LMFDB - Embedded newform 1875.2.b.h.1249.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 $ x^16 + 20x^14 + 156x^12 + 610x^10 + 1286x^8 + 1440x^6 + 761x^4 + 130*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 75)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.16
Root			$-1.53767i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.h.1249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53767i q^{2} +1.00000i q^{3} -4.43979 q^{4} -2.53767 q^{6} +1.04054i q^{7} -6.19138i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.53767i q^{2} +1.00000i q^{3} -4.43979 q^{4} -2.53767 q^{6} +1.04054i q^{7} -6.19138i q^{8} -1.00000 q^{9} +2.97101 q^{11} -4.43979i q^{12} -5.66922i q^{13} -2.64054 q^{14} +6.83213 q^{16} -5.08361i q^{17} -2.53767i q^{18} +5.37156 q^{19} -1.04054 q^{21} +7.53945i q^{22} -3.86039i q^{23} +6.19138 q^{24} +14.3866 q^{26} -1.00000i q^{27} -4.61976i q^{28} -0.679696 q^{29} +0.850111 q^{31} +4.95495i q^{32} +2.97101i q^{33} +12.9006 q^{34} +4.43979 q^{36} -1.61763i q^{37} +13.6313i q^{38} +5.66922 q^{39} +1.16529 q^{41} -2.64054i q^{42} -5.68601i q^{43} -13.1906 q^{44} +9.79640 q^{46} +3.28640i q^{47} +6.83213i q^{48} +5.91729 q^{49} +5.08361 q^{51} +25.1701i q^{52} -12.6861i q^{53} +2.53767 q^{54} +6.44235 q^{56} +5.37156i q^{57} -1.72485i q^{58} -3.21187 q^{59} -5.42093 q^{61} +2.15730i q^{62} -1.04054i q^{63} +1.09021 q^{64} -7.53945 q^{66} -0.929140i q^{67} +22.5702i q^{68} +3.86039 q^{69} -1.41358 q^{71} +6.19138i q^{72} -11.3234i q^{73} +4.10501 q^{74} -23.8486 q^{76} +3.09144i q^{77} +14.3866i q^{78} +1.44707 q^{79} +1.00000 q^{81} +2.95713i q^{82} +11.4756i q^{83} +4.61976 q^{84} +14.4292 q^{86} -0.679696i q^{87} -18.3946i q^{88} -9.07225 q^{89} +5.89903 q^{91} +17.1393i q^{92} +0.850111i q^{93} -8.33982 q^{94} -4.95495 q^{96} -6.02928i q^{97} +15.0161i q^{98} -2.97101 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	$ 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) $ 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.53767i	1.79441i	0.441618	+	0.897203i	$0.354405\pi$
$2$	2.53767i	1.79441i	−0.441618	+	0.897203i	$0.645595\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−4.43979	−2.21989
$5$	0	0
$5$	0	0
$6$	−2.53767	−1.03600
$7$	1.04054i	0.393285i	0.980475	+	0.196643i	$0.0630039\pi$
$7$	1.04054i	0.393285i	−0.980475	+	0.196643i	$0.936996\pi$
$8$	− 6.19138i	− 2.18898i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.97101	0.895792	0.447896	−	0.894086i	$-0.352173\pi$
$11$	2.97101	0.895792	0.447896	+	0.894086i	$0.352173\pi$
$12$	− 4.43979i	− 1.28166i
$13$	− 5.66922i	− 1.57236i	−0.617998	−	0.786180i	$-0.712056\pi$
$13$	− 5.66922i	− 1.57236i	0.617998	−	0.786180i	$-0.287944\pi$
$14$	−2.64054	−0.705714
$15$	0	0
$16$	6.83213	1.70803
$17$	− 5.08361i	− 1.23296i	−0.787372	−	0.616479i	$-0.788559\pi$
$17$	− 5.08361i	− 1.23296i	0.787372	−	0.616479i	$-0.211441\pi$
$18$	− 2.53767i	− 0.598135i
$19$	5.37156	1.23232	0.616160	−	0.787621i	$-0.288687\pi$
$19$	5.37156	1.23232	0.616160	+	0.787621i	$0.288687\pi$
$20$	0	0
$21$	−1.04054	−0.227063
$22$	7.53945i	1.60742i
$23$	− 3.86039i	− 0.804946i	−0.915432	−	0.402473i	$-0.868151\pi$
$23$	− 3.86039i	− 0.804946i	0.915432	−	0.402473i	$-0.131849\pi$
$24$	6.19138	1.26381
$25$	0	0
$26$	14.3866	2.82145
$27$	− 1.00000i	− 0.192450i
$28$	− 4.61976i	− 0.873052i
$29$	−0.679696	−0.126216	−0.0631082	−	0.998007i	$-0.520101\pi$
$29$	−0.679696	−0.126216	−0.0631082	+	0.998007i	$0.520101\pi$
$30$	0	0
$31$	0.850111	0.152684	0.0763422	−	0.997082i	$-0.475676\pi$
$31$	0.850111	0.152684	0.0763422	+	0.997082i	$0.475676\pi$
$32$	4.95495i	0.875921i
$33$	2.97101i	0.517186i
$34$	12.9006	2.21243
$35$	0	0
$36$	4.43979	0.739964
$37$	− 1.61763i	− 0.265936i	−0.991120	−	0.132968i	$-0.957549\pi$
$37$	− 1.61763i	− 0.265936i	0.991120	−	0.132968i	$-0.0424508\pi$
$38$	13.6313i	2.21128i
$39$	5.66922	0.907802
$40$	0	0
$41$	1.16529	0.181988	0.0909939	−	0.995851i	$-0.470996\pi$
$41$	1.16529	0.181988	0.0909939	+	0.995851i	$0.470996\pi$
$42$	− 2.64054i	− 0.407444i
$43$	− 5.68601i	− 0.867109i	−0.901127	−	0.433554i	$-0.857259\pi$
$43$	− 5.68601i	− 0.867109i	0.901127	−	0.433554i	$-0.142741\pi$
$44$	−13.1906	−1.98856
$45$	0	0
$46$	9.79640	1.44440
$47$	3.28640i	0.479371i	0.970851	+	0.239686i	$0.0770444\pi$
$47$	3.28640i	0.479371i	−0.970851	+	0.239686i	$0.922956\pi$
$48$	6.83213i	0.986133i
$49$	5.91729	0.845327
$50$	0	0
$51$	5.08361	0.711848
$52$	25.1701i	3.49047i
$53$	− 12.6861i	− 1.74257i	−0.490773	−	0.871287i	$-0.663286\pi$
$53$	− 12.6861i	− 1.74257i	0.490773	−	0.871287i	$-0.336714\pi$
$54$	2.53767	0.345334
$55$	0	0
$56$	6.44235	0.860896
$57$	5.37156i	0.711481i
$58$	− 1.72485i	− 0.226484i
$59$	−3.21187	−0.418150	−0.209075	−	0.977900i	$-0.567045\pi$
$59$	−3.21187	−0.418150	−0.209075	+	0.977900i	$0.567045\pi$
$60$	0	0
$61$	−5.42093	−0.694079	−0.347039	−	0.937851i	$-0.612813\pi$
$61$	−5.42093	−0.694079	−0.347039	+	0.937851i	$0.612813\pi$
$62$	2.15730i	0.273978i
$63$	− 1.04054i	− 0.131095i
$64$	1.09021	0.136276
$65$	0	0
$66$	−7.53945	−0.928042
$67$	− 0.929140i	− 0.113513i	−0.998388	−	0.0567563i	$-0.981924\pi$
$67$	− 0.929140i	− 0.113513i	0.998388	−	0.0567563i	$-0.0180758\pi$
$68$	22.5702i	2.73703i
$69$	3.86039	0.464736
$70$	0	0
$71$	−1.41358	−0.167761	−0.0838807	−	0.996476i	$-0.526731\pi$
$71$	−1.41358	−0.167761	−0.0838807	+	0.996476i	$0.526731\pi$
$72$	6.19138i	0.729661i
$73$	− 11.3234i	− 1.32530i	−0.748929	−	0.662650i	$-0.769432\pi$
$73$	− 11.3234i	− 1.32530i	0.748929	−	0.662650i	$-0.230568\pi$
$74$	4.10501	0.477198
$75$	0	0
$76$	−23.8486	−2.73562
$77$	3.09144i	0.352302i
$78$	14.3866i	1.62897i
$79$	1.44707	0.162809	0.0814043	−	0.996681i	$-0.474059\pi$
$79$	1.44707	0.162809	0.0814043	+	0.996681i	$0.474059\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.95713i	0.326560i
$83$	11.4756i	1.25961i	0.776752	+	0.629806i	$0.216866\pi$
$83$	11.4756i	1.25961i	−0.776752	+	0.629806i	$0.783134\pi$
$84$	4.61976	0.504057
$85$	0	0
$86$	14.4292	1.55594
$87$	− 0.679696i	− 0.0728711i
$88$	− 18.3946i	− 1.96087i
$89$	−9.07225	−0.961657	−0.480828	−	0.876815i	$-0.659664\pi$
$89$	−9.07225	−0.961657	−0.480828	+	0.876815i	$0.659664\pi$
$90$	0	0
$91$	5.89903	0.618386
$92$	17.1393i	1.78689i
$93$	0.850111i	0.0881524i
$94$	−8.33982	−0.860187
$95$	0	0
$96$	−4.95495	−0.505713
$97$	− 6.02928i	− 0.612181i	−0.952002	−	0.306091i	$-0.900979\pi$
$97$	− 6.02928i	− 0.612181i	0.952002	−	0.306091i	$-0.0990210\pi$
$98$	15.0161i	1.51686i
$99$	−2.97101	−0.298597
$100$	0	0
$101$	−15.3408	−1.52647	−0.763236	−	0.646120i	$-0.776390\pi$
$101$	−15.3408	−1.52647	−0.763236	+	0.646120i	$0.776390\pi$
$102$	12.9006i	1.27734i
$103$	12.0590i	1.18820i	0.804390	+	0.594102i	$0.202493\pi$
$103$	12.0590i	1.18820i	−0.804390	+	0.594102i	$0.797507\pi$
$104$	−35.1003	−3.44187
$105$	0	0
$106$	32.1933	3.12689
$107$	6.49787i	0.628173i	0.949394	+	0.314086i	$0.101698\pi$
$107$	6.49787i	0.628173i	−0.949394	+	0.314086i	$0.898302\pi$
$108$	4.43979i	0.427219i
$109$	2.31057	0.221313	0.110656	−	0.993859i	$-0.464705\pi$
$109$	2.31057	0.221313	0.110656	+	0.993859i	$0.464705\pi$
$110$	0	0
$111$	1.61763	0.153538
$112$	7.10907i	0.671744i
$113$	3.87281i	0.364323i	0.983269	+	0.182162i	$0.0583094\pi$
$113$	3.87281i	0.364323i	−0.983269	+	0.182162i	$0.941691\pi$
$114$	−13.6313	−1.27669
$115$	0	0
$116$	3.01771	0.280187
$117$	5.66922i	0.524120i
$118$	− 8.15067i	− 0.750330i
$119$	5.28968	0.484904
$120$	0	0
$121$	−2.17312	−0.197556
$122$	− 13.7565i	− 1.24546i
$123$	1.16529i	0.105071i
$124$	−3.77431	−0.338943
$125$	0	0
$126$	2.64054	0.235238
$127$	− 11.6938i	− 1.03765i	−0.854879	−	0.518827i	$-0.826369\pi$
$127$	− 11.6938i	− 1.03765i	0.854879	−	0.518827i	$-0.173631\pi$
$128$	12.6765i	1.12045i
$129$	5.68601	0.500625
$130$	0	0
$131$	−7.96210	−0.695652	−0.347826	−	0.937559i	$-0.613080\pi$
$131$	−7.96210	−0.695652	−0.347826	+	0.937559i	$0.613080\pi$
$132$	− 13.1906i	− 1.14810i
$133$	5.58930i	0.484654i
$134$	2.35785	0.203688
$135$	0	0
$136$	−31.4746	−2.69892
$137$	11.0513i	0.944176i	0.881551	+	0.472088i	$0.156500\pi$
$137$	11.0513i	0.944176i	−0.881551	+	0.472088i	$0.843500\pi$
$138$	9.79640i	0.833925i
$139$	12.2698	1.04071	0.520357	−	0.853949i	$-0.325799\pi$
$139$	12.2698	1.04071	0.520357	+	0.853949i	$0.325799\pi$
$140$	0	0
$141$	−3.28640	−0.276765
$142$	− 3.58721i	− 0.301032i
$143$	− 16.8433i	− 1.40851i
$144$	−6.83213	−0.569344
$145$	0	0
$146$	28.7350	2.37813
$147$	5.91729i	0.488050i
$148$	7.18192i	0.590350i
$149$	4.62832	0.379167	0.189584	−	0.981865i	$-0.439286\pi$
$149$	4.62832	0.379167	0.189584	+	0.981865i	$0.439286\pi$
$150$	0	0
$151$	−4.67249	−0.380242	−0.190121	−	0.981761i	$-0.560888\pi$
$151$	−4.67249	−0.380242	−0.190121	+	0.981761i	$0.560888\pi$
$152$	− 33.2574i	− 2.69753i
$153$	5.08361i	0.410986i
$154$	−7.84506	−0.632173
$155$	0	0
$156$	−25.1701	−2.01522
$157$	− 14.9726i	− 1.19494i	−0.801890	−	0.597472i	$-0.796172\pi$
$157$	− 14.9726i	− 1.19494i	0.801890	−	0.597472i	$-0.203828\pi$
$158$	3.67220i	0.292145i
$159$	12.6861	1.00608
$160$	0	0
$161$	4.01687	0.316574
$162$	2.53767i	0.199378i
$163$	11.9112i	0.932958i	0.884532	+	0.466479i	$0.154478\pi$
$163$	11.9112i	0.932958i	−0.884532	+	0.466479i	$0.845522\pi$
$164$	−5.17364	−0.403994
$165$	0	0
$166$	−29.1214	−2.26026
$167$	10.6081i	0.820881i	0.911887	+	0.410440i	$0.134625\pi$
$167$	10.6081i	0.820881i	−0.911887	+	0.410440i	$0.865375\pi$
$168$	6.44235i	0.497038i
$169$	−19.1401	−1.47231
$170$	0	0
$171$	−5.37156	−0.410774
$172$	25.2447i	1.92489i
$173$	− 1.25338i	− 0.0952928i	−0.998864	−	0.0476464i	$-0.984828\pi$
$173$	− 1.25338i	− 0.0952928i	0.998864	−	0.0476464i	$-0.0151721\pi$
$174$	1.72485	0.130760
$175$	0	0
$176$	20.2983	1.53004
$177$	− 3.21187i	− 0.241419i
$178$	− 23.0224i	− 1.72560i
$179$	20.8113	1.55551	0.777755	−	0.628567i	$-0.216358\pi$
$179$	20.8113	1.55551	0.777755	+	0.628567i	$0.216358\pi$
$180$	0	0
$181$	−6.92706	−0.514884	−0.257442	−	0.966294i	$-0.582880\pi$
$181$	−6.92706	−0.514884	−0.257442	+	0.966294i	$0.582880\pi$
$182$	14.9698i	1.10964i
$183$	− 5.42093i	− 0.400726i
$184$	−23.9011	−1.76201
$185$	0	0
$186$	−2.15730	−0.158181
$187$	− 15.1034i	− 1.10447i
$188$	− 14.5909i	− 1.06415i
$189$	1.04054	0.0756878
$190$	0	0
$191$	8.16415	0.590737	0.295369	−	0.955383i	$-0.404558\pi$
$191$	8.16415	0.590737	0.295369	+	0.955383i	$0.404558\pi$
$192$	1.09021i	0.0786788i
$193$	− 13.9629i	− 1.00507i	−0.864556	−	0.502537i	$-0.832400\pi$
$193$	− 13.9629i	− 1.00507i	0.864556	−	0.502537i	$-0.167600\pi$
$194$	15.3004	1.09850
$195$	0	0
$196$	−26.2715	−1.87653
$197$	− 5.76250i	− 0.410561i	−0.978703	−	0.205281i	$-0.934189\pi$
$197$	− 5.76250i	− 0.410561i	0.978703	−	0.205281i	$-0.0658107\pi$
$198$	− 7.53945i	− 0.535805i
$199$	26.5748	1.88384	0.941919	−	0.335841i	$-0.109020\pi$
$199$	26.5748	1.88384	0.941919	+	0.335841i	$0.109020\pi$
$200$	0	0
$201$	0.929140	0.0655365
$202$	− 38.9301i	− 2.73911i
$203$	− 0.707248i	− 0.0496391i
$204$	−22.5702	−1.58023
$205$	0	0
$206$	−30.6017	−2.13212
$207$	3.86039i	0.268315i
$208$	− 38.7329i	− 2.68564i
$209$	15.9589	1.10390
$210$	0	0
$211$	−26.4594	−1.82154	−0.910771	−	0.412912i	$-0.864512\pi$
$211$	−26.4594	−1.82154	−0.910771	+	0.412912i	$0.864512\pi$
$212$	56.3237i	3.86833i
$213$	− 1.41358i	− 0.0968571i
$214$	−16.4895	−1.12720
$215$	0	0
$216$	−6.19138	−0.421270
$217$	0.884570i	0.0600486i
$218$	5.86348i	0.397125i
$219$	11.3234	0.765163
$220$	0	0
$221$	−28.8201	−1.93865
$222$	4.10501i	0.275510i
$223$	− 27.4456i	− 1.83789i	−0.394383	−	0.918946i	$-0.629042\pi$
$223$	− 27.4456i	− 1.83789i	0.394383	−	0.918946i	$-0.370958\pi$
$224$	−5.15581	−0.344487
$225$	0	0
$226$	−9.82792	−0.653744
$227$	− 0.130161i	− 0.00863907i	−0.999991	−	0.00431954i	$-0.998625\pi$
$227$	− 0.130161i	− 0.00863907i	0.999991	−	0.00431954i	$-0.00137496\pi$
$228$	− 23.8486i	− 1.57941i
$229$	−2.82530	−0.186701	−0.0933504	−	0.995633i	$-0.529758\pi$
$229$	−2.82530	−0.186701	−0.0933504	+	0.995633i	$0.529758\pi$
$230$	0	0
$231$	−3.09144	−0.203402
$232$	4.20826i	0.276286i
$233$	7.98709i	0.523252i	0.965169	+	0.261626i	$0.0842586\pi$
$233$	7.98709i	0.523252i	−0.965169	+	0.261626i	$0.915741\pi$
$234$	−14.3866	−0.940484
$235$	0	0
$236$	14.2600	0.928247
$237$	1.44707i	0.0939976i
$238$	13.4235i	0.870115i
$239$	19.0619	1.23301	0.616506	−	0.787350i	$-0.288548\pi$
$239$	19.0619	1.23301	0.616506	+	0.787350i	$0.288548\pi$
$240$	0	0
$241$	21.1199	1.36045	0.680226	−	0.733002i	$-0.261882\pi$
$241$	21.1199	1.36045	0.680226	+	0.733002i	$0.261882\pi$
$242$	− 5.51466i	− 0.354496i
$243$	1.00000i	0.0641500i
$244$	24.0678	1.54078
$245$	0	0
$246$	−2.95713	−0.188540
$247$	− 30.4526i	− 1.93765i
$248$	− 5.26336i	− 0.334224i
$249$	−11.4756	−0.727238
$250$	0	0
$251$	30.2224	1.90762	0.953811	−	0.300408i	$-0.0971228\pi$
$251$	30.2224	1.90762	0.953811	+	0.300408i	$0.0971228\pi$
$252$	4.61976i	0.291017i
$253$	− 11.4692i	− 0.721065i
$254$	29.6750	1.86197
$255$	0	0
$256$	−29.9884	−1.87427
$257$	− 5.10215i	− 0.318263i	−0.987257	−	0.159132i	$-0.949131\pi$
$257$	− 5.10215i	− 0.318263i	0.987257	−	0.159132i	$-0.0508695\pi$
$258$	14.4292i	0.898325i
$259$	1.68320	0.104589
$260$	0	0
$261$	0.679696	0.0420721
$262$	− 20.2052i	− 1.24828i
$263$	6.41540i	0.395591i	0.980243	+	0.197795i	$0.0633782\pi$
$263$	6.41540i	0.395591i	−0.980243	+	0.197795i	$0.936622\pi$
$264$	18.3946	1.13211
$265$	0	0
$266$	−14.1838	−0.869666
$267$	− 9.07225i	− 0.555213i
$268$	4.12518i	0.251986i
$269$	17.4592	1.06450	0.532252	−	0.846586i	$-0.321346\pi$
$269$	17.4592	1.06450	0.532252	+	0.846586i	$0.321346\pi$
$270$	0	0
$271$	−0.0951857	−0.00578212	−0.00289106	−	0.999996i	$-0.500920\pi$
$271$	−0.0951857	−0.00578212	−0.00289106	+	0.999996i	$0.500920\pi$
$272$	− 34.7319i	− 2.10593i
$273$	5.89903i	0.357025i
$274$	−28.0446	−1.69424
$275$	0	0
$276$	−17.1393	−1.03166
$277$	18.4007i	1.10559i	0.833316	+	0.552796i	$0.186439\pi$
$277$	18.4007i	1.10559i	−0.833316	+	0.552796i	$0.813561\pi$
$278$	31.1369i	1.86747i
$279$	−0.850111	−0.0508948
$280$	0	0
$281$	17.9361	1.06998	0.534988	−	0.844860i	$-0.320316\pi$
$281$	17.9361	1.06998	0.534988	+	0.844860i	$0.320316\pi$
$282$	− 8.33982i	− 0.496629i
$283$	− 22.2399i	− 1.32203i	−0.750374	−	0.661014i	$-0.770126\pi$
$283$	− 22.2399i	− 1.32203i	0.750374	−	0.661014i	$-0.229874\pi$
$284$	6.27601	0.372413
$285$	0	0
$286$	42.7428	2.52743
$287$	1.21253i	0.0715732i
$288$	− 4.95495i	− 0.291974i
$289$	−8.84312	−0.520184
$290$	0	0
$291$	6.02928	0.353443
$292$	50.2734i	2.94203i
$293$	− 18.7316i	− 1.09431i	−0.837031	−	0.547155i	$-0.815711\pi$
$293$	− 18.7316i	− 1.09431i	0.837031	−	0.547155i	$-0.184289\pi$
$294$	−15.0161	−0.875759
$295$	0	0
$296$	−10.0154	−0.582130
$297$	− 2.97101i	− 0.172395i
$298$	11.7452i	0.680380i
$299$	−21.8854	−1.26566
$300$	0	0
$301$	5.91650	0.341021
$302$	− 11.8573i	− 0.682309i
$303$	− 15.3408i	− 0.881309i
$304$	36.6992	2.10484
$305$	0	0
$306$	−12.9006	−0.737475
$307$	7.03850i	0.401708i	0.979621	+	0.200854i	$0.0643717\pi$
$307$	7.03850i	0.401708i	−0.979621	+	0.200854i	$0.935628\pi$
$308$	− 13.7253i	− 0.782073i
$309$	−12.0590	−0.686010
$310$	0	0
$311$	−29.2790	−1.66026	−0.830130	−	0.557570i	$-0.811734\pi$
$311$	−29.2790	−1.66026	−0.830130	+	0.557570i	$0.811734\pi$
$312$	− 35.1003i	− 1.98716i
$313$	− 15.6354i	− 0.883764i	−0.897073	−	0.441882i	$-0.854311\pi$
$313$	− 15.6354i	− 0.883764i	0.897073	−	0.441882i	$-0.145689\pi$
$314$	37.9956	2.14422
$315$	0	0
$316$	−6.42470	−0.361418
$317$	6.24144i	0.350554i	0.984519	+	0.175277i	$0.0560821\pi$
$317$	6.24144i	0.350554i	−0.984519	+	0.175277i	$0.943918\pi$
$318$	32.1933i	1.80531i
$319$	−2.01938	−0.113064
$320$	0	0
$321$	−6.49787	−0.362676
$322$	10.1935i	0.568062i
$323$	− 27.3069i	− 1.51940i
$324$	−4.43979	−0.246655
$325$	0	0
$326$	−30.2268	−1.67411
$327$	2.31057i	0.127775i
$328$	− 7.21476i	− 0.398369i
$329$	−3.41962	−0.188530
$330$	0	0
$331$	21.4575	1.17941	0.589705	−	0.807619i	$-0.299244\pi$
$331$	21.4575	1.17941	0.589705	+	0.807619i	$0.299244\pi$
$332$	− 50.9493i	− 2.79621i
$333$	1.61763i	0.0886455i
$334$	−26.9199	−1.47299
$335$	0	0
$336$	−7.10907	−0.387832
$337$	34.7511i	1.89301i	0.322683	+	0.946507i	$0.395415\pi$
$337$	34.7511i	1.89301i	−0.322683	+	0.946507i	$0.604585\pi$
$338$	− 48.5712i	− 2.64193i
$339$	−3.87281	−0.210342
$340$	0	0
$341$	2.52569	0.136774
$342$	− 13.6313i	− 0.737094i
$343$	13.4409i	0.725740i
$344$	−35.2043	−1.89809
$345$	0	0
$346$	3.18067	0.170994
$347$	28.7241i	1.54199i	0.636841	+	0.770995i	$0.280241\pi$
$347$	28.7241i	1.54199i	−0.636841	+	0.770995i	$0.719759\pi$
$348$	3.01771i	0.161766i
$349$	12.2834	0.657515	0.328758	−	0.944414i	$-0.393370\pi$
$349$	12.2834	0.657515	0.328758	+	0.944414i	$0.393370\pi$
$350$	0	0
$351$	−5.66922	−0.302601
$352$	14.7212i	0.784643i
$353$	26.9779i	1.43589i	0.696100	+	0.717945i	$0.254917\pi$
$353$	26.9779i	1.43589i	−0.696100	+	0.717945i	$0.745083\pi$
$354$	8.15067	0.433203
$355$	0	0
$356$	40.2789	2.13478
$357$	5.28968i	0.279960i
$358$	52.8123i	2.79122i
$359$	−16.4243	−0.866839	−0.433419	−	0.901192i	$-0.642693\pi$
$359$	−16.4243	−0.866839	−0.433419	+	0.901192i	$0.642693\pi$
$360$	0	0
$361$	9.85366	0.518614
$362$	− 17.5786i	− 0.923912i
$363$	− 2.17312i	− 0.114059i
$364$	−26.1904	−1.37275
$365$	0	0
$366$	13.7565	0.719066
$367$	− 29.8953i	− 1.56052i	−0.625453	−	0.780262i	$-0.715086\pi$
$367$	− 29.8953i	− 1.56052i	0.625453	−	0.780262i	$-0.284914\pi$
$368$	− 26.3747i	− 1.37487i
$369$	−1.16529	−0.0606626
$370$	0	0
$371$	13.2004	0.685329
$372$	− 3.77431i	− 0.195689i
$373$	24.8307i	1.28568i	0.765999	+	0.642842i	$0.222245\pi$
$373$	24.8307i	1.28568i	−0.765999	+	0.642842i	$0.777755\pi$
$374$	38.3276	1.98187
$375$	0	0
$376$	20.3474	1.04934
$377$	3.85335i	0.198458i
$378$	2.64054i	0.135815i
$379$	−19.3966	−0.996338	−0.498169	−	0.867080i	$-0.665994\pi$
$379$	−19.3966	−0.996338	−0.498169	+	0.867080i	$0.665994\pi$
$380$	0	0
$381$	11.6938	0.599090
$382$	20.7179i	1.06002i
$383$	11.2508i	0.574891i	0.957797	+	0.287446i	$0.0928060\pi$
$383$	11.2508i	0.574891i	−0.957797	+	0.287446i	$0.907194\pi$
$384$	−12.6765	−0.646895
$385$	0	0
$386$	35.4334	1.80351
$387$	5.68601i	0.289036i
$388$	26.7687i	1.35898i
$389$	−14.9017	−0.755549	−0.377774	−	0.925898i	$-0.623310\pi$
$389$	−14.9017	−0.755549	−0.377774	+	0.925898i	$0.623310\pi$
$390$	0	0
$391$	−19.6247	−0.992464
$392$	− 36.6362i	− 1.85041i
$393$	− 7.96210i	− 0.401635i
$394$	14.6233	0.736713
$395$	0	0
$396$	13.1906	0.662854
$397$	24.0966i	1.20937i	0.796463	+	0.604687i	$0.206702\pi$
$397$	24.0966i	1.20937i	−0.796463	+	0.604687i	$0.793298\pi$
$398$	67.4382i	3.38037i
$399$	−5.58930	−0.279815
$400$	0	0
$401$	13.4580	0.672059	0.336030	−	0.941851i	$-0.390916\pi$
$401$	13.4580	0.672059	0.336030	+	0.941851i	$0.390916\pi$
$402$	2.35785i	0.117599i
$403$	− 4.81947i	− 0.240075i
$404$	68.1101	3.38860
$405$	0	0
$406$	1.79476	0.0890727
$407$	− 4.80598i	− 0.238224i
$408$	− 31.4746i	− 1.55822i
$409$	35.4737	1.75406	0.877030	−	0.480435i	$-0.159521\pi$
$409$	35.4737	1.75406	0.877030	+	0.480435i	$0.159521\pi$
$410$	0	0
$411$	−11.0513	−0.545120
$412$	− 53.5392i	− 2.63769i
$413$	− 3.34206i	− 0.164452i
$414$	−9.79640	−0.481467
$415$	0	0
$416$	28.0907	1.37726
$417$	12.2698i	0.600857i
$418$	40.4986i	1.98085i
$419$	−15.8120	−0.772466	−0.386233	−	0.922401i	$-0.626224\pi$
$419$	−15.8120	−0.772466	−0.386233	+	0.922401i	$0.626224\pi$
$420$	0	0
$421$	11.9813	0.583935	0.291967	−	0.956428i	$-0.405690\pi$
$421$	11.9813	0.583935	0.291967	+	0.956428i	$0.405690\pi$
$422$	− 67.1454i	− 3.26859i
$423$	− 3.28640i	− 0.159790i
$424$	−78.5447	−3.81447
$425$	0	0
$426$	3.58721	0.173801
$427$	− 5.64067i	− 0.272971i
$428$	− 28.8492i	− 1.39448i
$429$	16.8433	0.813202
$430$	0	0
$431$	14.6428	0.705317	0.352659	−	0.935752i	$-0.385278\pi$
$431$	14.6428	0.705317	0.352659	+	0.935752i	$0.385278\pi$
$432$	− 6.83213i	− 0.328711i
$433$	4.27293i	0.205344i	0.994715	+	0.102672i	$0.0327392\pi$
$433$	4.27293i	0.205344i	−0.994715	+	0.102672i	$0.967261\pi$
$434$	−2.24475	−0.107751
$435$	0	0
$436$	−10.2584	−0.491291
$437$	− 20.7363i	− 0.991952i
$438$	28.7350i	1.37301i
$439$	20.3073	0.969214	0.484607	−	0.874732i	$-0.338963\pi$
$439$	20.3073	0.969214	0.484607	+	0.874732i	$0.338963\pi$
$440$	0	0
$441$	−5.91729	−0.281776
$442$	− 73.1361i	− 3.47873i
$443$	19.0543i	0.905299i	0.891689	+	0.452649i	$0.149521\pi$
$443$	19.0543i	0.905299i	−0.891689	+	0.452649i	$0.850479\pi$
$444$	−7.18192	−0.340839
$445$	0	0
$446$	69.6479	3.29793
$447$	4.62832i	0.218912i
$448$	1.13440i	0.0535952i
$449$	−29.4793	−1.39122	−0.695608	−	0.718421i	$-0.744865\pi$
$449$	−29.4793	−1.39122	−0.695608	+	0.718421i	$0.744865\pi$
$450$	0	0
$451$	3.46209	0.163023
$452$	− 17.1944i	− 0.808759i
$453$	− 4.67249i	− 0.219533i
$454$	0.330305	0.0155020
$455$	0	0
$456$	33.2574	1.55742
$457$	− 28.3015i	− 1.32389i	−0.749553	−	0.661945i	$-0.769731\pi$
$457$	− 28.3015i	− 1.32389i	0.749553	−	0.661945i	$-0.230269\pi$
$458$	− 7.16968i	− 0.335017i
$459$	−5.08361	−0.237283
$460$	0	0
$461$	−16.5575	−0.771157	−0.385579	−	0.922675i	$-0.625998\pi$
$461$	−16.5575	−0.771157	−0.385579	+	0.922675i	$0.625998\pi$
$462$	− 7.84506i	− 0.364985i
$463$	9.20933i	0.427994i	0.976834	+	0.213997i	$0.0686483\pi$
$463$	9.20933i	0.427994i	−0.976834	+	0.213997i	$0.931352\pi$
$464$	−4.64377	−0.215582
$465$	0	0
$466$	−20.2686	−0.938926
$467$	− 10.6123i	− 0.491078i	−0.969387	−	0.245539i	$-0.921035\pi$
$467$	− 10.6123i	− 0.491078i	0.969387	−	0.245539i	$-0.0789650\pi$
$468$	− 25.1701i	− 1.16349i
$469$	0.966803	0.0446428
$470$	0	0
$471$	14.9726	0.689902
$472$	19.8859i	0.915323i
$473$	− 16.8932i	− 0.776749i
$474$	−3.67220	−0.168670
$475$	0	0
$476$	−23.4850	−1.07644
$477$	12.6861i	0.580858i
$478$	48.3729i	2.21253i
$479$	−33.6061	−1.53550	−0.767751	−	0.640749i	$-0.778624\pi$
$479$	−33.6061	−1.53550	−0.767751	+	0.640749i	$0.778624\pi$
$480$	0	0
$481$	−9.17069	−0.418147
$482$	53.5954i	2.44120i
$483$	4.01687i	0.182774i
$484$	9.64818	0.438554
$485$	0	0
$486$	−2.53767	−0.115111
$487$	− 29.2486i	− 1.32538i	−0.748894	−	0.662690i	$-0.769415\pi$
$487$	− 29.2486i	− 1.32538i	0.748894	−	0.662690i	$-0.230585\pi$
$488$	33.5630i	1.51933i
$489$	−11.9112	−0.538644
$490$	0	0
$491$	13.9549	0.629778	0.314889	−	0.949129i	$-0.398033\pi$
$491$	13.9549	0.629778	0.314889	+	0.949129i	$0.398033\pi$
$492$	− 5.17364i	− 0.233246i
$493$	3.45531i	0.155619i
$494$	77.2787	3.47693
$495$	0	0
$496$	5.80807	0.260790
$497$	− 1.47088i	− 0.0659782i
$498$	− 29.1214i	− 1.30496i
$499$	35.7533	1.60054	0.800268	−	0.599642i	$-0.204690\pi$
$499$	35.7533	1.60054	0.800268	+	0.599642i	$0.204690\pi$
$500$	0	0
$501$	−10.6081	−0.473936
$502$	76.6946i	3.42305i
$503$	− 11.6443i	− 0.519193i	−0.965717	−	0.259596i	$-0.916410\pi$
$503$	− 11.6443i	− 0.519193i	0.965717	−	0.259596i	$-0.0835895\pi$
$504$	−6.44235	−0.286965
$505$	0	0
$506$	29.1052	1.29388
$507$	− 19.1401i	− 0.850040i
$508$	51.9178i	2.30348i
$509$	8.53362	0.378246	0.189123	−	0.981953i	$-0.439436\pi$
$509$	8.53362	0.378246	0.189123	+	0.981953i	$0.439436\pi$
$510$	0	0
$511$	11.7824	0.521222
$512$	− 50.7478i	− 2.24276i
$513$	− 5.37156i	− 0.237160i
$514$	12.9476	0.571094
$515$	0	0
$516$	−25.2447	−1.11133
$517$	9.76393i	0.429417i
$518$	4.27141i	0.187675i
$519$	1.25338	0.0550173
$520$	0	0
$521$	1.26530	0.0554336	0.0277168	−	0.999616i	$-0.491176\pi$
$521$	1.26530	0.0554336	0.0277168	+	0.999616i	$0.491176\pi$
$522$	1.72485i	0.0754945i
$523$	27.2204i	1.19027i	0.803627	+	0.595133i	$0.202901\pi$
$523$	27.2204i	1.19027i	−0.803627	+	0.595133i	$0.797099\pi$
$524$	35.3500	1.54427
$525$	0	0
$526$	−16.2802	−0.709850
$527$	− 4.32163i	− 0.188253i
$528$	20.2983i	0.883370i
$529$	8.09742	0.352062
$530$	0	0
$531$	3.21187	0.139383
$532$	− 24.8153i	− 1.07588i
$533$	− 6.60629i	− 0.286150i
$534$	23.0224	0.996277
$535$	0	0
$536$	−5.75266	−0.248477
$537$	20.8113i	0.898074i
$538$	44.3057i	1.91015i
$539$	17.5803	0.757237
$540$	0	0
$541$	30.3830	1.30627	0.653134	−	0.757242i	$-0.273454\pi$
$541$	30.3830	1.30627	0.653134	+	0.757242i	$0.273454\pi$
$542$	− 0.241550i	− 0.0103755i
$543$	− 6.92706i	− 0.297269i
$544$	25.1891	1.07997
$545$	0	0
$546$	−14.9698	−0.640648
$547$	− 11.3768i	− 0.486439i	−0.969971	−	0.243219i	$-0.921797\pi$
$547$	− 11.3768i	− 0.486439i	0.969971	−	0.243219i	$-0.0782035\pi$
$548$	− 49.0654i	− 2.09597i
$549$	5.42093	0.231360
$550$	0	0
$551$	−3.65103	−0.155539
$552$	− 23.9011i	− 1.01730i
$553$	1.50573i	0.0640303i
$554$	−46.6950	−1.98388
$555$	0	0
$556$	−54.4755	−2.31028
$557$	45.7532i	1.93862i	0.245833	+	0.969312i	$0.420938\pi$
$557$	45.7532i	1.93862i	−0.245833	+	0.969312i	$0.579062\pi$
$558$	− 2.15730i	− 0.0913259i
$559$	−32.2353	−1.36341
$560$	0	0
$561$	15.1034	0.637668
$562$	45.5158i	1.91997i
$563$	8.94289i	0.376898i	0.982083	+	0.188449i	$0.0603460\pi$
$563$	8.94289i	0.376898i	−0.982083	+	0.188449i	$0.939654\pi$
$564$	14.5909	0.614389
$565$	0	0
$566$	56.4377	2.37225
$567$	1.04054i	0.0436984i
$568$	8.75203i	0.367227i
$569$	2.28908	0.0959632	0.0479816	−	0.998848i	$-0.484721\pi$
$569$	2.28908	0.0959632	0.0479816	+	0.998848i	$0.484721\pi$
$570$	0	0
$571$	−30.7595	−1.28725	−0.643623	−	0.765342i	$-0.722570\pi$
$571$	−30.7595	−1.28725	−0.643623	+	0.765342i	$0.722570\pi$
$572$	74.7806i	3.12674i
$573$	8.16415i	0.341062i
$574$	−3.07700	−0.128431
$575$	0	0
$576$	−1.09021	−0.0454252
$577$	− 6.59227i	− 0.274440i	−0.990541	−	0.137220i	$-0.956183\pi$
$577$	− 6.59227i	− 0.274440i	0.990541	−	0.137220i	$-0.0438167\pi$
$578$	− 22.4410i	− 0.933421i
$579$	13.9629	0.580280
$580$	0	0
$581$	−11.9408	−0.495387
$582$	15.3004i	0.634220i
$583$	− 37.6906i	− 1.56098i
$584$	−70.1073	−2.90106
$585$	0	0
$586$	47.5346	1.96364
$587$	− 5.53771i	− 0.228566i	−0.993448	−	0.114283i	$-0.963543\pi$
$587$	− 5.53771i	− 0.228566i	0.993448	−	0.114283i	$-0.0364570\pi$
$588$	− 26.2715i	− 1.08342i
$589$	4.56642	0.188156
$590$	0	0
$591$	5.76250	0.237038
$592$	− 11.0518i	− 0.454228i
$593$	1.88122i	0.0772524i	0.999254	+	0.0386262i	$0.0122982\pi$
$593$	1.88122i	0.0772524i	−0.999254	+	0.0386262i	$0.987702\pi$
$594$	7.53945	0.309347
$595$	0	0
$596$	−20.5488	−0.841710
$597$	26.5748i	1.08763i
$598$	− 55.5380i	− 2.27112i
$599$	−17.8272	−0.728400	−0.364200	−	0.931321i	$-0.618658\pi$
$599$	−17.8272	−0.728400	−0.364200	+	0.931321i	$0.618658\pi$
$600$	0	0
$601$	33.0994	1.35015	0.675077	−	0.737747i	$-0.264110\pi$
$601$	33.0994	1.35015	0.675077	+	0.737747i	$0.264110\pi$
$602$	15.0141i	0.611931i
$603$	0.929140i	0.0378375i
$604$	20.7449	0.844097
$605$	0	0
$606$	38.9301	1.58143
$607$	− 23.4603i	− 0.952226i	−0.879384	−	0.476113i	$-0.842045\pi$
$607$	− 23.4603i	− 0.952226i	0.879384	−	0.476113i	$-0.157955\pi$
$608$	26.6158i	1.07941i
$609$	0.707248	0.0286591
$610$	0	0
$611$	18.6314	0.753744
$612$	− 22.5702i	− 0.912345i
$613$	− 47.2281i	− 1.90752i	−0.300564	−	0.953762i	$-0.597175\pi$
$613$	− 47.2281i	− 1.90752i	0.300564	−	0.953762i	$-0.402825\pi$
$614$	−17.8614	−0.720828
$615$	0	0
$616$	19.1403	0.771184
$617$	16.2698i	0.654999i	0.944851	+	0.327499i	$0.106206\pi$
$617$	16.2698i	0.654999i	−0.944851	+	0.327499i	$0.893794\pi$
$618$	− 30.6017i	− 1.23098i
$619$	−35.4599	−1.42525	−0.712627	−	0.701543i	$-0.752495\pi$
$619$	−35.4599	−1.42525	−0.712627	+	0.701543i	$0.752495\pi$
$620$	0	0
$621$	−3.86039	−0.154912
$622$	− 74.3005i	− 2.97918i
$623$	− 9.44000i	− 0.378206i
$624$	38.7329	1.55056
$625$	0	0
$626$	39.6775	1.58583
$627$	15.9589i	0.637339i
$628$	66.4752i	2.65265i
$629$	−8.22339	−0.327888
$630$	0	0
$631$	−29.2364	−1.16388	−0.581941	−	0.813231i	$-0.697706\pi$
$631$	−29.2364	−1.16388	−0.581941	+	0.813231i	$0.697706\pi$
$632$	− 8.95939i	− 0.356385i
$633$	− 26.4594i	− 1.05167i
$634$	−15.8387	−0.629037
$635$	0	0
$636$	−56.3237	−2.23338
$637$	− 33.5464i	− 1.32916i
$638$	− 5.12453i	− 0.202882i
$639$	1.41358	0.0559205
$640$	0	0
$641$	−15.7160	−0.620745	−0.310373	−	0.950615i	$-0.600454\pi$
$641$	−15.7160	−0.620745	−0.310373	+	0.950615i	$0.600454\pi$
$642$	− 16.4895i	− 0.650788i
$643$	− 8.08055i	− 0.318666i	−0.987225	−	0.159333i	$-0.949066\pi$
$643$	− 8.08055i	− 0.318666i	0.987225	−	0.159333i	$-0.0509343\pi$
$644$	−17.8340	−0.702760
$645$	0	0
$646$	69.2961	2.72642
$647$	− 11.0193i	− 0.433214i	−0.976259	−	0.216607i	$-0.930501\pi$
$647$	− 11.0193i	− 0.433214i	0.976259	−	0.216607i	$-0.0694990\pi$
$648$	− 6.19138i	− 0.243220i
$649$	−9.54248	−0.374575
$650$	0	0
$651$	−0.884570	−0.0346691
$652$	− 52.8832i	− 2.07107i
$653$	− 31.0861i	− 1.21649i	−0.793748	−	0.608246i	$-0.791873\pi$
$653$	− 31.0861i	− 1.21649i	0.793748	−	0.608246i	$-0.208127\pi$
$654$	−5.86348	−0.229280
$655$	0	0
$656$	7.96142	0.310841
$657$	11.3234i	0.441767i
$658$	− 8.67788i	− 0.338299i
$659$	−32.4252	−1.26311	−0.631553	−	0.775333i	$-0.717582\pi$
$659$	−32.4252	−1.26311	−0.631553	+	0.775333i	$0.717582\pi$
$660$	0	0
$661$	15.6076	0.607067	0.303534	−	0.952821i	$-0.401834\pi$
$661$	15.6076	0.607067	0.303534	+	0.952821i	$0.401834\pi$
$662$	54.4521i	2.11634i
$663$	− 28.8201i	− 1.11928i
$664$	71.0499	2.75727
$665$	0	0
$666$	−4.10501	−0.159066
$667$	2.62389i	0.101597i
$668$	− 47.0978i	− 1.82227i
$669$	27.4456	1.06111
$670$	0	0
$671$	−16.1056	−0.621750
$672$	− 5.15581i	− 0.198890i
$673$	31.0271i	1.19601i	0.801494	+	0.598003i	$0.204039\pi$
$673$	31.0271i	1.19601i	−0.801494	+	0.598003i	$0.795961\pi$
$674$	−88.1870	−3.39684
$675$	0	0
$676$	84.9778	3.26838
$677$	23.0893i	0.887394i	0.896177	+	0.443697i	$0.146333\pi$
$677$	23.0893i	0.887394i	−0.896177	+	0.443697i	$0.853667\pi$
$678$	− 9.82792i	− 0.377439i
$679$	6.27369	0.240762
$680$	0	0
$681$	0.130161	0.00498777
$682$	6.40936i	0.245427i
$683$	− 41.3541i	− 1.58237i	−0.611577	−	0.791185i	$-0.709464\pi$
$683$	− 41.3541i	− 1.58237i	0.611577	−	0.791185i	$-0.290536\pi$
$684$	23.8486	0.911873
$685$	0	0
$686$	−34.1086	−1.30227
$687$	− 2.82530i	− 0.107792i
$688$	− 38.8476i	− 1.48105i
$689$	−71.9205	−2.73995
$690$	0	0
$691$	4.46909	0.170012	0.0850061	−	0.996380i	$-0.472909\pi$
$691$	4.46909	0.170012	0.0850061	+	0.996380i	$0.472909\pi$
$692$	5.56475i	0.211540i
$693$	− 3.09144i	− 0.117434i
$694$	−72.8924	−2.76696
$695$	0	0
$696$	−4.20826	−0.159514
$697$	− 5.92389i	− 0.224383i
$698$	31.1713i	1.17985i
$699$	−7.98709	−0.302099
$700$	0	0
$701$	−25.2265	−0.952791	−0.476396	−	0.879231i	$-0.658057\pi$
$701$	−25.2265	−0.952791	−0.476396	+	0.879231i	$0.658057\pi$
$702$	− 14.3866i	− 0.542988i
$703$	− 8.68919i	− 0.327719i
$704$	3.23901	0.122075
$705$	0	0
$706$	−68.4611	−2.57657
$707$	− 15.9627i	− 0.600339i
$708$	14.2600i	0.535924i
$709$	−1.55990	−0.0585834	−0.0292917	−	0.999571i	$-0.509325\pi$
$709$	−1.55990	−0.0585834	−0.0292917	+	0.999571i	$0.509325\pi$
$710$	0	0
$711$	−1.44707	−0.0542695
$712$	56.1698i	2.10505i
$713$	− 3.28176i	− 0.122903i
$714$	−13.4235	−0.502361
$715$	0	0
$716$	−92.3978	−3.45307
$717$	19.0619i	0.711880i
$718$	− 41.6794i	− 1.55546i
$719$	−28.9403	−1.07929	−0.539645	−	0.841893i	$-0.681442\pi$
$719$	−28.9403	−1.07929	−0.539645	+	0.841893i	$0.681442\pi$
$720$	0	0
$721$	−12.5478	−0.467304
$722$	25.0054i	0.930604i
$723$	21.1199i	0.785458i
$724$	30.7547	1.14299
$725$	0	0
$726$	5.51466	0.204668
$727$	− 36.9595i	− 1.37075i	−0.728190	−	0.685375i	$-0.759638\pi$
$727$	− 36.9595i	− 1.37075i	0.728190	−	0.685375i	$-0.240362\pi$
$728$	− 36.5231i	− 1.35364i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−28.9055	−1.06911
$732$	24.0678i	0.889570i
$733$	4.16161i	0.153713i	0.997042	+	0.0768564i	$0.0244883\pi$
$733$	4.16161i	0.153713i	−0.997042	+	0.0768564i	$0.975512\pi$
$734$	75.8646	2.80021
$735$	0	0
$736$	19.1280	0.705069
$737$	− 2.76048i	− 0.101684i
$738$	− 2.95713i	− 0.108853i
$739$	−18.9577	−0.697370	−0.348685	−	0.937240i	$-0.613372\pi$
$739$	−18.9577	−0.697370	−0.348685	+	0.937240i	$0.613372\pi$
$740$	0	0
$741$	30.4526	1.11870
$742$	33.4982i	1.22976i
$743$	11.7060i	0.429452i	0.976674	+	0.214726i	$0.0688859\pi$
$743$	11.7060i	0.429452i	−0.976674	+	0.214726i	$0.931114\pi$
$744$	5.26336	0.192964
$745$	0	0
$746$	−63.0122	−2.30704
$747$	− 11.4756i	− 0.419871i
$748$	67.0561i	2.45181i
$749$	−6.76126	−0.247051
$750$	0	0
$751$	−4.95672	−0.180873	−0.0904367	−	0.995902i	$-0.528826\pi$
$751$	−4.95672	−0.180873	−0.0904367	+	0.995902i	$0.528826\pi$
$752$	22.4531i	0.818782i
$753$	30.2224i	1.10137i
$754$	−9.77854	−0.356113
$755$	0	0
$756$	−4.61976	−0.168019
$757$	18.6020i	0.676101i	0.941128	+	0.338051i	$0.109768\pi$
$757$	18.6020i	0.676101i	−0.941128	+	0.338051i	$0.890232\pi$
$758$	− 49.2223i	− 1.78784i
$759$	11.4692	0.416307
$760$	0	0
$761$	−11.3585	−0.411747	−0.205873	−	0.978579i	$-0.566003\pi$
$761$	−11.3585	−0.411747	−0.205873	+	0.978579i	$0.566003\pi$
$762$	29.6750i	1.07501i
$763$	2.40423i	0.0870391i
$764$	−36.2471	−1.31137
$765$	0	0
$766$	−28.5510	−1.03159
$767$	18.2088i	0.657481i
$768$	− 29.9884i	− 1.08211i
$769$	−7.27477	−0.262335	−0.131168	−	0.991360i	$-0.541873\pi$
$769$	−7.27477	−0.262335	−0.131168	+	0.991360i	$0.541873\pi$
$770$	0	0
$771$	5.10215	0.183749
$772$	61.9925i	2.23116i
$773$	− 9.32288i	− 0.335321i	−0.985845	−	0.167660i	$-0.946379\pi$
$773$	− 9.32288i	− 0.335321i	0.985845	−	0.167660i	$-0.0536212\pi$
$774$	−14.4292	−0.518648
$775$	0	0
$776$	−37.3296	−1.34005
$777$	1.68320i	0.0603844i
$778$	− 37.8158i	− 1.35576i
$779$	6.25943	0.224267
$780$	0	0
$781$	−4.19977	−0.150279
$782$	− 49.8011i	− 1.78088i
$783$	0.679696i	0.0242904i
$784$	40.4277	1.44385
$785$	0	0
$786$	20.2052	0.720696
$787$	− 14.8658i	− 0.529907i	−0.964261	−	0.264953i	$-0.914643\pi$
$787$	− 14.8658i	− 0.529907i	0.964261	−	0.264953i	$-0.0853566\pi$
$788$	25.5843i	0.911402i
$789$	−6.41540	−0.228394
$790$	0	0
$791$	−4.02979	−0.143283
$792$	18.3946i	0.653625i
$793$	30.7324i	1.09134i
$794$	−61.1493	−2.17011
$795$	0	0
$796$	−117.986	−4.18192
$797$	9.57546i	0.339180i	0.985515	+	0.169590i	$0.0542444\pi$
$797$	9.57546i	0.339180i	−0.985515	+	0.169590i	$0.945756\pi$
$798$	− 14.1838i	− 0.502102i
$799$	16.7068	0.591044
$800$	0	0
$801$	9.07225	0.320552
$802$	34.1520i	1.20595i
$803$	− 33.6418i	− 1.18719i
$804$	−4.12518	−0.145484
$805$	0	0
$806$	12.2302	0.430792
$807$	17.4592i	0.614592i
$808$	94.9810i	3.34142i
$809$	−41.1436	−1.44653	−0.723266	−	0.690569i	$-0.757360\pi$
$809$	−41.1436	−1.44653	−0.723266	+	0.690569i	$0.757360\pi$
$810$	0	0
$811$	43.4398	1.52538	0.762688	−	0.646766i	$-0.223879\pi$
$811$	43.4398	1.52538	0.762688	+	0.646766i	$0.223879\pi$
$812$	3.14003i	0.110193i
$813$	− 0.0951857i	− 0.00333831i
$814$	12.1960	0.427470
$815$	0	0
$816$	34.7319	1.21586
$817$	− 30.5428i	− 1.06856i
$818$	90.0206i	3.14750i
$819$	−5.89903	−0.206129
$820$	0	0
$821$	17.4617	0.609416	0.304708	−	0.952446i	$-0.401441\pi$
$821$	17.4617	0.609416	0.304708	+	0.952446i	$0.401441\pi$
$822$	− 28.0446i	− 0.978167i
$823$	− 2.22456i	− 0.0775432i	−0.999248	−	0.0387716i	$-0.987656\pi$
$823$	− 2.22456i	− 0.0775432i	0.999248	−	0.0387716i	$-0.0123445\pi$
$824$	74.6616	2.60096
$825$	0	0
$826$	8.48106	0.295094
$827$	− 31.4550i	− 1.09380i	−0.837199	−	0.546898i	$-0.815808\pi$
$827$	− 31.4550i	− 1.09380i	0.837199	−	0.546898i	$-0.184192\pi$
$828$	− 17.1393i	− 0.595632i
$829$	−53.2947	−1.85100	−0.925500	−	0.378748i	$-0.876355\pi$
$829$	−53.2947	−1.85100	−0.925500	+	0.378748i	$0.876355\pi$
$830$	0	0
$831$	−18.4007	−0.638314
$832$	− 6.18061i	− 0.214274i
$833$	− 30.0812i	− 1.04225i
$834$	−31.1369	−1.07818
$835$	0	0
$836$	−70.8543	−2.45055
$837$	− 0.850111i	− 0.0293841i
$838$	− 40.1256i	− 1.38612i
$839$	8.37331	0.289079	0.144539	−	0.989499i	$-0.453830\pi$
$839$	8.37331	0.289079	0.144539	+	0.989499i	$0.453830\pi$
$840$	0	0
$841$	−28.5380	−0.984069
$842$	30.4047i	1.04782i
$843$	17.9361i	0.617750i
$844$	117.474	4.04363
$845$	0	0
$846$	8.33982	0.286729
$847$	− 2.26121i	− 0.0776960i
$848$	− 86.6733i	− 2.97637i
$849$	22.2399	0.763273
$850$	0	0
$851$	−6.24467	−0.214064
$852$	6.27601i	0.215013i
$853$	− 1.22370i	− 0.0418989i	−0.999781	−	0.0209494i	$-0.993331\pi$
$853$	− 1.22370i	− 0.0418989i	0.999781	−	0.0209494i	$-0.00666890\pi$
$854$	14.3142	0.489821
$855$	0	0
$856$	40.2308	1.37506
$857$	14.3684i	0.490816i	0.969420	+	0.245408i	$0.0789219\pi$
$857$	14.3684i	0.490816i	−0.969420	+	0.245408i	$0.921078\pi$
$858$	42.7428i	1.45921i
$859$	10.3090	0.351738	0.175869	−	0.984414i	$-0.443726\pi$
$859$	10.3090	0.351738	0.175869	+	0.984414i	$0.443726\pi$
$860$	0	0
$861$	−1.21253	−0.0413228
$862$	37.1586i	1.26563i
$863$	30.2724i	1.03048i	0.857045	+	0.515242i	$0.172298\pi$
$863$	30.2724i	1.03048i	−0.857045	+	0.515242i	$0.827702\pi$
$864$	4.95495	0.168571
$865$	0	0
$866$	−10.8433	−0.368470
$867$	− 8.84312i	− 0.300328i
$868$	− 3.92730i	− 0.133301i
$869$	4.29927	0.145843
$870$	0	0
$871$	−5.26750	−0.178482
$872$	− 14.3056i	− 0.484450i
$873$	6.02928i	0.204060i
$874$	52.6220	1.77996
$875$	0	0
$876$	−50.2734	−1.69858
$877$	− 30.6345i	− 1.03445i	−0.855848	−	0.517227i	$-0.826964\pi$
$877$	− 30.6345i	− 1.03445i	0.855848	−	0.517227i	$-0.173036\pi$
$878$	51.5333i	1.73916i
$879$	18.7316	0.631800
$880$	0	0
$881$	15.2211	0.512813	0.256406	−	0.966569i	$-0.417461\pi$
$881$	15.2211	0.512813	0.256406	+	0.966569i	$0.417461\pi$
$882$	− 15.0161i	− 0.505620i
$883$	40.1985i	1.35279i	0.736541	+	0.676393i	$0.236458\pi$
$883$	40.1985i	1.35279i	−0.736541	+	0.676393i	$0.763542\pi$
$884$	127.955	4.30360
$885$	0	0
$886$	−48.3537	−1.62447
$887$	31.8432i	1.06919i	0.845109	+	0.534594i	$0.179536\pi$
$887$	31.8432i	1.06919i	−0.845109	+	0.534594i	$0.820464\pi$
$888$	− 10.0154i	− 0.336093i
$889$	12.1678	0.408094
$890$	0	0
$891$	2.97101	0.0995325
$892$	121.853i	4.07993i
$893$	17.6531i	0.590739i
$894$	−11.7452	−0.392817
$895$	0	0
$896$	−13.1903	−0.440658
$897$	− 21.8854i	− 0.730732i
$898$	− 74.8089i	− 2.49641i
$899$	−0.577817	−0.0192713
$900$	0	0
$901$	−64.4914	−2.14852
$902$	8.78565i	0.292530i
$903$	5.91650i	0.196889i
$904$	23.9780	0.797498
$905$	0	0
$906$	11.8573	0.393931
$907$	28.8507i	0.957970i	0.877823	+	0.478985i	$0.158995\pi$
$907$	28.8507i	0.957970i	−0.877823	+	0.478985i	$0.841005\pi$
$908$	0.577886i	0.0191778i
$909$	15.3408	0.508824
$910$	0	0
$911$	48.3760	1.60277	0.801385	−	0.598149i	$-0.204097\pi$
$911$	48.3760	1.60277	0.801385	+	0.598149i	$0.204097\pi$
$912$	36.6992i	1.21523i
$913$	34.0941i	1.12835i
$914$	71.8200	2.37560
$915$	0	0
$916$	12.5437	0.414456
$917$	− 8.28485i	− 0.273590i
$918$	− 12.9006i	− 0.425782i
$919$	8.38022	0.276438	0.138219	−	0.990402i	$-0.455862\pi$
$919$	8.38022	0.276438	0.138219	+	0.990402i	$0.455862\pi$
$920$	0	0
$921$	−7.03850	−0.231926
$922$	− 42.0174i	− 1.38377i
$923$	8.01392i	0.263781i
$924$	13.7253	0.451530
$925$	0	0
$926$	−23.3703	−0.767995
$927$	− 12.0590i	− 0.396068i
$928$	− 3.36786i	− 0.110556i
$929$	−14.3126	−0.469580	−0.234790	−	0.972046i	$-0.575440\pi$
$929$	−14.3126	−0.469580	−0.234790	+	0.972046i	$0.575440\pi$
$930$	0	0
$931$	31.7851	1.04171
$932$	− 35.4610i	− 1.16156i
$933$	− 29.2790i	− 0.958552i
$934$	26.9305	0.881194
$935$	0	0
$936$	35.1003	1.14729
$937$	21.9244i	0.716240i	0.933675	+	0.358120i	$0.116582\pi$
$937$	21.9244i	0.716240i	−0.933675	+	0.358120i	$0.883418\pi$
$938$	2.45343i	0.0801074i
$939$	15.6354	0.510241
$940$	0	0
$941$	22.3934	0.730005	0.365003	−	0.931007i	$-0.381068\pi$
$941$	22.3934	0.730005	0.365003	+	0.931007i	$0.381068\pi$
$942$	37.9956i	1.23796i
$943$	− 4.49847i	− 0.146490i
$944$	−21.9439	−0.714213
$945$	0	0
$946$	42.8694	1.39380
$947$	− 43.5638i	− 1.41563i	−0.706397	−	0.707816i	$-0.749681\pi$
$947$	− 43.5638i	− 1.41563i	0.706397	−	0.707816i	$-0.250319\pi$
$948$	− 6.42470i	− 0.208665i
$949$	−64.1947	−2.08385
$950$	0	0
$951$	−6.24144	−0.202393
$952$	− 32.7504i	− 1.06145i
$953$	2.37169i	0.0768266i	0.999262	+	0.0384133i	$0.0122303\pi$
$953$	2.37169i	0.0768266i	−0.999262	+	0.0384133i	$0.987770\pi$
$954$	−32.1933	−1.04230
$955$	0	0
$956$	−84.6308	−2.73716
$957$	− 2.01938i	− 0.0652774i
$958$	− 85.2813i	− 2.75531i
$959$	−11.4993	−0.371331
$960$	0	0
$961$	−30.2773	−0.976687
$962$	− 23.2722i	− 0.750326i
$963$	− 6.49787i	− 0.209391i
$964$	−93.7678	−3.02006
$965$	0	0
$966$	−10.1935	−0.327971
$967$	19.7857i	0.636266i	0.948046	+	0.318133i	$0.103056\pi$
$967$	19.7857i	0.636266i	−0.948046	+	0.318133i	$0.896944\pi$
$968$	13.4546i	0.432447i
$969$	27.3069	0.877225
$970$	0	0
$971$	49.3838	1.58480	0.792401	−	0.610000i	$-0.208831\pi$
$971$	49.3838	1.58480	0.792401	+	0.610000i	$0.208831\pi$
$972$	− 4.43979i	− 0.142406i
$973$	12.7672i	0.409298i
$974$	74.2234	2.37827
$975$	0	0
$976$	−37.0365	−1.18551
$977$	47.7652i	1.52814i	0.645130	+	0.764072i	$0.276803\pi$
$977$	47.7652i	1.52814i	−0.645130	+	0.764072i	$0.723197\pi$
$978$	− 30.2268i	− 0.966545i
$979$	−26.9537	−0.861445
$980$	0	0
$981$	−2.31057	−0.0737709
$982$	35.4131i	1.13008i
$983$	51.9282i	1.65625i	0.560541	+	0.828127i	$0.310593\pi$
$983$	51.9282i	1.65625i	−0.560541	+	0.828127i	$0.689407\pi$
$984$	7.21476	0.229998
$985$	0	0
$986$	−8.76846	−0.279245
$987$	− 3.41962i	− 0.108848i
$988$	135.203i	4.30138i
$989$	−21.9502	−0.697976
$990$	0	0
$991$	−16.3790	−0.520297	−0.260148	−	0.965569i	$-0.583771\pi$
$991$	−16.3790	−0.520297	−0.260148	+	0.965569i	$0.583771\pi$
$992$	4.21226i	0.133739i
$993$	21.4575i	0.680933i
$994$	3.73262	0.118392
$995$	0	0
$996$	50.9493	1.61439
$997$	22.2515i	0.704713i	0.935866	+	0.352357i	$0.114620\pi$
$997$	22.2515i	0.704713i	−0.935866	+	0.352357i	$0.885380\pi$
$998$	90.7302i	2.87201i
$999$	−1.61763	−0.0511795

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.h.1249.16		16
5.2	odd	4		1875.2.a.m.1.1		8
5.3	odd	4		1875.2.a.p.1.8		8
5.4	even	2	inner	1875.2.b.h.1249.1		16
15.2	even	4		5625.2.a.bd.1.8		8
15.8	even	4		5625.2.a.t.1.1		8
25.3	odd	20		375.2.g.d.76.1		16
25.4	even	10		375.2.i.c.49.1		16
25.6	even	5		375.2.i.c.199.1		16
25.8	odd	20		375.2.g.d.301.1		16
25.17	odd	20		375.2.g.e.301.4		16
25.19	even	10		75.2.i.a.64.4	yes	16
25.21	even	5		75.2.i.a.34.4	✓	16
25.22	odd	20		375.2.g.e.76.4		16
75.44	odd	10		225.2.m.b.64.1		16
75.71	odd	10		225.2.m.b.109.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.34.4	✓	16	25.21	even	5
75.2.i.a.64.4	yes	16	25.19	even	10
225.2.m.b.64.1		16	75.44	odd	10
225.2.m.b.109.1		16	75.71	odd	10
375.2.g.d.76.1		16	25.3	odd	20
375.2.g.d.301.1		16	25.8	odd	20
375.2.g.e.76.4		16	25.22	odd	20
375.2.g.e.301.4		16	25.17	odd	20
375.2.i.c.49.1		16	25.4	even	10
375.2.i.c.199.1		16	25.6	even	5
1875.2.a.m.1.1		8	5.2	odd	4
1875.2.a.p.1.8		8	5.3	odd	4
1875.2.b.h.1249.1		16	5.4	even	2	inner
1875.2.b.h.1249.16		16	1.1	even	1	trivial
5625.2.a.t.1.1		8	15.8	even	4
5625.2.a.bd.1.8		8	15.2	even	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.53767i q^{2} +1.00000i q^{3} -4.43979 q^{4} -2.53767 q^{6} +1.04054i q^{7} -6.19138i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.53767i q^{2} +1.00000i q^{3} -4.43979 q^{4} -2.53767 q^{6} +1.04054i q^{7} -6.19138i q^{8} -1.00000 q^{9} +2.97101 q^{11} -4.43979i q^{12} -5.66922i q^{13} -2.64054 q^{14} +6.83213 q^{16} -5.08361i q^{17} -2.53767i q^{18} +5.37156 q^{19} -1.04054 q^{21} +7.53945i q^{22} -3.86039i q^{23} +6.19138 q^{24} +14.3866 q^{26} -1.00000i q^{27} -4.61976i q^{28} -0.679696 q^{29} +0.850111 q^{31} +4.95495i q^{32} +2.97101i q^{33} +12.9006 q^{34} +4.43979 q^{36} -1.61763i q^{37} +13.6313i q^{38} +5.66922 q^{39} +1.16529 q^{41} -2.64054i q^{42} -5.68601i q^{43} -13.1906 q^{44} +9.79640 q^{46} +3.28640i q^{47} +6.83213i q^{48} +5.91729 q^{49} +5.08361 q^{51} +25.1701i q^{52} -12.6861i q^{53} +2.53767 q^{54} +6.44235 q^{56} +5.37156i q^{57} -1.72485i q^{58} -3.21187 q^{59} -5.42093 q^{61} +2.15730i q^{62} -1.04054i q^{63} +1.09021 q^{64} -7.53945 q^{66} -0.929140i q^{67} +22.5702i q^{68} +3.86039 q^{69} -1.41358 q^{71} +6.19138i q^{72} -11.3234i q^{73} +4.10501 q^{74} -23.8486 q^{76} +3.09144i q^{77} +14.3866i q^{78} +1.44707 q^{79} +1.00000 q^{81} +2.95713i q^{82} +11.4756i q^{83} +4.61976 q^{84} +14.4292 q^{86} -0.679696i q^{87} -18.3946i q^{88} -9.07225 q^{89} +5.89903 q^{91} +17.1393i q^{92} +0.850111i q^{93} -8.33982 q^{94} -4.95495 q^{96} -6.02928i q^{97} +15.0161i q^{98} -2.97101 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9	\( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) 16 * q - 8 * q^4 - 8 * q^6 - 16 * q^9 + 4 * q^11 - 12 * q^14 + 28 * q^19 - 16 * q^21 + 24 * q^24 + 12 * q^26 - 4 * q^29 - 44 * q^31 + 24 * q^34 + 8 * q^36 + 32 * q^39 + 16 * q^41 - 44 * q^44 - 4 * q^46 - 32 * q^51 + 8 * q^54 + 60 * q^56 - 28 * q^59 - 40 * q^61 - 12 * q^64 - 24 * q^66 + 28 * q^69 + 32 * q^71 - 52 * q^74 - 32 * q^76 + 60 * q^79 + 16 * q^81 + 32 * q^84 + 64 * q^86 - 32 * q^89 - 24 * q^91 - 28 * q^94 + 4 * q^96 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more