	−5.37945	−0.277056
$378$	10.2784	0.528666
$379$	23.3923	1.20158	0.600791	−	0.799406i	$-0.294852\pi$
$379$	23.3923	1.20158	0.600791	+	0.799406i	$0.294852\pi$
$380$	0	0
$381$	−5.66025	−0.289984
$382$	−1.31268	−0.0671624
$383$	−9.48339	−0.484579	−0.242289	−	0.970204i	$-0.577898\pi$
$383$	−9.48339	−0.484579	−0.242289	+	0.970204i	$0.577898\pi$
$384$	5.85641	0.298858
$385$	0	0
$386$	−18.9282	−0.963420
$387$	−34.1170	−1.73426
$388$	0	0
$389$	13.8564	0.702548	0.351274	−	0.936273i	$-0.385749\pi$
$389$	13.8564	0.702548	0.351274	+	0.936273i	$0.385749\pi$
$390$	0	0
$391$	−4.19615	−0.212209
$392$	2.82843	0.142857
$393$	1.72878	0.0872054
$394$	−38.3923	−1.93418
$395$	0	0
$396$	0	0
$397$	−18.2832	−0.917610	−0.458805	−	0.888537i	$-0.651722\pi$
$397$	−18.2832	−0.917610	−0.458805	+	0.888537i	$0.651722\pi$
$398$	−12.1459	−0.608820
$399$	−4.05256	−0.202882
$400$	0	0
$401$	12.5885	0.628638	0.314319	−	0.949317i	$-0.398224\pi$
$401$	12.5885	0.628638	0.314319	+	0.949317i	$0.398224\pi$
$402$	1.79315	0.0894342
$403$	−2.44949	−0.122018
$404$	0	0
$405$	0	0
$406$	−7.60770	−0.377564
$407$	4.24264	0.210300
$408$	−4.34418	−0.215069
$409$	−2.58846	−0.127991	−0.0639955	−	0.997950i	$-0.520384\pi$
$409$	−2.58846	−0.127991	−0.0639955	+	0.997950i	$0.520384\pi$
$410$	0	0
$411$	6.51666	0.321443
$412$	0	0
$413$	24.3190	1.19666
$414$	5.46410	0.268546
$415$	0	0
$416$	0	0
$417$	6.31319	0.309158
$418$	−21.3891	−1.04617
$419$	−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	$0.653228\pi$
$422$	26.0106	1.26618
$423$	−23.9401	−1.16401
$424$	−3.60770	−0.175205
$425$	0	0
$426$	−5.66025	−0.274240
$427$	35.2538	1.70605
$428$	0	0
$429$	6.00000	0.289683
$430$	0	0
$431$	18.1244	0.873019	0.436510	−	0.899700i	$-0.356214\pi$
$431$	18.1244	0.873019	0.436510	+	0.899700i	$0.356214\pi$
$432$	11.8685	0.571024
$433$	32.8043	1.57647	0.788237	−	0.615371i	$-0.210994\pi$
$433$	32.8043	1.57647	0.788237	+	0.615371i	$0.210994\pi$
$434$	−3.46410	−0.166282
$435$	0	0
$436$	0	0
$437$	−4.52004	−0.216223
$438$	−6.03579	−0.288401
$439$	−3.39230	−0.161906	−0.0809529	−	0.996718i	$-0.525796\pi$
$439$	−3.39230	−0.161906	−0.0809529	+	0.996718i	$0.525796\pi$
$440$	0	0
$441$	2.73205	0.130098
$442$	−10.2784	−0.488895
$443$	−31.9449	−1.51775	−0.758874	−	0.651237i	$-0.774250\pi$
$443$	−31.9449	−1.51775	−0.758874	+	0.651237i	$0.774250\pi$
$444$	0	0
$445$	0	0
$446$	−28.7321	−1.36050
$447$	0.240237	0.0113628
$448$	−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
$451$	26.1962	1.23353
$452$	0	0
$453$	7.55154	0.354802
$454$	14.0000	0.657053
$455$	0	0
$456$	−4.67949	−0.219137
$457$	1.13681	0.0531778	0.0265889	−	0.999646i	$-0.491535\pi$
$457$	1.13681	0.0531778	0.0265889	+	0.999646i	$0.491535\pi$
$458$	8.76268	0.409453
$459$	−8.80385	−0.410928
$460$	0	0
$461$	−27.8038	−1.29495	−0.647477	−	0.762085i	$-0.724176\pi$
$461$	−27.8038	−1.29495	−0.647477	+	0.762085i	$0.724176\pi$
$462$	8.48528	0.394771
$463$	−18.9396	−0.880197	−0.440098	−	0.897950i	$-0.645056\pi$
$463$	−18.9396	−0.880197	−0.440098	+	0.897950i	$0.645056\pi$
$464$	−8.78461	−0.407815
$465$	0	0
$466$	−42.3923	−1.96379
$467$	−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
$471$	9.80385	0.451737
$472$	28.0812	1.29254
$473$	−59.0924	−2.71707
$474$	−3.07180	−0.141092
$475$	0	0
$476$	0	0
$477$	−3.48477	−0.159556
$478$	24.4949	1.12037
$479$	11.0718	0.505883	0.252942	−	0.967482i	$-0.418602\pi$
$479$	11.0718	0.505883	0.252942	+	0.967482i	$0.418602\pi$
$480$	0	0
$481$	2.19615	0.100136
$482$	20.6312	0.939725
$483$	1.79315	0.0815912
$484$	0	0
$485$	0	0
$486$	17.4641	0.792188
$487$	33.7009	1.52713	0.763567	−	0.645729i	$-0.223446\pi$
$487$	33.7009	1.52713	0.763567	+	0.645729i	$0.223446\pi$
$488$	40.7076	1.84275
$489$	−2.78461	−0.125924
$490$	0	0
$491$	34.6410	1.56333	0.781664	−	0.623700i	$-0.214371\pi$
$491$	34.6410	1.56333	0.781664	+	0.623700i	$0.214371\pi$
$492$	0	0
$493$	6.51626	0.293478
$494$	−11.0718	−0.498144
$495$	0	0
$496$	−4.00000	−0.179605
$497$	18.9396	0.849556
$498$	−10.6574	−0.477568
$499$	24.1962	1.08317	0.541584	−	0.840646i	$-0.317825\pi$
$499$	24.1962	1.08317	0.541584	+	0.840646i	$0.317825\pi$
$500$	0	0
$501$	3.14359	0.140445
$502$	−11.1106	−0.495892
$503$	−6.79367	−0.302915	−0.151457	−	0.988464i	$-0.548397\pi$
$503$	−6.79367	−0.302915	−0.151457	+	0.988464i	$0.548397\pi$
$504$	−18.9282	−0.843129
$505$	0	0
$506$	9.46410	0.420731
$507$	−3.62347	−0.160924
$508$	0	0
$509$	6.92820	0.307087	0.153544	−	0.988142i	$-0.450931\pi$
$509$	6.92820	0.307087	0.153544	+	0.988142i	$0.450931\pi$
$510$	0	0
$511$	20.1962	0.893425
$512$	−22.6274	−1.00000
$513$	−9.48339	−0.418702
$514$	2.00000	0.0882162
$515$	0	0
$516$	0	0
$517$	−41.4655	−1.82365
$518$	3.10583	0.136462
$519$	11.2679	0.494608
$520$	0	0
$521$	−31.0526	−1.36044	−0.680219	−	0.733009i	$-0.738115\pi$
$521$	−31.0526	−1.36044	−0.680219	+	0.733009i	$0.738115\pi$
$522$	−8.48528	−0.371391
$523$	−22.7661	−0.995493	−0.497746	−	0.867323i	$-0.665839\pi$
$523$	−22.7661	−0.995493	−0.497746	+	0.867323i	$0.665839\pi$
$524$	0	0
$525$	0	0
$526$	30.1962	1.31661
$527$	2.96713	0.129250
$528$	9.79796	0.426401
$529$	−21.0000	−0.913043
$530$	0	0
$531$	27.1244	1.17710
$532$	0	0
$533$	13.5601	0.587354
$534$	−10.3923	−0.449719
$535$	0	0
$536$	−6.92820	−0.299253
$537$	0.656339	0.0283231
$538$	−1.79315	−0.0773082
$539$	4.73205	0.203824
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	20.6312	0.886186
$543$	1.86748	0.0801413
$544$	0	0
$545$	0	0
$546$	4.39230	0.187973
$547$	8.66115	0.370324	0.185162	−	0.982708i	$-0.440719\pi$
$547$	8.66115	0.370324	0.185162	+	0.982708i	$0.440719\pi$
$548$	0	0
$549$	39.3205	1.67816
$550$	0	0
$551$	7.01924	0.299030
$552$	2.07055	0.0881286
$553$	10.2784	0.437083
$554$	10.9808	0.466528
$555$	0	0
$556$	0	0
$557$	−4.79744	−0.203274	−0.101637	−	0.994822i	$-0.532408\pi$
$557$	−4.79744	−0.203274	−0.101637	+	0.994822i	$0.532408\pi$
$558$	−3.86370	−0.163564
$559$	−30.5885	−1.29375
$560$	0	0
$561$	−7.26795	−0.306853
$562$	−41.2896	−1.74170
$563$	−30.8081	−1.29841	−0.649203	−	0.760615i	$-0.724897\pi$
$563$	−30.8081	−1.29841	−0.649203	+	0.760615i	$0.724897\pi$
$564$	0	0
$565$	0	0
$566$	−16.3923	−0.689020
$567$	−16.3142	−0.685133
$568$	21.8695	0.917626
$569$	−27.4641	−1.15136	−0.575678	−	0.817677i	$-0.695262\pi$
$569$	−27.4641	−1.15136	−0.575678	+	0.817677i	$0.695262\pi$
$570$	0	0
$571$	32.5885	1.36378	0.681892	−	0.731453i	$-0.261157\pi$
$571$	32.5885	1.36378	0.681892	+	0.731453i	$0.261157\pi$
$572$	0	0
$573$	−0.480473	−0.0200721
$574$	19.1769	0.800429
$575$	0	0
$576$	−21.8564	−0.910684
$577$	3.76217	0.156621	0.0783105	−	0.996929i	$-0.475047\pi$
$577$	3.76217	0.156621	0.0783105	+	0.996929i	$0.475047\pi$
$578$	−11.5911	−0.482127
$579$	−6.92820	−0.287926
$580$	0	0
$581$	35.6603	1.47944
$582$	−8.48528	−0.351726
$583$	−6.03579	−0.249977
$584$	23.3205	0.965009
$585$	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
$589$	3.19615	0.131695
$590$	0	0
$591$	−14.0526	−0.578045
$592$	3.58630	0.147396
$593$	−22.6274	−0.929197	−0.464598	−	0.885522i	$-0.653801\pi$
$593$	−22.6274	−0.929197	−0.464598	+	0.885522i	$0.653801\pi$
$594$	19.8564	0.814718
$595$	0	0
$596$	0	0
$597$	−4.44571	−0.181951
$598$	4.89898	0.200334
$599$	−3.46410	−0.141539	−0.0707697	−	0.997493i	$-0.522546\pi$
$599$	−3.46410	−0.141539	−0.0707697	+	0.997493i	$0.522546\pi$
$600$	0	0
$601$	−13.8038	−0.563071	−0.281535	−	0.959551i	$-0.590844\pi$
$601$	−13.8038	−0.563071	−0.281535	+	0.959551i	$0.590844\pi$
$602$	−43.2586	−1.76309
$603$	−6.69213	−0.272525
$604$	0	0
$605$	0	0
$606$	−0.588457	−0.0239044
$607$	47.9817	1.94752	0.973759	−	0.227581i	$-0.0730818\pi$
$607$	47.9817	1.94752	0.973759	+	0.227581i	$0.0730818\pi$
$608$	0	0
$609$	−2.78461	−0.112838
$610$	0	0
$611$	−21.4641	−0.868345
$612$	0	0
$613$	46.4288	1.87524	0.937621	−	0.347659i	$-0.113023\pi$
$613$	46.4288	1.87524	0.937621	+	0.347659i	$0.113023\pi$
$614$	20.5359	0.828761
$615$	0	0
$616$	−32.7846	−1.32093
$617$	4.52004	0.181970	0.0909850	−	0.995852i	$-0.470998\pi$
$617$	4.52004	0.181970	0.0909850	+	0.995852i	$0.470998\pi$
$618$	7.52433	0.302673
$619$	−6.39230	−0.256928	−0.128464	−	0.991714i	$-0.541005\pi$
$619$	−6.39230	−0.256928	−0.128464	+	0.991714i	$0.541005\pi$
$620$	0	0
$621$	4.19615	0.168386
$622$	−44.3954	−1.78009
$623$	34.7733	1.39316
$624$	5.07180	0.203034
$625$	0	0
$626$	14.8756	0.594550
$627$	−7.82894	−0.312658
$628$	0	0
$629$	−2.66025	−0.106071
$630$	0	0
$631$	−8.39230	−0.334092	−0.167046	−	0.985949i	$-0.553423\pi$
$631$	−8.39230	−0.334092	−0.167046	+	0.985949i	$0.553423\pi$
$632$	11.8685	0.472104
$633$	9.52056	0.378408
$634$	−32.3923	−1.28646
$635$	0	0
$636$	0	0
$637$	2.44949	0.0970523
$638$	−14.6969	−0.581857
$639$	21.1244	0.835667
$640$	0	0
$641$	9.46410	0.373810	0.186905	−	0.982378i	$-0.440154\pi$
$641$	9.46410	0.373810	0.186905	+	0.982378i	$0.440154\pi$
$642$	−4.34418	−0.171451
$643$	5.79555	0.228554	0.114277	−	0.993449i	$-0.463545\pi$
$643$	5.79555	0.228554	0.114277	+	0.993449i	$0.463545\pi$
$644$	0	0
$645$	0	0
$646$	13.4115	0.527670
$647$	−14.8356	−0.583249	−0.291625	−	0.956533i	$-0.594196\pi$
$647$	−14.8356	−0.583249	−0.291625	+	0.956533i	$0.594196\pi$
$648$	−18.8380	−0.740028
$649$	46.9808	1.84416
$650$	0	0
$651$	−1.26795	−0.0496948
$652$	0	0
$653$	−20.3538	−0.796505	−0.398253	−	0.917276i	$-0.630383\pi$
$653$	−20.3538	−0.796505	−0.398253	+	0.917276i	$0.630383\pi$
$654$	−12.8756	−0.503478
$655$	0	0
$656$	22.1436	0.864562
$657$	22.5259	0.878818
$658$	−30.3548	−1.18335
$659$	0.803848	0.0313135	0.0156567	−	0.999877i	$-0.495016\pi$
$659$	0.803848	0.0313135	0.0156567	+	0.999877i	$0.495016\pi$
$660$	0	0
$661$	−9.19615	−0.357689	−0.178844	−	0.983877i	$-0.557236\pi$
$661$	−9.19615	−0.357689	−0.178844	+	0.983877i	$0.557236\pi$
$662$	−3.38323	−0.131493
$663$	−3.76217	−0.146110
$664$	41.1769	1.59797
$665$	0	0
$666$	3.46410	0.134231
$667$	−3.10583	−0.120258
$668$	0	0
$669$	−10.5167	−0.406598
$670$	0	0
$671$	68.1051	2.62917
$672$	0	0
$673$	−18.6993	−0.720807	−0.360403	−	0.932797i	$-0.617361\pi$
$673$	−18.6993	−0.720807	−0.360403	+	0.932797i	$0.617361\pi$
$674$	12.5885	0.484889
$675$	0	0
$676$	0	0
$677$	17.6641	0.678885	0.339443	−	0.940627i	$-0.389762\pi$
$677$	17.6641	0.678885	0.339443	+	0.940627i	$0.389762\pi$
$678$	9.72363	0.373434
$679$	28.3923	1.08960
$680$	0	0
$681$	5.12436	0.196366
$682$	−6.69213	−0.256255
$683$	−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
$687$	3.20736	0.122369
$688$	−49.9507	−1.90435
$689$	−3.12436	−0.119028
$690$	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
$693$	−31.6675	−1.20295
$694$	−39.5692	−1.50203
$695$	0	0
$696$	−3.21539	−0.121879
$697$	−16.4257	−0.622168
$698$	33.0817	1.25216
$699$	−15.5167	−0.586894
$700$	0	0
$701$	−28.6410	−1.08176	−0.540878	−	0.841101i	$-0.681908\pi$
$701$	−28.6410	−1.08176	−0.540878	+	0.841101i	$0.681908\pi$
$702$	10.2784	0.387934
$703$	−2.86559	−0.108078
$704$	−37.8564	−1.42677
$705$	0	0
$706$	29.6077	1.11430
$707$	1.96902	0.0740525
$708$	0	0
$709$	24.7846	0.930806	0.465403	−	0.885099i	$-0.345909\pi$
$709$	24.7846	0.930806	0.465403	+	0.885099i	$0.345909\pi$
$710$	0	0
$711$	11.4641	0.429937
$712$	40.1528	1.50479
$713$	−1.41421	−0.0529627
$714$	−5.32051	−0.199115
$715$	0	0
$716$	0	0
$717$	8.96575	0.334832
$718$	−7.17260	−0.267679
$719$	28.6410	1.06813	0.534065	−	0.845444i	$-0.320664\pi$
$719$	28.6410	1.06813	0.534065	+	0.845444i	$0.320664\pi$
$720$	0	0
$721$	−25.1769	−0.937637
$722$	−12.4233	−0.462348
$723$	7.55154	0.280845
$724$	0	0
$725$	0	0
$726$	8.33975	0.309517
$727$	14.2165	0.527260	0.263630	−	0.964624i	$-0.415080\pi$
$727$	14.2165	0.527260	0.263630	+	0.964624i	$0.415080\pi$
$728$	−16.9706	−0.628971
$729$	−13.5885	−0.503276
$730$	0	0
$731$	37.0526	1.37044
$732$	0	0
$733$	−24.4949	−0.904740	−0.452370	−	0.891830i	$-0.649421\pi$
$733$	−24.4949	−0.904740	−0.452370	+	0.891830i	$0.649421\pi$
$734$	−7.51666	−0.277445
$735$	0	0
$736$	0	0
$737$	−11.5911	−0.426964
$738$	21.3891	0.787343
$739$	18.7846	0.691003	0.345502	−	0.938418i	$-0.387709\pi$
$739$	18.7846	0.691003	0.345502	+	0.938418i	$0.387709\pi$
$740$	0	0
$741$	−4.05256	−0.148874
$742$	−4.41851	−0.162208
$743$	38.0179	1.39474	0.697370	−	0.716711i	$-0.254353\pi$
$743$	38.0179	1.39474	0.697370	+	0.716711i	$0.254353\pi$
$744$	−1.46410	−0.0536766
$745$	0	0
$746$	14.7846	0.541303
$747$	39.7738	1.45525
$748$	0	0
$749$	14.5359	0.531130
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	−35.0507	−1.27817
$753$	−4.06678	−0.148202
$754$	−7.60770	−0.277056
$755$	0	0
$756$	0	0
$757$	33.7009	1.22488	0.612440	−	0.790517i	$-0.290188\pi$
$757$	33.7009	1.22488	0.612440	+	0.790517i	$0.290188\pi$
$758$	33.0817	1.20158
$759$	3.46410	0.125739
$760$	0	0
$761$	21.7128	0.787089	0.393544	−	0.919306i	$-0.371249\pi$
$761$	21.7128	0.787089	0.393544	+	0.919306i	$0.371249\pi$
$762$	−8.00481	−0.289984
$763$	43.0827	1.55970
$764$	0	0
$765$	0	0
$766$	−13.4115	−0.484579
$767$	24.3190	0.878109
$768$	0	0
$769$	−2.21539	−0.0798890	−0.0399445	−	0.999202i	$-0.512718\pi$
$769$	−2.21539	−0.0798890	−0.0399445	+	0.999202i	$0.512718\pi$
$770$	0	0
$771$	0.732051	0.0263642
$772$	0	0
$773$	−18.3848	−0.661254	−0.330627	−	0.943761i	$-0.607260\pi$
$773$	−18.3848	−0.661254	−0.330627	+	0.943761i	$0.607260\pi$
$774$	−48.2487	−1.73426
$775$	0	0
$776$	32.7846	1.17690
$777$	1.13681	0.0407829
$778$	19.5959	0.702548
$779$	−17.6936	−0.633938
$780$	0	0
$781$	36.5885	1.30924
$782$	−5.93426	−0.212209
$783$	−6.51626	−0.232872
$784$	4.00000	0.142857
$785$	0	0
$786$	2.44486	0.0872054
$787$	26.1122	0.930799	0.465399	−	0.885101i	$-0.345911\pi$
$787$	26.1122	0.930799	0.465399	+	0.885101i	$0.345911\pi$
$788$	0	0
$789$	11.0526	0.393482
$790$	0	0
$791$	−32.5359	−1.15684
$792$	−36.5665	−1.29933
$793$	35.2538	1.25190
$794$	−25.8564	−0.917610
$795$	0	0
$796$	0	0
$797$	−27.9797	−0.991091	−0.495545	−	0.868582i	$-0.665032\pi$
$797$	−27.9797	−0.991091	−0.495545	+	0.868582i	$0.665032\pi$
$798$	−5.73118	−0.202882
$799$	26.0000	0.919814
$800$	0	0
$801$	38.7846	1.37039
$802$	17.8028	0.628638
$803$	39.0160	1.37684
$804$	0	0
$805$	0	0
$806$	−3.46410	−0.122018
$807$	−0.656339	−0.0231042
$808$	2.27362	0.0799858
$809$	−6.33975	−0.222894	−0.111447	−	0.993770i	$-0.535548\pi$
$809$	−6.33975	−0.222894	−0.111447	+	0.993770i	$0.535548\pi$
$810$	0	0
$811$	43.7846	1.53749	0.768743	−	0.639558i	$-0.220883\pi$
$811$	43.7846	1.53749	0.768743	+	0.639558i	$0.220883\pi$
$812$	0	0
$813$	7.55154	0.264844
$814$	6.00000	0.210300
$815$	0	0
$816$	−6.14359	−0.215069
$817$	39.9125	1.39636
$818$	−3.66063	−0.127991
$819$	−16.3923	−0.572793
$820$	0	0
$821$	−51.3731	−1.79293	−0.896466	−	0.443112i	$-0.853874\pi$
$821$	−51.3731	−1.79293	−0.896466	+	0.443112i	$0.853874\pi$
$822$	9.21595	0.321443
$823$	40.7448	1.42027	0.710136	−	0.704064i	$-0.248633\pi$
$823$	40.7448	1.42027	0.710136	+	0.704064i	$0.248633\pi$
$824$	−29.0718	−1.01276
$825$	0	0
$826$	34.3923	1.19666
$827$	−29.8372	−1.03754	−0.518770	−	0.854914i	$-0.673610\pi$
$827$	−29.8372	−1.03754	−0.518770	+	0.854914i	$0.673610\pi$
$828$	0	0
$829$	30.7846	1.06919	0.534597	−	0.845107i	$-0.320463\pi$
$829$	30.7846	1.06919	0.534597	+	0.845107i	$0.320463\pi$
$830$	0	0
$831$	4.01924	0.139426
$832$	−19.5959	−0.679366
$833$	−2.96713	−0.102805
$834$	8.92820	0.309158
$835$	0	0
$836$	0	0
$837$	−2.96713	−0.102559
$838$	−21.2132	−0.732798
$839$	13.9808	0.482670	0.241335	−	0.970442i	$-0.422415\pi$
$839$	13.9808	0.482670	0.241335	+	0.970442i	$0.422415\pi$
$840$	0	0
$841$	−24.1769	−0.833687
$842$	−26.8701	−0.926003
$843$	−15.1130	−0.520521
$844$	0	0
$845$	0	0
$846$	−33.8564	−1.16401
$847$	−27.9053	−0.958839
$848$	−5.10205	−0.175205
$849$	−6.00000	−0.205919
$850$	0	0
$851$	1.26795	0.0434647
$852$	0	0
$853$	−23.1822	−0.793744	−0.396872	−	0.917874i	$-0.629904\pi$
$853$	−23.1822	−0.793744	−0.396872	+	0.917874i	$0.629904\pi$
$854$	49.8564	1.70605
$855$	0	0
$856$	16.7846	0.573686
$857$	3.68784	0.125974	0.0629871	−	0.998014i	$-0.479937\pi$
$857$	3.68784	0.125974	0.0629871	+	0.998014i	$0.479937\pi$
$858$	8.48528	0.289683
$859$	−51.7654	−1.76621	−0.883106	−	0.469174i	$-0.844552\pi$
$859$	−51.7654	−1.76621	−0.883106	+	0.469174i	$0.844552\pi$
$860$	0	0
$861$	7.01924	0.239215
$862$	25.6317	0.873019
$863$	−43.1199	−1.46782	−0.733909	−	0.679247i	$-0.762306\pi$
$863$	−43.1199	−1.46782	−0.733909	+	0.679247i	$0.762306\pi$
$864$	0	0
$865$	0	0
$866$	46.3923	1.57647
$867$	−4.24264	−0.144088
$868$	0	0
$869$	19.8564	0.673582
$870$	0	0
$871$	−6.00000	−0.203302
$872$	49.7477	1.68467
$873$	31.6675	1.07178
$874$	−6.39230	−0.216223
$875$	0	0
$876$	0	0
$877$	−17.6269	−0.595218	−0.297609	−	0.954688i	$-0.596189\pi$
$877$	−17.6269	−0.595218	−0.297609	+	0.954688i	$0.596189\pi$
$878$	−4.79744	−0.161906
$879$	−0.732051	−0.0246915
$880$	0	0
$881$	4.05256	0.136534	0.0682671	−	0.997667i	$-0.478253\pi$
$881$	4.05256	0.136534	0.0682671	+	0.997667i	$0.478253\pi$
$882$	3.86370	0.130098
$883$	30.1146	1.01344	0.506718	−	0.862112i	$-0.330858\pi$
$883$	30.1146	1.01344	0.506718	+	0.862112i	$0.330858\pi$
$884$	0	0
$885$	0	0
$886$	−45.1769	−1.51775
$887$	−4.79744	−0.161082	−0.0805412	−	0.996751i	$-0.525665\pi$
$887$	−4.79744	−0.161082	−0.0805412	+	0.996751i	$0.525665\pi$
$888$	1.31268	0.0440506
$889$	26.7846	0.898327
$890$	0	0
$891$	−31.5167	−1.05585
$892$	0	0
$893$	28.0069	0.937214
$894$	0.339746	0.0113628
$895$	0	0
$896$	−27.7128	−0.925820
$897$	1.79315	0.0598716
$898$	−25.4558	−0.849473
$899$	2.19615	0.0732458
$900$	0	0
$901$	3.78461	0.126084
$902$	37.0470	1.23353
$903$	−15.8338	−0.526914
$904$	−37.5692	−1.24953
$905$	0	0
$906$	10.6795	0.354802
$907$	−11.5911	−0.384876	−0.192438	−	0.981309i	$-0.561640\pi$
$907$	−11.5911	−0.384876	−0.192438	+	0.981309i	$0.561640\pi$
$908$	0	0
$909$	2.19615	0.0728418
$910$	0	0
$911$	3.12436	0.103515	0.0517573	−	0.998660i	$-0.483518\pi$
$911$	3.12436	0.103515	0.0517573	+	0.998660i	$0.483518\pi$
$912$	−6.61780	−0.219137
$913$	68.8903	2.27994
$914$	1.60770	0.0531778
$915$	0	0
$916$	0	0
$917$	−8.18067	−0.270150
$918$	−12.4505	−0.410928
$919$	31.5885	1.04201	0.521004	−	0.853555i	$-0.325558\pi$
$919$	31.5885	1.04201	0.521004	+	0.853555i	$0.325558\pi$
$920$	0	0
$921$	7.51666	0.247682
$922$	−39.3206	−1.29495
$923$	18.9396	0.623404
$924$	0	0
$925$	0	0
$926$	−26.7846	−0.880197
$927$	−28.0812	−0.922308
$928$	0	0
$929$	16.3923	0.537814	0.268907	−	0.963166i	$-0.413338\pi$
$929$	16.3923	0.537814	0.268907	+	0.963166i	$0.413338\pi$
$930$	0	0
$931$	−3.19615	−0.104750
$932$	0	0
$933$	−16.2499	−0.531996
$934$	−28.0000	−0.916188
$935$	0	0
$936$	−18.9282	−0.618688
$937$	−20.2523	−0.661612	−0.330806	−	0.943699i	$-0.607321\pi$
$937$	−20.2523	−0.661612	−0.330806	+	0.943699i	$0.607321\pi$
$938$	−8.48528	−0.277054
$939$	5.44486	0.177686
$940$	0	0
$941$	6.24871	0.203702	0.101851	−	0.994800i	$-0.467523\pi$
$941$	6.24871	0.203702	0.101851	+	0.994800i	$0.467523\pi$
$942$	13.8647	0.451737
$943$	7.82894	0.254945
$944$	39.7128	1.29254
$945$	0	0
$946$	−83.5692	−2.71707
$947$	11.1478	0.362255	0.181127	−	0.983460i	$-0.442025\pi$
$947$	11.1478	0.362255	0.181127	+	0.983460i	$0.442025\pi$
$948$	0	0
$949$	20.1962	0.655595
$950$	0	0
$951$	−11.8564	−0.384470
$952$	20.5569	0.666252
$953$	12.8666	0.416791	0.208395	−	0.978045i	$-0.433176\pi$
$953$	12.8666	0.416791	0.208395	+	0.978045i	$0.433176\pi$
$954$	−4.92820	−0.159556
$955$	0	0
$956$	0	0
$957$	−5.37945	−0.173893
$958$	15.6579	0.505883
$959$	−30.8372	−0.995784
$960$	0	0
$961$	1.00000	0.0322581
$962$	3.10583	0.100136
$963$	16.2127	0.522447
$964$	0	0
$965$	0	0
$966$	2.53590	0.0815912
$967$	9.97382	0.320737	0.160368	−	0.987057i	$-0.448732\pi$
$967$	9.97382	0.320737	0.160368	+	0.987057i	$0.448732\pi$
$968$	−32.2223	−1.03566
$969$	4.90897	0.157699
$970$	0	0
$971$	−32.3205	−1.03721	−0.518607	−	0.855013i	$-0.673549\pi$
$971$	−32.3205	−1.03721	−0.518607	+	0.855013i	$0.673549\pi$
$972$	0	0
$973$	−29.8744	−0.957728
$974$	47.6603	1.52713
$975$	0	0
$976$	57.5692	1.84275
$977$	13.0053	0.416077	0.208039	−	0.978121i	$-0.433292\pi$
$977$	13.0053	0.416077	0.208039	+	0.978121i	$0.433292\pi$
$978$	−3.93803	−0.125924
$979$	67.1769	2.14698
$980$	0	0
$981$	48.0526	1.53420
$982$	48.9898	1.56333
$983$	30.2533	0.964930	0.482465	−	0.875915i	$-0.339742\pi$
$983$	30.2533	0.964930	0.482465	+	0.875915i	$0.339742\pi$
$984$	8.10512	0.258382
$985$	0	0
$986$	9.21539	0.293478
$987$	−11.1106	−0.353655
$988$	0	0
$989$	−17.6603	−0.561563
$990$	0	0
$991$	−34.5885	−1.09874	−0.549369	−	0.835580i	$-0.685132\pi$
$991$	−34.5885	−1.09874	−0.549369	+	0.835580i	$0.685132\pi$
$992$	0	0
$993$	−1.23835	−0.0392978
$994$	26.7846	0.849556
$995$	0	0
$996$	0	0
$997$	26.7685	0.847768	0.423884	−	0.905717i	$-0.360666\pi$
$997$	26.7685	0.847768	0.423884	+	0.905717i	$0.360666\pi$
$998$	34.2185	1.08317
$999$	2.66025	0.0841667

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.a.f.1.4		4
3.2	odd	2		6975.2.a.bk.1.1		4
5.2	odd	4		155.2.b.a.94.3	yes	4
5.3	odd	4		155.2.b.a.94.2	✓	4
5.4	even	2	inner	775.2.a.f.1.1		4
15.2	even	4		1395.2.c.c.559.1		4
15.8	even	4		1395.2.c.c.559.3		4
15.14	odd	2		6975.2.a.bk.1.4		4
20.3	even	4		2480.2.d.b.1489.2		4
20.7	even	4		2480.2.d.b.1489.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.b.a.94.2	✓	4	5.3	odd	4
155.2.b.a.94.3	yes	4	5.2	odd	4
775.2.a.f.1.1		4	5.4	even	2	inner
775.2.a.f.1.4		4	1.1	even	1	trivial
1395.2.c.c.559.1		4	15.2	even	4
1395.2.c.c.559.3		4	15.8	even	4
2480.2.d.b.1489.2		4	20.3	even	4
2480.2.d.b.1489.3		4	20.7	even	4
6975.2.a.bk.1.1		4	3.2	odd	2
6975.2.a.bk.1.4		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)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +0.517638 q^{3} +0.732051 q^{6} -2.44949 q^{7} -2.82843 q^{8} -2.73205 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +0.517638 q^{3} +0.732051 q^{6} -2.44949 q^{7} -2.82843 q^{8} -2.73205 q^{9} -4.73205 q^{11} -2.44949 q^{13} -3.46410 q^{14} -4.00000 q^{16} +2.96713 q^{17} -3.86370 q^{18} +3.19615 q^{19} -1.26795 q^{21} -6.69213 q^{22} -1.41421 q^{23} -1.46410 q^{24} -3.46410 q^{26} -2.96713 q^{27} +2.19615 q^{29} +1.00000 q^{31} -2.44949 q^{33} +4.19615 q^{34} -0.896575 q^{37} +4.52004 q^{38} -1.26795 q^{39} -5.53590 q^{41} -1.79315 q^{42} +12.4877 q^{43} -2.00000 q^{46} +8.76268 q^{47} -2.07055 q^{48} -1.00000 q^{49} +1.53590 q^{51} +1.27551 q^{53} -4.19615 q^{54} +6.92820 q^{56} +1.65445 q^{57} +3.10583 q^{58} -9.92820 q^{59} -14.3923 q^{61} +1.41421 q^{62} +6.69213 q^{63} +8.00000 q^{64} -3.46410 q^{66} +2.44949 q^{67} -0.732051 q^{69} -7.73205 q^{71} +7.72741 q^{72} -8.24504 q^{73} -1.26795 q^{74} +11.5911 q^{77} -1.79315 q^{78} -4.19615 q^{79} +6.66025 q^{81} -7.82894 q^{82} -14.5582 q^{83} +17.6603 q^{86} +1.13681 q^{87} +13.3843 q^{88} -14.1962 q^{89} +6.00000 q^{91} +0.517638 q^{93} +12.3923 q^{94} -11.5911 q^{97} -1.41421 q^{98} +12.9282 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more