LMFDB - Embedded newform 775.2.b.e.249.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [775,2,Mod(249,775)]

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([1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("775.249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 775 = 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	775.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$6.18840615665$
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} - 2x^{5} + 28x^{4} - 12x^{3} + 2x^{2} + 8x + 16 $ x^8 - 2x^5 + 28x^4 - 12x^3 + 2x^2 + 8*x + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 155)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			249.3
Root			$1.48716 - 1.48716i$ of defining polynomial
Character	$\chi$	$=$	775.249
Dual form			775.2.b.e.249.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.62946i q^{2} +3.05273i q^{3} -0.655151 q^{4} +4.97431 q^{6} -2.97431i q^{7} -2.19138i q^{8} -6.31916 q^{9} +O(q^{10})$	\(q-1.62946i q^{2} +3.05273i q^{3} -0.655151 q^{4} +4.97431 q^{6} -2.97431i q^{7} -2.19138i q^{8} -6.31916 q^{9} -1.71539 q^{11} -2.00000i q^{12} -6.97431i q^{13} -4.84653 q^{14} -4.88108 q^{16} -2.76812i q^{17} +10.2968i q^{18} -5.47600 q^{19} +9.07977 q^{21} +2.79516i q^{22} -0.127779i q^{23} +6.68970 q^{24} -11.3644 q^{26} -10.1325i q^{27} +1.94862i q^{28} +7.07977 q^{29} +1.00000 q^{31} +3.57078i q^{32} -5.23661i q^{33} -4.51054 q^{34} +4.14001 q^{36} +8.02704i q^{37} +8.92294i q^{38} +21.2907 q^{39} +4.50168 q^{41} -14.7952i q^{42} -4.76812i q^{43} +1.12384 q^{44} -0.208211 q^{46} +2.10546i q^{47} -14.9006i q^{48} -1.84653 q^{49} +8.45031 q^{51} +4.56923i q^{52} -5.20957i q^{53} -16.5105 q^{54} -6.51785 q^{56} -16.7167i q^{57} -11.5362i q^{58} -13.2968 q^{59} -0.105460 q^{61} -1.62946i q^{62} +18.7952i q^{63} -3.94371 q^{64} -8.53286 q^{66} -2.28461i q^{67} +1.81353i q^{68} +0.390075 q^{69} +5.31916 q^{71} +13.8477i q^{72} +0.640336i q^{73} +13.0798 q^{74} +3.58761 q^{76} +5.10209i q^{77} -34.6924i q^{78} -4.92294 q^{79} +11.9743 q^{81} -7.33533i q^{82} -1.87021i q^{83} -5.94862 q^{84} -7.76947 q^{86} +21.6126i q^{87} +3.75906i q^{88} +11.4922 q^{89} -20.7438 q^{91} +0.0837146i q^{92} +3.05273i q^{93} +3.43077 q^{94} -10.9006 q^{96} +6.84653i q^{97} +3.00886i q^{98} +10.8398 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 18 q^{4} + 16 q^{6} - 14 q^{9}+O(q^{10}) $ 8 * q - 18 * q^4 + 16 * q^6 - 14 * q^9	$ 8 q - 18 q^{4} + 16 q^{6} - 14 q^{9} - 12 q^{11} - 16 q^{14} + 22 q^{16} - 10 q^{19} + 4 q^{21} + 28 q^{24} - 24 q^{26} - 12 q^{29} + 8 q^{31} + 36 q^{34} - 50 q^{36} + 12 q^{39} + 26 q^{41} - 40 q^{44} - 52 q^{46} + 8 q^{49} + 10 q^{51} - 60 q^{54} - 8 q^{56} - 26 q^{59} + 44 q^{61} - 94 q^{64} - 40 q^{66} - 40 q^{69} + 6 q^{71} + 36 q^{74} + 28 q^{76} + 32 q^{79} + 72 q^{81} + 32 q^{86} + 24 q^{89} - 48 q^{91} + 24 q^{94} + 28 q^{96} + 104 q^{99}+O(q^{100}) $ 8 * q - 18 * q^4 + 16 * q^6 - 14 * q^9 - 12 * q^11 - 16 * q^14 + 22 * q^16 - 10 * q^19 + 4 * q^21 + 28 * q^24 - 24 * q^26 - 12 * q^29 + 8 * q^31 + 36 * q^34 - 50 * q^36 + 12 * q^39 + 26 * q^41 - 40 * q^44 - 52 * q^46 + 8 * q^49 + 10 * q^51 - 60 * q^54 - 8 * q^56 - 26 * q^59 + 44 * q^61 - 94 * q^64 - 40 * q^66 - 40 * q^69 + 6 * q^71 + 36 * q^74 + 28 * q^76 + 32 * q^79 + 72 * q^81 + 32 * q^86 + 24 * q^89 - 48 * q^91 + 24 * q^94 + 28 * q^96 + 104 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$.

$n$	$251$	$652$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.62946i	− 1.15220i	−0.817378	−	0.576102i	$-0.804573\pi$
$2$	− 1.62946i	− 1.15220i	0.817378	−	0.576102i	$-0.195427\pi$
$3$	3.05273i	1.76249i	0.472656	+	0.881247i	$0.343295\pi$
$3$	3.05273i	1.76249i	−0.472656	+	0.881247i	$0.656705\pi$
$4$	−0.655151	−0.327576
$5$	0	0
$5$	0	0
$6$	4.97431	2.03075
$7$	− 2.97431i	− 1.12418i	−0.827075	−	0.562092i	$-0.809997\pi$
$7$	− 2.97431i	− 1.12418i	0.827075	−	0.562092i	$-0.190003\pi$
$8$	− 2.19138i	− 0.774771i
$9$	−6.31916	−2.10639
$10$	0	0
$11$	−1.71539	−0.517208	−0.258604	−	0.965983i	$-0.583262\pi$
$11$	−1.71539	−0.517208	−0.258604	+	0.965983i	$0.583262\pi$
$12$	− 2.00000i	− 0.577350i
$13$	− 6.97431i	− 1.93433i	−0.254157	−	0.967163i	$-0.581798\pi$
$13$	− 6.97431i	− 1.93433i	0.254157	−	0.967163i	$-0.418202\pi$
$14$	−4.84653	−1.29529
$15$	0	0
$16$	−4.88108	−1.22027
$17$	− 2.76812i	− 0.671367i	−0.941975	−	0.335683i	$-0.891033\pi$
$17$	− 2.76812i	− 0.671367i	0.941975	−	0.335683i	$-0.108967\pi$
$18$	10.2968i	2.42699i
$19$	−5.47600	−1.25628	−0.628140	−	0.778100i	$-0.716183\pi$
$19$	−5.47600	−1.25628	−0.628140	+	0.778100i	$0.716183\pi$
$20$	0	0
$21$	9.07977	1.98137
$22$	2.79516i	0.595930i
$23$	− 0.127779i	− 0.0266438i	−0.999911	−	0.0133219i	$-0.995759\pi$
$23$	− 0.127779i	− 0.0266438i	0.999911	−	0.0133219i	$-0.00424061\pi$
$24$	6.68970	1.36553
$25$	0	0
$26$	−11.3644	−2.22874
$27$	− 10.1325i	− 1.95000i
$28$	1.94862i	0.368255i
$29$	7.07977	1.31468	0.657340	−	0.753594i	$-0.271681\pi$
$29$	7.07977	1.31468	0.657340	+	0.753594i	$0.271681\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	3.57078i	0.631230i
$33$	− 5.23661i	− 0.911576i
$34$	−4.51054	−0.773552
$35$	0	0
$36$	4.14001	0.690001
$37$	8.02704i	1.31964i	0.751425	+	0.659819i	$0.229367\pi$
$37$	8.02704i	1.31964i	−0.751425	+	0.659819i	$0.770633\pi$
$38$	8.92294i	1.44749i
$39$	21.2907	3.40924
$40$	0	0
$41$	4.50168	0.703045	0.351522	−	0.936179i	$-0.385664\pi$
$41$	4.50168	0.703045	0.351522	+	0.936179i	$0.385664\pi$
$42$	− 14.7952i	− 2.28294i
$43$	− 4.76812i	− 0.727131i	−0.931569	−	0.363565i	$-0.881559\pi$
$43$	− 4.76812i	− 0.727131i	0.931569	−	0.363565i	$-0.118441\pi$
$44$	1.12384	0.169425
$45$	0	0
$46$	−0.208211	−0.0306991
$47$	2.10546i	0.307113i	0.988140	+	0.153556i	$0.0490727\pi$
$47$	2.10546i	0.307113i	−0.988140	+	0.153556i	$0.950927\pi$
$48$	− 14.9006i	− 2.15072i
$49$	−1.84653	−0.263790
$50$	0	0
$51$	8.45031	1.18328
$52$	4.56923i	0.633638i
$53$	− 5.20957i	− 0.715589i	−0.933800	−	0.357794i	$-0.883529\pi$
$53$	− 5.20957i	− 0.715589i	0.933800	−	0.357794i	$-0.116471\pi$
$54$	−16.5105	−2.24680
$55$	0	0
$56$	−6.51785	−0.870985
$57$	− 16.7167i	− 2.21419i
$58$	− 11.5362i	− 1.51478i
$59$	−13.2968	−1.73110	−0.865551	−	0.500821i	$-0.833031\pi$
$59$	−13.2968	−1.73110	−0.865551	+	0.500821i	$0.833031\pi$
$60$	0	0
$61$	−0.105460	−0.0135028	−0.00675140	−	0.999977i	$-0.502149\pi$
$61$	−0.105460	−0.0135028	−0.00675140	+	0.999977i	$0.502149\pi$
$62$	− 1.62946i	− 0.206942i
$63$	18.7952i	2.36797i
$64$	−3.94371	−0.492964
$65$	0	0
$66$	−8.53286	−1.05032
$67$	− 2.28461i	− 0.279110i	−0.990214	−	0.139555i	$-0.955433\pi$
$67$	− 2.28461i	− 0.279110i	0.990214	−	0.139555i	$-0.0445672\pi$
$68$	1.81353i	0.219923i
$69$	0.390075	0.0469595
$70$	0	0
$71$	5.31916	0.631268	0.315634	−	0.948881i	$-0.397783\pi$
$71$	5.31916	0.631268	0.315634	+	0.948881i	$0.397783\pi$
$72$	13.8477i	1.63197i
$73$	0.640336i	0.0749457i	0.999298	+	0.0374728i	$0.0119308\pi$
$73$	0.640336i	0.0749457i	−0.999298	+	0.0374728i	$0.988069\pi$
$74$	13.0798	1.52049
$75$	0	0
$76$	3.58761	0.411527
$77$	5.10209i	0.581437i
$78$	− 34.6924i	− 3.92814i
$79$	−4.92294	−0.553874	−0.276937	−	0.960888i	$-0.589319\pi$
$79$	−4.92294	−0.553874	−0.276937	+	0.960888i	$0.589319\pi$
$80$	0	0
$81$	11.9743	1.33048
$82$	− 7.33533i	− 0.810052i
$83$	− 1.87021i	− 0.205282i	−0.994718	−	0.102641i	$-0.967271\pi$
$83$	− 1.87021i	− 0.205282i	0.994718	−	0.102641i	$-0.0327293\pi$
$84$	−5.94862	−0.649048
$85$	0	0
$86$	−7.76947	−0.837803
$87$	21.6126i	2.31712i
$88$	3.75906i	0.400718i
$89$	11.4922	1.21817	0.609084	−	0.793106i	$-0.291537\pi$
$89$	11.4922	1.21817	0.609084	+	0.793106i	$0.291537\pi$
$90$	0	0
$91$	−20.7438	−2.17454
$92$	0.0837146i	0.00872785i
$93$	3.05273i	0.316553i
$94$	3.43077	0.353857
$95$	0	0
$96$	−10.9006	−1.11254
$97$	6.84653i	0.695160i	0.937650	+	0.347580i	$0.112997\pi$
$97$	6.84653i	0.695160i	−0.937650	+	0.347580i	$0.887003\pi$
$98$	3.00886i	0.303941i
$99$	10.8398	1.08944
$100$	0	0
$101$	18.2712	1.81805	0.909024	−	0.416744i	$-0.136829\pi$
$101$	18.2712	1.81805	0.909024	+	0.416744i	$0.136829\pi$
$102$	− 13.7695i	− 1.36338i
$103$	1.20484i	0.118717i	0.998237	+	0.0593583i	$0.0189055\pi$
$103$	1.20484i	0.118717i	−0.998237	+	0.0593583i	$0.981095\pi$
$104$	−15.2834	−1.49866
$105$	0	0
$106$	−8.48880	−0.824505
$107$	− 5.79179i	− 0.559913i	−0.960013	−	0.279957i	$-0.909680\pi$
$107$	− 5.79179i	− 0.559913i	0.960013	−	0.279957i	$-0.0903201\pi$
$108$	6.63832i	0.638773i
$109$	5.42462	0.519584	0.259792	−	0.965665i	$-0.416346\pi$
$109$	5.42462	0.519584	0.259792	+	0.965665i	$0.416346\pi$
$110$	0	0
$111$	−24.5044	−2.32585
$112$	14.5179i	1.37181i
$113$	2.61329i	0.245838i	0.992417	+	0.122919i	$0.0392255\pi$
$113$	2.61329i	0.245838i	−0.992417	+	0.122919i	$0.960774\pi$
$114$	−27.2393	−2.55120
$115$	0	0
$116$	−4.63832	−0.430657
$117$	44.0718i	4.07444i
$118$	21.6667i	1.99458i
$119$	−8.23324	−0.754740
$120$	0	0
$121$	−8.05745	−0.732496
$122$	0.171843i	0.0155580i
$123$	13.7424i	1.23911i
$124$	−0.655151	−0.0588343
$125$	0	0
$126$	30.6260	2.72838
$127$	− 6.66738i	− 0.591634i	−0.955245	−	0.295817i	$-0.904408\pi$
$127$	− 6.66738i	− 0.591634i	0.955245	−	0.295817i	$-0.0955919\pi$
$128$	13.5677i	1.19923i
$129$	14.5558	1.28156
$130$	0	0
$131$	14.8336	1.29602	0.648011	−	0.761631i	$-0.275601\pi$
$131$	14.8336	1.29602	0.648011	+	0.761631i	$0.275601\pi$
$132$	3.43077i	0.298610i
$133$	16.2873i	1.41229i
$134$	−3.72270	−0.321592
$135$	0	0
$136$	−6.06600	−0.520155
$137$	− 6.10074i	− 0.521221i	−0.965444	−	0.260611i	$-0.916076\pi$
$137$	− 6.10074i	− 0.521221i	0.965444	−	0.260611i	$-0.0839239\pi$
$138$	− 0.635613i	− 0.0541070i
$139$	−0.112771	−0.00956514	−0.00478257	−	0.999989i	$-0.501522\pi$
$139$	−0.112771	−0.00956514	−0.00478257	+	0.999989i	$0.501522\pi$
$140$	0	0
$141$	−6.42740	−0.541285
$142$	− 8.66738i	− 0.727350i
$143$	11.9636i	1.00045i
$144$	30.8443	2.57036
$145$	0	0
$146$	1.04340	0.0863528
$147$	− 5.63697i	− 0.464929i
$148$	− 5.25893i	− 0.432281i
$149$	7.98654	0.654283	0.327141	−	0.944975i	$-0.393915\pi$
$149$	7.98654	0.654283	0.327141	+	0.944975i	$0.393915\pi$
$150$	0	0
$151$	−20.8566	−1.69728	−0.848641	−	0.528969i	$-0.822579\pi$
$151$	−20.8566	−1.69728	−0.848641	+	0.528969i	$0.822579\pi$
$152$	12.0000i	0.973329i
$153$	17.4922i	1.41416i
$154$	8.31367	0.669935
$155$	0	0
$156$	−13.9486	−1.11678
$157$	15.0798i	1.20350i	0.798686	+	0.601748i	$0.205529\pi$
$157$	15.0798i	1.20350i	−0.798686	+	0.601748i	$0.794471\pi$
$158$	8.02175i	0.638176i
$159$	15.9034	1.26122
$160$	0	0
$161$	−0.380055	−0.0299525
$162$	− 19.5117i	− 1.53298i
$163$	− 4.27730i	− 0.335024i	−0.985870	−	0.167512i	$-0.946427\pi$
$163$	− 4.27730i	− 0.335024i	0.985870	−	0.167512i	$-0.0535733\pi$
$164$	−2.94928	−0.230300
$165$	0	0
$166$	−3.04743	−0.236527
$167$	− 14.5599i	− 1.12668i	−0.826225	−	0.563340i	$-0.809516\pi$
$167$	− 14.5599i	− 1.12668i	0.826225	−	0.563340i	$-0.190484\pi$
$168$	− 19.8972i	− 1.53511i
$169$	−35.6410	−2.74162
$170$	0	0
$171$	34.6037	2.64621
$172$	3.12384i	0.238190i
$173$	− 23.5463i	− 1.79019i	−0.445877	−	0.895094i	$-0.647108\pi$
$173$	− 23.5463i	− 1.79019i	0.445877	−	0.895094i	$-0.352892\pi$
$174$	35.2170	2.66979
$175$	0	0
$176$	8.37293	0.631133
$177$	− 40.5917i	− 3.05106i
$178$	− 18.7261i	− 1.40358i
$179$	8.86885	0.662889	0.331445	−	0.943475i	$-0.392464\pi$
$179$	8.86885	0.662889	0.331445	+	0.943475i	$0.392464\pi$
$180$	0	0
$181$	13.1088	0.974372	0.487186	−	0.873298i	$-0.338023\pi$
$181$	13.1088	0.974372	0.487186	+	0.873298i	$0.338023\pi$
$182$	33.8012i	2.50551i
$183$	− 0.321941i	− 0.0237986i
$184$	−0.280013	−0.0206428
$185$	0	0
$186$	4.97431	0.364734
$187$	4.74838i	0.347236i
$188$	− 1.37939i	− 0.100603i
$189$	−30.1372	−2.19216
$190$	0	0
$191$	8.53286	0.617416	0.308708	−	0.951157i	$-0.400103\pi$
$191$	8.53286	0.617416	0.308708	+	0.951157i	$0.400103\pi$
$192$	− 12.0391i	− 0.868846i
$193$	− 16.1595i	− 1.16319i	−0.813479	−	0.581595i	$-0.802429\pi$
$193$	− 16.1595i	− 1.16319i	0.813479	−	0.581595i	$-0.197571\pi$
$194$	11.1562	0.800967
$195$	0	0
$196$	1.20976	0.0864113
$197$	− 1.02569i	− 0.0730772i	−0.999332	−	0.0365386i	$-0.988367\pi$
$197$	− 1.02569i	− 0.0730772i	0.999332	−	0.0365386i	$-0.0116332\pi$
$198$	− 17.6631i	− 1.25526i
$199$	−26.6701	−1.89059	−0.945296	−	0.326213i	$-0.894227\pi$
$199$	−26.6701	−1.89059	−0.945296	+	0.326213i	$0.894227\pi$
$200$	0	0
$201$	6.97431	0.491930
$202$	− 29.7722i	− 2.09476i
$203$	− 21.0575i	− 1.47794i
$204$	−5.53623	−0.387614
$205$	0	0
$206$	1.96325	0.136786
$207$	0.807456i	0.0561221i
$208$	34.0422i	2.36040i
$209$	9.39344	0.649758
$210$	0	0
$211$	15.0061	1.03306	0.516531	−	0.856269i	$-0.327223\pi$
$211$	15.0061	1.03306	0.516531	+	0.856269i	$0.327223\pi$
$212$	3.41305i	0.234409i
$213$	16.2380i	1.11261i
$214$	−9.43751	−0.645135
$215$	0	0
$216$	−22.2042	−1.51080
$217$	− 2.97431i	− 0.201909i
$218$	− 8.83922i	− 0.598668i
$219$	−1.95477	−0.132091
$220$	0	0
$221$	−19.3057	−1.29864
$222$	39.9290i	2.67986i
$223$	8.36574i	0.560211i	0.959969	+	0.280106i	$0.0903695\pi$
$223$	8.36574i	0.560211i	−0.959969	+	0.280106i	$0.909630\pi$
$224$	10.6206	0.709619
$225$	0	0
$226$	4.25827	0.283256
$227$	14.7147i	0.976651i	0.872662	+	0.488325i	$0.162392\pi$
$227$	14.7147i	0.976651i	−0.872662	+	0.488325i	$0.837608\pi$
$228$	10.9520i	0.725313i
$229$	18.4659	1.22026	0.610131	−	0.792301i	$-0.291117\pi$
$229$	18.4659	1.22026	0.610131	+	0.792301i	$0.291117\pi$
$230$	0	0
$231$	−15.5753	−1.02478
$232$	− 15.5145i	− 1.01858i
$233$	0.0736944i	0.00482788i	0.999997	+	0.00241394i	$0.000768382\pi$
$233$	0.0736944i	0.00482788i	−0.999997	+	0.00241394i	$0.999232\pi$
$234$	71.8134	4.69459
$235$	0	0
$236$	8.71144	0.567067
$237$	− 15.0284i	− 0.976199i
$238$	13.4158i	0.869615i
$239$	−20.0758	−1.29860	−0.649299	−	0.760533i	$-0.724938\pi$
$239$	−20.0758	−1.29860	−0.649299	+	0.760533i	$0.724938\pi$
$240$	0	0
$241$	−3.49217	−0.224950	−0.112475	−	0.993655i	$-0.535878\pi$
$241$	−3.49217	−0.224950	−0.112475	+	0.993655i	$0.535878\pi$
$242$	13.1293i	0.843985i
$243$	6.15684i	0.394961i
$244$	0.0690924	0.00442319
$245$	0	0
$246$	22.3928	1.42771
$247$	38.1913i	2.43005i
$248$	− 2.19138i	− 0.139153i
$249$	5.70924	0.361808
$250$	0	0
$251$	18.9520	1.19624	0.598120	−	0.801407i	$-0.295915\pi$
$251$	18.9520	1.19624	0.598120	+	0.801407i	$0.295915\pi$
$252$	− 12.3137i	− 0.775688i
$253$	0.219190i	0.0137804i
$254$	−10.8642	−0.681684
$255$	0	0
$256$	14.2206	0.888789
$257$	6.73376i	0.420041i	0.977697	+	0.210020i	$0.0673530\pi$
$257$	6.73376i	0.420041i	−0.977697	+	0.210020i	$0.932647\pi$
$258$	− 23.7181i	− 1.47662i
$259$	23.8749	1.48352
$260$	0	0
$261$	−44.7382	−2.76923
$262$	− 24.1709i	− 1.49328i
$263$	13.5342i	0.834556i	0.908779	+	0.417278i	$0.137016\pi$
$263$	13.5342i	0.834556i	−0.908779	+	0.417278i	$0.862984\pi$
$264$	−11.4754	−0.706263
$265$	0	0
$266$	26.5396	1.62725
$267$	35.0825i	2.14701i
$268$	1.49677i	0.0914297i
$269$	7.04340	0.429444	0.214722	−	0.976675i	$-0.431115\pi$
$269$	7.04340	0.429444	0.214722	+	0.976675i	$0.431115\pi$
$270$	0	0
$271$	11.8208	0.718065	0.359033	−	0.933325i	$-0.383107\pi$
$271$	11.8208	0.718065	0.359033	+	0.933325i	$0.383107\pi$
$272$	13.5114i	0.819248i
$273$	− 63.3252i	− 3.83261i
$274$	−9.94093	−0.600553
$275$	0	0
$276$	−0.255558	−0.0153828
$277$	− 22.9350i	− 1.37803i	−0.724748	−	0.689014i	$-0.758044\pi$
$277$	− 22.9350i	− 1.37803i	0.724748	−	0.689014i	$-0.241956\pi$
$278$	0.183757i	0.0110210i
$279$	−6.31916	−0.378318
$280$	0	0
$281$	−14.4603	−0.862631	−0.431315	−	0.902201i	$-0.641950\pi$
$281$	−14.4603	−0.862631	−0.431315	+	0.902201i	$0.641950\pi$
$282$	10.4732i	0.623671i
$283$	27.7985i	1.65245i	0.563340	+	0.826225i	$0.309516\pi$
$283$	27.7985i	1.65245i	−0.563340	+	0.826225i	$0.690484\pi$
$284$	−3.48486	−0.206788
$285$	0	0
$286$	19.4943	1.15272
$287$	− 13.3894i	− 0.790352i
$288$	− 22.5643i	− 1.32961i
$289$	9.33754	0.549267
$290$	0	0
$291$	−20.9006	−1.22522
$292$	− 0.419517i	− 0.0245504i
$293$	2.95930i	0.172884i	0.996257	+	0.0864422i	$0.0275498\pi$
$293$	2.95930i	0.172884i	−0.996257	+	0.0864422i	$0.972450\pi$
$294$	−9.18523	−0.535694
$295$	0	0
$296$	17.5903	1.02242
$297$	17.3811i	1.00856i
$298$	− 13.0138i	− 0.753868i
$299$	−0.891171	−0.0515377
$300$	0	0
$301$	−14.1819	−0.817429
$302$	33.9850i	1.95562i
$303$	55.7769i	3.20430i
$304$	26.7288	1.53300
$305$	0	0
$306$	28.5028	1.62940
$307$	− 10.6160i	− 0.605887i	−0.953008	−	0.302944i	$-0.902031\pi$
$307$	− 10.6160i	− 0.605887i	0.953008	−	0.302944i	$-0.0979694\pi$
$308$	− 3.34264i	− 0.190465i
$309$	−3.67806	−0.209237
$310$	0	0
$311$	15.6579	0.887876	0.443938	−	0.896058i	$-0.353581\pi$
$311$	15.6579	0.887876	0.443938	+	0.896058i	$0.353581\pi$
$312$	− 46.6560i	− 2.64138i
$313$	22.2994i	1.26043i	0.776419	+	0.630217i	$0.217034\pi$
$313$	22.2994i	1.26043i	−0.776419	+	0.630217i	$0.782966\pi$
$314$	24.5719	1.38667
$315$	0	0
$316$	3.22527	0.181436
$317$	7.84990i	0.440894i	0.975399	+	0.220447i	$0.0707517\pi$
$317$	7.84990i	0.440894i	−0.975399	+	0.220447i	$0.929248\pi$
$318$	− 25.9140i	− 1.45319i
$319$	−12.1445	−0.679964
$320$	0	0
$321$	17.6808	0.986844
$322$	0.619285i	0.0345114i
$323$	15.1582i	0.843424i
$324$	−7.84499	−0.435833
$325$	0	0
$326$	−6.96971	−0.386017
$327$	16.5599i	0.915765i
$328$	− 9.86491i	− 0.544699i
$329$	6.26230	0.345252
$330$	0	0
$331$	−15.6199	−0.858550	−0.429275	−	0.903174i	$-0.641231\pi$
$331$	−15.6199	−0.858550	−0.429275	+	0.903174i	$0.641231\pi$
$332$	1.22527i	0.0672453i
$333$	− 50.7242i	− 2.77967i
$334$	−23.7248	−1.29817
$335$	0	0
$336$	−44.3191	−2.41780
$337$	− 13.6660i	− 0.744436i	−0.928145	−	0.372218i	$-0.878597\pi$
$337$	− 13.6660i	− 0.744436i	0.928145	−	0.372218i	$-0.121403\pi$
$338$	58.0758i	3.15890i
$339$	−7.97768	−0.433288
$340$	0	0
$341$	−1.71539	−0.0928933
$342$	− 56.3855i	− 3.04898i
$343$	− 15.3280i	− 0.827635i
$344$	−10.4488	−0.563359
$345$	0	0
$346$	−38.3678	−2.06266
$347$	17.7181i	0.951157i	0.879673	+	0.475579i	$0.157761\pi$
$347$	17.7181i	0.951157i	−0.879673	+	0.475579i	$0.842239\pi$
$348$	− 14.1595i	− 0.759031i
$349$	−20.0529	−1.07341	−0.536704	−	0.843770i	$-0.680331\pi$
$349$	−20.0529	−1.07341	−0.536704	+	0.843770i	$0.680331\pi$
$350$	0	0
$351$	−70.6672	−3.77194
$352$	− 6.12526i	− 0.326477i
$353$	− 2.40508i	− 0.128010i	−0.997950	−	0.0640048i	$-0.979613\pi$
$353$	− 2.40508i	− 0.128010i	0.997950	−	0.0640048i	$-0.0203873\pi$
$354$	−66.1426	−3.51544
$355$	0	0
$356$	−7.52911	−0.399042
$357$	− 25.1339i	− 1.33022i
$358$	− 14.4515i	− 0.763784i
$359$	−10.9843	−0.579731	−0.289865	−	0.957067i	$-0.593611\pi$
$359$	−10.9843	−0.579731	−0.289865	+	0.957067i	$0.593611\pi$
$360$	0	0
$361$	10.9865	0.578239
$362$	− 21.3604i	− 1.12268i
$363$	− 24.5972i	− 1.29102i
$364$	13.5903	0.712326
$365$	0	0
$366$	−0.524592	−0.0274209
$367$	19.2196i	1.00325i	0.865084	+	0.501627i	$0.167265\pi$
$367$	19.2196i	1.00325i	−0.865084	+	0.501627i	$0.832735\pi$
$368$	0.623700i	0.0325126i
$369$	−28.4469	−1.48088
$370$	0	0
$371$	−15.4949	−0.804454
$372$	− 2.00000i	− 0.103695i
$373$	− 20.9079i	− 1.08257i	−0.840839	−	0.541286i	$-0.817938\pi$
$373$	− 20.9079i	− 1.08257i	0.840839	−	0.541286i	$-0.182062\pi$
$374$	7.73732	0.400087
$375$	0	0
$376$	4.61387	0.237942
$377$	− 49.3765i	− 2.54302i
$378$	49.1075i	2.52582i
$379$	27.2996	1.40228	0.701142	−	0.713022i	$-0.252674\pi$
$379$	27.2996	1.40228	0.701142	+	0.713022i	$0.252674\pi$
$380$	0	0
$381$	20.3537	1.04275
$382$	− 13.9040i	− 0.711390i
$383$	25.5883i	1.30750i	0.756710	+	0.653751i	$0.226805\pi$
$383$	25.5883i	1.30750i	−0.756710	+	0.653751i	$0.773195\pi$
$384$	−41.4185	−2.11363
$385$	0	0
$386$	−26.3314	−1.34023
$387$	30.1305i	1.53162i
$388$	− 4.48551i	− 0.227718i
$389$	6.23267	0.316009	0.158004	−	0.987438i	$-0.449494\pi$
$389$	6.23267	0.316009	0.158004	+	0.987438i	$0.449494\pi$
$390$	0	0
$391$	−0.353707	−0.0178877
$392$	4.04646i	0.204377i
$393$	45.2831i	2.28423i
$394$	−1.67132	−0.0841999
$395$	0	0
$396$	−7.10171	−0.356874
$397$	− 8.29231i	− 0.416179i	−0.978110	−	0.208090i	$-0.933275\pi$
$397$	− 8.29231i	− 0.416179i	0.978110	−	0.208090i	$-0.0667246\pi$
$398$	43.4579i	2.17835i
$399$	−49.7208	−2.48915
$400$	0	0
$401$	29.8263	1.48945	0.744726	−	0.667370i	$-0.232580\pi$
$401$	29.8263	1.48945	0.744726	+	0.667370i	$0.232580\pi$
$402$	− 11.3644i	− 0.566804i
$403$	− 6.97431i	− 0.347415i
$404$	−11.9704	−0.595548
$405$	0	0
$406$	−34.3124	−1.70289
$407$	− 13.7695i	− 0.682527i
$408$	− 18.5179i	− 0.916770i
$409$	−7.18252	−0.355153	−0.177576	−	0.984107i	$-0.556826\pi$
$409$	−7.18252	−0.355153	−0.177576	+	0.984107i	$0.556826\pi$
$410$	0	0
$411$	18.6239	0.918649
$412$	− 0.789354i	− 0.0388887i
$413$	39.5490i	1.94608i
$414$	1.31572	0.0646641
$415$	0	0
$416$	24.9037	1.22100
$417$	− 0.344260i	− 0.0168585i
$418$	− 15.3063i	− 0.748654i
$419$	17.7997	0.869572	0.434786	−	0.900534i	$-0.356824\pi$
$419$	17.7997	0.869572	0.434786	+	0.900534i	$0.356824\pi$
$420$	0	0
$421$	−31.1941	−1.52031	−0.760153	−	0.649744i	$-0.774876\pi$
$421$	−31.1941	−1.52031	−0.760153	+	0.649744i	$0.774876\pi$
$422$	− 24.4519i	− 1.19030i
$423$	− 13.3047i	− 0.646899i
$424$	−11.4161	−0.554417
$425$	0	0
$426$	26.4592	1.28195
$427$	0.313671i	0.0151796i
$428$	3.79450i	0.183414i
$429$	−36.5217	−1.76329
$430$	0	0
$431$	30.7526	1.48130	0.740651	−	0.671890i	$-0.234517\pi$
$431$	30.7526	1.48130	0.740651	+	0.671890i	$0.234517\pi$
$432$	49.4575i	2.37953i
$433$	− 15.7991i	− 0.759256i	−0.925139	−	0.379628i	$-0.876052\pi$
$433$	− 15.7991i	− 0.759256i	0.925139	−	0.379628i	$-0.123948\pi$
$434$	−4.84653	−0.232641
$435$	0	0
$436$	−3.55395	−0.170203
$437$	0.699718i	0.0334720i
$438$	3.18523i	0.152196i
$439$	24.4503	1.16695	0.583475	−	0.812131i	$-0.301693\pi$
$439$	24.4503	1.16695	0.583475	+	0.812131i	$0.301693\pi$
$440$	0	0
$441$	11.6685	0.555645
$442$	31.4579i	1.49630i
$443$	− 5.28067i	− 0.250892i	−0.992100	−	0.125446i	$-0.959964\pi$
$443$	− 5.28067i	− 0.250892i	0.992100	−	0.125446i	$-0.0400362\pi$
$444$	16.0541	0.761893
$445$	0	0
$446$	13.6317	0.645478
$447$	24.3807i	1.15317i
$448$	11.7298i	0.554182i
$449$	−21.4308	−1.01138	−0.505690	−	0.862715i	$-0.668762\pi$
$449$	−21.4308	−1.01138	−0.505690	+	0.862715i	$0.668762\pi$
$450$	0	0
$451$	−7.72212	−0.363621
$452$	− 1.71210i	− 0.0805305i
$453$	− 63.6694i	− 2.99145i
$454$	23.9771	1.12530
$455$	0	0
$456$	−36.6328	−1.71549
$457$	− 25.1920i	− 1.17843i	−0.807976	−	0.589215i	$-0.799437\pi$
$457$	− 25.1920i	− 1.17843i	0.807976	−	0.589215i	$-0.200563\pi$
$458$	− 30.0895i	− 1.40599i
$459$	−28.0479	−1.30917
$460$	0	0
$461$	19.0975	0.889459	0.444729	−	0.895665i	$-0.353300\pi$
$461$	19.0975	0.889459	0.444729	+	0.895665i	$0.353300\pi$
$462$	25.3794i	1.18076i
$463$	− 28.8933i	− 1.34279i	−0.741102	−	0.671393i	$-0.765696\pi$
$463$	− 28.8933i	− 1.34279i	0.741102	−	0.671393i	$-0.234304\pi$
$464$	−34.5569	−1.60427
$465$	0	0
$466$	0.120082	0.00556271
$467$	− 9.34938i	− 0.432638i	−0.976323	−	0.216319i	$-0.930595\pi$
$467$	− 9.34938i	− 0.432638i	0.976323	−	0.216319i	$-0.0694051\pi$
$468$	− 28.8737i	− 1.33469i
$469$	−6.79516	−0.313771
$470$	0	0
$471$	−46.0345	−2.12116
$472$	29.1385i	1.34121i
$473$	8.17915i	0.376078i
$474$	−24.4882	−1.12478
$475$	0	0
$476$	5.39402	0.247234
$477$	32.9201i	1.50731i
$478$	32.7128i	1.49625i
$479$	−27.8309	−1.27162	−0.635812	−	0.771844i	$-0.719335\pi$
$479$	−27.8309	−1.27162	−0.635812	+	0.771844i	$0.719335\pi$
$480$	0	0
$481$	55.9831	2.55261
$482$	5.69036i	0.259189i
$483$	− 1.16020i	− 0.0527911i
$484$	5.27885	0.239948
$485$	0	0
$486$	10.0323	0.455076
$487$	3.08910i	0.139980i	0.997548	+	0.0699902i	$0.0222968\pi$
$487$	3.08910i	0.139980i	−0.997548	+	0.0699902i	$0.977703\pi$
$488$	0.231103i	0.0104616i
$489$	13.0575	0.590478
$490$	0	0
$491$	2.65062	0.119621	0.0598104	−	0.998210i	$-0.480950\pi$
$491$	2.65062	0.119621	0.0598104	+	0.998210i	$0.480950\pi$
$492$	− 9.00337i	− 0.405903i
$493$	− 19.5976i	− 0.882633i
$494$	62.2313	2.79992
$495$	0	0
$496$	−4.88108	−0.219167
$497$	− 15.8208i	− 0.709662i
$498$	− 9.30299i	− 0.416877i
$499$	2.46320	0.110268	0.0551339	−	0.998479i	$-0.482441\pi$
$499$	2.46320	0.110268	0.0551339	+	0.998479i	$0.482441\pi$
$500$	0	0
$501$	44.4475	1.98577
$502$	− 30.8816i	− 1.37831i
$503$	− 10.2846i	− 0.458568i	−0.973360	−	0.229284i	$-0.926361\pi$
$503$	− 10.2846i	− 0.458568i	0.973360	−	0.229284i	$-0.0736385\pi$
$504$	41.1874	1.83463
$505$	0	0
$506$	0.357163	0.0158778
$507$	− 108.802i	− 4.83209i
$508$	4.36814i	0.193805i
$509$	42.7097	1.89308	0.946538	−	0.322593i	$-0.104554\pi$
$509$	42.7097	1.89308	0.946538	+	0.322593i	$0.104554\pi$
$510$	0	0
$511$	1.90456	0.0842528
$512$	3.96337i	0.175158i
$513$	55.4855i	2.44975i
$514$	10.9724	0.483973
$515$	0	0
$516$	−9.53623	−0.419809
$517$	− 3.61168i	− 0.158841i
$518$	− 38.9033i	− 1.70931i
$519$	71.8803	3.15520
$520$	0	0
$521$	−31.6142	−1.38504	−0.692521	−	0.721398i	$-0.743500\pi$
$521$	−31.6142	−1.38504	−0.692521	+	0.721398i	$0.743500\pi$
$522$	72.8993i	3.19072i
$523$	− 33.7769i	− 1.47696i	−0.674275	−	0.738480i	$-0.735544\pi$
$523$	− 33.7769i	− 1.47696i	0.674275	−	0.738480i	$-0.264456\pi$
$524$	−9.71828	−0.424545
$525$	0	0
$526$	22.0535	0.961579
$527$	− 2.76812i	− 0.120581i
$528$	25.5603i	1.11237i
$529$	22.9837	0.999290
$530$	0	0
$531$	84.0249	3.64637
$532$	− 10.6707i	− 0.462632i
$533$	− 31.3962i	− 1.35992i
$534$	57.1656	2.47380
$535$	0	0
$536$	−5.00646	−0.216246
$537$	27.0742i	1.16834i
$538$	− 11.4770i	− 0.494807i
$539$	3.16752	0.136435
$540$	0	0
$541$	15.6350	0.672199	0.336100	−	0.941826i	$-0.390892\pi$
$541$	15.6350	0.672199	0.336100	+	0.941826i	$0.390892\pi$
$542$	− 19.2616i	− 0.827358i
$543$	40.0177i	1.71732i
$544$	9.88432	0.423787
$545$	0	0
$546$	−103.186	−4.41596
$547$	23.5289i	1.00602i	0.864279	+	0.503012i	$0.167775\pi$
$547$	23.5289i	1.00602i	−0.864279	+	0.503012i	$0.832225\pi$
$548$	3.99691i	0.170739i
$549$	0.666420	0.0284421
$550$	0	0
$551$	−38.7688	−1.65161
$552$	− 0.854803i	− 0.0363828i
$553$	14.6424i	0.622656i
$554$	−37.3717	−1.58777
$555$	0	0
$556$	0.0738822	0.00313331
$557$	− 6.90793i	− 0.292698i	−0.989233	−	0.146349i	$-0.953248\pi$
$557$	− 6.90793i	− 0.292698i	0.989233	−	0.146349i	$-0.0467523\pi$
$558$	10.2968i	0.435900i
$559$	−33.2543	−1.40651
$560$	0	0
$561$	−14.4955	−0.612002
$562$	23.5626i	0.993927i
$563$	17.0351i	0.717945i	0.933348	+	0.358973i	$0.116873\pi$
$563$	17.0351i	0.717945i	−0.933348	+	0.358973i	$0.883127\pi$
$564$	4.21092	0.177312
$565$	0	0
$566$	45.2967	1.90396
$567$	− 35.6153i	− 1.49570i
$568$	− 11.6563i	− 0.489088i
$569$	−36.6801	−1.53771	−0.768855	−	0.639423i	$-0.779173\pi$
$569$	−36.6801	−1.53771	−0.768855	+	0.639423i	$0.779173\pi$
$570$	0	0
$571$	7.24335	0.303125	0.151562	−	0.988448i	$-0.451570\pi$
$571$	7.24335	0.303125	0.151562	+	0.988448i	$0.451570\pi$
$572$	− 7.83799i	− 0.327723i
$573$	26.0485i	1.08819i
$574$	−21.8176	−0.910647
$575$	0	0
$576$	24.9209	1.03837
$577$	− 10.0981i	− 0.420391i	−0.977659	−	0.210196i	$-0.932590\pi$
$577$	− 10.0981i	− 0.420391i	0.977659	−	0.210196i	$-0.0674101\pi$
$578$	− 15.2152i	− 0.632868i
$579$	49.3307	2.05012
$580$	0	0
$581$	−5.56258	−0.230775
$582$	34.0568i	1.41170i
$583$	8.93641i	0.370108i
$584$	1.40322	0.0580657
$585$	0	0
$586$	4.82208	0.199198
$587$	− 21.9290i	− 0.905107i	−0.891737	−	0.452554i	$-0.850513\pi$
$587$	− 21.9290i	− 0.905107i	0.891737	−	0.452554i	$-0.149487\pi$
$588$	3.69307i	0.152299i
$589$	−5.47600	−0.225635
$590$	0	0
$591$	3.13115	0.128798
$592$	− 39.1806i	− 1.61031i
$593$	21.5212i	0.883771i	0.897072	+	0.441885i	$0.145690\pi$
$593$	21.5212i	0.883771i	−0.897072	+	0.441885i	$0.854310\pi$
$594$	28.3219	1.16206
$595$	0	0
$596$	−5.23239	−0.214327
$597$	− 81.4166i	− 3.33216i
$598$	1.45213i	0.0593820i
$599$	3.72270	0.152105	0.0760526	−	0.997104i	$-0.475768\pi$
$599$	3.72270	0.152105	0.0760526	+	0.997104i	$0.475768\pi$
$600$	0	0
$601$	−32.2142	−1.31404	−0.657022	−	0.753871i	$-0.728184\pi$
$601$	−32.2142	−1.31404	−0.657022	+	0.753871i	$0.728184\pi$
$602$	23.1088i	0.941846i
$603$	14.4368i	0.587914i
$604$	13.6642	0.555988
$605$	0	0
$606$	90.8864	3.69201
$607$	32.9324i	1.33668i	0.743854	+	0.668342i	$0.232996\pi$
$607$	32.9324i	1.33668i	−0.743854	+	0.668342i	$0.767004\pi$
$608$	− 19.5536i	− 0.793002i
$609$	64.2827	2.60487
$610$	0	0
$611$	14.6841	0.594057
$612$	− 11.4600i	− 0.463244i
$613$	− 19.8619i	− 0.802216i	−0.916031	−	0.401108i	$-0.868625\pi$
$613$	− 19.8619i	− 0.802216i	0.916031	−	0.401108i	$-0.131375\pi$
$614$	−17.2984	−0.698106
$615$	0	0
$616$	11.1806	0.450480
$617$	− 14.1251i	− 0.568654i	−0.958727	−	0.284327i	$-0.908230\pi$
$617$	− 14.1251i	− 0.568654i	0.958727	−	0.284327i	$-0.0917701\pi$
$618$	5.99326i	0.241084i
$619$	30.6478	1.23184	0.615919	−	0.787810i	$-0.288785\pi$
$619$	30.6478	1.23184	0.615919	+	0.787810i	$0.288785\pi$
$620$	0	0
$621$	−1.29472	−0.0519554
$622$	− 25.5139i	− 1.02301i
$623$	− 34.1813i	− 1.36944i
$624$	−103.922	−4.16019
$625$	0	0
$626$	36.3360	1.45228
$627$	28.6756i	1.14520i
$628$	− 9.87953i	− 0.394236i
$629$	22.2198	0.885961
$630$	0	0
$631$	−4.67198	−0.185989	−0.0929943	−	0.995667i	$-0.529644\pi$
$631$	−4.67198	−0.185989	−0.0929943	+	0.995667i	$0.529644\pi$
$632$	10.7880i	0.429125i
$633$	45.8095i	1.82076i
$634$	12.7911	0.508001
$635$	0	0
$636$	−10.4191	−0.413145
$637$	12.8783i	0.510257i
$638$	19.7891i	0.783457i
$639$	−33.6126	−1.32970
$640$	0	0
$641$	−0.427402	−0.0168813	−0.00844067	−	0.999964i	$-0.502687\pi$
$641$	−0.427402	−0.0168813	−0.00844067	+	0.999964i	$0.502687\pi$
$642$	− 28.8102i	− 1.13705i
$643$	37.4755i	1.47789i	0.673765	+	0.738945i	$0.264676\pi$
$643$	37.4755i	1.47789i	−0.673765	+	0.738945i	$0.735324\pi$
$644$	0.248993	0.00981171
$645$	0	0
$646$	24.6997	0.971797
$647$	− 7.60791i	− 0.299098i	−0.988754	−	0.149549i	$-0.952218\pi$
$647$	− 7.60791i	− 0.299098i	0.988754	−	0.149549i	$-0.0477821\pi$
$648$	− 26.2403i	− 1.03082i
$649$	22.8092	0.895340
$650$	0	0
$651$	9.07977	0.355864
$652$	2.80228i	0.109746i
$653$	9.97308i	0.390277i	0.980776	+	0.195138i	$0.0625156\pi$
$653$	9.97308i	0.390277i	−0.980776	+	0.195138i	$0.937484\pi$
$654$	26.9838	1.05515
$655$	0	0
$656$	−21.9731	−0.857905
$657$	− 4.04639i	− 0.157865i
$658$	− 10.2042i	− 0.397800i
$659$	26.4130	1.02890	0.514452	−	0.857519i	$-0.327995\pi$
$659$	26.4130	1.02890	0.514452	+	0.857519i	$0.327995\pi$
$660$	0	0
$661$	25.4910	0.991485	0.495743	−	0.868470i	$-0.334896\pi$
$661$	25.4910	0.991485	0.495743	+	0.868470i	$0.334896\pi$
$662$	25.4521i	0.989225i
$663$	− 58.9351i	− 2.28885i
$664$	−4.09834	−0.159046
$665$	0	0
$666$	−82.6532	−3.20275
$667$	− 0.904647i	− 0.0350281i
$668$	9.53894i	0.369073i
$669$	−25.5384	−0.987370
$670$	0	0
$671$	0.180905	0.00698375
$672$	32.4218i	1.25070i
$673$	− 30.2988i	− 1.16793i	−0.811778	−	0.583966i	$-0.801500\pi$
$673$	− 30.2988i	− 1.16793i	0.811778	−	0.583966i	$-0.198500\pi$
$674$	−22.2683	−0.857742
$675$	0	0
$676$	23.3503	0.898087
$677$	16.1111i	0.619202i	0.950867	+	0.309601i	$0.100195\pi$
$677$	16.1111i	0.619202i	−0.950867	+	0.309601i	$0.899805\pi$
$678$	12.9993i	0.499237i
$679$	20.3637	0.781488
$680$	0	0
$681$	−44.9201	−1.72134
$682$	2.79516i	0.107032i
$683$	4.68912i	0.179424i	0.995968	+	0.0897122i	$0.0285947\pi$
$683$	4.68912i	0.179424i	−0.995968	+	0.0897122i	$0.971405\pi$
$684$	−22.6707	−0.866834
$685$	0	0
$686$	−24.9764	−0.953605
$687$	56.3714i	2.15070i
$688$	23.2735i	0.887296i
$689$	−36.3331	−1.38418
$690$	0	0
$691$	−19.1696	−0.729247	−0.364624	−	0.931155i	$-0.618802\pi$
$691$	−19.1696	−0.729247	−0.364624	+	0.931155i	$0.618802\pi$
$692$	15.4264i	0.586422i
$693$	− 32.2409i	− 1.22473i
$694$	28.8710	1.09593
$695$	0	0
$696$	47.3615	1.79523
$697$	− 12.4612i	− 0.472001i
$698$	32.6755i	1.23679i
$699$	−0.224969	−0.00850912
$700$	0	0
$701$	30.9764	1.16996	0.584982	−	0.811046i	$-0.301102\pi$
$701$	30.9764	1.16996	0.584982	+	0.811046i	$0.301102\pi$
$702$	115.150i	4.34604i
$703$	− 43.9561i	− 1.65783i
$704$	6.76498	0.254965
$705$	0	0
$706$	−3.91899	−0.147493
$707$	− 54.3441i	− 2.04382i
$708$	26.5937i	0.999452i
$709$	17.6781	0.663913	0.331957	−	0.943295i	$-0.392291\pi$
$709$	17.6781	0.663913	0.331957	+	0.943295i	$0.392291\pi$
$710$	0	0
$711$	31.1088	1.16667
$712$	− 25.1837i	− 0.943800i
$713$	− 0.127779i	− 0.00478536i
$714$	−40.9547	−1.53269
$715$	0	0
$716$	−5.81044	−0.217146
$717$	− 61.2861i	− 2.28877i
$718$	17.8986i	0.667969i
$719$	−0.0363678	−0.00135629	−0.000678145	−	1.00000i	$-0.500216\pi$
$719$	−0.0363678	−0.00135629	−0.000678145		1.00000i	$0.500216\pi$
$720$	0	0
$721$	3.58358	0.133459
$722$	− 17.9022i	− 0.666250i
$723$	− 10.6606i	− 0.396473i
$724$	−8.58827	−0.319180
$725$	0	0
$726$	−40.0803	−1.48752
$727$	49.9904i	1.85404i	0.375010	+	0.927021i	$0.377639\pi$
$727$	49.9904i	1.85404i	−0.375010	+	0.927021i	$0.622361\pi$
$728$	45.4575i	1.68477i
$729$	17.1278	0.634362
$730$	0	0
$731$	−13.1987	−0.488171
$732$	0.210920i	0.00779584i
$733$	17.0602i	0.630131i	0.949070	+	0.315066i	$0.102027\pi$
$733$	17.0602i	0.630131i	−0.949070	+	0.315066i	$0.897973\pi$
$734$	31.3176	1.15595
$735$	0	0
$736$	0.456270	0.0168184
$737$	3.91899i	0.144358i
$738$	46.3531i	1.70628i
$739$	11.6240	0.427595	0.213797	−	0.976878i	$-0.431417\pi$
$739$	11.6240	0.427595	0.213797	+	0.976878i	$0.431417\pi$
$740$	0	0
$741$	−116.588	−4.28296
$742$	25.2483i	0.926895i
$743$	− 40.8980i	− 1.50040i	−0.661209	−	0.750202i	$-0.729956\pi$
$743$	− 40.8980i	− 1.50040i	0.661209	−	0.750202i	$-0.270044\pi$
$744$	6.68970	0.245256
$745$	0	0
$746$	−34.0687	−1.24734
$747$	11.8181i	0.432403i
$748$	− 3.11091i	− 0.113746i
$749$	−17.2266	−0.629446
$750$	0	0
$751$	−9.73770	−0.355334	−0.177667	−	0.984091i	$-0.556855\pi$
$751$	−9.73770	−0.355334	−0.177667	+	0.984091i	$0.556855\pi$
$752$	− 10.2769i	− 0.374761i
$753$	57.8553i	2.10837i
$754$	−80.4573	−2.93008
$755$	0	0
$756$	19.7444	0.718098
$757$	15.7898i	0.573889i	0.957947	+	0.286945i	$0.0926396\pi$
$757$	15.7898i	0.573889i	−0.957947	+	0.286945i	$0.907360\pi$
$758$	− 44.4836i	− 1.61572i
$759$	−0.669129	−0.0242878
$760$	0	0
$761$	−50.7315	−1.83901	−0.919507	−	0.393073i	$-0.871412\pi$
$761$	−50.7315	−1.83901	−0.919507	+	0.393073i	$0.871412\pi$
$762$	− 33.1656i	− 1.20146i
$763$	− 16.1345i	− 0.584109i
$764$	−5.59032	−0.202251
$765$	0	0
$766$	41.6952	1.50651
$767$	92.7363i	3.34851i
$768$	43.4117i	1.56649i
$769$	3.91071	0.141024	0.0705119	−	0.997511i	$-0.477537\pi$
$769$	3.91071	0.141024	0.0705119	+	0.997511i	$0.477537\pi$
$770$	0	0
$771$	−20.5564	−0.740319
$772$	10.5869i	0.381033i
$773$	26.2213i	0.943116i	0.881835	+	0.471558i	$0.156308\pi$
$773$	26.2213i	0.943116i	−0.881835	+	0.471558i	$0.843692\pi$
$774$	49.0965	1.76474
$775$	0	0
$776$	15.0034	0.538590
$777$	72.8837i	2.61469i
$778$	− 10.1559i	− 0.364107i
$779$	−24.6512	−0.883221
$780$	0	0
$781$	−9.12441	−0.326497
$782$	0.576353i	0.0206103i
$783$	− 71.7358i	− 2.56363i
$784$	9.01307	0.321896
$785$	0	0
$786$	73.7872	2.63190
$787$	− 12.6377i	− 0.450487i	−0.974302	−	0.225244i	$-0.927682\pi$
$787$	− 12.6377i	− 0.450487i	0.974302	−	0.225244i	$-0.0723178\pi$
$788$	0.671981i	0.0239383i
$789$	−41.3163	−1.47090
$790$	0	0
$791$	7.77275	0.276367
$792$	− 23.7541i	− 0.844066i
$793$	0.735512i	0.0261188i
$794$	−13.5120	−0.479524
$795$	0	0
$796$	17.4729	0.619312
$797$	23.0461i	0.816335i	0.912907	+	0.408168i	$0.133832\pi$
$797$	23.0461i	0.816335i	−0.912907	+	0.408168i	$0.866168\pi$
$798$	81.0182i	2.86801i
$799$	5.82816	0.206185
$800$	0	0
$801$	−72.6208	−2.56593
$802$	− 48.6008i	− 1.71615i
$803$	− 1.09842i	− 0.0387625i
$804$	−4.56923	−0.161144
$805$	0	0
$806$	−11.3644	−0.400293
$807$	21.5016i	0.756893i
$808$	− 40.0391i	− 1.40857i
$809$	−19.1161	−0.672088	−0.336044	−	0.941846i	$-0.609089\pi$
$809$	−19.1161	−0.672088	−0.336044	+	0.941846i	$0.609089\pi$
$810$	0	0
$811$	−18.5288	−0.650636	−0.325318	−	0.945605i	$-0.605471\pi$
$811$	−18.5288	−0.650636	−0.325318	+	0.945605i	$0.605471\pi$
$812$	13.7958i	0.484138i
$813$	36.0859i	1.26559i
$814$	−22.4368	−0.786411
$815$	0	0
$816$	−41.2466	−1.44392
$817$	26.1102i	0.913480i
$818$	11.7037i	0.409209i
$819$	131.083	4.58042
$820$	0	0
$821$	−36.0522	−1.25823	−0.629115	−	0.777312i	$-0.716583\pi$
$821$	−36.0522	−1.25823	−0.629115	+	0.777312i	$0.716583\pi$
$822$	− 30.3470i	− 1.05847i
$823$	− 17.3204i	− 0.603753i	−0.953347	−	0.301876i	$-0.902387\pi$
$823$	− 17.3204i	− 0.603753i	0.953347	−	0.301876i	$-0.0976130\pi$
$824$	2.64027	0.0919782
$825$	0	0
$826$	64.4436	2.24228
$827$	− 23.1732i	− 0.805811i	−0.915242	−	0.402906i	$-0.868000\pi$
$827$	− 23.1732i	− 0.805811i	0.915242	−	0.402906i	$-0.132000\pi$
$828$	− 0.529006i	− 0.0183842i
$829$	−23.0938	−0.802082	−0.401041	−	0.916060i	$-0.631351\pi$
$829$	−23.0938	−0.802082	−0.401041	+	0.916060i	$0.631351\pi$
$830$	0	0
$831$	70.0143	2.42877
$832$	27.5047i	0.953552i
$833$	5.11142i	0.177100i
$834$	−0.560960	−0.0194244
$835$	0	0
$836$	−6.15413	−0.212845
$837$	− 10.1325i	− 0.350231i
$838$	− 29.0039i	− 1.00192i
$839$	43.7883	1.51174	0.755871	−	0.654721i	$-0.227214\pi$
$839$	43.7883	1.51174	0.755871	+	0.654721i	$0.227214\pi$
$840$	0	0
$841$	21.1232	0.728385
$842$	50.8296i	1.75170i
$843$	− 44.1435i	− 1.52038i
$844$	−9.83125	−0.338406
$845$	0	0
$846$	−21.6796	−0.745360
$847$	23.9654i	0.823460i
$848$	25.4283i	0.873212i
$849$	−84.8614	−2.91244
$850$	0	0
$851$	1.02569	0.0351601
$852$	− 10.6383i	− 0.364463i
$853$	− 30.5569i	− 1.04625i	−0.852256	−	0.523125i	$-0.824766\pi$
$853$	− 30.5569i	− 1.04625i	0.852256	−	0.523125i	$-0.175234\pi$
$854$	0.511116	0.0174900
$855$	0	0
$856$	−12.6920	−0.433804
$857$	54.5160i	1.86223i	0.364726	+	0.931115i	$0.381163\pi$
$857$	54.5160i	1.86223i	−0.364726	+	0.931115i	$0.618837\pi$
$858$	59.5108i	2.03167i
$859$	49.3771	1.68473	0.842363	−	0.538911i	$-0.181164\pi$
$859$	49.3771	1.68473	0.842363	+	0.538911i	$0.181164\pi$
$860$	0	0
$861$	40.8743	1.39299
$862$	− 50.1103i	− 1.70676i
$863$	29.6647i	1.00980i	0.863178	+	0.504899i	$0.168470\pi$
$863$	29.6647i	1.00980i	−0.863178	+	0.504899i	$0.831530\pi$
$864$	36.1809	1.23090
$865$	0	0
$866$	−25.7441	−0.874819
$867$	28.5050i	0.968080i
$868$	1.94862i	0.0661406i
$869$	8.44473	0.286468
$870$	0	0
$871$	−15.9336	−0.539890
$872$	− 11.8874i	− 0.402559i
$873$	− 43.2643i	− 1.46428i
$874$	1.14016	0.0385666
$875$	0	0
$876$	1.28067	0.0432699
$877$	− 28.6701i	− 0.968120i	−0.875035	−	0.484060i	$-0.839162\pi$
$877$	− 28.6701i	− 0.968120i	0.875035	−	0.484060i	$-0.160838\pi$
$878$	− 39.8409i	− 1.34456i
$879$	−9.03396	−0.304708
$880$	0	0
$881$	26.8689	0.905235	0.452617	−	0.891705i	$-0.350490\pi$
$881$	26.8689	0.905235	0.452617	+	0.891705i	$0.350490\pi$
$882$	− 19.0135i	− 0.640217i
$883$	− 21.4596i	− 0.722172i	−0.932533	−	0.361086i	$-0.882406\pi$
$883$	− 21.4596i	− 0.722172i	0.932533	−	0.361086i	$-0.117594\pi$
$884$	12.6482	0.425403
$885$	0	0
$886$	−8.60466	−0.289079
$887$	− 7.94131i	− 0.266643i	−0.991073	−	0.133322i	$-0.957436\pi$
$887$	− 7.94131i	− 0.266643i	0.991073	−	0.133322i	$-0.0425643\pi$
$888$	53.6985i	1.80200i
$889$	−19.8309	−0.665106
$890$	0	0
$891$	−20.5406	−0.688135
$892$	− 5.48083i	− 0.183512i
$893$	− 11.5295i	− 0.385820i
$894$	39.7275	1.32869
$895$	0	0
$896$	40.3545	1.34815
$897$	− 2.72050i	− 0.0908350i
$898$	34.9207i	1.16532i
$899$	7.07977	0.236124
$900$	0	0
$901$	−14.4207	−0.480422
$902$	12.5829i	0.418965i
$903$	− 43.2934i	− 1.44071i
$904$	5.72673	0.190468
$905$	0	0
$906$	−103.747	−3.44676
$907$	− 45.4658i	− 1.50967i	−0.655916	−	0.754834i	$-0.727717\pi$
$907$	− 45.4658i	− 1.50967i	0.655916	−	0.754834i	$-0.272283\pi$
$908$	− 9.64037i	− 0.319927i
$909$	−115.458	−3.82951
$910$	0	0
$911$	−33.6098	−1.11354	−0.556771	−	0.830666i	$-0.687960\pi$
$911$	−33.6098	−1.11354	−0.556771	+	0.830666i	$0.687960\pi$
$912$	81.5957i	2.70190i
$913$	3.20812i	0.106173i
$914$	−41.0494	−1.35779
$915$	0	0
$916$	−12.0980	−0.399728
$917$	− 44.1199i	− 1.45697i
$918$	45.7031i	1.50843i
$919$	24.3699	0.803888	0.401944	−	0.915664i	$-0.368335\pi$
$919$	24.3699	0.803888	0.401944	+	0.915664i	$0.368335\pi$
$920$	0	0
$921$	32.4078	1.06787
$922$	− 31.1187i	− 1.02484i
$923$	− 37.0975i	− 1.22108i
$924$	10.2042	0.335693
$925$	0	0
$926$	−47.0806	−1.54716
$927$	− 7.61359i	− 0.250063i
$928$	25.2803i	0.829866i
$929$	9.10480	0.298719	0.149359	−	0.988783i	$-0.452279\pi$
$929$	9.10480	0.298719	0.149359	+	0.988783i	$0.452279\pi$
$930$	0	0
$931$	10.1116	0.331395
$932$	− 0.0482810i	− 0.00158150i
$933$	47.7992i	1.56488i
$934$	−15.2345	−0.498487
$935$	0	0
$936$	96.5781	3.15676
$937$	− 1.74387i	− 0.0569697i	−0.999594	−	0.0284849i	$-0.990932\pi$
$937$	− 1.74387i	− 0.0569697i	0.999594	−	0.0284849i	$-0.00906824\pi$
$938$	11.0725i	0.361529i
$939$	−68.0739	−2.22151
$940$	0	0
$941$	44.3041	1.44427	0.722136	−	0.691751i	$-0.243161\pi$
$941$	44.3041	1.44427	0.722136	+	0.691751i	$0.243161\pi$
$942$	75.0115i	2.44401i
$943$	− 0.575221i	− 0.0187318i
$944$	64.9029	2.11241
$945$	0	0
$946$	13.3276	0.433319
$947$	17.9601i	0.583624i	0.956476	+	0.291812i	$0.0942582\pi$
$947$	17.9601i	0.583624i	−0.956476	+	0.291812i	$0.905742\pi$
$948$	9.84587i	0.319779i
$949$	4.46591	0.144969
$950$	0	0
$951$	−23.9636	−0.777074
$952$	18.0422i	0.584750i
$953$	21.7642i	0.705011i	0.935810	+	0.352505i	$0.114670\pi$
$953$	21.7642i	0.705011i	−0.935810	+	0.352505i	$0.885330\pi$
$954$	53.6421	1.73673
$955$	0	0
$956$	13.1527	0.425389
$957$	− 37.0740i	− 1.19843i
$958$	45.3494i	1.46517i
$959$	−18.1455	−0.585949
$960$	0	0
$961$	1.00000	0.0322581
$962$	− 91.2224i	− 2.94113i
$963$	36.5992i	1.17939i
$964$	2.28790	0.0736882
$965$	0	0
$966$	−1.89051	−0.0608262
$967$	− 29.2152i	− 0.939499i	−0.882800	−	0.469750i	$-0.844344\pi$
$967$	− 29.2152i	− 0.939499i	0.882800	−	0.469750i	$-0.155656\pi$
$968$	17.6570i	0.567516i
$969$	−46.2739	−1.48653
$970$	0	0
$971$	−4.11489	−0.132053	−0.0660266	−	0.997818i	$-0.521032\pi$
$971$	−4.11489	−0.132053	−0.0660266	+	0.997818i	$0.521032\pi$
$972$	− 4.03366i	− 0.129380i
$973$	0.335417i	0.0107530i
$974$	5.03357	0.161286
$975$	0	0
$976$	0.514759	0.0164770
$977$	− 42.0412i	− 1.34502i	−0.740089	−	0.672509i	$-0.765217\pi$
$977$	− 42.0412i	− 1.34502i	0.740089	−	0.672509i	$-0.234783\pi$
$978$	− 21.2766i	− 0.680352i
$979$	−19.7135	−0.630046
$980$	0	0
$981$	−34.2791	−1.09445
$982$	− 4.31909i	− 0.137828i
$983$	− 44.2806i	− 1.41233i	−0.708047	−	0.706166i	$-0.750423\pi$
$983$	− 44.2806i	− 1.41233i	0.708047	−	0.706166i	$-0.249577\pi$
$984$	30.1149	0.960028
$985$	0	0
$986$	−31.9336	−1.01697
$987$	19.1171i	0.608504i
$988$	− 25.0211i	− 0.796027i
$989$	−0.609265	−0.0193735
$990$	0	0
$991$	8.37142	0.265927	0.132964	−	0.991121i	$-0.457551\pi$
$991$	8.37142	0.265927	0.132964	+	0.991121i	$0.457551\pi$
$992$	3.57078i	0.113372i
$993$	− 47.6835i	− 1.51319i
$994$	−25.7795	−0.817676
$995$	0	0
$996$	−3.74041	−0.118520
$997$	53.8984i	1.70698i	0.521109	+	0.853490i	$0.325518\pi$
$997$	53.8984i	1.70698i	−0.521109	+	0.853490i	$0.674482\pi$
$998$	− 4.01369i	− 0.127051i
$999$	81.3340	2.57329

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	775.2.b.e.249.3		8
5.2	odd	4		775.2.a.g.1.3		4
5.3	odd	4		155.2.a.d.1.2	✓	4
5.4	even	2	inner	775.2.b.e.249.6		8
15.2	even	4		6975.2.a.bj.1.2		4
15.8	even	4		1395.2.a.m.1.3		4
20.3	even	4		2480.2.a.z.1.4		4
35.13	even	4		7595.2.a.q.1.2		4
40.3	even	4		9920.2.a.cd.1.1		4
40.13	odd	4		9920.2.a.ch.1.4		4
155.123	even	4		4805.2.a.j.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
155.2.a.d.1.2	✓	4	5.3	odd	4
775.2.a.g.1.3		4	5.2	odd	4
775.2.b.e.249.3		8	1.1	even	1	trivial
775.2.b.e.249.6		8	5.4	even	2	inner
1395.2.a.m.1.3		4	15.8	even	4
2480.2.a.z.1.4		4	20.3	even	4
4805.2.a.j.1.2		4	155.123	even	4
6975.2.a.bj.1.2		4	15.2	even	4
7595.2.a.q.1.2		4	35.13	even	4
9920.2.a.cd.1.1		4	40.3	even	4
9920.2.a.ch.1.4		4	40.13	odd	4

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.62946i q^{2} +3.05273i q^{3} -0.655151 q^{4} +4.97431 q^{6} -2.97431i q^{7} -2.19138i q^{8} -6.31916 q^{9} +O(q^{10})\)	\(q-1.62946i q^{2} +3.05273i q^{3} -0.655151 q^{4} +4.97431 q^{6} -2.97431i q^{7} -2.19138i q^{8} -6.31916 q^{9} -1.71539 q^{11} -2.00000i q^{12} -6.97431i q^{13} -4.84653 q^{14} -4.88108 q^{16} -2.76812i q^{17} +10.2968i q^{18} -5.47600 q^{19} +9.07977 q^{21} +2.79516i q^{22} -0.127779i q^{23} +6.68970 q^{24} -11.3644 q^{26} -10.1325i q^{27} +1.94862i q^{28} +7.07977 q^{29} +1.00000 q^{31} +3.57078i q^{32} -5.23661i q^{33} -4.51054 q^{34} +4.14001 q^{36} +8.02704i q^{37} +8.92294i q^{38} +21.2907 q^{39} +4.50168 q^{41} -14.7952i q^{42} -4.76812i q^{43} +1.12384 q^{44} -0.208211 q^{46} +2.10546i q^{47} -14.9006i q^{48} -1.84653 q^{49} +8.45031 q^{51} +4.56923i q^{52} -5.20957i q^{53} -16.5105 q^{54} -6.51785 q^{56} -16.7167i q^{57} -11.5362i q^{58} -13.2968 q^{59} -0.105460 q^{61} -1.62946i q^{62} +18.7952i q^{63} -3.94371 q^{64} -8.53286 q^{66} -2.28461i q^{67} +1.81353i q^{68} +0.390075 q^{69} +5.31916 q^{71} +13.8477i q^{72} +0.640336i q^{73} +13.0798 q^{74} +3.58761 q^{76} +5.10209i q^{77} -34.6924i q^{78} -4.92294 q^{79} +11.9743 q^{81} -7.33533i q^{82} -1.87021i q^{83} -5.94862 q^{84} -7.76947 q^{86} +21.6126i q^{87} +3.75906i q^{88} +11.4922 q^{89} -20.7438 q^{91} +0.0837146i q^{92} +3.05273i q^{93} +3.43077 q^{94} -10.9006 q^{96} +6.84653i q^{97} +3.00886i q^{98} +10.8398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 18 q^{4} + 16 q^{6} - 14 q^{9}+O(q^{10}) \) 8 * q - 18 * q^4 + 16 * q^6 - 14 * q^9	\( 8 q - 18 q^{4} + 16 q^{6} - 14 q^{9} - 12 q^{11} - 16 q^{14} + 22 q^{16} - 10 q^{19} + 4 q^{21} + 28 q^{24} - 24 q^{26} - 12 q^{29} + 8 q^{31} + 36 q^{34} - 50 q^{36} + 12 q^{39} + 26 q^{41} - 40 q^{44} - 52 q^{46} + 8 q^{49} + 10 q^{51} - 60 q^{54} - 8 q^{56} - 26 q^{59} + 44 q^{61} - 94 q^{64} - 40 q^{66} - 40 q^{69} + 6 q^{71} + 36 q^{74} + 28 q^{76} + 32 q^{79} + 72 q^{81} + 32 q^{86} + 24 q^{89} - 48 q^{91} + 24 q^{94} + 28 q^{96} + 104 q^{99}+O(q^{100}) \) 8 * q - 18 * q^4 + 16 * q^6 - 14 * q^9 - 12 * q^11 - 16 * q^14 + 22 * q^16 - 10 * q^19 + 4 * q^21 + 28 * q^24 - 24 * q^26 - 12 * q^29 + 8 * q^31 + 36 * q^34 - 50 * q^36 + 12 * q^39 + 26 * q^41 - 40 * q^44 - 52 * q^46 + 8 * q^49 + 10 * q^51 - 60 * q^54 - 8 * q^56 - 26 * q^59 + 44 * q^61 - 94 * q^64 - 40 * q^66 - 40 * q^69 + 6 * q^71 + 36 * q^74 + 28 * q^76 + 32 * q^79 + 72 * q^81 + 32 * q^86 + 24 * q^89 - 48 * q^91 + 24 * q^94 + 28 * q^96 + 104 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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