LMFDB - Embedded newform 2175.2.c.p.349.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(349,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.c (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:	$17.3674624396$
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} + 28x^{14} + 308x^{12} + 1671x^{10} + 4568x^{8} + 5616x^{6} + 2105x^{4} + 256x^{2} + 4 $ x^16 + 28x^14 + 308x^12 + 1671x^10 + 4568x^8 + 5616x^6 + 2105x^4 + 256*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.2
Root			$-2.59587i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.p.349.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59587i q^{2} +1.00000i q^{3} -4.73855 q^{4} +2.59587 q^{6} +3.93826i q^{7} +7.10893i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.59587i q^{2} +1.00000i q^{3} -4.73855 q^{4} +2.59587 q^{6} +3.93826i q^{7} +7.10893i q^{8} -1.00000 q^{9} +2.24645 q^{11} -4.73855i q^{12} +2.09464i q^{13} +10.2232 q^{14} +8.97678 q^{16} -6.95310i q^{17} +2.59587i q^{18} -4.29591 q^{19} -3.93826 q^{21} -5.83149i q^{22} -5.19955i q^{23} -7.10893 q^{24} +5.43743 q^{26} -1.00000i q^{27} -18.6617i q^{28} -1.00000 q^{29} +2.25035 q^{31} -9.08471i q^{32} +2.24645i q^{33} -18.0494 q^{34} +4.73855 q^{36} +10.5280i q^{37} +11.1516i q^{38} -2.09464 q^{39} -2.66629 q^{41} +10.2232i q^{42} +6.03681i q^{43} -10.6449 q^{44} -13.4974 q^{46} +7.74829i q^{47} +8.97678i q^{48} -8.50991 q^{49} +6.95310 q^{51} -9.92559i q^{52} -1.41309i q^{53} -2.59587 q^{54} -27.9968 q^{56} -4.29591i q^{57} +2.59587i q^{58} -5.76800 q^{59} -10.9192 q^{61} -5.84162i q^{62} -3.93826i q^{63} -5.62918 q^{64} +5.83149 q^{66} +11.7471i q^{67} +32.9476i q^{68} +5.19955 q^{69} -4.55691 q^{71} -7.10893i q^{72} +12.3564i q^{73} +27.3294 q^{74} +20.3564 q^{76} +8.84710i q^{77} +5.43743i q^{78} -14.4098 q^{79} +1.00000 q^{81} +6.92135i q^{82} -3.81272i q^{83} +18.6617 q^{84} +15.6708 q^{86} -1.00000i q^{87} +15.9699i q^{88} +2.07340 q^{89} -8.24926 q^{91} +24.6383i q^{92} +2.25035i q^{93} +20.1136 q^{94} +9.08471 q^{96} -12.0068i q^{97} +22.0906i q^{98} -2.24645 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10}) $ 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9	$ 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9} + 12 q^{11} - 18 q^{14} + 64 q^{16} - 4 q^{21} - 6 q^{24} + 36 q^{26} - 16 q^{29} + 16 q^{31} + 26 q^{34} + 24 q^{36} + 12 q^{39} + 4 q^{41} + 30 q^{44} + 48 q^{46} - 76 q^{49} + 24 q^{51} - 4 q^{54} + 116 q^{56} - 36 q^{59} + 24 q^{61} - 42 q^{64} - 6 q^{66} - 28 q^{69} + 48 q^{71} + 44 q^{74} - 20 q^{79} + 16 q^{81} + 28 q^{84} + 16 q^{86} - 68 q^{89} + 52 q^{91} + 86 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) $ 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9 + 12 * q^11 - 18 * q^14 + 64 * q^16 - 4 * q^21 - 6 * q^24 + 36 * q^26 - 16 * q^29 + 16 * q^31 + 26 * q^34 + 24 * q^36 + 12 * q^39 + 4 * q^41 + 30 * q^44 + 48 * q^46 - 76 * q^49 + 24 * q^51 - 4 * q^54 + 116 * q^56 - 36 * q^59 + 24 * q^61 - 42 * q^64 - 6 * q^66 - 28 * q^69 + 48 * q^71 + 44 * q^74 - 20 * q^79 + 16 * q^81 + 28 * q^84 + 16 * q^86 - 68 * q^89 + 52 * q^91 + 86 * q^94 - 4 * q^96 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 2.59587i	− 1.83556i	−0.397090	−	0.917779i	$-0.629980\pi$
$2$	− 2.59587i	− 1.83556i	0.397090	−	0.917779i	$-0.370020\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−4.73855	−2.36928
$5$	0	0
$5$	0	0
$6$	2.59587	1.05976
$7$	3.93826i	1.48852i	0.667888	+	0.744262i	$0.267198\pi$
$7$	3.93826i	1.48852i	−0.667888	+	0.744262i	$0.732802\pi$
$8$	7.10893i	2.51339i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.24645	0.677329	0.338665	−	0.940907i	$-0.390025\pi$
$11$	2.24645	0.677329	0.338665	+	0.940907i	$0.390025\pi$
$12$	− 4.73855i	− 1.36790i
$13$	2.09464i	0.580950i	0.956883	+	0.290475i	$0.0938133\pi$
$13$	2.09464i	0.580950i	−0.956883	+	0.290475i	$0.906187\pi$
$14$	10.2232	2.73227
$15$	0	0
$16$	8.97678	2.24420
$17$	− 6.95310i	− 1.68637i	−0.537620	−	0.843187i	$-0.680676\pi$
$17$	− 6.95310i	− 1.68637i	0.537620	−	0.843187i	$-0.319324\pi$
$18$	2.59587i	0.611853i
$19$	−4.29591	−0.985549	−0.492774	−	0.870157i	$-0.664017\pi$
$19$	−4.29591	−0.985549	−0.492774	+	0.870157i	$0.664017\pi$
$20$	0	0
$21$	−3.93826	−0.859399
$22$	− 5.83149i	− 1.24328i
$23$	− 5.19955i	− 1.08418i	−0.840320	−	0.542090i	$-0.817633\pi$
$23$	− 5.19955i	− 1.08418i	0.840320	−	0.542090i	$-0.182367\pi$
$24$	−7.10893	−1.45111
$25$	0	0
$26$	5.43743	1.06637
$27$	− 1.00000i	− 0.192450i
$28$	− 18.6617i	− 3.52672i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	2.25035	0.404175	0.202087	−	0.979367i	$-0.435227\pi$
$31$	2.25035	0.404175	0.202087	+	0.979367i	$0.435227\pi$
$32$	− 9.08471i	− 1.60596i
$33$	2.24645i	0.391056i
$34$	−18.0494	−3.09544
$35$	0	0
$36$	4.73855	0.789759
$37$	10.5280i	1.73079i	0.501086	+	0.865397i	$0.332934\pi$
$37$	10.5280i	1.73079i	−0.501086	+	0.865397i	$0.667066\pi$
$38$	11.1516i	1.80903i
$39$	−2.09464	−0.335412
$40$	0	0
$41$	−2.66629	−0.416405	−0.208202	−	0.978086i	$-0.566761\pi$
$41$	−2.66629	−0.416405	−0.208202	+	0.978086i	$0.566761\pi$
$42$	10.2232i	1.57748i
$43$	6.03681i	0.920605i	0.887762	+	0.460302i	$0.152259\pi$
$43$	6.03681i	0.920605i	−0.887762	+	0.460302i	$0.847741\pi$
$44$	−10.6449	−1.60478
$45$	0	0
$46$	−13.4974	−1.99008
$47$	7.74829i	1.13020i	0.825021	+	0.565102i	$0.191163\pi$
$47$	7.74829i	1.13020i	−0.825021	+	0.565102i	$0.808837\pi$
$48$	8.97678i	1.29569i
$49$	−8.50991	−1.21570
$50$	0	0
$51$	6.95310	0.973629
$52$	− 9.92559i	− 1.37643i
$53$	− 1.41309i	− 0.194103i	−0.995279	−	0.0970513i	$-0.969059\pi$
$53$	− 1.41309i	− 0.194103i	0.995279	−	0.0970513i	$-0.0309411\pi$
$54$	−2.59587	−0.353253
$55$	0	0
$56$	−27.9968	−3.74124
$57$	− 4.29591i	− 0.569007i
$58$	2.59587i	0.340855i
$59$	−5.76800	−0.750929	−0.375465	−	0.926837i	$-0.622517\pi$
$59$	−5.76800	−0.750929	−0.375465	+	0.926837i	$0.622517\pi$
$60$	0	0
$61$	−10.9192	−1.39806	−0.699030	−	0.715092i	$-0.746385\pi$
$61$	−10.9192	−1.39806	−0.699030	+	0.715092i	$0.746385\pi$
$62$	− 5.84162i	− 0.741887i
$63$	− 3.93826i	− 0.496174i
$64$	−5.62918	−0.703647
$65$	0	0
$66$	5.83149	0.717807
$67$	11.7471i	1.43514i	0.696487	+	0.717569i	$0.254745\pi$
$67$	11.7471i	1.43514i	−0.696487	+	0.717569i	$0.745255\pi$
$68$	32.9476i	3.99549i
$69$	5.19955	0.625952
$70$	0	0
$71$	−4.55691	−0.540806	−0.270403	−	0.962747i	$-0.587157\pi$
$71$	−4.55691	−0.540806	−0.270403	+	0.962747i	$0.587157\pi$
$72$	− 7.10893i	− 0.837796i
$73$	12.3564i	1.44621i	0.690740	+	0.723103i	$0.257285\pi$
$73$	12.3564i	1.44621i	−0.690740	+	0.723103i	$0.742715\pi$
$74$	27.3294	3.17698
$75$	0	0
$76$	20.3564	2.33504
$77$	8.84710i	1.00822i
$78$	5.43743i	0.615668i
$79$	−14.4098	−1.62122	−0.810612	−	0.585584i	$-0.800865\pi$
$79$	−14.4098	−1.62122	−0.810612	+	0.585584i	$0.800865\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.92135i	0.764335i
$83$	− 3.81272i	− 0.418501i	−0.977862	−	0.209250i	$-0.932898\pi$
$83$	− 3.81272i	− 0.418501i	0.977862	−	0.209250i	$-0.0671024\pi$
$84$	18.6617	2.03615
$85$	0	0
$86$	15.6708	1.68982
$87$	− 1.00000i	− 0.107211i
$88$	15.9699i	1.70239i
$89$	2.07340	0.219780	0.109890	−	0.993944i	$-0.464950\pi$
$89$	2.07340	0.219780	0.109890	+	0.993944i	$0.464950\pi$
$90$	0	0
$91$	−8.24926	−0.864757
$92$	24.6383i	2.56872i
$93$	2.25035i	0.233350i
$94$	20.1136	2.07456
$95$	0	0
$96$	9.08471	0.927204
$97$	− 12.0068i	− 1.21911i	−0.792745	−	0.609553i	$-0.791349\pi$
$97$	− 12.0068i	− 1.21911i	0.792745	−	0.609553i	$-0.208651\pi$
$98$	22.0906i	2.23149i
$99$	−2.24645	−0.225776
$100$	0	0
$101$	−15.8642	−1.57855	−0.789276	−	0.614039i	$-0.789544\pi$
$101$	−15.8642	−1.57855	−0.789276	+	0.614039i	$0.789544\pi$
$102$	− 18.0494i	− 1.78715i
$103$	− 10.3119i	− 1.01606i	−0.861339	−	0.508031i	$-0.830373\pi$
$103$	− 10.3119i	− 1.01606i	0.861339	−	0.508031i	$-0.169627\pi$
$104$	−14.8907	−1.46015
$105$	0	0
$106$	−3.66820	−0.356287
$107$	− 4.47645i	− 0.432755i	−0.976310	−	0.216378i	$-0.930576\pi$
$107$	− 4.47645i	− 0.432755i	0.976310	−	0.216378i	$-0.0694242\pi$
$108$	4.73855i	0.455967i
$109$	−13.4783	−1.29098	−0.645492	−	0.763767i	$-0.723348\pi$
$109$	−13.4783	−1.29098	−0.645492	+	0.763767i	$0.723348\pi$
$110$	0	0
$111$	−10.5280	−0.999275
$112$	35.3529i	3.34054i
$113$	− 11.4525i	− 1.07736i	−0.842511	−	0.538679i	$-0.818923\pi$
$113$	− 11.4525i	− 1.07736i	0.842511	−	0.538679i	$-0.181077\pi$
$114$	−11.1516	−1.04445
$115$	0	0
$116$	4.73855	0.439964
$117$	− 2.09464i	− 0.193650i
$118$	14.9730i	1.37838i
$119$	27.3831	2.51021
$120$	0	0
$121$	−5.95347	−0.541225
$122$	28.3449i	2.56622i
$123$	− 2.66629i	− 0.240411i
$124$	−10.6634	−0.957602
$125$	0	0
$126$	−10.2232	−0.910757
$127$	3.23499i	0.287059i	0.989646	+	0.143529i	$0.0458451\pi$
$127$	3.23499i	0.287059i	−0.989646	+	0.143529i	$0.954155\pi$
$128$	− 3.55679i	− 0.314378i
$129$	−6.03681	−0.531511
$130$	0	0
$131$	−13.3567	−1.16698	−0.583489	−	0.812121i	$-0.698313\pi$
$131$	−13.3567	−1.16698	−0.583489	+	0.812121i	$0.698313\pi$
$132$	− 10.6449i	− 0.926521i
$133$	− 16.9184i	− 1.46701i
$134$	30.4940	2.63428
$135$	0	0
$136$	49.4291	4.23851
$137$	− 4.35642i	− 0.372194i	−0.982531	−	0.186097i	$-0.940416\pi$
$137$	− 4.35642i	− 0.372194i	0.982531	−	0.186097i	$-0.0595839\pi$
$138$	− 13.4974i	− 1.14897i
$139$	18.4171	1.56212	0.781061	−	0.624455i	$-0.214679\pi$
$139$	18.4171	1.56212	0.781061	+	0.624455i	$0.214679\pi$
$140$	0	0
$141$	−7.74829	−0.652523
$142$	11.8292i	0.992681i
$143$	4.70551i	0.393495i
$144$	−8.97678	−0.748065
$145$	0	0
$146$	32.0756	2.65460
$147$	− 8.50991i	− 0.701885i
$148$	− 49.8876i	− 4.10073i
$149$	−17.6427	−1.44535	−0.722674	−	0.691189i	$-0.757087\pi$
$149$	−17.6427	−1.44535	−0.722674	+	0.691189i	$0.757087\pi$
$150$	0	0
$151$	4.02381	0.327453	0.163726	−	0.986506i	$-0.447649\pi$
$151$	4.02381	0.327453	0.163726	+	0.986506i	$0.447649\pi$
$152$	− 30.5393i	− 2.47707i
$153$	6.95310i	0.562125i
$154$	22.9659	1.85065
$155$	0	0
$156$	9.92559	0.794683
$157$	7.70119i	0.614622i	0.951609	+	0.307311i	$0.0994292\pi$
$157$	7.70119i	0.614622i	−0.951609	+	0.307311i	$0.900571\pi$
$158$	37.4059i	2.97585i
$159$	1.41309	0.112065
$160$	0	0
$161$	20.4772	1.61383
$162$	− 2.59587i	− 0.203951i
$163$	− 13.6829i	− 1.07173i	−0.844303	−	0.535866i	$-0.819985\pi$
$163$	− 13.6829i	− 1.07173i	0.844303	−	0.535866i	$-0.180015\pi$
$164$	12.6344	0.986578
$165$	0	0
$166$	−9.89734	−0.768183
$167$	24.9607i	1.93151i	0.259449	+	0.965757i	$0.416459\pi$
$167$	24.9607i	1.93151i	−0.259449	+	0.965757i	$0.583541\pi$
$168$	− 27.9968i	− 2.16000i
$169$	8.61246	0.662497
$170$	0	0
$171$	4.29591	0.328516
$172$	− 28.6057i	− 2.18117i
$173$	19.9187i	1.51439i	0.653191	+	0.757194i	$0.273430\pi$
$173$	19.9187i	1.51439i	−0.653191	+	0.757194i	$0.726570\pi$
$174$	−2.59587	−0.196793
$175$	0	0
$176$	20.1659	1.52006
$177$	− 5.76800i	− 0.433549i
$178$	− 5.38227i	− 0.403418i
$179$	−10.8816	−0.813329	−0.406664	−	0.913578i	$-0.633308\pi$
$179$	−10.8816	−0.813329	−0.406664	+	0.913578i	$0.633308\pi$
$180$	0	0
$181$	−13.9372	−1.03594	−0.517972	−	0.855398i	$-0.673313\pi$
$181$	−13.9372	−1.03594	−0.517972	+	0.855398i	$0.673313\pi$
$182$	21.4140i	1.58731i
$183$	− 10.9192i	− 0.807171i
$184$	36.9632	2.72497
$185$	0	0
$186$	5.84162	0.428329
$187$	− 15.6198i	− 1.14223i
$188$	− 36.7157i	− 2.67777i
$189$	3.93826	0.286466
$190$	0	0
$191$	13.3458	0.965670	0.482835	−	0.875711i	$-0.339607\pi$
$191$	13.3458	0.965670	0.482835	+	0.875711i	$0.339607\pi$
$192$	− 5.62918i	− 0.406251i
$193$	15.6337i	1.12534i	0.826683	+	0.562668i	$0.190225\pi$
$193$	15.6337i	1.12534i	−0.826683	+	0.562668i	$0.809775\pi$
$194$	−31.1681	−2.23774
$195$	0	0
$196$	40.3247	2.88033
$197$	20.1140i	1.43306i	0.697556	+	0.716530i	$0.254271\pi$
$197$	20.1140i	1.43306i	−0.697556	+	0.716530i	$0.745729\pi$
$198$	5.83149i	0.414426i
$199$	9.27232	0.657297	0.328649	−	0.944452i	$-0.393407\pi$
$199$	9.27232	0.657297	0.328649	+	0.944452i	$0.393407\pi$
$200$	0	0
$201$	−11.7471	−0.828578
$202$	41.1816i	2.89752i
$203$	− 3.93826i	− 0.276412i
$204$	−32.9476	−2.30680
$205$	0	0
$206$	−26.7684	−1.86504
$207$	5.19955i	0.361394i
$208$	18.8032i	1.30376i
$209$	−9.65053	−0.667541
$210$	0	0
$211$	−0.228998	−0.0157649	−0.00788245	−	0.999969i	$-0.502509\pi$
$211$	−0.228998	−0.0157649	−0.00788245	+	0.999969i	$0.502509\pi$
$212$	6.69600i	0.459883i
$213$	− 4.55691i	− 0.312235i
$214$	−11.6203	−0.794348
$215$	0	0
$216$	7.10893	0.483702
$217$	8.86247i	0.601624i
$218$	34.9879i	2.36968i
$219$	−12.3564	−0.834968
$220$	0	0
$221$	14.5643	0.979699
$222$	27.3294i	1.83423i
$223$	− 25.1117i	− 1.68160i	−0.541342	−	0.840802i	$-0.682084\pi$
$223$	− 25.1117i	− 1.68160i	0.541342	−	0.840802i	$-0.317916\pi$
$224$	35.7780	2.39052
$225$	0	0
$226$	−29.7292	−1.97756
$227$	4.98706i	0.331003i	0.986210	+	0.165501i	$0.0529242\pi$
$227$	4.98706i	0.331003i	−0.986210	+	0.165501i	$0.947076\pi$
$228$	20.3564i	1.34813i
$229$	−1.60591	−0.106122	−0.0530608	−	0.998591i	$-0.516898\pi$
$229$	−1.60591	−0.106122	−0.0530608	+	0.998591i	$0.516898\pi$
$230$	0	0
$231$	−8.84710	−0.582096
$232$	− 7.10893i	− 0.466724i
$233$	6.65588i	0.436041i	0.975944	+	0.218021i	$0.0699600\pi$
$233$	6.65588i	0.436041i	−0.975944	+	0.218021i	$0.930040\pi$
$234$	−5.43743	−0.355456
$235$	0	0
$236$	27.3320	1.77916
$237$	− 14.4098i	− 0.936014i
$238$	− 71.0831i	− 4.60763i
$239$	−2.87227	−0.185792	−0.0928959	−	0.995676i	$-0.529612\pi$
$239$	−2.87227	−0.185792	−0.0928959	+	0.995676i	$0.529612\pi$
$240$	0	0
$241$	15.2768	0.984068	0.492034	−	0.870576i	$-0.336254\pi$
$241$	15.2768	0.984068	0.492034	+	0.870576i	$0.336254\pi$
$242$	15.4545i	0.993450i
$243$	1.00000i	0.0641500i
$244$	51.7412	3.31239
$245$	0	0
$246$	−6.92135	−0.441289
$247$	− 8.99840i	− 0.572554i
$248$	15.9976i	1.01585i
$249$	3.81272	0.241621
$250$	0	0
$251$	10.8782	0.686627	0.343314	−	0.939221i	$-0.388451\pi$
$251$	10.8782	0.686627	0.343314	+	0.939221i	$0.388451\pi$
$252$	18.6617i	1.17557i
$253$	− 11.6805i	− 0.734348i
$254$	8.39761	0.526913
$255$	0	0
$256$	−20.4913	−1.28071
$257$	18.0748i	1.12748i	0.825953	+	0.563739i	$0.190637\pi$
$257$	18.0748i	1.12748i	−0.825953	+	0.563739i	$0.809363\pi$
$258$	15.6708i	0.975621i
$259$	−41.4621	−2.57633
$260$	0	0
$261$	1.00000	0.0618984
$262$	34.6722i	2.14206i
$263$	− 18.4454i	− 1.13739i	−0.822547	−	0.568697i	$-0.807448\pi$
$263$	− 18.4454i	− 1.13739i	0.822547	−	0.568697i	$-0.192552\pi$
$264$	−15.9699	−0.982876
$265$	0	0
$266$	−43.9180	−2.69279
$267$	2.07340i	0.126890i
$268$	− 55.6643i	− 3.40024i
$269$	−11.6702	−0.711544	−0.355772	−	0.934573i	$-0.615782\pi$
$269$	−11.6702	−0.711544	−0.355772	+	0.934573i	$0.615782\pi$
$270$	0	0
$271$	20.2501	1.23011	0.615054	−	0.788485i	$-0.289134\pi$
$271$	20.2501	1.23011	0.615054	+	0.788485i	$0.289134\pi$
$272$	− 62.4165i	− 3.78455i
$273$	− 8.24926i	− 0.499268i
$274$	−11.3087	−0.683184
$275$	0	0
$276$	−24.6383	−1.48305
$277$	− 3.66952i	− 0.220480i	−0.993905	−	0.110240i	$-0.964838\pi$
$277$	− 3.66952i	− 0.220480i	0.993905	−	0.110240i	$-0.0351620\pi$
$278$	− 47.8085i	− 2.86737i
$279$	−2.25035	−0.134725
$280$	0	0
$281$	22.8056	1.36047	0.680233	−	0.732996i	$-0.261879\pi$
$281$	22.8056	1.36047	0.680233	+	0.732996i	$0.261879\pi$
$282$	20.1136i	1.19775i
$283$	− 2.38657i	− 0.141867i	−0.997481	−	0.0709335i	$-0.977402\pi$
$283$	− 2.38657i	− 0.141867i	0.997481	−	0.0709335i	$-0.0225978\pi$
$284$	21.5932	1.28132
$285$	0	0
$286$	12.2149	0.722282
$287$	− 10.5006i	− 0.619828i
$288$	9.08471i	0.535321i
$289$	−31.3456	−1.84386
$290$	0	0
$291$	12.0068	0.703852
$292$	− 58.5514i	− 3.42646i
$293$	19.4843i	1.13828i	0.822239	+	0.569142i	$0.192725\pi$
$293$	19.4843i	1.13828i	−0.822239	+	0.569142i	$0.807275\pi$
$294$	−22.0906	−1.28835
$295$	0	0
$296$	−74.8430	−4.35016
$297$	− 2.24645i	− 0.130352i
$298$	45.7982i	2.65302i
$299$	10.8912	0.629855
$300$	0	0
$301$	−23.7745	−1.37034
$302$	− 10.4453i	− 0.601059i
$303$	− 15.8642i	− 0.911377i
$304$	−38.5634	−2.21176
$305$	0	0
$306$	18.0494	1.03181
$307$	0.527333i	0.0300965i	0.999887	+	0.0150482i	$0.00479018\pi$
$307$	0.527333i	0.0300965i	−0.999887	+	0.0150482i	$0.995210\pi$
$308$	− 41.9225i	− 2.38875i
$309$	10.3119	0.586624
$310$	0	0
$311$	−7.89624	−0.447755	−0.223877	−	0.974617i	$-0.571871\pi$
$311$	−7.89624	−0.447755	−0.223877	+	0.974617i	$0.571871\pi$
$312$	− 14.8907i	− 0.843019i
$313$	10.0064i	0.565597i	0.959179	+	0.282799i	$0.0912628\pi$
$313$	10.0064i	0.565597i	−0.959179	+	0.282799i	$0.908737\pi$
$314$	19.9913	1.12818
$315$	0	0
$316$	68.2814	3.84113
$317$	− 32.5401i	− 1.82763i	−0.406125	−	0.913817i	$-0.633120\pi$
$317$	− 32.5401i	− 1.82763i	0.406125	−	0.913817i	$-0.366880\pi$
$318$	− 3.66820i	− 0.205702i
$319$	−2.24645	−0.125777
$320$	0	0
$321$	4.47645	0.249851
$322$	− 53.1562i	− 2.96228i
$323$	29.8699i	1.66200i
$324$	−4.73855	−0.263253
$325$	0	0
$326$	−35.5192	−1.96723
$327$	− 13.4783i	− 0.745350i
$328$	− 18.9545i	− 1.04659i
$329$	−30.5148	−1.68233
$330$	0	0
$331$	22.9411	1.26096	0.630480	−	0.776206i	$-0.282858\pi$
$331$	22.9411	1.26096	0.630480	+	0.776206i	$0.282858\pi$
$332$	18.0668i	0.991544i
$333$	− 10.5280i	− 0.576932i
$334$	64.7947	3.54541
$335$	0	0
$336$	−35.3529	−1.92866
$337$	− 16.0039i	− 0.871789i	−0.899998	−	0.435894i	$-0.856432\pi$
$337$	− 16.0039i	− 0.871789i	0.899998	−	0.435894i	$-0.143568\pi$
$338$	− 22.3569i	− 1.21605i
$339$	11.4525	0.622013
$340$	0	0
$341$	5.05529	0.273760
$342$	− 11.1516i	− 0.603011i
$343$	− 5.94642i	− 0.321076i
$344$	−42.9153	−2.31384
$345$	0	0
$346$	51.7063	2.77975
$347$	− 22.9990i	− 1.23465i	−0.786707	−	0.617326i	$-0.788216\pi$
$347$	− 22.9990i	− 1.23465i	0.786707	−	0.617326i	$-0.211784\pi$
$348$	4.73855i	0.254013i
$349$	−24.3108	−1.30133	−0.650664	−	0.759366i	$-0.725509\pi$
$349$	−24.3108	−1.30133	−0.650664	+	0.759366i	$0.725509\pi$
$350$	0	0
$351$	2.09464	0.111804
$352$	− 20.4083i	− 1.08777i
$353$	− 29.2685i	− 1.55780i	−0.627146	−	0.778902i	$-0.715777\pi$
$353$	− 29.2685i	− 1.55780i	0.627146	−	0.778902i	$-0.284223\pi$
$354$	−14.9730	−0.795805
$355$	0	0
$356$	−9.82490	−0.520719
$357$	27.3831i	1.44927i
$358$	28.2472i	1.49291i
$359$	−18.5090	−0.976868	−0.488434	−	0.872601i	$-0.662432\pi$
$359$	−18.5090	−0.976868	−0.488434	+	0.872601i	$0.662432\pi$
$360$	0	0
$361$	−0.545179	−0.0286936
$362$	36.1792i	1.90154i
$363$	− 5.95347i	− 0.312476i
$364$	39.0896	2.04885
$365$	0	0
$366$	−28.3449	−1.48161
$367$	− 0.896398i	− 0.0467916i	−0.999726	−	0.0233958i	$-0.992552\pi$
$367$	− 0.896398i	− 0.0467916i	0.999726	−	0.0233958i	$-0.00744779\pi$
$368$	− 46.6752i	− 2.43311i
$369$	2.66629	0.138802
$370$	0	0
$371$	5.56511	0.288926
$372$	− 10.6634i	− 0.552872i
$373$	4.64559i	0.240540i	0.992741	+	0.120270i	$0.0383760\pi$
$373$	4.64559i	0.240540i	−0.992741	+	0.120270i	$0.961624\pi$
$374$	−40.5469	−2.09663
$375$	0	0
$376$	−55.0821	−2.84064
$377$	− 2.09464i	− 0.107880i
$378$	− 10.2232i	− 0.525826i
$379$	32.4863	1.66871	0.834356	−	0.551226i	$-0.185840\pi$
$379$	32.4863	1.66871	0.834356	+	0.551226i	$0.185840\pi$
$380$	0	0
$381$	−3.23499	−0.165733
$382$	− 34.6441i	− 1.77254i
$383$	9.48311i	0.484564i	0.970206	+	0.242282i	$0.0778959\pi$
$383$	9.48311i	0.484564i	−0.970206	+	0.242282i	$0.922104\pi$
$384$	3.55679	0.181507
$385$	0	0
$386$	40.5830	2.06562
$387$	− 6.03681i	− 0.306868i
$388$	56.8949i	2.88840i
$389$	−33.9509	−1.72138	−0.860689	−	0.509132i	$-0.829967\pi$
$389$	−33.9509	−1.72138	−0.860689	+	0.509132i	$0.829967\pi$
$390$	0	0
$391$	−36.1530	−1.82833
$392$	− 60.4964i	− 3.05553i
$393$	− 13.3567i	− 0.673755i
$394$	52.2132	2.63047
$395$	0	0
$396$	10.6449	0.534927
$397$	25.0892i	1.25919i	0.776923	+	0.629596i	$0.216780\pi$
$397$	25.0892i	1.25919i	−0.776923	+	0.629596i	$0.783220\pi$
$398$	− 24.0698i	− 1.20651i
$399$	16.9184	0.846980
$400$	0	0
$401$	−4.85814	−0.242604	−0.121302	−	0.992616i	$-0.538707\pi$
$401$	−4.85814	−0.242604	−0.121302	+	0.992616i	$0.538707\pi$
$402$	30.4940i	1.52090i
$403$	4.71368i	0.234805i
$404$	75.1736	3.74003
$405$	0	0
$406$	−10.2232	−0.507370
$407$	23.6506i	1.17232i
$408$	49.4291i	2.44711i
$409$	−14.6604	−0.724911	−0.362456	−	0.932001i	$-0.618062\pi$
$409$	−14.6604	−0.724911	−0.362456	+	0.932001i	$0.618062\pi$
$410$	0	0
$411$	4.35642	0.214886
$412$	48.8635i	2.40733i
$413$	− 22.7159i	− 1.11778i
$414$	13.4974	0.663359
$415$	0	0
$416$	19.0292	0.932985
$417$	18.4171i	0.901891i
$418$	25.0515i	1.22531i
$419$	18.4123	0.899502	0.449751	−	0.893154i	$-0.351513\pi$
$419$	18.4123	0.899502	0.449751	+	0.893154i	$0.351513\pi$
$420$	0	0
$421$	29.1489	1.42063	0.710316	−	0.703883i	$-0.248552\pi$
$421$	29.1489	1.42063	0.710316	+	0.703883i	$0.248552\pi$
$422$	0.594451i	0.0289374i
$423$	− 7.74829i	− 0.376735i
$424$	10.0456	0.487855
$425$	0	0
$426$	−11.8292	−0.573125
$427$	− 43.0027i	− 2.08105i
$428$	21.2119i	1.02532i
$429$	−4.70551	−0.227184
$430$	0	0
$431$	−2.05315	−0.0988966	−0.0494483	−	0.998777i	$-0.515746\pi$
$431$	−2.05315	−0.0988966	−0.0494483	+	0.998777i	$0.515746\pi$
$432$	− 8.97678i	− 0.431896i
$433$	− 18.7820i	− 0.902603i	−0.892371	−	0.451302i	$-0.850960\pi$
$433$	− 18.7820i	− 0.902603i	0.892371	−	0.451302i	$-0.149040\pi$
$434$	23.0058	1.10432
$435$	0	0
$436$	63.8675	3.05870
$437$	22.3368i	1.06851i
$438$	32.0756i	1.53263i
$439$	33.7889	1.61266	0.806329	−	0.591467i	$-0.201451\pi$
$439$	33.7889	1.61266	0.806329	+	0.591467i	$0.201451\pi$
$440$	0	0
$441$	8.50991	0.405234
$442$	− 37.8070i	− 1.79830i
$443$	24.2962i	1.15435i	0.816622	+	0.577173i	$0.195844\pi$
$443$	24.2962i	1.15435i	−0.816622	+	0.577173i	$0.804156\pi$
$444$	49.8876	2.36756
$445$	0	0
$446$	−65.1868	−3.08668
$447$	− 17.6427i	− 0.834472i
$448$	− 22.1692i	− 1.04740i
$449$	21.0976	0.995656	0.497828	−	0.867276i	$-0.334131\pi$
$449$	21.0976	0.995656	0.497828	+	0.867276i	$0.334131\pi$
$450$	0	0
$451$	−5.98968	−0.282043
$452$	54.2682i	2.55256i
$453$	4.02381i	0.189055i
$454$	12.9458	0.607575
$455$	0	0
$456$	30.5393	1.43013
$457$	− 8.70983i	− 0.407429i	−0.979030	−	0.203714i	$-0.934699\pi$
$457$	− 8.70983i	− 0.407429i	0.979030	−	0.203714i	$-0.0653014\pi$
$458$	4.16874i	0.194793i
$459$	−6.95310	−0.324543
$460$	0	0
$461$	29.5424	1.37592	0.687962	−	0.725746i	$-0.258505\pi$
$461$	29.5424	1.37592	0.687962	+	0.725746i	$0.258505\pi$
$462$	22.9659i	1.06847i
$463$	37.3067i	1.73379i	0.498493	+	0.866894i	$0.333887\pi$
$463$	37.3067i	1.73379i	−0.498493	+	0.866894i	$0.666113\pi$
$464$	−8.97678	−0.416737
$465$	0	0
$466$	17.2778	0.800379
$467$	− 30.2144i	− 1.39816i	−0.715045	−	0.699079i	$-0.753594\pi$
$467$	− 30.2144i	− 1.39816i	0.715045	−	0.699079i	$-0.246406\pi$
$468$	9.92559i	0.458810i
$469$	−46.2632	−2.13624
$470$	0	0
$471$	−7.70119	−0.354852
$472$	− 41.0043i	− 1.88738i
$473$	13.5614i	0.623553i
$474$	−37.4059	−1.71811
$475$	0	0
$476$	−129.756	−5.94738
$477$	1.41309i	0.0647009i
$478$	7.45605i	0.341032i
$479$	−39.1511	−1.78886	−0.894430	−	0.447209i	$-0.852418\pi$
$479$	−39.1511	−1.78886	−0.894430	+	0.447209i	$0.852418\pi$
$480$	0	0
$481$	−22.0524	−1.00551
$482$	− 39.6567i	− 1.80632i
$483$	20.4772i	0.931744i
$484$	28.2108	1.28231
$485$	0	0
$486$	2.59587	0.117751
$487$	23.9182i	1.08384i	0.840430	+	0.541919i	$0.182302\pi$
$487$	23.9182i	1.08384i	−0.840430	+	0.541919i	$0.817698\pi$
$488$	− 77.6239i	− 3.51387i
$489$	13.6829	0.618764
$490$	0	0
$491$	4.40974	0.199009	0.0995044	−	0.995037i	$-0.468274\pi$
$491$	4.40974	0.199009	0.0995044	+	0.995037i	$0.468274\pi$
$492$	12.6344i	0.569601i
$493$	6.95310i	0.313152i
$494$	−23.3587	−1.05096
$495$	0	0
$496$	20.2009	0.907047
$497$	− 17.9463i	− 0.805002i
$498$	− 9.89734i	− 0.443510i
$499$	37.0446	1.65834	0.829171	−	0.558995i	$-0.188813\pi$
$499$	37.0446	1.65834	0.829171	+	0.558995i	$0.188813\pi$
$500$	0	0
$501$	−24.9607	−1.11516
$502$	− 28.2385i	− 1.26034i
$503$	− 12.6691i	− 0.564886i	−0.959284	−	0.282443i	$-0.908855\pi$
$503$	− 12.6691i	− 0.564886i	0.959284	−	0.282443i	$-0.0911448\pi$
$504$	27.9968	1.24708
$505$	0	0
$506$	−30.3211	−1.34794
$507$	8.61246i	0.382493i
$508$	− 15.3292i	− 0.680121i
$509$	11.7078	0.518939	0.259469	−	0.965751i	$-0.416452\pi$
$509$	11.7078	0.518939	0.259469	+	0.965751i	$0.416452\pi$
$510$	0	0
$511$	−48.6627	−2.15271
$512$	46.0793i	2.03644i
$513$	4.29591i	0.189669i
$514$	46.9200	2.06955
$515$	0	0
$516$	28.6057	1.25930
$517$	17.4061i	0.765520i
$518$	107.630i	4.72900i
$519$	−19.9187	−0.874332
$520$	0	0
$521$	0.0559550	0.00245143	0.00122572	−	0.999999i	$-0.499610\pi$
$521$	0.0559550	0.00245143	0.00122572	+	0.999999i	$0.499610\pi$
$522$	− 2.59587i	− 0.113618i
$523$	− 0.256669i	− 0.0112234i	−0.999984	−	0.00561168i	$-0.998214\pi$
$523$	− 0.256669i	− 0.0112234i	0.999984	−	0.00561168i	$-0.00178626\pi$
$524$	63.2913	2.76489
$525$	0	0
$526$	−47.8819	−2.08775
$527$	− 15.6469i	− 0.681590i
$528$	20.1659i	0.877607i
$529$	−4.03530	−0.175448
$530$	0	0
$531$	5.76800	0.250310
$532$	80.1688i	3.47576i
$533$	− 5.58493i	− 0.241910i
$534$	5.38227	0.232914
$535$	0	0
$536$	−83.5095	−3.60706
$537$	− 10.8816i	− 0.469576i
$538$	30.2943i	1.30608i
$539$	−19.1171	−0.823430
$540$	0	0
$541$	1.88887	0.0812090	0.0406045	−	0.999175i	$-0.487072\pi$
$541$	1.88887	0.0812090	0.0406045	+	0.999175i	$0.487072\pi$
$542$	− 52.5668i	− 2.25794i
$543$	− 13.9372i	− 0.598102i
$544$	−63.1669	−2.70826
$545$	0	0
$546$	−21.4140	−0.916436
$547$	0.193341i	0.00826668i	0.999991	+	0.00413334i	$0.00131569\pi$
$547$	0.193341i	0.00826668i	−0.999991	+	0.00413334i	$0.998684\pi$
$548$	20.6431i	0.881831i
$549$	10.9192	0.466020
$550$	0	0
$551$	4.29591	0.183012
$552$	36.9632i	1.57326i
$553$	− 56.7494i	− 2.41323i
$554$	−9.52561	−0.404704
$555$	0	0
$556$	−87.2706	−3.70110
$557$	26.4282i	1.11980i	0.828561	+	0.559899i	$0.189160\pi$
$557$	26.4282i	1.11980i	−0.828561	+	0.559899i	$0.810840\pi$
$558$	5.84162i	0.247296i
$559$	−12.6450	−0.534825
$560$	0	0
$561$	15.6198	0.659467
$562$	− 59.2003i	− 2.49722i
$563$	− 6.18961i	− 0.260861i	−0.991457	−	0.130430i	$-0.958364\pi$
$563$	− 6.18961i	− 0.260861i	0.991457	−	0.130430i	$-0.0416359\pi$
$564$	36.7157	1.54601
$565$	0	0
$566$	−6.19523	−0.260405
$567$	3.93826i	0.165391i
$568$	− 32.3948i	− 1.35926i
$569$	15.9034	0.666705	0.333353	−	0.942802i	$-0.391820\pi$
$569$	15.9034	0.666705	0.333353	+	0.942802i	$0.391820\pi$
$570$	0	0
$571$	−16.8555	−0.705380	−0.352690	−	0.935740i	$-0.614733\pi$
$571$	−16.8555	−0.705380	−0.352690	+	0.935740i	$0.614733\pi$
$572$	− 22.2973i	− 0.932297i
$573$	13.3458i	0.557530i
$574$	−27.2581	−1.13773
$575$	0	0
$576$	5.62918	0.234549
$577$	23.1237i	0.962650i	0.876542	+	0.481325i	$0.159844\pi$
$577$	23.1237i	0.962650i	−0.876542	+	0.481325i	$0.840156\pi$
$578$	81.3692i	3.38451i
$579$	−15.6337	−0.649713
$580$	0	0
$581$	15.0155	0.622948
$582$	− 31.1681i	− 1.29196i
$583$	− 3.17443i	− 0.131471i
$584$	−87.8408	−3.63488
$585$	0	0
$586$	50.5787	2.08939
$587$	7.80902i	0.322313i	0.986929	+	0.161156i	$0.0515224\pi$
$587$	7.80902i	0.322313i	−0.986929	+	0.161156i	$0.948478\pi$
$588$	40.3247i	1.66296i
$589$	−9.66730	−0.398334
$590$	0	0
$591$	−20.1140	−0.827377
$592$	94.5077i	3.88424i
$593$	19.5804i	0.804069i	0.915625	+	0.402034i	$0.131697\pi$
$593$	19.5804i	0.804069i	−0.915625	+	0.402034i	$0.868303\pi$
$594$	−5.83149	−0.239269
$595$	0	0
$596$	83.6009	3.42443
$597$	9.27232i	0.379491i
$598$	− 28.2722i	− 1.15614i
$599$	−30.2322	−1.23525	−0.617627	−	0.786471i	$-0.711906\pi$
$599$	−30.2322	−1.23525	−0.617627	+	0.786471i	$0.711906\pi$
$600$	0	0
$601$	33.1180	1.35091	0.675457	−	0.737400i	$-0.263947\pi$
$601$	33.1180	1.35091	0.675457	+	0.737400i	$0.263947\pi$
$602$	61.7157i	2.51534i
$603$	− 11.7471i	− 0.478380i
$604$	−19.0670	−0.775826
$605$	0	0
$606$	−41.1816	−1.67289
$607$	23.4711i	0.952661i	0.879266	+	0.476330i	$0.158033\pi$
$607$	23.4711i	0.952661i	−0.879266	+	0.476330i	$0.841967\pi$
$608$	39.0271i	1.58276i
$609$	3.93826	0.159586
$610$	0	0
$611$	−16.2299	−0.656592
$612$	− 32.9476i	− 1.33183i
$613$	24.1560i	0.975650i	0.872941	+	0.487825i	$0.162210\pi$
$613$	24.1560i	0.975650i	−0.872941	+	0.487825i	$0.837790\pi$
$614$	1.36889	0.0552439
$615$	0	0
$616$	−62.8935	−2.53405
$617$	47.7509i	1.92238i	0.275890	+	0.961189i	$0.411027\pi$
$617$	47.7509i	1.92238i	−0.275890	+	0.961189i	$0.588973\pi$
$618$	− 26.7684i	− 1.07678i
$619$	−0.779735	−0.0313402	−0.0156701	−	0.999877i	$-0.504988\pi$
$619$	−0.779735	−0.0313402	−0.0156701	+	0.999877i	$0.504988\pi$
$620$	0	0
$621$	−5.19955	−0.208651
$622$	20.4976i	0.821880i
$623$	8.16558i	0.327147i
$624$	−18.8032	−0.752729
$625$	0	0
$626$	25.9754	1.03819
$627$	− 9.65053i	− 0.385405i
$628$	− 36.4925i	− 1.45621i
$629$	73.2023	2.91877
$630$	0	0
$631$	17.8120	0.709083	0.354542	−	0.935040i	$-0.384637\pi$
$631$	17.8120	0.709083	0.354542	+	0.935040i	$0.384637\pi$
$632$	− 102.438i	− 4.07476i
$633$	− 0.228998i	− 0.00910187i
$634$	−84.4700	−3.35473
$635$	0	0
$636$	−6.69600	−0.265514
$637$	− 17.8252i	− 0.706262i
$638$	5.83149i	0.230871i
$639$	4.55691	0.180269
$640$	0	0
$641$	29.7647	1.17563	0.587817	−	0.808994i	$-0.299987\pi$
$641$	29.7647	1.17563	0.587817	+	0.808994i	$0.299987\pi$
$642$	− 11.6203i	− 0.458617i
$643$	− 41.4049i	− 1.63285i	−0.577451	−	0.816426i	$-0.695952\pi$
$643$	− 41.4049i	− 1.63285i	0.577451	−	0.816426i	$-0.304048\pi$
$644$	−97.0322	−3.82361
$645$	0	0
$646$	77.5384	3.05071
$647$	− 14.6074i	− 0.574275i	−0.957889	−	0.287137i	$-0.907296\pi$
$647$	− 14.6074i	− 0.574275i	0.957889	−	0.287137i	$-0.0927036\pi$
$648$	7.10893i	0.279265i
$649$	−12.9575	−0.508627
$650$	0	0
$651$	−8.86247	−0.347348
$652$	64.8374i	2.53923i
$653$	30.5647i	1.19609i	0.801462	+	0.598045i	$0.204056\pi$
$653$	30.5647i	1.19609i	−0.801462	+	0.598045i	$0.795944\pi$
$654$	−34.9879	−1.36813
$655$	0	0
$656$	−23.9347	−0.934493
$657$	− 12.3564i	− 0.482069i
$658$	79.2125i	3.08802i
$659$	−22.1893	−0.864371	−0.432186	−	0.901785i	$-0.642257\pi$
$659$	−22.1893	−0.864371	−0.432186	+	0.901785i	$0.642257\pi$
$660$	0	0
$661$	−9.85731	−0.383405	−0.191702	−	0.981453i	$-0.561401\pi$
$661$	−9.85731	−0.383405	−0.191702	+	0.981453i	$0.561401\pi$
$662$	− 59.5523i	− 2.31457i
$663$	14.5643i	0.565630i
$664$	27.1044	1.05185
$665$	0	0
$666$	−27.3294	−1.05899
$667$	5.19955i	0.201327i
$668$	− 118.277i	− 4.57629i
$669$	25.1117	0.970875
$670$	0	0
$671$	−24.5294	−0.946948
$672$	35.7780i	1.38016i
$673$	− 25.3057i	− 0.975463i	−0.872994	−	0.487732i	$-0.837824\pi$
$673$	− 25.3057i	− 0.975463i	0.872994	−	0.487732i	$-0.162176\pi$
$674$	−41.5441	−1.60022
$675$	0	0
$676$	−40.8106	−1.56964
$677$	3.67306i	0.141167i	0.997506	+	0.0705836i	$0.0224862\pi$
$677$	3.67306i	0.141167i	−0.997506	+	0.0705836i	$0.977514\pi$
$678$	− 29.7292i	− 1.14174i
$679$	47.2860	1.81467
$680$	0	0
$681$	−4.98706	−0.191104
$682$	− 13.1229i	− 0.502502i
$683$	17.3807i	0.665053i	0.943094	+	0.332527i	$0.107901\pi$
$683$	17.3807i	0.665053i	−0.943094	+	0.332527i	$0.892099\pi$
$684$	−20.3564	−0.778346
$685$	0	0
$686$	−15.4361	−0.589354
$687$	− 1.60591i	− 0.0612694i
$688$	54.1911i	2.06602i
$689$	2.95992	0.112764
$690$	0	0
$691$	−28.5585	−1.08642	−0.543209	−	0.839598i	$-0.682791\pi$
$691$	−28.5585	−1.08642	−0.543209	+	0.839598i	$0.682791\pi$
$692$	− 94.3856i	− 3.58800i
$693$	− 8.84710i	− 0.336074i
$694$	−59.7025	−2.26628
$695$	0	0
$696$	7.10893	0.269463
$697$	18.5390i	0.702214i
$698$	63.1078i	2.38866i
$699$	−6.65588	−0.251749
$700$	0	0
$701$	21.5401	0.813557	0.406779	−	0.913527i	$-0.366652\pi$
$701$	21.5401	0.813557	0.406779	+	0.913527i	$0.366652\pi$
$702$	− 5.43743i	− 0.205223i
$703$	− 45.2274i	− 1.70578i
$704$	−12.6457	−0.476601
$705$	0	0
$706$	−75.9772	−2.85944
$707$	− 62.4776i	− 2.34971i
$708$	27.3320i	1.02720i
$709$	2.29948	0.0863587	0.0431793	−	0.999067i	$-0.486251\pi$
$709$	2.29948	0.0863587	0.0431793	+	0.999067i	$0.486251\pi$
$710$	0	0
$711$	14.4098	0.540408
$712$	14.7396i	0.552391i
$713$	− 11.7008i	− 0.438199i
$714$	71.0831	2.66022
$715$	0	0
$716$	51.5630	1.92700
$717$	− 2.87227i	− 0.107267i
$718$	48.0470i	1.79310i
$719$	−13.6814	−0.510232	−0.255116	−	0.966910i	$-0.582114\pi$
$719$	−13.6814	−0.510232	−0.255116	+	0.966910i	$0.582114\pi$
$720$	0	0
$721$	40.6110	1.51243
$722$	1.41521i	0.0526688i
$723$	15.2768i	0.568152i
$724$	66.0422	2.45444
$725$	0	0
$726$	−15.4545	−0.573569
$727$	− 50.0451i	− 1.85607i	−0.372492	−	0.928035i	$-0.621497\pi$
$727$	− 50.0451i	− 1.85607i	0.372492	−	0.928035i	$-0.378503\pi$
$728$	− 58.6435i	− 2.17347i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	41.9745	1.55248
$732$	51.7412i	1.91241i
$733$	38.7734i	1.43213i	0.698034	+	0.716065i	$0.254058\pi$
$733$	38.7734i	1.43213i	−0.698034	+	0.716065i	$0.745942\pi$
$734$	−2.32694	−0.0858887
$735$	0	0
$736$	−47.2364	−1.74116
$737$	26.3893i	0.972062i
$738$	− 6.92135i	− 0.254778i
$739$	18.0707	0.664742	0.332371	−	0.943149i	$-0.392151\pi$
$739$	18.0707	0.664742	0.332371	+	0.943149i	$0.392151\pi$
$740$	0	0
$741$	8.99840	0.330564
$742$	− 14.4463i	− 0.530341i
$743$	− 20.9864i	− 0.769917i	−0.922934	−	0.384959i	$-0.874216\pi$
$743$	− 20.9864i	− 0.769917i	0.922934	−	0.384959i	$-0.125784\pi$
$744$	−15.9976	−0.586500
$745$	0	0
$746$	12.0594	0.441525
$747$	3.81272i	0.139500i
$748$	74.0151i	2.70626i
$749$	17.6294	0.644166
$750$	0	0
$751$	−20.3274	−0.741758	−0.370879	−	0.928681i	$-0.620944\pi$
$751$	−20.3274	−0.741758	−0.370879	+	0.928681i	$0.620944\pi$
$752$	69.5547i	2.53640i
$753$	10.8782i	0.396424i
$754$	−5.43743	−0.198020
$755$	0	0
$756$	−18.6617	−0.678718
$757$	− 0.0654846i	− 0.00238008i	−0.999999	−	0.00119004i	$-0.999621\pi$
$757$	− 0.0654846i	− 0.00238008i	0.999999	−	0.00119004i	$-0.000378801\pi$
$758$	− 84.3304i	− 3.06302i
$759$	11.6805	0.423976
$760$	0	0
$761$	−28.8872	−1.04716	−0.523580	−	0.851976i	$-0.675404\pi$
$761$	−28.8872	−1.04716	−0.523580	+	0.851976i	$0.675404\pi$
$762$	8.39761i	0.304213i
$763$	− 53.0810i	− 1.92166i
$764$	−63.2399	−2.28794
$765$	0	0
$766$	24.6169	0.889446
$767$	− 12.0819i	− 0.436252i
$768$	− 20.4913i	− 0.739417i
$769$	10.0813	0.363541	0.181771	−	0.983341i	$-0.441817\pi$
$769$	10.0813	0.363541	0.181771	+	0.983341i	$0.441817\pi$
$770$	0	0
$771$	−18.0748	−0.650949
$772$	− 74.0809i	− 2.66623i
$773$	− 20.5353i	− 0.738602i	−0.929310	−	0.369301i	$-0.879597\pi$
$773$	− 20.5353i	− 0.738602i	0.929310	−	0.369301i	$-0.120403\pi$
$774$	−15.6708	−0.563275
$775$	0	0
$776$	85.3556	3.06409
$777$	− 41.4621i	− 1.48744i
$778$	88.1321i	3.15969i
$779$	11.4541	0.410387
$780$	0	0
$781$	−10.2369	−0.366304
$782$	93.8485i	3.35602i
$783$	1.00000i	0.0357371i
$784$	−76.3916	−2.72827
$785$	0	0
$786$	−34.6722	−1.23672
$787$	46.0601i	1.64186i	0.571026	+	0.820932i	$0.306545\pi$
$787$	46.0601i	1.64186i	−0.571026	+	0.820932i	$0.693455\pi$
$788$	− 95.3110i	− 3.39531i
$789$	18.4454	0.656674
$790$	0	0
$791$	45.1029	1.60367
$792$	− 15.9699i	− 0.567464i
$793$	− 22.8718i	− 0.812203i
$794$	65.1284	2.31132
$795$	0	0
$796$	−43.9374	−1.55732
$797$	− 37.4745i	− 1.32742i	−0.747992	−	0.663708i	$-0.768982\pi$
$797$	− 37.4745i	− 1.32742i	0.747992	−	0.663708i	$-0.231018\pi$
$798$	− 43.9180i	− 1.55468i
$799$	53.8746	1.90595
$800$	0	0
$801$	−2.07340	−0.0732599
$802$	12.6111i	0.445314i
$803$	27.7580i	0.979558i
$804$	55.6643	1.96313
$805$	0	0
$806$	12.2361	0.430999
$807$	− 11.6702i	− 0.410810i
$808$	− 112.778i	− 3.96751i
$809$	−29.9284	−1.05223	−0.526113	−	0.850414i	$-0.676351\pi$
$809$	−29.9284	−1.05223	−0.526113	+	0.850414i	$0.676351\pi$
$810$	0	0
$811$	−49.1738	−1.72673	−0.863363	−	0.504583i	$-0.831646\pi$
$811$	−49.1738	−1.72673	−0.863363	+	0.504583i	$0.831646\pi$
$812$	18.6617i	0.654896i
$813$	20.2501i	0.710204i
$814$	61.3940	2.15186
$815$	0	0
$816$	62.4165	2.18501
$817$	− 25.9336i	− 0.907301i
$818$	38.0566i	1.33062i
$819$	8.24926	0.288252
$820$	0	0
$821$	33.9715	1.18561	0.592807	−	0.805345i	$-0.298020\pi$
$821$	33.9715	1.18561	0.592807	+	0.805345i	$0.298020\pi$
$822$	− 11.3087i	− 0.394437i
$823$	33.8272i	1.17914i	0.807716	+	0.589572i	$0.200703\pi$
$823$	33.8272i	1.17914i	−0.807716	+	0.589572i	$0.799297\pi$
$824$	73.3067	2.55376
$825$	0	0
$826$	−58.9676	−2.05174
$827$	− 35.1083i	− 1.22083i	−0.792080	−	0.610417i	$-0.791002\pi$
$827$	− 35.1083i	− 1.22083i	0.792080	−	0.610417i	$-0.208998\pi$
$828$	− 24.6383i	− 0.856241i
$829$	−15.6701	−0.544245	−0.272122	−	0.962263i	$-0.587726\pi$
$829$	−15.6701	−0.544245	−0.272122	+	0.962263i	$0.587726\pi$
$830$	0	0
$831$	3.66952	0.127294
$832$	− 11.7911i	− 0.408784i
$833$	59.1702i	2.05013i
$834$	47.8085	1.65547
$835$	0	0
$836$	45.7296	1.58159
$837$	− 2.25035i	− 0.0777835i
$838$	− 47.7961i	− 1.65109i
$839$	16.5131	0.570095	0.285047	−	0.958513i	$-0.407991\pi$
$839$	16.5131	0.570095	0.285047	+	0.958513i	$0.407991\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 75.6669i	− 2.60765i
$843$	22.8056i	0.785466i
$844$	1.08512	0.0373514
$845$	0	0
$846$	−20.1136	−0.691518
$847$	− 23.4463i	− 0.805626i
$848$	− 12.6850i	− 0.435604i
$849$	2.38657	0.0819069
$850$	0	0
$851$	54.7409	1.87649
$852$	21.5932i	0.739770i
$853$	− 16.3926i	− 0.561271i	−0.959814	−	0.280636i	$-0.909455\pi$
$853$	− 16.3926i	− 0.561271i	0.959814	−	0.280636i	$-0.0905452\pi$
$854$	−111.629	−3.81988
$855$	0	0
$856$	31.8228	1.08768
$857$	− 9.83745i	− 0.336041i	−0.985784	−	0.168020i	$-0.946263\pi$
$857$	− 9.83745i	− 0.336041i	0.985784	−	0.168020i	$-0.0537375\pi$
$858$	12.2149i	0.417010i
$859$	−39.8257	−1.35883	−0.679417	−	0.733752i	$-0.737767\pi$
$859$	−39.8257	−1.35883	−0.679417	+	0.733752i	$0.737767\pi$
$860$	0	0
$861$	10.5006	0.357858
$862$	5.32971i	0.181530i
$863$	15.6707i	0.533438i	0.963774	+	0.266719i	$0.0859395\pi$
$863$	15.6707i	0.533438i	−0.963774	+	0.266719i	$0.914060\pi$
$864$	−9.08471	−0.309068
$865$	0	0
$866$	−48.7556	−1.65678
$867$	− 31.3456i	− 1.06455i
$868$	− 41.9953i	− 1.42541i
$869$	−32.3708	−1.09810
$870$	0	0
$871$	−24.6060	−0.833744
$872$	− 95.8162i	− 3.24474i
$873$	12.0068i	0.406369i
$874$	57.9834	1.96132
$875$	0	0
$876$	58.5514	1.97827
$877$	21.2819i	0.718640i	0.933214	+	0.359320i	$0.116991\pi$
$877$	21.2819i	0.718640i	−0.933214	+	0.359320i	$0.883009\pi$
$878$	− 87.7118i	− 2.96013i
$879$	−19.4843	−0.657189
$880$	0	0
$881$	29.1037	0.980529	0.490265	−	0.871574i	$-0.336900\pi$
$881$	29.1037	0.980529	0.490265	+	0.871574i	$0.336900\pi$
$882$	− 22.0906i	− 0.743830i
$883$	− 4.03715i	− 0.135861i	−0.997690	−	0.0679305i	$-0.978360\pi$
$883$	− 4.03715i	− 0.135861i	0.997690	−	0.0679305i	$-0.0216396\pi$
$884$	−69.0136	−2.32118
$885$	0	0
$886$	63.0698	2.11887
$887$	2.85498i	0.0958608i	0.998851	+	0.0479304i	$0.0152626\pi$
$887$	2.85498i	0.0958608i	−0.998851	+	0.0479304i	$0.984737\pi$
$888$	− 74.8430i	− 2.51157i
$889$	−12.7402	−0.427293
$890$	0	0
$891$	2.24645	0.0752588
$892$	118.993i	3.98419i
$893$	− 33.2859i	− 1.11387i
$894$	−45.7982	−1.53172
$895$	0	0
$896$	14.0076	0.467960
$897$	10.8912i	0.363647i
$898$	− 54.7666i	− 1.82758i
$899$	−2.25035	−0.0750534
$900$	0	0
$901$	−9.82535	−0.327330
$902$	15.5485i	0.517707i
$903$	− 23.7745i	− 0.791167i
$904$	81.4150	2.70782
$905$	0	0
$906$	10.4453	0.347022
$907$	3.08757i	0.102521i	0.998685	+	0.0512605i	$0.0163239\pi$
$907$	3.08757i	0.102521i	−0.998685	+	0.0512605i	$0.983676\pi$
$908$	− 23.6314i	− 0.784237i
$909$	15.8642	0.526184
$910$	0	0
$911$	34.1306	1.13080	0.565398	−	0.824818i	$-0.308723\pi$
$911$	34.1306	1.13080	0.565398	+	0.824818i	$0.308723\pi$
$912$	− 38.5634i	− 1.27696i
$913$	− 8.56508i	− 0.283463i
$914$	−22.6096	−0.747860
$915$	0	0
$916$	7.60970	0.251432
$917$	− 52.6021i	− 1.73707i
$918$	18.0494i	0.595718i
$919$	−16.9603	−0.559468	−0.279734	−	0.960078i	$-0.590246\pi$
$919$	−16.9603	−0.559468	−0.279734	+	0.960078i	$0.590246\pi$
$920$	0	0
$921$	−0.527333	−0.0173762
$922$	− 76.6882i	− 2.52559i
$923$	− 9.54511i	− 0.314181i
$924$	41.9225	1.37915
$925$	0	0
$926$	96.8433	3.18247
$927$	10.3119i	0.338688i
$928$	9.08471i	0.298220i
$929$	−5.63831	−0.184987	−0.0924935	−	0.995713i	$-0.529484\pi$
$929$	−5.63831	−0.184987	−0.0924935	+	0.995713i	$0.529484\pi$
$930$	0	0
$931$	36.5578	1.19813
$932$	− 31.5392i	− 1.03310i
$933$	− 7.89624i	− 0.258511i
$934$	−78.4328	−2.56640
$935$	0	0
$936$	14.8907	0.486718
$937$	− 32.0790i	− 1.04798i	−0.851726	−	0.523988i	$-0.824444\pi$
$937$	− 32.0790i	− 1.04798i	0.851726	−	0.523988i	$-0.175556\pi$
$938$	120.093i	3.92119i
$939$	−10.0064	−0.326548
$940$	0	0
$941$	31.2028	1.01718	0.508592	−	0.861008i	$-0.330166\pi$
$941$	31.2028	1.01718	0.508592	+	0.861008i	$0.330166\pi$
$942$	19.9913i	0.651352i
$943$	13.8635i	0.451458i
$944$	−51.7781	−1.68523
$945$	0	0
$946$	35.2036	1.14457
$947$	27.7302i	0.901110i	0.892749	+	0.450555i	$0.148774\pi$
$947$	27.7302i	0.901110i	−0.892749	+	0.450555i	$0.851226\pi$
$948$	68.2814i	2.21768i
$949$	−25.8822	−0.840173
$950$	0	0
$951$	32.5401	1.05519
$952$	194.665i	6.30913i
$953$	23.9481i	0.775755i	0.921711	+	0.387877i	$0.126792\pi$
$953$	23.9481i	0.775755i	−0.921711	+	0.387877i	$0.873208\pi$
$954$	3.66820	0.118762
$955$	0	0
$956$	13.6104	0.440192
$957$	− 2.24645i	− 0.0726173i
$958$	101.631i	3.28356i
$959$	17.1567	0.554020
$960$	0	0
$961$	−25.9359	−0.836643
$962$	57.2453i	1.84566i
$963$	4.47645i	0.144252i
$964$	−72.3902	−2.33153
$965$	0	0
$966$	53.1562	1.71027
$967$	43.6821i	1.40472i	0.711821	+	0.702361i	$0.247871\pi$
$967$	43.6821i	1.40472i	−0.711821	+	0.702361i	$0.752129\pi$
$968$	− 42.3228i	− 1.36031i
$969$	−29.8699	−0.959559
$970$	0	0
$971$	1.12406	0.0360729	0.0180365	−	0.999837i	$-0.494259\pi$
$971$	1.12406	0.0360729	0.0180365	+	0.999837i	$0.494259\pi$
$972$	− 4.73855i	− 0.151989i
$973$	72.5315i	2.32525i
$974$	62.0887	1.98945
$975$	0	0
$976$	−98.0193	−3.13752
$977$	18.8819i	0.604086i	0.953294	+	0.302043i	$0.0976686\pi$
$977$	18.8819i	0.604086i	−0.953294	+	0.302043i	$0.902331\pi$
$978$	− 35.5192i	− 1.13578i
$979$	4.65778	0.148863
$980$	0	0
$981$	13.4783	0.430328
$982$	− 11.4471i	− 0.365292i
$983$	− 23.5807i	− 0.752109i	−0.926598	−	0.376054i	$-0.877281\pi$
$983$	− 23.5807i	− 0.752109i	0.926598	−	0.376054i	$-0.122719\pi$
$984$	18.9545	0.604247
$985$	0	0
$986$	18.0494	0.574809
$987$	− 30.5148i	− 0.971296i
$988$	42.6394i	1.35654i
$989$	31.3887	0.998102
$990$	0	0
$991$	−31.3818	−0.996877	−0.498438	−	0.866925i	$-0.666093\pi$
$991$	−31.3818	−0.996877	−0.498438	+	0.866925i	$0.666093\pi$
$992$	− 20.4438i	− 0.649090i
$993$	22.9411i	0.728015i
$994$	−46.5864	−1.47763
$995$	0	0
$996$	−18.0668	−0.572468
$997$	− 45.3442i	− 1.43606i	−0.696010	−	0.718032i	$-0.745043\pi$
$997$	− 45.3442i	− 1.43606i	0.696010	−	0.718032i	$-0.254957\pi$
$998$	− 96.1629i	− 3.04398i
$999$	10.5280	0.333092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.p.349.2		16
5.2	odd	4		2175.2.a.bd.1.8	yes	8
5.3	odd	4		2175.2.a.bc.1.1	✓	8
5.4	even	2	inner	2175.2.c.p.349.15		16
15.2	even	4		6525.2.a.by.1.1		8
15.8	even	4		6525.2.a.bz.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.1	✓	8	5.3	odd	4
2175.2.a.bd.1.8	yes	8	5.2	odd	4
2175.2.c.p.349.2		16	1.1	even	1	trivial
2175.2.c.p.349.15		16	5.4	even	2	inner
6525.2.a.by.1.1		8	15.2	even	4
6525.2.a.bz.1.8		8	15.8	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:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.59587i q^{2} +1.00000i q^{3} -4.73855 q^{4} +2.59587 q^{6} +3.93826i q^{7} +7.10893i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.59587i q^{2} +1.00000i q^{3} -4.73855 q^{4} +2.59587 q^{6} +3.93826i q^{7} +7.10893i q^{8} -1.00000 q^{9} +2.24645 q^{11} -4.73855i q^{12} +2.09464i q^{13} +10.2232 q^{14} +8.97678 q^{16} -6.95310i q^{17} +2.59587i q^{18} -4.29591 q^{19} -3.93826 q^{21} -5.83149i q^{22} -5.19955i q^{23} -7.10893 q^{24} +5.43743 q^{26} -1.00000i q^{27} -18.6617i q^{28} -1.00000 q^{29} +2.25035 q^{31} -9.08471i q^{32} +2.24645i q^{33} -18.0494 q^{34} +4.73855 q^{36} +10.5280i q^{37} +11.1516i q^{38} -2.09464 q^{39} -2.66629 q^{41} +10.2232i q^{42} +6.03681i q^{43} -10.6449 q^{44} -13.4974 q^{46} +7.74829i q^{47} +8.97678i q^{48} -8.50991 q^{49} +6.95310 q^{51} -9.92559i q^{52} -1.41309i q^{53} -2.59587 q^{54} -27.9968 q^{56} -4.29591i q^{57} +2.59587i q^{58} -5.76800 q^{59} -10.9192 q^{61} -5.84162i q^{62} -3.93826i q^{63} -5.62918 q^{64} +5.83149 q^{66} +11.7471i q^{67} +32.9476i q^{68} +5.19955 q^{69} -4.55691 q^{71} -7.10893i q^{72} +12.3564i q^{73} +27.3294 q^{74} +20.3564 q^{76} +8.84710i q^{77} +5.43743i q^{78} -14.4098 q^{79} +1.00000 q^{81} +6.92135i q^{82} -3.81272i q^{83} +18.6617 q^{84} +15.6708 q^{86} -1.00000i q^{87} +15.9699i q^{88} +2.07340 q^{89} -8.24926 q^{91} +24.6383i q^{92} +2.25035i q^{93} +20.1136 q^{94} +9.08471 q^{96} -12.0068i q^{97} +22.0906i q^{98} -2.24645 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10}) \) 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9	\( 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9} + 12 q^{11} - 18 q^{14} + 64 q^{16} - 4 q^{21} - 6 q^{24} + 36 q^{26} - 16 q^{29} + 16 q^{31} + 26 q^{34} + 24 q^{36} + 12 q^{39} + 4 q^{41} + 30 q^{44} + 48 q^{46} - 76 q^{49} + 24 q^{51} - 4 q^{54} + 116 q^{56} - 36 q^{59} + 24 q^{61} - 42 q^{64} - 6 q^{66} - 28 q^{69} + 48 q^{71} + 44 q^{74} - 20 q^{79} + 16 q^{81} + 28 q^{84} + 16 q^{86} - 68 q^{89} + 52 q^{91} + 86 q^{94} - 4 q^{96} - 12 q^{99}+O(q^{100}) \) 16 * q - 24 * q^4 + 4 * q^6 - 16 * q^9 + 12 * q^11 - 18 * q^14 + 64 * q^16 - 4 * q^21 - 6 * q^24 + 36 * q^26 - 16 * q^29 + 16 * q^31 + 26 * q^34 + 24 * q^36 + 12 * q^39 + 4 * q^41 + 30 * q^44 + 48 * q^46 - 76 * q^49 + 24 * q^51 - 4 * q^54 + 116 * q^56 - 36 * q^59 + 24 * q^61 - 42 * q^64 - 6 * q^66 - 28 * q^69 + 48 * q^71 + 44 * q^74 - 20 * q^79 + 16 * q^81 + 28 * q^84 + 16 * q^86 - 68 * q^89 + 52 * q^91 + 86 * q^94 - 4 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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