LMFDB - Embedded newform 2175.2.c.p.349.3

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			349.3
Root			$-2.57789i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.p.349.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57789i q^{2} +1.00000i q^{3} -4.64553 q^{4} +2.57789 q^{6} -4.69867i q^{7} +6.81989i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.57789i q^{2} +1.00000i q^{3} -4.64553 q^{4} +2.57789 q^{6} -4.69867i q^{7} +6.81989i q^{8} -1.00000 q^{9} -3.11503 q^{11} -4.64553i q^{12} -5.07945i q^{13} -12.1127 q^{14} +8.28989 q^{16} -1.40020i q^{17} +2.57789i q^{18} +3.76514 q^{19} +4.69867 q^{21} +8.03022i q^{22} +5.71483i q^{23} -6.81989 q^{24} -13.0943 q^{26} -1.00000i q^{27} +21.8278i q^{28} -1.00000 q^{29} -2.23218 q^{31} -7.73067i q^{32} -3.11503i q^{33} -3.60957 q^{34} +4.64553 q^{36} -5.79088i q^{37} -9.70614i q^{38} +5.07945 q^{39} -10.6968 q^{41} -12.1127i q^{42} -8.89527i q^{43} +14.4710 q^{44} +14.7322 q^{46} -3.62785i q^{47} +8.28989i q^{48} -15.0775 q^{49} +1.40020 q^{51} +23.5967i q^{52} -0.948260i q^{53} -2.57789 q^{54} +32.0445 q^{56} +3.76514i q^{57} +2.57789i q^{58} +8.53886 q^{59} -6.21467 q^{61} +5.75432i q^{62} +4.69867i q^{63} -3.34905 q^{64} -8.03022 q^{66} +13.9099i q^{67} +6.50468i q^{68} -5.71483 q^{69} +5.88726 q^{71} -6.81989i q^{72} +7.08398i q^{73} -14.9283 q^{74} -17.4911 q^{76} +14.6365i q^{77} -13.0943i q^{78} -7.31672 q^{79} +1.00000 q^{81} +27.5751i q^{82} +13.6579i q^{83} -21.8278 q^{84} -22.9311 q^{86} -1.00000i q^{87} -21.2442i q^{88} +7.98017 q^{89} -23.8667 q^{91} -26.5484i q^{92} -2.23218i q^{93} -9.35221 q^{94} +7.73067 q^{96} +13.8343i q^{97} +38.8683i q^{98} +3.11503 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	− 2.57789i	− 1.82285i	−0.411471	−	0.911423i	$-0.634985\pi$
$2$	− 2.57789i	− 1.82285i	0.411471	−	0.911423i	$-0.365015\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−4.64553	−2.32277
$5$	0	0
$5$	0	0
$6$	2.57789	1.05242
$7$	− 4.69867i	− 1.77593i	−0.459909	−	0.887966i	$-0.652118\pi$
$7$	− 4.69867i	− 1.77593i	0.459909	−	0.887966i	$-0.347882\pi$
$8$	6.81989i	2.41120i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.11503	−0.939218	−0.469609	−	0.882875i	$-0.655605\pi$
$11$	−3.11503	−0.939218	−0.469609	+	0.882875i	$0.655605\pi$
$12$	− 4.64553i	− 1.34105i
$13$	− 5.07945i	− 1.40879i	−0.709810	−	0.704393i	$-0.751219\pi$
$13$	− 5.07945i	− 1.40879i	0.709810	−	0.704393i	$-0.248781\pi$
$14$	−12.1127	−3.23725
$15$	0	0
$16$	8.28989	2.07247
$17$	− 1.40020i	− 0.339599i	−0.985479	−	0.169799i	$-0.945688\pi$
$17$	− 1.40020i	− 0.339599i	0.985479	−	0.169799i	$-0.0543120\pi$
$18$	2.57789i	0.607615i
$19$	3.76514	0.863783	0.431892	−	0.901925i	$-0.357846\pi$
$19$	3.76514	0.863783	0.431892	+	0.901925i	$0.357846\pi$
$20$	0	0
$21$	4.69867	1.02533
$22$	8.03022i	1.71205i
$23$	5.71483i	1.19162i	0.803124	+	0.595812i	$0.203170\pi$
$23$	5.71483i	1.19162i	−0.803124	+	0.595812i	$0.796830\pi$
$24$	−6.81989	−1.39211
$25$	0	0
$26$	−13.0943	−2.56800
$27$	− 1.00000i	− 0.192450i
$28$	21.8278i	4.12507i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.23218	−0.400911	−0.200456	−	0.979703i	$-0.564242\pi$
$31$	−2.23218	−0.400911	−0.200456	+	0.979703i	$0.564242\pi$
$32$	− 7.73067i	− 1.36660i
$33$	− 3.11503i	− 0.542258i
$34$	−3.60957	−0.619036
$35$	0	0
$36$	4.64553	0.774255
$37$	− 5.79088i	− 0.952015i	−0.879441	−	0.476008i	$-0.842083\pi$
$37$	− 5.79088i	− 0.952015i	0.879441	−	0.476008i	$-0.157917\pi$
$38$	− 9.70614i	− 1.57454i
$39$	5.07945	0.813363
$40$	0	0
$41$	−10.6968	−1.67055	−0.835276	−	0.549830i	$-0.814692\pi$
$41$	−10.6968	−1.67055	−0.835276	+	0.549830i	$0.814692\pi$
$42$	− 12.1127i	− 1.86903i
$43$	− 8.89527i	− 1.35652i	−0.734824	−	0.678258i	$-0.762735\pi$
$43$	− 8.89527i	− 1.35652i	0.734824	−	0.678258i	$-0.237265\pi$
$44$	14.4710	2.18158
$45$	0	0
$46$	14.7322	2.17215
$47$	− 3.62785i	− 0.529176i	−0.964362	−	0.264588i	$-0.914764\pi$
$47$	− 3.62785i	− 0.529176i	0.964362	−	0.264588i	$-0.0852360\pi$
$48$	8.28989i	1.19654i
$49$	−15.0775	−2.15393
$50$	0	0
$51$	1.40020	0.196067
$52$	23.5967i	3.27228i
$53$	− 0.948260i	− 0.130254i	−0.997877	−	0.0651268i	$-0.979255\pi$
$53$	− 0.948260i	− 0.130254i	0.997877	−	0.0651268i	$-0.0207452\pi$
$54$	−2.57789	−0.350807
$55$	0	0
$56$	32.0445	4.28212
$57$	3.76514i	0.498706i
$58$	2.57789i	0.338494i
$59$	8.53886	1.11166	0.555832	−	0.831294i	$-0.312400\pi$
$59$	8.53886	1.11166	0.555832	+	0.831294i	$0.312400\pi$
$60$	0	0
$61$	−6.21467	−0.795707	−0.397853	−	0.917449i	$-0.630245\pi$
$61$	−6.21467	−0.795707	−0.397853	+	0.917449i	$0.630245\pi$
$62$	5.75432i	0.730799i
$63$	4.69867i	0.591977i
$64$	−3.34905	−0.418631
$65$	0	0
$66$	−8.03022	−0.988452
$67$	13.9099i	1.69937i	0.527291	+	0.849685i	$0.323208\pi$
$67$	13.9099i	1.69937i	−0.527291	+	0.849685i	$0.676792\pi$
$68$	6.50468i	0.788808i
$69$	−5.71483	−0.687985
$70$	0	0
$71$	5.88726	0.698690	0.349345	−	0.936994i	$-0.386404\pi$
$71$	5.88726	0.698690	0.349345	+	0.936994i	$0.386404\pi$
$72$	− 6.81989i	− 0.803732i
$73$	7.08398i	0.829117i	0.910023	+	0.414559i	$0.136064\pi$
$73$	7.08398i	0.829117i	−0.910023	+	0.414559i	$0.863936\pi$
$74$	−14.9283	−1.73538
$75$	0	0
$76$	−17.4911	−2.00637
$77$	14.6365i	1.66799i
$78$	− 13.0943i	− 1.48263i
$79$	−7.31672	−0.823195	−0.411598	−	0.911366i	$-0.635029\pi$
$79$	−7.31672	−0.823195	−0.411598	+	0.911366i	$0.635029\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	27.5751i	3.04516i
$83$	13.6579i	1.49915i	0.661917	+	0.749577i	$0.269743\pi$
$83$	13.6579i	1.49915i	−0.661917	+	0.749577i	$0.730257\pi$
$84$	−21.8278	−2.38161
$85$	0	0
$86$	−22.9311	−2.47272
$87$	− 1.00000i	− 0.107211i
$88$	− 21.2442i	− 2.26464i
$89$	7.98017	0.845897	0.422948	−	0.906154i	$-0.360995\pi$
$89$	7.98017	0.845897	0.422948	+	0.906154i	$0.360995\pi$
$90$	0	0
$91$	−23.8667	−2.50191
$92$	− 26.5484i	− 2.76786i
$93$	− 2.23218i	− 0.231466i
$94$	−9.35221	−0.964607
$95$	0	0
$96$	7.73067	0.789008
$97$	13.8343i	1.40466i	0.711853	+	0.702329i	$0.247856\pi$
$97$	13.8343i	1.40466i	−0.711853	+	0.702329i	$0.752144\pi$
$98$	38.8683i	3.92629i
$99$	3.11503	0.313073
$100$	0	0
$101$	12.7461	1.26829	0.634144	−	0.773215i	$-0.281353\pi$
$101$	12.7461	1.26829	0.634144	+	0.773215i	$0.281353\pi$
$102$	− 3.60957i	− 0.357401i
$103$	8.20406i	0.808370i	0.914677	+	0.404185i	$0.132445\pi$
$103$	8.20406i	0.808370i	−0.914677	+	0.404185i	$0.867555\pi$
$104$	34.6413	3.39686
$105$	0	0
$106$	−2.44451	−0.237432
$107$	− 3.28873i	− 0.317933i	−0.987284	−	0.158967i	$-0.949184\pi$
$107$	− 3.28873i	− 0.317933i	0.987284	−	0.158967i	$-0.0508162\pi$
$108$	4.64553i	0.447016i
$109$	0.246732	0.0236326	0.0118163	−	0.999930i	$-0.496239\pi$
$109$	0.246732	0.0236326	0.0118163	+	0.999930i	$0.496239\pi$
$110$	0	0
$111$	5.79088	0.549646
$112$	− 38.9515i	− 3.68057i
$113$	14.4273i	1.35721i	0.734505	+	0.678603i	$0.237414\pi$
$113$	14.4273i	1.35721i	−0.734505	+	0.678603i	$0.762586\pi$
$114$	9.70614	0.909063
$115$	0	0
$116$	4.64553	0.431327
$117$	5.07945i	0.469595i
$118$	− 22.0123i	− 2.02639i
$119$	−6.57909	−0.603104
$120$	0	0
$121$	−1.29657	−0.117870
$122$	16.0207i	1.45045i
$123$	− 10.6968i	− 0.964494i
$124$	10.3697	0.931223
$125$	0	0
$126$	12.1127	1.07908
$127$	− 21.1707i	− 1.87860i	−0.343101	−	0.939299i	$-0.611477\pi$
$127$	− 21.1707i	− 1.87860i	0.343101	−	0.939299i	$-0.388523\pi$
$128$	− 6.82785i	− 0.603503i
$129$	8.89527	0.783185
$130$	0	0
$131$	−8.18994	−0.715559	−0.357779	−	0.933806i	$-0.616466\pi$
$131$	−8.18994	−0.715559	−0.357779	+	0.933806i	$0.616466\pi$
$132$	14.4710i	1.25954i
$133$	− 17.6912i	− 1.53402i
$134$	35.8583	3.09769
$135$	0	0
$136$	9.54923	0.818840
$137$	17.4054i	1.48704i	0.668714	+	0.743520i	$0.266845\pi$
$137$	17.4054i	1.48704i	−0.668714	+	0.743520i	$0.733155\pi$
$138$	14.7322i	1.25409i
$139$	6.81900	0.578380	0.289190	−	0.957272i	$-0.406614\pi$
$139$	6.81900	0.578380	0.289190	+	0.957272i	$0.406614\pi$
$140$	0	0
$141$	3.62785	0.305520
$142$	− 15.1767i	− 1.27360i
$143$	15.8226i	1.32316i
$144$	−8.28989	−0.690825
$145$	0	0
$146$	18.2617	1.51135
$147$	− 15.0775i	− 1.24357i
$148$	26.9017i	2.21131i
$149$	−23.7335	−1.94432	−0.972162	−	0.234311i	$-0.924717\pi$
$149$	−23.7335	−1.94432	−0.972162	+	0.234311i	$0.924717\pi$
$150$	0	0
$151$	−17.3095	−1.40863	−0.704315	−	0.709888i	$-0.748746\pi$
$151$	−17.3095	−1.40863	−0.704315	+	0.709888i	$0.748746\pi$
$152$	25.6779i	2.08275i
$153$	1.40020i	0.113200i
$154$	37.7314	3.04048
$155$	0	0
$156$	−23.5967	−1.88925
$157$	10.0537i	0.802375i	0.915996	+	0.401188i	$0.131402\pi$
$157$	10.0537i	0.802375i	−0.915996	+	0.401188i	$0.868598\pi$
$158$	18.8617i	1.50056i
$159$	0.948260	0.0752019
$160$	0	0
$161$	26.8521	2.11624
$162$	− 2.57789i	− 0.202538i
$163$	− 12.2602i	− 0.960291i	−0.877189	−	0.480145i	$-0.840584\pi$
$163$	− 12.2602i	− 0.960291i	0.877189	−	0.480145i	$-0.159416\pi$
$164$	49.6921	3.88030
$165$	0	0
$166$	35.2087	2.73273
$167$	− 13.5990i	− 1.05232i	−0.850386	−	0.526160i	$-0.823631\pi$
$167$	− 13.5990i	− 1.05232i	0.850386	−	0.526160i	$-0.176369\pi$
$168$	32.0445i	2.47228i
$169$	−12.8008	−0.984677
$170$	0	0
$171$	−3.76514	−0.287928
$172$	41.3233i	3.15087i
$173$	− 8.35226i	− 0.635011i	−0.948256	−	0.317505i	$-0.897155\pi$
$173$	− 8.35226i	− 0.635011i	0.948256	−	0.317505i	$-0.102845\pi$
$174$	−2.57789	−0.195430
$175$	0	0
$176$	−25.8233	−1.94650
$177$	8.53886i	0.641820i
$178$	− 20.5720i	− 1.54194i
$179$	−26.5865	−1.98717	−0.993583	−	0.113109i	$-0.963919\pi$
$179$	−26.5865	−1.98717	−0.993583	+	0.113109i	$0.963919\pi$
$180$	0	0
$181$	3.72547	0.276912	0.138456	−	0.990369i	$-0.455786\pi$
$181$	3.72547	0.276912	0.138456	+	0.990369i	$0.455786\pi$
$182$	61.5257i	4.56059i
$183$	− 6.21467i	− 0.459401i
$184$	−38.9745	−2.87324
$185$	0	0
$186$	−5.75432	−0.421927
$187$	4.36167i	0.318957i
$188$	16.8533i	1.22915i
$189$	−4.69867	−0.341778
$190$	0	0
$191$	6.76433	0.489450	0.244725	−	0.969593i	$-0.421302\pi$
$191$	6.76433	0.489450	0.244725	+	0.969593i	$0.421302\pi$
$192$	− 3.34905i	− 0.241697i
$193$	− 0.140715i	− 0.0101289i	−0.999987	−	0.00506444i	$-0.998388\pi$
$193$	− 0.140715i	− 0.0101289i	0.999987	−	0.00506444i	$-0.00161207\pi$
$194$	35.6633	2.56047
$195$	0	0
$196$	70.0431	5.00308
$197$	− 1.48273i	− 0.105640i	−0.998604	−	0.0528200i	$-0.983179\pi$
$197$	− 1.48273i	− 0.105640i	0.998604	−	0.0528200i	$-0.0168210\pi$
$198$	− 8.03022i	− 0.570683i
$199$	−25.8626	−1.83335	−0.916677	−	0.399628i	$-0.869139\pi$
$199$	−25.8626	−1.83335	−0.916677	+	0.399628i	$0.869139\pi$
$200$	0	0
$201$	−13.9099	−0.981132
$202$	− 32.8582i	− 2.31189i
$203$	4.69867i	0.329782i
$204$	−6.50468	−0.455419
$205$	0	0
$206$	21.1492	1.47353
$207$	− 5.71483i	− 0.397208i
$208$	− 42.1081i	− 2.91967i
$209$	−11.7285	−0.811281
$210$	0	0
$211$	23.3428	1.60698	0.803491	−	0.595317i	$-0.202974\pi$
$211$	23.3428	1.60698	0.803491	+	0.595317i	$0.202974\pi$
$212$	4.40517i	0.302548i
$213$	5.88726i	0.403389i
$214$	−8.47799	−0.579543
$215$	0	0
$216$	6.81989	0.464035
$217$	10.4883i	0.711991i
$218$	− 0.636049i	− 0.0430787i
$219$	−7.08398	−0.478691
$220$	0	0
$221$	−7.11225	−0.478422
$222$	− 14.9283i	− 1.00192i
$223$	− 22.4256i	− 1.50173i	−0.660455	−	0.750865i	$-0.729637\pi$
$223$	− 22.4256i	− 1.50173i	0.660455	−	0.750865i	$-0.270363\pi$
$224$	−36.3239	−2.42699
$225$	0	0
$226$	37.1920	2.47398
$227$	− 4.07327i	− 0.270353i	−0.990822	−	0.135176i	$-0.956840\pi$
$227$	− 4.07327i	− 0.270353i	0.990822	−	0.135176i	$-0.0431601\pi$
$228$	− 17.4911i	− 1.15838i
$229$	15.7170	1.03861	0.519303	−	0.854590i	$-0.326192\pi$
$229$	15.7170	1.03861	0.519303	+	0.854590i	$0.326192\pi$
$230$	0	0
$231$	−14.6365	−0.963012
$232$	− 6.81989i	− 0.447748i
$233$	− 16.4568i	− 1.07812i	−0.842268	−	0.539059i	$-0.818780\pi$
$233$	− 16.4568i	− 1.07812i	0.842268	−	0.539059i	$-0.181220\pi$
$234$	13.0943	0.856000
$235$	0	0
$236$	−39.6675	−2.58214
$237$	− 7.31672i	− 0.475272i
$238$	16.9602i	1.09937i
$239$	−3.18702	−0.206151	−0.103075	−	0.994674i	$-0.532868\pi$
$239$	−3.18702	−0.206151	−0.103075	+	0.994674i	$0.532868\pi$
$240$	0	0
$241$	11.1482	0.718118	0.359059	−	0.933315i	$-0.383098\pi$
$241$	11.1482	0.718118	0.359059	+	0.933315i	$0.383098\pi$
$242$	3.34243i	0.214859i
$243$	1.00000i	0.0641500i
$244$	28.8704	1.84824
$245$	0	0
$246$	−27.5751	−1.75812
$247$	− 19.1249i	− 1.21689i
$248$	− 15.2232i	− 0.966676i
$249$	−13.6579	−0.865537
$250$	0	0
$251$	−3.23388	−0.204121	−0.102060	−	0.994778i	$-0.532543\pi$
$251$	−3.23388	−0.204121	−0.102060	+	0.994778i	$0.532543\pi$
$252$	− 21.8278i	− 1.37502i
$253$	− 17.8019i	− 1.11919i
$254$	−54.5758	−3.42439
$255$	0	0
$256$	−24.2996	−1.51872
$257$	3.96761i	0.247493i	0.992314	+	0.123746i	$0.0394909\pi$
$257$	3.96761i	0.247493i	−0.992314	+	0.123746i	$0.960509\pi$
$258$	− 22.9311i	− 1.42763i
$259$	−27.2095	−1.69071
$260$	0	0
$261$	1.00000	0.0618984
$262$	21.1128i	1.30435i
$263$	− 16.6865i	− 1.02893i	−0.857510	−	0.514467i	$-0.827990\pi$
$263$	− 16.6865i	− 1.02893i	0.857510	−	0.514467i	$-0.172010\pi$
$264$	21.2442	1.30749
$265$	0	0
$266$	−45.6060	−2.79628
$267$	7.98017i	0.488379i
$268$	− 64.6191i	− 3.94724i
$269$	−7.94329	−0.484311	−0.242155	−	0.970237i	$-0.577854\pi$
$269$	−7.94329	−0.484311	−0.242155	+	0.970237i	$0.577854\pi$
$270$	0	0
$271$	−17.3087	−1.05143	−0.525714	−	0.850662i	$-0.676202\pi$
$271$	−17.3087	−1.05143	−0.525714	+	0.850662i	$0.676202\pi$
$272$	− 11.6075i	− 0.703810i
$273$	− 23.8667i	− 1.44448i
$274$	44.8691	2.71064
$275$	0	0
$276$	26.5484	1.59803
$277$	10.2791i	0.617612i	0.951125	+	0.308806i	$0.0999294\pi$
$277$	10.2791i	0.617612i	−0.951125	+	0.308806i	$0.900071\pi$
$278$	− 17.5786i	− 1.05430i
$279$	2.23218	0.133637
$280$	0	0
$281$	4.57869	0.273142	0.136571	−	0.990630i	$-0.456392\pi$
$281$	4.57869	0.273142	0.136571	+	0.990630i	$0.456392\pi$
$282$	− 9.35221i	− 0.556916i
$283$	− 19.2431i	− 1.14388i	−0.820294	−	0.571941i	$-0.806190\pi$
$283$	− 19.2431i	− 1.14388i	0.820294	−	0.571941i	$-0.193810\pi$
$284$	−27.3495	−1.62289
$285$	0	0
$286$	40.7891	2.41191
$287$	50.2606i	2.96679i
$288$	7.73067i	0.455534i
$289$	15.0394	0.884673
$290$	0	0
$291$	−13.8343	−0.810979
$292$	− 32.9089i	− 1.92584i
$293$	− 14.6045i	− 0.853203i	−0.904440	−	0.426602i	$-0.859711\pi$
$293$	− 14.6045i	− 0.853203i	0.904440	−	0.426602i	$-0.140289\pi$
$294$	−38.8683	−2.26684
$295$	0	0
$296$	39.4932	2.29550
$297$	3.11503i	0.180753i
$298$	61.1824i	3.54420i
$299$	29.0282	1.67874
$300$	0	0
$301$	−41.7960	−2.40908
$302$	44.6221i	2.56771i
$303$	12.7461i	0.732246i
$304$	31.2127	1.79017
$305$	0	0
$306$	3.60957	0.206345
$307$	12.3324i	0.703850i	0.936028	+	0.351925i	$0.114473\pi$
$307$	12.3324i	0.703850i	−0.936028	+	0.351925i	$0.885527\pi$
$308$	− 67.9944i	− 3.87434i
$309$	−8.20406	−0.466713
$310$	0	0
$311$	12.3084	0.697943	0.348971	−	0.937133i	$-0.386531\pi$
$311$	12.3084	0.697943	0.348971	+	0.937133i	$0.386531\pi$
$312$	34.6413i	1.96118i
$313$	− 16.7306i	− 0.945672i	−0.881150	−	0.472836i	$-0.843230\pi$
$313$	− 16.7306i	− 0.945672i	0.881150	−	0.472836i	$-0.156770\pi$
$314$	25.9174	1.46261
$315$	0	0
$316$	33.9901	1.91209
$317$	− 10.9429i	− 0.614617i	−0.951610	−	0.307308i	$-0.900572\pi$
$317$	− 10.9429i	− 0.614617i	0.951610	−	0.307308i	$-0.0994283\pi$
$318$	− 2.44451i	− 0.137081i
$319$	3.11503	0.174408
$320$	0	0
$321$	3.28873	0.183559
$322$	− 69.2219i	− 3.85759i
$323$	− 5.27196i	− 0.293340i
$324$	−4.64553	−0.258085
$325$	0	0
$326$	−31.6054	−1.75046
$327$	0.246732i	0.0136443i
$328$	− 72.9508i	− 4.02803i
$329$	−17.0461	−0.939781
$330$	0	0
$331$	−19.1431	−1.05220	−0.526099	−	0.850423i	$-0.676346\pi$
$331$	−19.1431	−1.05220	−0.526099	+	0.850423i	$0.676346\pi$
$332$	− 63.4484i	− 3.48218i
$333$	5.79088i	0.317338i
$334$	−35.0567	−1.91822
$335$	0	0
$336$	38.9515	2.12498
$337$	− 17.0294i	− 0.927650i	−0.885927	−	0.463825i	$-0.846477\pi$
$337$	− 17.0294i	− 0.927650i	0.885927	−	0.463825i	$-0.153523\pi$
$338$	32.9991i	1.79491i
$339$	−14.4273	−0.783583
$340$	0	0
$341$	6.95331	0.376543
$342$	9.70614i	0.524848i
$343$	37.9537i	2.04931i
$344$	60.6648	3.27083
$345$	0	0
$346$	−21.5312	−1.15753
$347$	− 17.2564i	− 0.926373i	−0.886261	−	0.463186i	$-0.846706\pi$
$347$	− 17.2564i	− 0.926373i	0.886261	−	0.463186i	$-0.153294\pi$
$348$	4.64553i	0.249027i
$349$	−13.3969	−0.717118	−0.358559	−	0.933507i	$-0.616732\pi$
$349$	−13.3969	−0.717118	−0.358559	+	0.933507i	$0.616732\pi$
$350$	0	0
$351$	−5.07945	−0.271121
$352$	24.0813i	1.28354i
$353$	− 2.04594i	− 0.108894i	−0.998517	−	0.0544472i	$-0.982660\pi$
$353$	− 2.04594i	− 0.108894i	0.998517	−	0.0544472i	$-0.0173397\pi$
$354$	22.0123	1.16994
$355$	0	0
$356$	−37.0721	−1.96482
$357$	− 6.57909i	− 0.348202i
$358$	68.5370i	3.62229i
$359$	25.2336	1.33178	0.665889	−	0.746051i	$-0.268053\pi$
$359$	25.2336	1.33178	0.665889	+	0.746051i	$0.268053\pi$
$360$	0	0
$361$	−4.82368	−0.253878
$362$	− 9.60387i	− 0.504768i
$363$	− 1.29657i	− 0.0680525i
$364$	110.873	5.81134
$365$	0	0
$366$	−16.0207	−0.837418
$367$	− 9.13285i	− 0.476731i	−0.971176	−	0.238366i	$-0.923388\pi$
$367$	− 9.13285i	− 0.476731i	0.971176	−	0.238366i	$-0.0766116\pi$
$368$	47.3753i	2.46961i
$369$	10.6968	0.556851
$370$	0	0
$371$	−4.45557	−0.231321
$372$	10.3697i	0.537642i
$373$	− 5.41148i	− 0.280196i	−0.990138	−	0.140098i	$-0.955258\pi$
$373$	− 5.41148i	− 0.280196i	0.990138	−	0.140098i	$-0.0447418\pi$
$374$	11.2439	0.581410
$375$	0	0
$376$	24.7416	1.27595
$377$	5.07945i	0.261605i
$378$	12.1127i	0.623009i
$379$	−19.8284	−1.01852	−0.509258	−	0.860614i	$-0.670080\pi$
$379$	−19.8284	−1.01852	−0.509258	+	0.860614i	$0.670080\pi$
$380$	0	0
$381$	21.1707	1.08461
$382$	− 17.4377i	− 0.892191i
$383$	− 14.8919i	− 0.760941i	−0.924793	−	0.380470i	$-0.875762\pi$
$383$	− 14.8919i	− 0.760941i	0.924793	−	0.380470i	$-0.124238\pi$
$384$	6.82785	0.348432
$385$	0	0
$386$	−0.362748	−0.0184634
$387$	8.89527i	0.452172i
$388$	− 64.2675i	− 3.26269i
$389$	15.7680	0.799470	0.399735	−	0.916631i	$-0.369102\pi$
$389$	15.7680	0.799470	0.399735	+	0.916631i	$0.369102\pi$
$390$	0	0
$391$	8.00192	0.404674
$392$	− 102.827i	− 5.19356i
$393$	− 8.18994i	− 0.413128i
$394$	−3.82232	−0.192566
$395$	0	0
$396$	−14.4710	−0.727194
$397$	− 26.5318i	− 1.33159i	−0.746134	−	0.665796i	$-0.768092\pi$
$397$	− 26.5318i	− 1.33159i	0.746134	−	0.665796i	$-0.231908\pi$
$398$	66.6711i	3.34192i
$399$	17.6912	0.885667
$400$	0	0
$401$	9.71205	0.484997	0.242498	−	0.970152i	$-0.422033\pi$
$401$	9.71205	0.484997	0.242498	+	0.970152i	$0.422033\pi$
$402$	35.8583i	1.78845i
$403$	11.3382i	0.564798i
$404$	−59.2126	−2.94593
$405$	0	0
$406$	12.1127	0.601142
$407$	18.0388i	0.894150i
$408$	9.54923i	0.472757i
$409$	17.4468	0.862691	0.431346	−	0.902187i	$-0.358039\pi$
$409$	17.4468	0.862691	0.431346	+	0.902187i	$0.358039\pi$
$410$	0	0
$411$	−17.4054	−0.858543
$412$	− 38.1122i	− 1.87765i
$413$	− 40.1213i	− 1.97424i
$414$	−14.7322	−0.724049
$415$	0	0
$416$	−39.2675	−1.92525
$417$	6.81900i	0.333928i
$418$	30.2349i	1.47884i
$419$	−26.0588	−1.27306	−0.636529	−	0.771253i	$-0.719630\pi$
$419$	−26.0588	−1.27306	−0.636529	+	0.771253i	$0.719630\pi$
$420$	0	0
$421$	2.66465	0.129867	0.0649337	−	0.997890i	$-0.479316\pi$
$421$	2.66465	0.129867	0.0649337	+	0.997890i	$0.479316\pi$
$422$	− 60.1751i	− 2.92928i
$423$	3.62785i	0.176392i
$424$	6.46704	0.314067
$425$	0	0
$426$	15.1767	0.735315
$427$	29.2007i	1.41312i
$428$	15.2779i	0.738485i
$429$	−15.8226	−0.763925
$430$	0	0
$431$	−0.150532	−0.00725086	−0.00362543	−	0.999993i	$-0.501154\pi$
$431$	−0.150532	−0.00725086	−0.00362543	+	0.999993i	$0.501154\pi$
$432$	− 8.28989i	− 0.398848i
$433$	33.2461i	1.59770i	0.601527	+	0.798852i	$0.294559\pi$
$433$	33.2461i	1.59770i	−0.601527	+	0.798852i	$0.705441\pi$
$434$	27.0377	1.29785
$435$	0	0
$436$	−1.14620	−0.0548931
$437$	21.5172i	1.02931i
$438$	18.2617i	0.872580i
$439$	−18.1320	−0.865391	−0.432695	−	0.901540i	$-0.642437\pi$
$439$	−18.1320	−0.865391	−0.432695	+	0.901540i	$0.642437\pi$
$440$	0	0
$441$	15.0775	0.717978
$442$	18.3346i	0.872089i
$443$	− 34.8312i	− 1.65488i	−0.561554	−	0.827440i	$-0.689796\pi$
$443$	− 34.8312i	− 1.65488i	0.561554	−	0.827440i	$-0.310204\pi$
$444$	−26.9017	−1.27670
$445$	0	0
$446$	−57.8108	−2.73742
$447$	− 23.7335i	− 1.12256i
$448$	15.7361i	0.743460i
$449$	−36.1516	−1.70610	−0.853049	−	0.521830i	$-0.825249\pi$
$449$	−36.1516	−1.70610	−0.853049	+	0.521830i	$0.825249\pi$
$450$	0	0
$451$	33.3207	1.56901
$452$	− 67.0225i	− 3.15247i
$453$	− 17.3095i	− 0.813273i
$454$	−10.5005	−0.492811
$455$	0	0
$456$	−25.6779	−1.20248
$457$	− 8.93775i	− 0.418090i	−0.977906	−	0.209045i	$-0.932964\pi$
$457$	− 8.93775i	− 0.418090i	0.977906	−	0.209045i	$-0.0670356\pi$
$458$	− 40.5166i	− 1.89322i
$459$	−1.40020	−0.0653558
$460$	0	0
$461$	34.6733	1.61490	0.807449	−	0.589937i	$-0.200848\pi$
$461$	34.6733	1.61490	0.807449	+	0.589937i	$0.200848\pi$
$462$	37.7314i	1.75542i
$463$	− 30.9412i	− 1.43796i	−0.695030	−	0.718981i	$-0.744609\pi$
$463$	− 30.9412i	− 1.43796i	0.695030	−	0.718981i	$-0.255391\pi$
$464$	−8.28989	−0.384849
$465$	0	0
$466$	−42.4237	−1.96524
$467$	5.40920i	0.250308i	0.992137	+	0.125154i	$0.0399425\pi$
$467$	5.40920i	0.250308i	−0.992137	+	0.125154i	$0.960057\pi$
$468$	− 23.5967i	− 1.09076i
$469$	65.3583	3.01797
$470$	0	0
$471$	−10.0537	−0.463251
$472$	58.2341i	2.68044i
$473$	27.7091i	1.27406i
$474$	−18.8617	−0.866348
$475$	0	0
$476$	30.5634	1.40087
$477$	0.948260i	0.0434179i
$478$	8.21578i	0.375781i
$479$	−13.4270	−0.613494	−0.306747	−	0.951791i	$-0.599241\pi$
$479$	−13.4270	−0.613494	−0.306747	+	0.951791i	$0.599241\pi$
$480$	0	0
$481$	−29.4145	−1.34119
$482$	− 28.7389i	− 1.30902i
$483$	26.8521i	1.22181i
$484$	6.02327	0.273785
$485$	0	0
$486$	2.57789	0.116936
$487$	− 24.8746i	− 1.12718i	−0.826055	−	0.563589i	$-0.809420\pi$
$487$	− 24.8746i	− 1.12718i	0.826055	−	0.563589i	$-0.190580\pi$
$488$	− 42.3834i	− 1.91861i
$489$	12.2602	0.554424
$490$	0	0
$491$	20.6717	0.932902	0.466451	−	0.884547i	$-0.345532\pi$
$491$	20.6717	0.932902	0.466451	+	0.884547i	$0.345532\pi$
$492$	49.6921i	2.24029i
$493$	1.40020i	0.0630619i
$494$	−49.3018	−2.21819
$495$	0	0
$496$	−18.5045	−0.830878
$497$	− 27.6623i	− 1.24083i
$498$	35.2087i	1.57774i
$499$	−21.2182	−0.949856	−0.474928	−	0.880025i	$-0.657526\pi$
$499$	−21.2182	−0.949856	−0.474928	+	0.880025i	$0.657526\pi$
$500$	0	0
$501$	13.5990	0.607557
$502$	8.33660i	0.372081i
$503$	35.9602i	1.60339i	0.597736	+	0.801693i	$0.296067\pi$
$503$	35.9602i	1.60339i	−0.597736	+	0.801693i	$0.703933\pi$
$504$	−32.0445	−1.42737
$505$	0	0
$506$	−45.8913	−2.04012
$507$	− 12.8008i	− 0.568504i
$508$	98.3492i	4.36354i
$509$	−21.3861	−0.947924	−0.473962	−	0.880545i	$-0.657177\pi$
$509$	−21.3861	−0.947924	−0.473962	+	0.880545i	$0.657177\pi$
$510$	0	0
$511$	33.2853	1.47246
$512$	48.9860i	2.16489i
$513$	− 3.76514i	− 0.166235i
$514$	10.2281	0.451141
$515$	0	0
$516$	−41.3233	−1.81915
$517$	11.3009i	0.497012i
$518$	70.1431i	3.08191i
$519$	8.35226	0.366624
$520$	0	0
$521$	−6.56775	−0.287738	−0.143869	−	0.989597i	$-0.545954\pi$
$521$	−6.56775	−0.287738	−0.143869	+	0.989597i	$0.545954\pi$
$522$	− 2.57789i	− 0.112831i
$523$	21.1811i	0.926184i	0.886310	+	0.463092i	$0.153260\pi$
$523$	21.1811i	0.926184i	−0.886310	+	0.463092i	$0.846740\pi$
$524$	38.0466	1.66207
$525$	0	0
$526$	−43.0160	−1.87559
$527$	3.12550i	0.136149i
$528$	− 25.8233i	− 1.12381i
$529$	−9.65929	−0.419969
$530$	0	0
$531$	−8.53886	−0.370555
$532$	82.1850i	3.56317i
$533$	54.3336i	2.35345i
$534$	20.5720	0.890239
$535$	0	0
$536$	−94.8643	−4.09752
$537$	− 26.5865i	− 1.14729i
$538$	20.4769i	0.882824i
$539$	46.9670	2.02301
$540$	0	0
$541$	−9.98542	−0.429307	−0.214653	−	0.976690i	$-0.568862\pi$
$541$	−9.98542	−0.429307	−0.214653	+	0.976690i	$0.568862\pi$
$542$	44.6199i	1.91659i
$543$	3.72547i	0.159875i
$544$	−10.8245	−0.464097
$545$	0	0
$546$	−61.5257	−2.63306
$547$	− 1.01143i	− 0.0432457i	−0.999766	−	0.0216228i	$-0.993117\pi$
$547$	− 1.01143i	− 0.0432457i	0.999766	−	0.0216228i	$-0.00688330\pi$
$548$	− 80.8571i	− 3.45404i
$549$	6.21467	0.265236
$550$	0	0
$551$	−3.76514	−0.160401
$552$	− 38.9745i	− 1.65887i
$553$	34.3789i	1.46194i
$554$	26.4985	1.12581
$555$	0	0
$556$	−31.6779	−1.34344
$557$	30.4618i	1.29071i	0.763885	+	0.645353i	$0.223290\pi$
$557$	30.4618i	1.29071i	−0.763885	+	0.645353i	$0.776710\pi$
$558$	− 5.75432i	− 0.243600i
$559$	−45.1831	−1.91104
$560$	0	0
$561$	−4.36167	−0.184150
$562$	− 11.8034i	− 0.497895i
$563$	− 2.49954i	− 0.105343i	−0.998612	−	0.0526716i	$-0.983226\pi$
$563$	− 2.49954i	− 0.105343i	0.998612	−	0.0526716i	$-0.0167736\pi$
$564$	−16.8533	−0.709651
$565$	0	0
$566$	−49.6066	−2.08512
$567$	− 4.69867i	− 0.197326i
$568$	40.1505i	1.68468i
$569$	0.611166	0.0256214	0.0128107	−	0.999918i	$-0.495922\pi$
$569$	0.611166	0.0256214	0.0128107	+	0.999918i	$0.495922\pi$
$570$	0	0
$571$	−0.527376	−0.0220700	−0.0110350	−	0.999939i	$-0.503513\pi$
$571$	−0.527376	−0.0220700	−0.0110350	+	0.999939i	$0.503513\pi$
$572$	− 73.5046i	− 3.07338i
$573$	6.76433i	0.282584i
$574$	129.566	5.40800
$575$	0	0
$576$	3.34905	0.139544
$577$	24.3938i	1.01553i	0.861496	+	0.507764i	$0.169528\pi$
$577$	24.3938i	1.01553i	−0.861496	+	0.507764i	$0.830472\pi$
$578$	− 38.7700i	− 1.61262i
$579$	0.140715	0.00584791
$580$	0	0
$581$	64.1742	2.66239
$582$	35.6633i	1.47829i
$583$	2.95386i	0.122336i
$584$	−48.3120	−1.99917
$585$	0	0
$586$	−37.6488	−1.55526
$587$	28.4529i	1.17438i	0.809450	+	0.587188i	$0.199765\pi$
$587$	28.4529i	1.17438i	−0.809450	+	0.587188i	$0.800235\pi$
$588$	70.0431i	2.88853i
$589$	−8.40448	−0.346301
$590$	0	0
$591$	1.48273	0.0609913
$592$	− 48.0058i	− 1.97303i
$593$	2.96753i	0.121862i	0.998142	+	0.0609310i	$0.0194070\pi$
$593$	2.96753i	0.121862i	−0.998142	+	0.0609310i	$0.980593\pi$
$594$	8.03022	0.329484
$595$	0	0
$596$	110.255	4.51621
$597$	− 25.8626i	− 1.05849i
$598$	− 74.8316i	− 3.06009i
$599$	−21.9546	−0.897040	−0.448520	−	0.893773i	$-0.648049\pi$
$599$	−21.9546	−0.897040	−0.448520	+	0.893773i	$0.648049\pi$
$600$	0	0
$601$	−21.4446	−0.874745	−0.437372	−	0.899280i	$-0.644091\pi$
$601$	−21.4446	−0.874745	−0.437372	+	0.899280i	$0.644091\pi$
$602$	107.746i	4.39138i
$603$	− 13.9099i	− 0.566457i
$604$	80.4120	3.27192
$605$	0	0
$606$	32.8582	1.33477
$607$	− 48.0114i	− 1.94872i	−0.224988	−	0.974361i	$-0.572234\pi$
$607$	− 48.0114i	− 1.94872i	0.224988	−	0.974361i	$-0.427766\pi$
$608$	− 29.1071i	− 1.18045i
$609$	−4.69867	−0.190400
$610$	0	0
$611$	−18.4275	−0.745496
$612$	− 6.50468i	− 0.262936i
$613$	37.9093i	1.53114i	0.643351	+	0.765571i	$0.277544\pi$
$613$	37.9093i	1.53114i	−0.643351	+	0.765571i	$0.722456\pi$
$614$	31.7917	1.28301
$615$	0	0
$616$	−99.8195	−4.02184
$617$	− 8.29772i	− 0.334054i	−0.985952	−	0.167027i	$-0.946583\pi$
$617$	− 8.29772i	− 0.334054i	0.985952	−	0.167027i	$-0.0534166\pi$
$618$	21.1492i	0.850745i
$619$	15.6135	0.627559	0.313779	−	0.949496i	$-0.398405\pi$
$619$	15.6135	0.627559	0.313779	+	0.949496i	$0.398405\pi$
$620$	0	0
$621$	5.71483	0.229328
$622$	− 31.7296i	− 1.27224i
$623$	− 37.4962i	− 1.50225i
$624$	42.1081	1.68567
$625$	0	0
$626$	−43.1298	−1.72381
$627$	− 11.7285i	− 0.468393i
$628$	− 46.7049i	− 1.86373i
$629$	−8.10841	−0.323303
$630$	0	0
$631$	10.7848	0.429336	0.214668	−	0.976687i	$-0.431133\pi$
$631$	10.7848	0.429336	0.214668	+	0.976687i	$0.431133\pi$
$632$	− 49.8993i	− 1.98489i
$633$	23.3428i	0.927791i
$634$	−28.2097	−1.12035
$635$	0	0
$636$	−4.40517	−0.174676
$637$	76.5856i	3.03443i
$638$	− 8.03022i	− 0.317919i
$639$	−5.88726	−0.232897
$640$	0	0
$641$	16.1226	0.636805	0.318403	−	0.947956i	$-0.396854\pi$
$641$	16.1226	0.636805	0.318403	+	0.947956i	$0.396854\pi$
$642$	− 8.47799i	− 0.334599i
$643$	− 25.0313i	− 0.987137i	−0.869707	−	0.493569i	$-0.835692\pi$
$643$	− 25.0313i	− 0.987137i	0.869707	−	0.493569i	$-0.164308\pi$
$644$	−124.742	−4.91554
$645$	0	0
$646$	−13.5906	−0.534713
$647$	− 34.9798i	− 1.37520i	−0.726091	−	0.687599i	$-0.758665\pi$
$647$	− 34.9798i	− 1.37520i	0.726091	−	0.687599i	$-0.241335\pi$
$648$	6.81989i	0.267911i
$649$	−26.5988	−1.04409
$650$	0	0
$651$	−10.4883	−0.411068
$652$	56.9550i	2.23053i
$653$	22.0250i	0.861904i	0.902375	+	0.430952i	$0.141822\pi$
$653$	22.0250i	0.861904i	−0.902375	+	0.430952i	$0.858178\pi$
$654$	0.636049	0.0248715
$655$	0	0
$656$	−88.6750	−3.46218
$657$	− 7.08398i	− 0.276372i
$658$	43.9430i	1.71308i
$659$	32.9279	1.28269	0.641345	−	0.767253i	$-0.278377\pi$
$659$	32.9279	1.28269	0.641345	+	0.767253i	$0.278377\pi$
$660$	0	0
$661$	9.78801	0.380710	0.190355	−	0.981715i	$-0.439036\pi$
$661$	9.78801	0.380710	0.190355	+	0.981715i	$0.439036\pi$
$662$	49.3488i	1.91799i
$663$	− 7.11225i	− 0.276217i
$664$	−93.1457	−3.61475
$665$	0	0
$666$	14.9283	0.578459
$667$	− 5.71483i	− 0.221279i
$668$	63.1745i	2.44429i
$669$	22.4256	0.867025
$670$	0	0
$671$	19.3589	0.747342
$672$	− 36.3239i	− 1.40122i
$673$	− 25.3634i	− 0.977689i	−0.872371	−	0.488844i	$-0.837419\pi$
$673$	− 25.3634i	− 0.977689i	0.872371	−	0.488844i	$-0.162581\pi$
$674$	−43.8999	−1.69096
$675$	0	0
$676$	59.4665	2.28717
$677$	19.7896i	0.760578i	0.924868	+	0.380289i	$0.124175\pi$
$677$	19.7896i	0.760578i	−0.924868	+	0.380289i	$0.875825\pi$
$678$	37.1920i	1.42835i
$679$	65.0027	2.49458
$680$	0	0
$681$	4.07327	0.156088
$682$	− 17.9249i	− 0.686380i
$683$	44.9656i	1.72056i	0.509820	+	0.860281i	$0.329712\pi$
$683$	44.9656i	1.72056i	−0.509820	+	0.860281i	$0.670288\pi$
$684$	17.4911	0.668789
$685$	0	0
$686$	97.8406	3.73557
$687$	15.7170i	0.599640i
$688$	− 73.7409i	− 2.81134i
$689$	−4.81664	−0.183499
$690$	0	0
$691$	−30.7758	−1.17077	−0.585384	−	0.810756i	$-0.699056\pi$
$691$	−30.7758	−1.17077	−0.585384	+	0.810756i	$0.699056\pi$
$692$	38.8007i	1.47498i
$693$	− 14.6365i	− 0.555995i
$694$	−44.4852	−1.68863
$695$	0	0
$696$	6.81989	0.258507
$697$	14.9776i	0.567318i
$698$	34.5357i	1.30720i
$699$	16.4568	0.622452
$700$	0	0
$701$	−23.2459	−0.877984	−0.438992	−	0.898491i	$-0.644664\pi$
$701$	−23.2459	−0.877984	−0.438992	+	0.898491i	$0.644664\pi$
$702$	13.0943i	0.494212i
$703$	− 21.8035i	− 0.822335i
$704$	10.4324	0.393186
$705$	0	0
$706$	−5.27422	−0.198498
$707$	− 59.8899i	− 2.25239i
$708$	− 39.6675i	− 1.49080i
$709$	−7.86251	−0.295283	−0.147641	−	0.989041i	$-0.547168\pi$
$709$	−7.86251	−0.295283	−0.147641	+	0.989041i	$0.547168\pi$
$710$	0	0
$711$	7.31672	0.274398
$712$	54.4239i	2.03962i
$713$	− 12.7565i	− 0.477736i
$714$	−16.9602	−0.634719
$715$	0	0
$716$	123.508	4.61572
$717$	− 3.18702i	− 0.119021i
$718$	− 65.0495i	− 2.42762i
$719$	−35.0267	−1.30628	−0.653138	−	0.757239i	$-0.726548\pi$
$719$	−35.0267	−1.30628	−0.653138	+	0.757239i	$0.726548\pi$
$720$	0	0
$721$	38.5482	1.43561
$722$	12.4349i	0.462781i
$723$	11.1482i	0.414606i
$724$	−17.3068	−0.643202
$725$	0	0
$726$	−3.34243	−0.124049
$727$	− 22.4518i	− 0.832691i	−0.909206	−	0.416346i	$-0.863311\pi$
$727$	− 22.4518i	− 0.832691i	0.909206	−	0.416346i	$-0.136689\pi$
$728$	− 162.768i	− 6.03259i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−12.4552	−0.460671
$732$	28.8704i	1.06708i
$733$	− 15.0028i	− 0.554140i	−0.960850	−	0.277070i	$-0.910637\pi$
$733$	− 15.0028i	− 0.554140i	0.960850	−	0.277070i	$-0.0893635\pi$
$734$	−23.5435	−0.869007
$735$	0	0
$736$	44.1795	1.62848
$737$	− 43.3299i	− 1.59608i
$738$	− 27.5751i	− 1.01505i
$739$	33.4801	1.23159	0.615793	−	0.787908i	$-0.288836\pi$
$739$	33.4801	1.23159	0.615793	+	0.787908i	$0.288836\pi$
$740$	0	0
$741$	19.1249	0.702569
$742$	11.4860i	0.421663i
$743$	6.75081i	0.247663i	0.992303	+	0.123832i	$0.0395183\pi$
$743$	6.75081i	0.247663i	−0.992303	+	0.123832i	$0.960482\pi$
$744$	15.2232	0.558111
$745$	0	0
$746$	−13.9502	−0.510754
$747$	− 13.6579i	− 0.499718i
$748$	− 20.2623i	− 0.740863i
$749$	−15.4527	−0.564628
$750$	0	0
$751$	28.0482	1.02349	0.511747	−	0.859136i	$-0.328999\pi$
$751$	28.0482	1.02349	0.511747	+	0.859136i	$0.328999\pi$
$752$	− 30.0745i	− 1.09670i
$753$	− 3.23388i	− 0.117849i
$754$	13.0943	0.476865
$755$	0	0
$756$	21.8278	0.793871
$757$	− 33.1698i	− 1.20558i	−0.797901	−	0.602789i	$-0.794056\pi$
$757$	− 33.1698i	− 1.20558i	0.797901	−	0.602789i	$-0.205944\pi$
$758$	51.1155i	1.85660i
$759$	17.8019	0.646167
$760$	0	0
$761$	−12.0982	−0.438560	−0.219280	−	0.975662i	$-0.570371\pi$
$761$	−12.0982	−0.438560	−0.219280	+	0.975662i	$0.570371\pi$
$762$	− 54.5758i	− 1.97707i
$763$	− 1.15931i	− 0.0419700i
$764$	−31.4239	−1.13688
$765$	0	0
$766$	−38.3897	−1.38708
$767$	− 43.3727i	− 1.56610i
$768$	− 24.2996i	− 0.876835i
$769$	42.5811	1.53551	0.767756	−	0.640742i	$-0.221373\pi$
$769$	42.5811	1.53551	0.767756	+	0.640742i	$0.221373\pi$
$770$	0	0
$771$	−3.96761	−0.142890
$772$	0.653695i	0.0235270i
$773$	6.87342i	0.247220i	0.992331	+	0.123610i	$0.0394471\pi$
$773$	6.87342i	0.247220i	−0.992331	+	0.123610i	$0.960553\pi$
$774$	22.9311	0.824240
$775$	0	0
$776$	−94.3483	−3.38691
$777$	− 27.2095i	− 0.976134i
$778$	− 40.6482i	− 1.45731i
$779$	−40.2748	−1.44300
$780$	0	0
$781$	−18.3390	−0.656222
$782$	− 20.6281i	− 0.737659i
$783$	1.00000i	0.0357371i
$784$	−124.991	−4.46397
$785$	0	0
$786$	−21.1128	−0.753068
$787$	4.51935i	0.161097i	0.996751	+	0.0805487i	$0.0256673\pi$
$787$	4.51935i	0.161097i	−0.996751	+	0.0805487i	$0.974333\pi$
$788$	6.88806i	0.245377i
$789$	16.6865	0.594056
$790$	0	0
$791$	67.7892	2.41031
$792$	21.2442i	0.754879i
$793$	31.5671i	1.12098i
$794$	−68.3961	−2.42729
$795$	0	0
$796$	120.146	4.25845
$797$	− 13.9670i	− 0.494736i	−0.968922	−	0.247368i	$-0.920434\pi$
$797$	− 13.9670i	− 0.494736i	0.968922	−	0.247368i	$-0.0795656\pi$
$798$	− 45.6060i	− 1.61443i
$799$	−5.07972	−0.179708
$800$	0	0
$801$	−7.98017	−0.281966
$802$	− 25.0366i	− 0.884074i
$803$	− 22.0668i	− 0.778722i
$804$	64.6191	2.27894
$805$	0	0
$806$	29.2288	1.02954
$807$	− 7.94329i	− 0.279617i
$808$	86.9273i	3.05809i
$809$	−8.18148	−0.287646	−0.143823	−	0.989603i	$-0.545940\pi$
$809$	−8.18148	−0.287646	−0.143823	+	0.989603i	$0.545940\pi$
$810$	0	0
$811$	−44.4011	−1.55913	−0.779566	−	0.626320i	$-0.784560\pi$
$811$	−44.4011	−1.55913	−0.779566	+	0.626320i	$0.784560\pi$
$812$	− 21.8278i	− 0.766007i
$813$	− 17.3087i	− 0.607042i
$814$	46.5021	1.62990
$815$	0	0
$816$	11.6075	0.406345
$817$	− 33.4920i	− 1.17174i
$818$	− 44.9761i	− 1.57255i
$819$	23.8667	0.833969
$820$	0	0
$821$	8.73119	0.304721	0.152360	−	0.988325i	$-0.451313\pi$
$821$	8.73119	0.304721	0.152360	+	0.988325i	$0.451313\pi$
$822$	44.8691i	1.56499i
$823$	− 39.1844i	− 1.36588i	−0.730473	−	0.682942i	$-0.760700\pi$
$823$	− 39.1844i	− 1.36588i	0.730473	−	0.682942i	$-0.239300\pi$
$824$	−55.9508	−1.94914
$825$	0	0
$826$	−103.428	−3.59873
$827$	− 6.33428i	− 0.220264i	−0.993917	−	0.110132i	$-0.964873\pi$
$827$	− 6.33428i	− 0.220264i	0.993917	−	0.110132i	$-0.0351274\pi$
$828$	26.5484i	0.922621i
$829$	−1.36706	−0.0474798	−0.0237399	−	0.999718i	$-0.507557\pi$
$829$	−1.36706	−0.0474798	−0.0237399	+	0.999718i	$0.507557\pi$
$830$	0	0
$831$	−10.2791	−0.356579
$832$	17.0113i	0.589762i
$833$	21.1116i	0.731473i
$834$	17.5786	0.608699
$835$	0	0
$836$	54.4853	1.88441
$837$	2.23218i	0.0771554i
$838$	67.1769i	2.32059i
$839$	55.7902	1.92609	0.963046	−	0.269339i	$-0.0868051\pi$
$839$	55.7902	1.92609	0.963046	+	0.269339i	$0.0868051\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	− 6.86919i	− 0.236728i
$843$	4.57869i	0.157698i
$844$	−108.439	−3.73264
$845$	0	0
$846$	9.35221	0.321536
$847$	6.09218i	0.209330i
$848$	− 7.86098i	− 0.269947i
$849$	19.2431	0.660421
$850$	0	0
$851$	33.0939	1.13444
$852$	− 27.3495i	− 0.936977i
$853$	6.42812i	0.220095i	0.993926	+	0.110047i	$0.0351003\pi$
$853$	6.42812i	0.220095i	−0.993926	+	0.110047i	$0.964900\pi$
$854$	75.2762	2.57590
$855$	0	0
$856$	22.4288	0.766600
$857$	− 35.0274i	− 1.19651i	−0.801304	−	0.598257i	$-0.795860\pi$
$857$	− 35.0274i	− 1.19651i	0.801304	−	0.598257i	$-0.204140\pi$
$858$	40.7891i	1.39252i
$859$	−18.4412	−0.629207	−0.314603	−	0.949223i	$-0.601872\pi$
$859$	−18.4412	−0.629207	−0.314603	+	0.949223i	$0.601872\pi$
$860$	0	0
$861$	−50.2606	−1.71288
$862$	0.388055i	0.0132172i
$863$	7.23911i	0.246422i	0.992380	+	0.123211i	$0.0393192\pi$
$863$	7.23911i	0.246422i	−0.992380	+	0.123211i	$0.960681\pi$
$864$	−7.73067	−0.263003
$865$	0	0
$866$	85.7048	2.91237
$867$	15.0394i	0.510766i
$868$	− 48.7237i	− 1.65379i
$869$	22.7918	0.773160
$870$	0	0
$871$	70.6549	2.39405
$872$	1.68269i	0.0569830i
$873$	− 13.8343i	− 0.468219i
$874$	55.4689	1.87626
$875$	0	0
$876$	32.9089	1.11189
$877$	− 24.6324i	− 0.831777i	−0.909416	−	0.415889i	$-0.863471\pi$
$877$	− 24.6324i	− 0.831777i	0.909416	−	0.415889i	$-0.136529\pi$
$878$	46.7422i	1.57747i
$879$	14.6045	0.492597
$880$	0	0
$881$	26.2145	0.883190	0.441595	−	0.897214i	$-0.354413\pi$
$881$	26.2145	0.883190	0.441595	+	0.897214i	$0.354413\pi$
$882$	− 38.8683i	− 1.30876i
$883$	10.7656i	0.362290i	0.983456	+	0.181145i	$0.0579803\pi$
$883$	10.7656i	0.362290i	−0.983456	+	0.181145i	$0.942020\pi$
$884$	33.0402	1.11126
$885$	0	0
$886$	−89.7911	−3.01659
$887$	6.27796i	0.210793i	0.994430	+	0.105397i	$0.0336112\pi$
$887$	6.27796i	0.210793i	−0.994430	+	0.105397i	$0.966389\pi$
$888$	39.4932i	1.32531i
$889$	−99.4743	−3.33626
$890$	0	0
$891$	−3.11503	−0.104358
$892$	104.179i	3.48817i
$893$	− 13.6594i	− 0.457094i
$894$	−61.1824	−2.04625
$895$	0	0
$896$	−32.0818	−1.07178
$897$	29.0282i	0.969223i
$898$	93.1949i	3.10995i
$899$	2.23218	0.0744474
$900$	0	0
$901$	−1.32776	−0.0442340
$902$	− 85.8973i	− 2.86007i
$903$	− 41.7960i	− 1.39088i
$904$	−98.3927	−3.27249
$905$	0	0
$906$	−44.6221	−1.48247
$907$	21.7954i	0.723704i	0.932235	+	0.361852i	$0.117855\pi$
$907$	21.7954i	0.723704i	−0.932235	+	0.361852i	$0.882145\pi$
$908$	18.9225i	0.627966i
$909$	−12.7461	−0.422763
$910$	0	0
$911$	−5.56422	−0.184351	−0.0921753	−	0.995743i	$-0.529382\pi$
$911$	−5.56422	−0.184351	−0.0921753	+	0.995743i	$0.529382\pi$
$912$	31.2127i	1.03355i
$913$	− 42.5449i	− 1.40803i
$914$	−23.0406	−0.762114
$915$	0	0
$916$	−73.0136	−2.41244
$917$	38.4819i	1.27078i
$918$	3.60957i	0.119134i
$919$	7.76705	0.256211	0.128106	−	0.991761i	$-0.459110\pi$
$919$	7.76705	0.256211	0.128106	+	0.991761i	$0.459110\pi$
$920$	0	0
$921$	−12.3324	−0.406368
$922$	− 89.3841i	− 2.94371i
$923$	− 29.9041i	− 0.984304i
$924$	67.9944	2.23685
$925$	0	0
$926$	−79.7632	−2.62118
$927$	− 8.20406i	− 0.269457i
$928$	7.73067i	0.253772i
$929$	11.3588	0.372672	0.186336	−	0.982486i	$-0.440339\pi$
$929$	11.3588	0.372672	0.186336	+	0.982486i	$0.440339\pi$
$930$	0	0
$931$	−56.7691	−1.86053
$932$	76.4503i	2.50421i
$933$	12.3084i	0.402957i
$934$	13.9443	0.456273
$935$	0	0
$936$	−34.6413	−1.13229
$937$	32.7813i	1.07092i	0.844561	+	0.535460i	$0.179862\pi$
$937$	32.7813i	1.07092i	−0.844561	+	0.535460i	$0.820138\pi$
$938$	− 168.487i	− 5.50128i
$939$	16.7306	0.545984
$940$	0	0
$941$	−13.3074	−0.433808	−0.216904	−	0.976193i	$-0.569596\pi$
$941$	−13.3074	−0.433808	−0.216904	+	0.976193i	$0.569596\pi$
$942$	25.9174i	0.844436i
$943$	− 61.1302i	− 1.99067i
$944$	70.7862	2.30390
$945$	0	0
$946$	71.4310	2.32242
$947$	31.9463i	1.03811i	0.854740	+	0.519057i	$0.173717\pi$
$947$	31.9463i	1.03811i	−0.854740	+	0.519057i	$0.826283\pi$
$948$	33.9901i	1.10395i
$949$	35.9827	1.16805
$950$	0	0
$951$	10.9429	0.354849
$952$	− 44.8687i	− 1.45420i
$953$	53.5304i	1.73402i	0.498292	+	0.867009i	$0.333961\pi$
$953$	53.5304i	1.73402i	−0.498292	+	0.867009i	$0.666039\pi$
$954$	2.44451	0.0791440
$955$	0	0
$956$	14.8054	0.478840
$957$	3.11503i	0.100695i
$958$	34.6133i	1.11831i
$959$	81.7821	2.64088
$960$	0	0
$961$	−26.0174	−0.839270
$962$	75.8274i	2.44477i
$963$	3.28873i	0.105978i
$964$	−51.7893	−1.66802
$965$	0	0
$966$	69.2219	2.22718
$967$	16.9490i	0.545044i	0.962150	+	0.272522i	$0.0878578\pi$
$967$	16.9490i	0.545044i	−0.962150	+	0.272522i	$0.912142\pi$
$968$	− 8.84250i	− 0.284209i
$969$	5.27196	0.169360
$970$	0	0
$971$	−28.5476	−0.916135	−0.458067	−	0.888917i	$-0.651458\pi$
$971$	−28.5476	−0.916135	−0.458067	+	0.888917i	$0.651458\pi$
$972$	− 4.64553i	− 0.149005i
$973$	− 32.0402i	− 1.02716i
$974$	−64.1242	−2.05467
$975$	0	0
$976$	−51.5189	−1.64908
$977$	− 30.8519i	− 0.987040i	−0.869734	−	0.493520i	$-0.835710\pi$
$977$	− 30.8519i	− 0.987040i	0.869734	−	0.493520i	$-0.164290\pi$
$978$	− 31.6054i	− 1.01063i
$979$	−24.8585	−0.794481
$980$	0	0
$981$	−0.246732	−0.00787755
$982$	− 53.2895i	− 1.70054i
$983$	− 25.9915i	− 0.829000i	−0.910049	−	0.414500i	$-0.863956\pi$
$983$	− 25.9915i	− 0.829000i	0.910049	−	0.414500i	$-0.136044\pi$
$984$	72.9508	2.32559
$985$	0	0
$986$	3.60957	0.114952
$987$	− 17.0461i	− 0.542583i
$988$	88.8451i	2.82654i
$989$	50.8350	1.61646
$990$	0	0
$991$	−3.17860	−0.100972	−0.0504858	−	0.998725i	$-0.516077\pi$
$991$	−3.17860	−0.100972	−0.0504858	+	0.998725i	$0.516077\pi$
$992$	17.2562i	0.547886i
$993$	− 19.1431i	− 0.607487i
$994$	−71.3105	−2.26183
$995$	0	0
$996$	63.4484	2.01044
$997$	31.6331i	1.00183i	0.865496	+	0.500916i	$0.167003\pi$
$997$	31.6331i	1.00183i	−0.865496	+	0.500916i	$0.832997\pi$
$998$	54.6982i	1.73144i
$999$	−5.79088	−0.183215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.p.349.3		16
5.2	odd	4		2175.2.a.bd.1.7	yes	8
5.3	odd	4		2175.2.a.bc.1.2	✓	8
5.4	even	2	inner	2175.2.c.p.349.14		16
15.2	even	4		6525.2.a.by.1.2		8
15.8	even	4		6525.2.a.bz.1.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.bc.1.2	✓	8	5.3	odd	4
2175.2.a.bd.1.7	yes	8	5.2	odd	4
2175.2.c.p.349.3		16	1.1	even	1	trivial
2175.2.c.p.349.14		16	5.4	even	2	inner
6525.2.a.by.1.2		8	15.2	even	4
6525.2.a.bz.1.7		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)

\(f(q)\)	\(=\)	\(q-2.57789i q^{2} +1.00000i q^{3} -4.64553 q^{4} +2.57789 q^{6} -4.69867i q^{7} +6.81989i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.57789i q^{2} +1.00000i q^{3} -4.64553 q^{4} +2.57789 q^{6} -4.69867i q^{7} +6.81989i q^{8} -1.00000 q^{9} -3.11503 q^{11} -4.64553i q^{12} -5.07945i q^{13} -12.1127 q^{14} +8.28989 q^{16} -1.40020i q^{17} +2.57789i q^{18} +3.76514 q^{19} +4.69867 q^{21} +8.03022i q^{22} +5.71483i q^{23} -6.81989 q^{24} -13.0943 q^{26} -1.00000i q^{27} +21.8278i q^{28} -1.00000 q^{29} -2.23218 q^{31} -7.73067i q^{32} -3.11503i q^{33} -3.60957 q^{34} +4.64553 q^{36} -5.79088i q^{37} -9.70614i q^{38} +5.07945 q^{39} -10.6968 q^{41} -12.1127i q^{42} -8.89527i q^{43} +14.4710 q^{44} +14.7322 q^{46} -3.62785i q^{47} +8.28989i q^{48} -15.0775 q^{49} +1.40020 q^{51} +23.5967i q^{52} -0.948260i q^{53} -2.57789 q^{54} +32.0445 q^{56} +3.76514i q^{57} +2.57789i q^{58} +8.53886 q^{59} -6.21467 q^{61} +5.75432i q^{62} +4.69867i q^{63} -3.34905 q^{64} -8.03022 q^{66} +13.9099i q^{67} +6.50468i q^{68} -5.71483 q^{69} +5.88726 q^{71} -6.81989i q^{72} +7.08398i q^{73} -14.9283 q^{74} -17.4911 q^{76} +14.6365i q^{77} -13.0943i q^{78} -7.31672 q^{79} +1.00000 q^{81} +27.5751i q^{82} +13.6579i q^{83} -21.8278 q^{84} -22.9311 q^{86} -1.00000i q^{87} -21.2442i q^{88} +7.98017 q^{89} -23.8667 q^{91} -26.5484i q^{92} -2.23218i q^{93} -9.35221 q^{94} +7.73067 q^{96} +13.8343i q^{97} +38.8683i q^{98} +3.11503 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 24 q^{4} + 4 q^{6} - 16 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more