LMFDB - Embedded newform 1725.2.b.u.1174.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1725,2,Mod(1174,1725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1725, 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("1725.1174");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1725 = 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1725.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.7741943487$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 $ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 345)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1174.1
Root			$-0.671462 - 1.24464i$ of defining polynomial
Character	$\chi$	$=$	1725.1174
Dual form			1725.2.b.u.1174.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34292i q^{2} -1.00000i q^{3} -3.48929 q^{4} -2.34292 q^{6} +4.48929i q^{7} +3.48929i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-2.34292i q^{2} -1.00000i q^{3} -3.48929 q^{4} -2.34292 q^{6} +4.48929i q^{7} +3.48929i q^{8} -1.00000 q^{9} -1.14637 q^{11} +3.48929i q^{12} -0.853635i q^{13} +10.5181 q^{14} +1.19656 q^{16} +1.34292i q^{17} +2.34292i q^{18} +3.83221 q^{19} +4.48929 q^{21} +2.68585i q^{22} -1.00000i q^{23} +3.48929 q^{24} -2.00000 q^{26} +1.00000i q^{27} -15.6644i q^{28} +8.02877 q^{29} +2.19656 q^{31} +4.17513i q^{32} +1.14637i q^{33} +3.14637 q^{34} +3.48929 q^{36} -2.48929i q^{37} -8.97858i q^{38} -0.853635 q^{39} +11.3001 q^{41} -10.5181i q^{42} -10.6858i q^{43} +4.00000 q^{44} -2.34292 q^{46} +1.53948i q^{47} -1.19656i q^{48} -13.1537 q^{49} +1.34292 q^{51} +2.97858i q^{52} -4.02877i q^{53} +2.34292 q^{54} -15.6644 q^{56} -3.83221i q^{57} -18.8108i q^{58} +15.0073 q^{59} -5.83221 q^{61} -5.14637i q^{62} -4.48929i q^{63} +12.1751 q^{64} +2.68585 q^{66} -11.5682i q^{67} -4.68585i q^{68} -1.00000 q^{69} +4.32150 q^{71} -3.48929i q^{72} +13.1035i q^{73} -5.83221 q^{74} -13.3717 q^{76} -5.14637i q^{77} +2.00000i q^{78} -0.585462 q^{79} +1.00000 q^{81} -26.4752i q^{82} -5.63565i q^{83} -15.6644 q^{84} -25.0361 q^{86} -8.02877i q^{87} -4.00000i q^{88} +0.100384 q^{89} +3.83221 q^{91} +3.48929i q^{92} -2.19656i q^{93} +3.60688 q^{94} +4.17513 q^{96} +11.5640i q^{97} +30.8181i q^{98} +1.14637 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^4 - 2 * q^6 - 6 * q^9	$ 6 q - 6 q^{4} - 2 q^{6} - 6 q^{9} - 4 q^{11} + 12 q^{14} - 2 q^{16} - 4 q^{19} + 12 q^{21} + 6 q^{24} - 12 q^{26} + 12 q^{29} + 4 q^{31} + 16 q^{34} + 6 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} - 2 q^{46} - 10 q^{49} - 4 q^{51} + 2 q^{54} - 40 q^{56} + 24 q^{59} - 8 q^{61} + 34 q^{64} - 8 q^{66} - 6 q^{69} - 16 q^{71} - 8 q^{74} - 32 q^{76} + 8 q^{79} + 6 q^{81} - 40 q^{84} - 48 q^{86} - 12 q^{89} - 4 q^{91} + 40 q^{94} - 14 q^{96} + 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^4 - 2 * q^6 - 6 * q^9 - 4 * q^11 + 12 * q^14 - 2 * q^16 - 4 * q^19 + 12 * q^21 + 6 * q^24 - 12 * q^26 + 12 * q^29 + 4 * q^31 + 16 * q^34 + 6 * q^36 - 8 * q^39 - 4 * q^41 + 24 * q^44 - 2 * q^46 - 10 * q^49 - 4 * q^51 + 2 * q^54 - 40 * q^56 + 24 * q^59 - 8 * q^61 + 34 * q^64 - 8 * q^66 - 6 * q^69 - 16 * q^71 - 8 * q^74 - 32 * q^76 + 8 * q^79 + 6 * q^81 - 40 * q^84 - 48 * q^86 - 12 * q^89 - 4 * q^91 + 40 * q^94 - 14 * q^96 + 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times$.

$n$	$277$	$1151$	$1201$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	− 2.34292i	− 1.65670i	−0.560213	−	0.828348i	$-0.689281\pi$
$2$	− 2.34292i	− 1.65670i	0.560213	−	0.828348i	$-0.310719\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−3.48929	−1.74464
$5$	0	0
$5$	0	0
$6$	−2.34292	−0.956494
$7$	4.48929i	1.69679i	0.529362	+	0.848396i	$0.322431\pi$
$7$	4.48929i	1.69679i	−0.529362	+	0.848396i	$0.677569\pi$
$8$	3.48929i	1.23365i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.14637	−0.345642	−0.172821	−	0.984953i	$-0.555288\pi$
$11$	−1.14637	−0.345642	−0.172821	+	0.984953i	$0.555288\pi$
$12$	3.48929i	1.00727i
$13$	− 0.853635i	− 0.236756i	−0.992969	−	0.118378i	$-0.962231\pi$
$13$	− 0.853635i	− 0.236756i	0.992969	−	0.118378i	$-0.0377694\pi$
$14$	10.5181	2.81107
$15$	0	0
$16$	1.19656	0.299139
$17$	1.34292i	0.325707i	0.986650	+	0.162853i	$0.0520697\pi$
$17$	1.34292i	0.325707i	−0.986650	+	0.162853i	$0.947930\pi$
$18$	2.34292i	0.552232i
$19$	3.83221	0.879170	0.439585	−	0.898201i	$-0.355126\pi$
$19$	3.83221	0.879170	0.439585	+	0.898201i	$0.355126\pi$
$20$	0	0
$21$	4.48929	0.979643
$22$	2.68585i	0.572624i
$23$	− 1.00000i	− 0.208514i
$23$	− 1.00000i	− 0.208514i
$24$	3.48929	0.712248
$25$	0	0
$26$	−2.00000	−0.392232
$27$	1.00000i	0.192450i
$28$	− 15.6644i	− 2.96030i
$29$	8.02877	1.49091	0.745453	−	0.666559i	$-0.232233\pi$
$29$	8.02877	1.49091	0.745453	+	0.666559i	$0.232233\pi$
$30$	0	0
$31$	2.19656	0.394513	0.197257	−	0.980352i	$-0.436797\pi$
$31$	2.19656	0.394513	0.197257	+	0.980352i	$0.436797\pi$
$32$	4.17513i	0.738067i
$33$	1.14637i	0.199557i
$34$	3.14637	0.539597
$35$	0	0
$36$	3.48929	0.581548
$37$	− 2.48929i	− 0.409237i	−0.978842	−	0.204618i	$-0.934405\pi$
$37$	− 2.48929i	− 0.409237i	0.978842	−	0.204618i	$-0.0655953\pi$
$38$	− 8.97858i	− 1.45652i
$39$	−0.853635	−0.136691
$40$	0	0
$41$	11.3001	1.76478	0.882388	−	0.470523i	$-0.155935\pi$
$41$	11.3001	1.76478	0.882388	+	0.470523i	$0.155935\pi$
$42$	− 10.5181i	− 1.62297i
$43$	− 10.6858i	− 1.62958i	−0.579759	−	0.814788i	$-0.696853\pi$
$43$	− 10.6858i	− 1.62958i	0.579759	−	0.814788i	$-0.303147\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−2.34292	−0.345445
$47$	1.53948i	0.224556i	0.993677	+	0.112278i	$0.0358148\pi$
$47$	1.53948i	0.224556i	−0.993677	+	0.112278i	$0.964185\pi$
$48$	− 1.19656i	− 0.172708i
$49$	−13.1537	−1.87910
$50$	0	0
$51$	1.34292	0.188047
$52$	2.97858i	0.413054i
$53$	− 4.02877i	− 0.553394i	−0.960957	−	0.276697i	$-0.910760\pi$
$53$	− 4.02877i	− 0.553394i	0.960957	−	0.276697i	$-0.0892398\pi$
$54$	2.34292	0.318831
$55$	0	0
$56$	−15.6644	−2.09325
$57$	− 3.83221i	− 0.507589i
$58$	− 18.8108i	− 2.46998i
$59$	15.0073	1.95379	0.976895	−	0.213720i	$-0.0685579\pi$
$59$	15.0073	1.95379	0.976895	+	0.213720i	$0.0685579\pi$
$60$	0	0
$61$	−5.83221	−0.746738	−0.373369	−	0.927683i	$-0.621797\pi$
$61$	−5.83221	−0.746738	−0.373369	+	0.927683i	$0.621797\pi$
$62$	− 5.14637i	− 0.653589i
$63$	− 4.48929i	− 0.565597i
$64$	12.1751	1.52189
$65$	0	0
$66$	2.68585	0.330605
$67$	− 11.5682i	− 1.41329i	−0.707570	−	0.706643i	$-0.750209\pi$
$67$	− 11.5682i	− 1.41329i	0.707570	−	0.706643i	$-0.249791\pi$
$68$	− 4.68585i	− 0.568242i
$69$	−1.00000	−0.120386
$70$	0	0
$71$	4.32150	0.512868	0.256434	−	0.966562i	$-0.417452\pi$
$71$	4.32150	0.512868	0.256434	+	0.966562i	$0.417452\pi$
$72$	− 3.48929i	− 0.411217i
$73$	13.1035i	1.53365i	0.641856	+	0.766825i	$0.278165\pi$
$73$	13.1035i	1.53365i	−0.641856	+	0.766825i	$0.721835\pi$
$74$	−5.83221	−0.677981
$75$	0	0
$76$	−13.3717	−1.53384
$77$	− 5.14637i	− 0.586483i
$78$	2.00000i	0.226455i
$79$	−0.585462	−0.0658696	−0.0329348	−	0.999458i	$-0.510485\pi$
$79$	−0.585462	−0.0658696	−0.0329348	+	0.999458i	$0.510485\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 26.4752i	− 2.92370i
$83$	− 5.63565i	− 0.618593i	−0.950966	−	0.309297i	$-0.899906\pi$
$83$	− 5.63565i	− 0.618593i	0.950966	−	0.309297i	$-0.100094\pi$
$84$	−15.6644	−1.70913
$85$	0	0
$86$	−25.0361	−2.69971
$87$	− 8.02877i	− 0.860774i
$88$	− 4.00000i	− 0.426401i
$89$	0.100384	0.0106407	0.00532035	−	0.999986i	$-0.498306\pi$
$89$	0.100384	0.0106407	0.00532035	+	0.999986i	$0.498306\pi$
$90$	0	0
$91$	3.83221	0.401725
$92$	3.48929i	0.363783i
$93$	− 2.19656i	− 0.227772i
$94$	3.60688	0.372022
$95$	0	0
$96$	4.17513	0.426123
$97$	11.5640i	1.17415i	0.809532	+	0.587075i	$0.199721\pi$
$97$	11.5640i	1.17415i	−0.809532	+	0.587075i	$0.800279\pi$
$98$	30.8181i	3.11310i
$99$	1.14637	0.115214
$100$	0	0
$101$	11.6932	1.16352	0.581758	−	0.813362i	$-0.302365\pi$
$101$	11.6932	1.16352	0.581758	+	0.813362i	$0.302365\pi$
$102$	− 3.14637i	− 0.311537i
$103$	19.7220i	1.94326i	0.236501	+	0.971631i	$0.423999\pi$
$103$	19.7220i	1.94326i	−0.236501	+	0.971631i	$0.576001\pi$
$104$	2.97858	0.292074
$105$	0	0
$106$	−9.43910	−0.916806
$107$	5.44331i	0.526224i	0.964765	+	0.263112i	$0.0847489\pi$
$107$	5.44331i	0.526224i	−0.964765	+	0.263112i	$0.915251\pi$
$108$	− 3.48929i	− 0.335757i
$109$	12.5181	1.19901	0.599506	−	0.800370i	$-0.295364\pi$
$109$	12.5181	1.19901	0.599506	+	0.800370i	$0.295364\pi$
$110$	0	0
$111$	−2.48929	−0.236273
$112$	5.37169i	0.507577i
$113$	14.1292i	1.32916i	0.747218	+	0.664579i	$0.231389\pi$
$113$	14.1292i	1.32916i	−0.747218	+	0.664579i	$0.768611\pi$
$114$	−8.97858	−0.840921
$115$	0	0
$116$	−28.0147	−2.60110
$117$	0.853635i	0.0789185i
$118$	− 35.1611i	− 3.23684i
$119$	−6.02877	−0.552656
$120$	0	0
$121$	−9.68585	−0.880531
$122$	13.6644i	1.23712i
$123$	− 11.3001i	− 1.01889i
$124$	−7.66442	−0.688286
$125$	0	0
$126$	−10.5181	−0.937023
$127$	5.48194i	0.486444i	0.969971	+	0.243222i	$0.0782043\pi$
$127$	5.48194i	0.486444i	−0.969971	+	0.243222i	$0.921796\pi$
$128$	− 20.1751i	− 1.78325i
$129$	−10.6858	−0.940836
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	− 4.00000i	− 0.348155i
$133$	17.2039i	1.49177i
$134$	−27.1035	−2.34139
$135$	0	0
$136$	−4.68585	−0.401808
$137$	11.7648i	1.00514i	0.864538	+	0.502568i	$0.167611\pi$
$137$	11.7648i	1.00514i	−0.864538	+	0.502568i	$0.832389\pi$
$138$	2.34292i	0.199443i
$139$	−8.15371	−0.691589	−0.345794	−	0.938310i	$-0.612391\pi$
$139$	−8.15371	−0.691589	−0.345794	+	0.938310i	$0.612391\pi$
$140$	0	0
$141$	1.53948	0.129648
$142$	− 10.1249i	− 0.849666i
$143$	0.978577i	0.0818327i
$144$	−1.19656	−0.0997131
$145$	0	0
$146$	30.7005	2.54079
$147$	13.1537i	1.08490i
$148$	8.68585i	0.713972i
$149$	3.73183	0.305723	0.152862	−	0.988248i	$-0.451151\pi$
$149$	3.73183	0.305723	0.152862	+	0.988248i	$0.451151\pi$
$150$	0	0
$151$	16.6430	1.35439	0.677194	−	0.735804i	$-0.263196\pi$
$151$	16.6430	1.35439	0.677194	+	0.735804i	$0.263196\pi$
$152$	13.3717i	1.08459i
$153$	− 1.34292i	− 0.108569i
$154$	−12.0575	−0.971624
$155$	0	0
$156$	2.97858	0.238477
$157$	− 13.2327i	− 1.05608i	−0.849219	−	0.528041i	$-0.822927\pi$
$157$	− 13.2327i	− 1.05608i	0.849219	−	0.528041i	$-0.177073\pi$
$158$	1.37169i	0.109126i
$159$	−4.02877	−0.319502
$160$	0	0
$161$	4.48929	0.353806
$162$	− 2.34292i	− 0.184077i
$163$	− 16.3931i	− 1.28401i	−0.766701	−	0.642004i	$-0.778103\pi$
$163$	− 16.3931i	− 1.28401i	0.766701	−	0.642004i	$-0.221897\pi$
$164$	−39.4292	−3.07891
$165$	0	0
$166$	−13.2039	−1.02482
$167$	3.04598i	0.235705i	0.993031	+	0.117853i	$0.0376010\pi$
$167$	3.04598i	0.235705i	−0.993031	+	0.117853i	$0.962399\pi$
$168$	15.6644i	1.20854i
$169$	12.2713	0.943947
$170$	0	0
$171$	−3.83221	−0.293057
$172$	37.2860i	2.84303i
$173$	24.9357i	1.89583i	0.318526	+	0.947914i	$0.396812\pi$
$173$	24.9357i	1.89583i	−0.318526	+	0.947914i	$0.603188\pi$
$174$	−18.8108	−1.42604
$175$	0	0
$176$	−1.37169	−0.103395
$177$	− 15.0073i	− 1.12802i
$178$	− 0.235192i	− 0.0176284i
$179$	6.68585	0.499724	0.249862	−	0.968282i	$-0.419615\pi$
$179$	6.68585	0.499724	0.249862	+	0.968282i	$0.419615\pi$
$180$	0	0
$181$	−20.6002	−1.53120	−0.765599	−	0.643318i	$-0.777557\pi$
$181$	−20.6002	−1.53120	−0.765599	+	0.643318i	$0.777557\pi$
$182$	− 8.97858i	− 0.665536i
$183$	5.83221i	0.431129i
$184$	3.48929	0.257234
$185$	0	0
$186$	−5.14637	−0.377350
$187$	− 1.53948i	− 0.112578i
$188$	− 5.37169i	− 0.391771i
$189$	−4.48929	−0.326548
$190$	0	0
$191$	4.95402	0.358460	0.179230	−	0.983807i	$-0.442639\pi$
$191$	4.95402	0.358460	0.179230	+	0.983807i	$0.442639\pi$
$192$	− 12.1751i	− 0.878665i
$193$	− 6.05754i	− 0.436031i	−0.975945	−	0.218016i	$-0.930042\pi$
$193$	− 6.05754i	− 0.436031i	0.975945	−	0.218016i	$-0.0699584\pi$
$194$	27.0937	1.94521
$195$	0	0
$196$	45.8971	3.27836
$197$	− 24.4078i	− 1.73898i	−0.493947	−	0.869492i	$-0.664446\pi$
$197$	− 24.4078i	− 1.73898i	0.493947	−	0.869492i	$-0.335554\pi$
$198$	− 2.68585i	− 0.190875i
$199$	3.85677	0.273399	0.136700	−	0.990613i	$-0.456350\pi$
$199$	3.85677	0.273399	0.136700	+	0.990613i	$0.456350\pi$
$200$	0	0
$201$	−11.5682	−0.815961
$202$	− 27.3963i	− 1.92759i
$203$	36.0435i	2.52976i
$204$	−4.68585	−0.328075
$205$	0	0
$206$	46.2070	3.21940
$207$	1.00000i	0.0695048i
$208$	− 1.02142i	− 0.0708229i
$209$	−4.39312	−0.303878
$210$	0	0
$211$	−0.824865	−0.0567861	−0.0283930	−	0.999597i	$-0.509039\pi$
$211$	−0.824865	−0.0567861	−0.0283930	+	0.999597i	$0.509039\pi$
$212$	14.0575i	0.965476i
$213$	− 4.32150i	− 0.296104i
$214$	12.7533	0.871794
$215$	0	0
$216$	−3.48929	−0.237416
$217$	9.86098i	0.669407i
$218$	− 29.3288i	− 1.98640i
$219$	13.1035	0.885454
$220$	0	0
$221$	1.14637	0.0771129
$222$	5.83221i	0.391432i
$223$	− 5.56404i	− 0.372596i	−0.982493	−	0.186298i	$-0.940351\pi$
$223$	− 5.56404i	− 0.372596i	0.982493	−	0.186298i	$-0.0596489\pi$
$224$	−18.7434	−1.25235
$225$	0	0
$226$	33.1035	2.20201
$227$	16.5855i	1.10082i	0.834896	+	0.550408i	$0.185528\pi$
$227$	16.5855i	1.10082i	−0.834896	+	0.550408i	$0.814472\pi$
$228$	13.3717i	0.885562i
$229$	−6.64300	−0.438982	−0.219491	−	0.975615i	$-0.570440\pi$
$229$	−6.64300	−0.438982	−0.219491	+	0.975615i	$0.570440\pi$
$230$	0	0
$231$	−5.14637	−0.338606
$232$	28.0147i	1.83925i
$233$	14.3931i	0.942924i	0.881886	+	0.471462i	$0.156274\pi$
$233$	14.3931i	0.942924i	−0.881886	+	0.471462i	$0.843726\pi$
$234$	2.00000	0.130744
$235$	0	0
$236$	−52.3650	−3.40867
$237$	0.585462i	0.0380298i
$238$	14.1249i	0.915584i
$239$	13.1077	0.847869	0.423934	−	0.905693i	$-0.360649\pi$
$239$	13.1077	0.847869	0.423934	+	0.905693i	$0.360649\pi$
$240$	0	0
$241$	12.5181	0.806359	0.403179	−	0.915121i	$-0.367905\pi$
$241$	12.5181	0.806359	0.403179	+	0.915121i	$0.367905\pi$
$242$	22.6932i	1.45877i
$243$	− 1.00000i	− 0.0641500i
$244$	20.3503	1.30279
$245$	0	0
$246$	−26.4752	−1.68800
$247$	− 3.27131i	− 0.208148i
$248$	7.66442i	0.486691i
$249$	−5.63565	−0.357145
$250$	0	0
$251$	−26.3503	−1.66321	−0.831607	−	0.555364i	$-0.812579\pi$
$251$	−26.3503	−1.66321	−0.831607	+	0.555364i	$0.812579\pi$
$252$	15.6644i	0.986766i
$253$	1.14637i	0.0720714i
$254$	12.8438	0.805890
$255$	0	0
$256$	−22.9185	−1.43241
$257$	− 27.0116i	− 1.68493i	−0.538747	−	0.842467i	$-0.681102\pi$
$257$	− 27.0116i	− 1.68493i	0.538747	−	0.842467i	$-0.318898\pi$
$258$	25.0361i	1.55868i
$259$	11.1751	0.694389
$260$	0	0
$261$	−8.02877	−0.496968
$262$	0	0
$263$	− 21.8855i	− 1.34952i	−0.738037	−	0.674760i	$-0.764247\pi$
$263$	− 21.8855i	− 1.34952i	0.738037	−	0.674760i	$-0.235753\pi$
$264$	−4.00000	−0.246183
$265$	0	0
$266$	40.3074	2.47141
$267$	− 0.100384i	− 0.00614341i
$268$	40.3650i	2.46568i
$269$	−3.69319	−0.225178	−0.112589	−	0.993642i	$-0.535914\pi$
$269$	−3.69319	−0.225178	−0.112589	+	0.993642i	$0.535914\pi$
$270$	0	0
$271$	0.288520	0.0175264	0.00876318	−	0.999962i	$-0.497211\pi$
$271$	0.288520	0.0175264	0.00876318	+	0.999962i	$0.497211\pi$
$272$	1.60688i	0.0974317i
$273$	− 3.83221i	− 0.231936i
$274$	27.5640	1.66520
$275$	0	0
$276$	3.48929	0.210030
$277$	21.3288i	1.28153i	0.767739	+	0.640763i	$0.221382\pi$
$277$	21.3288i	1.28153i	−0.767739	+	0.640763i	$0.778618\pi$
$278$	19.1035i	1.14575i
$279$	−2.19656	−0.131504
$280$	0	0
$281$	−26.6676	−1.59085	−0.795427	−	0.606050i	$-0.792753\pi$
$281$	−26.6676	−1.59085	−0.795427	+	0.606050i	$0.792753\pi$
$282$	− 3.60688i	− 0.214787i
$283$	− 25.9185i	− 1.54070i	−0.637624	−	0.770348i	$-0.720082\pi$
$283$	− 25.9185i	− 1.54070i	0.637624	−	0.770348i	$-0.279918\pi$
$284$	−15.0790	−0.894772
$285$	0	0
$286$	2.29273	0.135572
$287$	50.7293i	2.99446i
$288$	− 4.17513i	− 0.246022i
$289$	15.1966	0.893915
$290$	0	0
$291$	11.5640	0.677896
$292$	− 45.7220i	− 2.67568i
$293$	− 13.6503i	− 0.797462i	−0.917068	−	0.398731i	$-0.869451\pi$
$293$	− 13.6503i	− 0.797462i	0.917068	−	0.398731i	$-0.130549\pi$
$294$	30.8181	1.79735
$295$	0	0
$296$	8.68585	0.504855
$297$	− 1.14637i	− 0.0665189i
$298$	− 8.74338i	− 0.506491i
$299$	−0.853635	−0.0493670
$300$	0	0
$301$	47.9718	2.76505
$302$	− 38.9933i	− 2.24381i
$303$	− 11.6932i	− 0.671756i
$304$	4.58546	0.262994
$305$	0	0
$306$	−3.14637	−0.179866
$307$	− 9.78937i	− 0.558709i	−0.960188	−	0.279354i	$-0.909880\pi$
$307$	− 9.78937i	− 0.558709i	0.960188	−	0.279354i	$-0.0901204\pi$
$308$	17.9572i	1.02320i
$309$	19.7220	1.12194
$310$	0	0
$311$	−3.32885	−0.188762	−0.0943808	−	0.995536i	$-0.530087\pi$
$311$	−3.32885	−0.188762	−0.0943808	+	0.995536i	$0.530087\pi$
$312$	− 2.97858i	− 0.168629i
$313$	18.8824i	1.06730i	0.845707	+	0.533648i	$0.179179\pi$
$313$	18.8824i	1.06730i	−0.845707	+	0.533648i	$0.820821\pi$
$314$	−31.0031	−1.74961
$315$	0	0
$316$	2.04285	0.114919
$317$	− 8.06740i	− 0.453111i	−0.973998	−	0.226555i	$-0.927254\pi$
$317$	− 8.06740i	− 0.453111i	0.973998	−	0.226555i	$-0.0727464\pi$
$318$	9.43910i	0.529318i
$319$	−9.20390	−0.515320
$320$	0	0
$321$	5.44331	0.303816
$322$	− 10.5181i	− 0.586148i
$323$	5.14637i	0.286351i
$324$	−3.48929	−0.193849
$325$	0	0
$326$	−38.4078	−2.12721
$327$	− 12.5181i	− 0.692250i
$328$	39.4292i	2.17712i
$329$	−6.91117	−0.381025
$330$	0	0
$331$	−26.6388	−1.46420	−0.732100	−	0.681197i	$-0.761460\pi$
$331$	−26.6388	−1.46420	−0.732100	+	0.681197i	$0.761460\pi$
$332$	19.6644i	1.07923i
$333$	2.48929i	0.136412i
$334$	7.13650	0.390492
$335$	0	0
$336$	5.37169	0.293050
$337$	− 8.54262i	− 0.465346i	−0.972555	−	0.232673i	$-0.925253\pi$
$337$	− 8.54262i	− 0.465346i	0.972555	−	0.232673i	$-0.0747472\pi$
$338$	− 28.7507i	− 1.56383i
$339$	14.1292	0.767390
$340$	0	0
$341$	−2.51806	−0.136360
$342$	8.97858i	0.485506i
$343$	− 27.6258i	− 1.49165i
$344$	37.2860	2.01033
$345$	0	0
$346$	58.4225	3.14081
$347$	− 16.2499i	− 0.872340i	−0.899864	−	0.436170i	$-0.856335\pi$
$347$	− 16.2499i	− 0.872340i	0.899864	−	0.436170i	$-0.143665\pi$
$348$	28.0147i	1.50175i
$349$	−23.1898	−1.24132	−0.620662	−	0.784079i	$-0.713136\pi$
$349$	−23.1898	−1.24132	−0.620662	+	0.784079i	$0.713136\pi$
$350$	0	0
$351$	0.853635	0.0455636
$352$	− 4.78623i	− 0.255107i
$353$	7.48194i	0.398224i	0.979977	+	0.199112i	$0.0638057\pi$
$353$	7.48194i	0.398224i	−0.979977	+	0.199112i	$0.936194\pi$
$354$	−35.1611	−1.86879
$355$	0	0
$356$	−0.350269	−0.0185642
$357$	6.02877i	0.319076i
$358$	− 15.6644i	− 0.827890i
$359$	−1.87506	−0.0989617	−0.0494809	−	0.998775i	$-0.515757\pi$
$359$	−1.87506	−0.0989617	−0.0494809	+	0.998775i	$0.515757\pi$
$360$	0	0
$361$	−4.31415	−0.227061
$362$	48.2646i	2.53673i
$363$	9.68585i	0.508375i
$364$	−13.3717	−0.700867
$365$	0	0
$366$	13.6644	0.714251
$367$	− 4.68164i	− 0.244379i	−0.992507	−	0.122190i	$-0.961008\pi$
$367$	− 4.68164i	− 0.244379i	0.992507	−	0.122190i	$-0.0389916\pi$
$368$	− 1.19656i	− 0.0623749i
$369$	−11.3001	−0.588259
$370$	0	0
$371$	18.0863	0.938994
$372$	7.66442i	0.397382i
$373$	− 18.6430i	− 0.965298i	−0.875814	−	0.482649i	$-0.839675\pi$
$373$	− 18.6430i	− 0.965298i	0.875814	−	0.482649i	$-0.160325\pi$
$374$	−3.60688	−0.186508
$375$	0	0
$376$	−5.37169	−0.277024
$377$	− 6.85363i	− 0.352980i
$378$	10.5181i	0.540991i
$379$	−29.8715	−1.53439	−0.767197	−	0.641412i	$-0.778349\pi$
$379$	−29.8715	−1.53439	−0.767197	+	0.641412i	$0.778349\pi$
$380$	0	0
$381$	5.48194	0.280848
$382$	− 11.6069i	− 0.593860i
$383$	− 20.5714i	− 1.05115i	−0.850748	−	0.525574i	$-0.823850\pi$
$383$	− 20.5714i	− 1.05115i	0.850748	−	0.525574i	$-0.176150\pi$
$384$	−20.1751	−1.02956
$385$	0	0
$386$	−14.1923	−0.722371
$387$	10.6858i	0.543192i
$388$	− 40.3503i	− 2.04847i
$389$	2.33558	0.118418	0.0592092	−	0.998246i	$-0.481142\pi$
$389$	2.33558	0.118418	0.0592092	+	0.998246i	$0.481142\pi$
$390$	0	0
$391$	1.34292	0.0679145
$392$	− 45.8971i	− 2.31815i
$393$	0	0
$394$	−57.1856	−2.88097
$395$	0	0
$396$	−4.00000	−0.201008
$397$	29.1365i	1.46232i	0.682207	+	0.731160i	$0.261020\pi$
$397$	29.1365i	1.46232i	−0.682207	+	0.731160i	$0.738980\pi$
$398$	− 9.03612i	− 0.452940i
$399$	17.2039	0.861272
$400$	0	0
$401$	−14.8353	−0.740842	−0.370421	−	0.928864i	$-0.620787\pi$
$401$	−14.8353	−0.740842	−0.370421	+	0.928864i	$0.620787\pi$
$402$	27.1035i	1.35180i
$403$	− 1.87506i	− 0.0934033i
$404$	−40.8009	−2.02992
$405$	0	0
$406$	84.4471	4.19104
$407$	2.85363i	0.141449i
$408$	4.68585i	0.231984i
$409$	38.0189	1.87991	0.939957	−	0.341293i	$-0.110865\pi$
$409$	38.0189	1.87991	0.939957	+	0.341293i	$0.110865\pi$
$410$	0	0
$411$	11.7648	0.580315
$412$	− 68.8156i	− 3.39030i
$413$	67.3723i	3.31517i
$414$	2.34292	0.115148
$415$	0	0
$416$	3.56404	0.174741
$417$	8.15371i	0.399289i
$418$	10.2927i	0.503434i
$419$	−21.8469	−1.06729	−0.533646	−	0.845708i	$-0.679178\pi$
$419$	−21.8469	−1.06729	−0.533646	+	0.845708i	$0.679178\pi$
$420$	0	0
$421$	21.4391	1.04488	0.522439	−	0.852677i	$-0.325022\pi$
$421$	21.4391	1.04488	0.522439	+	0.852677i	$0.325022\pi$
$422$	1.93260i	0.0940773i
$423$	− 1.53948i	− 0.0748521i
$424$	14.0575	0.682694
$425$	0	0
$426$	−10.1249	−0.490555
$427$	− 26.1825i	− 1.26706i
$428$	− 18.9933i	− 0.918074i
$429$	0.978577	0.0472461
$430$	0	0
$431$	29.8223	1.43649	0.718246	−	0.695789i	$-0.244945\pi$
$431$	29.8223	1.43649	0.718246	+	0.695789i	$0.244945\pi$
$432$	1.19656i	0.0575694i
$433$	0.925249i	0.0444647i	0.999753	+	0.0222323i	$0.00707735\pi$
$433$	0.925249i	0.0444647i	−0.999753	+	0.0222323i	$0.992923\pi$
$434$	23.1035	1.10900
$435$	0	0
$436$	−43.6791	−2.09185
$437$	− 3.83221i	− 0.183320i
$438$	− 30.7005i	− 1.46693i
$439$	21.6791	1.03469	0.517344	−	0.855778i	$-0.326921\pi$
$439$	21.6791	1.03469	0.517344	+	0.855778i	$0.326921\pi$
$440$	0	0
$441$	13.1537	0.626367
$442$	− 2.68585i	− 0.127753i
$443$	− 5.20390i	− 0.247245i	−0.992329	−	0.123622i	$-0.960549\pi$
$443$	− 5.20390i	− 0.247245i	0.992329	−	0.123622i	$-0.0394512\pi$
$444$	8.68585	0.412212
$445$	0	0
$446$	−13.0361	−0.617278
$447$	− 3.73183i	− 0.176509i
$448$	54.6577i	2.58233i
$449$	10.8578	0.512413	0.256207	−	0.966622i	$-0.417527\pi$
$449$	10.8578	0.512413	0.256207	+	0.966622i	$0.417527\pi$
$450$	0	0
$451$	−12.9540	−0.609981
$452$	− 49.3007i	− 2.31891i
$453$	− 16.6430i	− 0.781956i
$454$	38.8585	1.82372
$455$	0	0
$456$	13.3717	0.626187
$457$	27.2755i	1.27589i	0.770080	+	0.637947i	$0.220216\pi$
$457$	27.2755i	1.27589i	−0.770080	+	0.637947i	$0.779784\pi$
$458$	15.5640i	0.727260i
$459$	−1.34292	−0.0626823
$460$	0	0
$461$	20.2070	0.941136	0.470568	−	0.882364i	$-0.344049\pi$
$461$	20.2070	0.941136	0.470568	+	0.882364i	$0.344049\pi$
$462$	12.0575i	0.560967i
$463$	− 8.03298i	− 0.373324i	−0.982424	−	0.186662i	$-0.940233\pi$
$463$	− 8.03298i	− 0.373324i	0.982424	−	0.186662i	$-0.0597670\pi$
$464$	9.60688	0.445988
$465$	0	0
$466$	33.7220	1.56214
$467$	− 19.4580i	− 0.900409i	−0.892926	−	0.450204i	$-0.851351\pi$
$467$	− 19.4580i	− 0.900409i	0.892926	−	0.450204i	$-0.148649\pi$
$468$	− 2.97858i	− 0.137685i
$469$	51.9332	2.39805
$470$	0	0
$471$	−13.2327	−0.609729
$472$	52.3650i	2.41029i
$473$	12.2499i	0.563250i
$474$	1.37169	0.0630039
$475$	0	0
$476$	21.0361	0.964189
$477$	4.02877i	0.184465i
$478$	− 30.7104i	− 1.40466i
$479$	−28.9540	−1.32294	−0.661471	−	0.749970i	$-0.730068\pi$
$479$	−28.9540	−1.32294	−0.661471	+	0.749970i	$0.730068\pi$
$480$	0	0
$481$	−2.12494	−0.0968890
$482$	− 29.3288i	− 1.33589i
$483$	− 4.48929i	− 0.204270i
$484$	33.7967	1.53621
$485$	0	0
$486$	−2.34292	−0.106277
$487$	20.8108i	0.943027i	0.881859	+	0.471513i	$0.156292\pi$
$487$	20.8108i	0.943027i	−0.881859	+	0.471513i	$0.843708\pi$
$488$	− 20.3503i	− 0.921213i
$489$	−16.3931	−0.741322
$490$	0	0
$491$	35.1140	1.58467	0.792336	−	0.610084i	$-0.208865\pi$
$491$	35.1140	1.58467	0.792336	+	0.610084i	$0.208865\pi$
$492$	39.4292i	1.77761i
$493$	10.7820i	0.485598i
$494$	−7.66442	−0.344839
$495$	0	0
$496$	2.62831	0.118014
$497$	19.4005i	0.870230i
$498$	13.2039i	0.591681i
$499$	7.31836	0.327615	0.163807	−	0.986492i	$-0.447622\pi$
$499$	7.31836	0.327615	0.163807	+	0.986492i	$0.447622\pi$
$500$	0	0
$501$	3.04598	0.136084
$502$	61.7367i	2.75544i
$503$	22.0006i	0.980959i	0.871453	+	0.490479i	$0.163178\pi$
$503$	22.0006i	0.980959i	−0.871453	+	0.490479i	$0.836822\pi$
$504$	15.6644	0.697749
$505$	0	0
$506$	2.68585	0.119400
$507$	− 12.2713i	− 0.544988i
$508$	− 19.1281i	− 0.848671i
$509$	35.4783	1.57255	0.786275	−	0.617877i	$-0.212007\pi$
$509$	35.4783	1.57255	0.786275	+	0.617877i	$0.212007\pi$
$510$	0	0
$511$	−58.8255	−2.60229
$512$	13.3461i	0.589818i
$513$	3.83221i	0.169196i
$514$	−63.2860	−2.79143
$515$	0	0
$516$	37.2860	1.64142
$517$	− 1.76481i	− 0.0776161i
$518$	− 26.1825i	− 1.15039i
$519$	24.9357	1.09456
$520$	0	0
$521$	21.4966	0.941785	0.470892	−	0.882191i	$-0.343932\pi$
$521$	21.4966	0.941785	0.470892	+	0.882191i	$0.343932\pi$
$522$	18.8108i	0.823326i
$523$	11.3288i	0.495376i	0.968840	+	0.247688i	$0.0796708\pi$
$523$	11.3288i	0.495376i	−0.968840	+	0.247688i	$0.920329\pi$
$524$	0	0
$525$	0	0
$526$	−51.2761	−2.23575
$527$	2.94981i	0.128496i
$528$	1.37169i	0.0596952i
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	−15.0073	−0.651263
$532$	− 60.0294i	− 2.60260i
$533$	− 9.64614i	− 0.417821i
$534$	−0.235192	−0.0101778
$535$	0	0
$536$	40.3650	1.74350
$537$	− 6.68585i	− 0.288516i
$538$	8.65287i	0.373052i
$539$	15.0790	0.649497
$540$	0	0
$541$	−7.67912	−0.330151	−0.165075	−	0.986281i	$-0.552787\pi$
$541$	−7.67912	−0.330151	−0.165075	+	0.986281i	$0.552787\pi$
$542$	− 0.675981i	− 0.0290358i
$543$	20.6002i	0.884037i
$544$	−5.60688	−0.240393
$545$	0	0
$546$	−8.97858	−0.384248
$547$	31.5296i	1.34811i	0.738682	+	0.674054i	$0.235449\pi$
$547$	31.5296i	1.34811i	−0.738682	+	0.674054i	$0.764551\pi$
$548$	− 41.0508i	− 1.75360i
$549$	5.83221	0.248913
$550$	0	0
$551$	30.7679	1.31076
$552$	− 3.48929i	− 0.148514i
$553$	− 2.62831i	− 0.111767i
$554$	49.9718	2.12310
$555$	0	0
$556$	28.4507	1.20658
$557$	− 10.1292i	− 0.429186i	−0.976704	−	0.214593i	$-0.931157\pi$
$557$	− 10.1292i	− 0.429186i	0.976704	−	0.214593i	$-0.0688425\pi$
$558$	5.14637i	0.217863i
$559$	−9.12181	−0.385811
$560$	0	0
$561$	−1.53948	−0.0649969
$562$	62.4800i	2.63556i
$563$	− 40.7146i	− 1.71592i	−0.513719	−	0.857958i	$-0.671733\pi$
$563$	− 40.7146i	− 1.71592i	0.513719	−	0.857958i	$-0.328267\pi$
$564$	−5.37169	−0.226189
$565$	0	0
$566$	−60.7251	−2.55247
$567$	4.48929i	0.188532i
$568$	15.0790i	0.632699i
$569$	−1.66442	−0.0697763	−0.0348881	−	0.999391i	$-0.511107\pi$
$569$	−1.66442	−0.0697763	−0.0348881	+	0.999391i	$0.511107\pi$
$570$	0	0
$571$	−41.8469	−1.75124	−0.875619	−	0.483002i	$-0.839546\pi$
$571$	−41.8469	−1.75124	−0.875619	+	0.483002i	$0.839546\pi$
$572$	− 3.41454i	− 0.142769i
$573$	− 4.95402i	− 0.206957i
$574$	118.855	4.96091
$575$	0	0
$576$	−12.1751	−0.507297
$577$	21.6069i	0.899506i	0.893153	+	0.449753i	$0.148488\pi$
$577$	21.6069i	0.899506i	−0.893153	+	0.449753i	$0.851512\pi$
$578$	− 35.6044i	− 1.48095i
$579$	−6.05754	−0.251743
$580$	0	0
$581$	25.3001	1.04962
$582$	− 27.0937i	− 1.12307i
$583$	4.61844i	0.191276i
$584$	−45.7220	−1.89199
$585$	0	0
$586$	−31.9817	−1.32115
$587$	0.393115i	0.0162256i	0.999967	+	0.00811280i	$0.00258241\pi$
$587$	0.393115i	0.0162256i	−0.999967	+	0.00811280i	$0.997418\pi$
$588$	− 45.8971i	− 1.89276i
$589$	8.41767	0.346844
$590$	0	0
$591$	−24.4078	−1.00400
$592$	− 2.97858i	− 0.122419i
$593$	8.31729i	0.341550i	0.985310	+	0.170775i	$0.0546271\pi$
$593$	8.31729i	0.341550i	−0.985310	+	0.170775i	$0.945373\pi$
$594$	−2.68585	−0.110202
$595$	0	0
$596$	−13.0214	−0.533378
$597$	− 3.85677i	− 0.157847i
$598$	2.00000i	0.0817861i
$599$	−9.03612	−0.369206	−0.184603	−	0.982813i	$-0.559100\pi$
$599$	−9.03612	−0.369206	−0.184603	+	0.982813i	$0.559100\pi$
$600$	0	0
$601$	−27.6602	−1.12828	−0.564142	−	0.825678i	$-0.690793\pi$
$601$	−27.6602	−1.12828	−0.564142	+	0.825678i	$0.690793\pi$
$602$	− 112.394i	− 4.58085i
$603$	11.5682i	0.471096i
$604$	−58.0722	−2.36293
$605$	0	0
$606$	−27.3963	−1.11290
$607$	29.6461i	1.20330i	0.798760	+	0.601650i	$0.205490\pi$
$607$	29.6461i	1.20330i	−0.798760	+	0.601650i	$0.794510\pi$
$608$	16.0000i	0.648886i
$609$	36.0435	1.46055
$610$	0	0
$611$	1.31415	0.0531650
$612$	4.68585i	0.189414i
$613$	− 39.4868i	− 1.59486i	−0.603414	−	0.797428i	$-0.706193\pi$
$613$	− 39.4868i	− 1.59486i	0.603414	−	0.797428i	$-0.293807\pi$
$614$	−22.9357	−0.925611
$615$	0	0
$616$	17.9572	0.723514
$617$	16.8641i	0.678924i	0.940620	+	0.339462i	$0.110245\pi$
$617$	16.8641i	0.678924i	−0.940620	+	0.339462i	$0.889755\pi$
$618$	− 46.2070i	− 1.85872i
$619$	−2.74338	−0.110266	−0.0551330	−	0.998479i	$-0.517558\pi$
$619$	−2.74338	−0.110266	−0.0551330	+	0.998479i	$0.517558\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	7.79923i	0.312721i
$623$	0.450654i	0.0180551i
$624$	−1.02142	−0.0408896
$625$	0	0
$626$	44.2400	1.76819
$627$	4.39312i	0.175444i
$628$	46.1726i	1.84249i
$629$	3.34292	0.133291
$630$	0	0
$631$	−9.93260	−0.395410	−0.197705	−	0.980262i	$-0.563349\pi$
$631$	−9.93260	−0.395410	−0.197705	+	0.980262i	$0.563349\pi$
$632$	− 2.04285i	− 0.0812600i
$633$	0.824865i	0.0327855i
$634$	−18.9013	−0.750667
$635$	0	0
$636$	14.0575	0.557418
$637$	11.2285i	0.444888i
$638$	21.5640i	0.853728i
$639$	−4.32150	−0.170956
$640$	0	0
$641$	−10.6184	−0.419403	−0.209702	−	0.977765i	$-0.567249\pi$
$641$	−10.6184	−0.419403	−0.209702	+	0.977765i	$0.567249\pi$
$642$	− 12.7533i	− 0.503331i
$643$	− 20.5959i	− 0.812225i	−0.913823	−	0.406112i	$-0.866884\pi$
$643$	− 20.5959i	− 0.812225i	0.913823	−	0.406112i	$-0.133116\pi$
$644$	−15.6644	−0.617265
$645$	0	0
$646$	12.0575	0.474398
$647$	− 6.71883i	− 0.264144i	−0.991240	−	0.132072i	$-0.957837\pi$
$647$	− 6.71883i	− 0.264144i	0.991240	−	0.132072i	$-0.0421631\pi$
$648$	3.48929i	0.137072i
$649$	−17.2039	−0.675312
$650$	0	0
$651$	9.86098	0.386482
$652$	57.2003i	2.24014i
$653$	4.07583i	0.159499i	0.996815	+	0.0797497i	$0.0254121\pi$
$653$	4.07583i	0.159499i	−0.996815	+	0.0797497i	$0.974588\pi$
$654$	−29.3288	−1.14685
$655$	0	0
$656$	13.5212	0.527914
$657$	− 13.1035i	− 0.511217i
$658$	16.1923i	0.631243i
$659$	29.0607	1.13204	0.566022	−	0.824390i	$-0.308482\pi$
$659$	29.0607	1.13204	0.566022	+	0.824390i	$0.308482\pi$
$660$	0	0
$661$	31.2369	1.21497	0.607487	−	0.794330i	$-0.292178\pi$
$661$	31.2369	1.21497	0.607487	+	0.794330i	$0.292178\pi$
$662$	62.4126i	2.42574i
$663$	− 1.14637i	− 0.0445211i
$664$	19.6644	0.763128
$665$	0	0
$666$	5.83221	0.225994
$667$	− 8.02877i	− 0.310875i
$668$	− 10.6283i	− 0.411222i
$669$	−5.56404	−0.215118
$670$	0	0
$671$	6.68585	0.258104
$672$	18.7434i	0.723042i
$673$	− 9.74652i	− 0.375701i	−0.982198	−	0.187850i	$-0.939848\pi$
$673$	− 9.74652i	− 0.375701i	0.982198	−	0.187850i	$-0.0601520\pi$
$674$	−20.0147	−0.770937
$675$	0	0
$676$	−42.8181	−1.64685
$677$	6.80031i	0.261357i	0.991425	+	0.130679i	$0.0417156\pi$
$677$	6.80031i	0.261357i	−0.991425	+	0.130679i	$0.958284\pi$
$678$	− 33.1035i	− 1.27133i
$679$	−51.9143	−1.99229
$680$	0	0
$681$	16.5855	0.635556
$682$	5.89962i	0.225908i
$683$	41.5029i	1.58806i	0.607876	+	0.794032i	$0.292022\pi$
$683$	41.5029i	1.58806i	−0.607876	+	0.794032i	$0.707978\pi$
$684$	13.3717	0.511279
$685$	0	0
$686$	−64.7251	−2.47122
$687$	6.64300i	0.253446i
$688$	− 12.7862i	− 0.487470i
$689$	−3.43910	−0.131019
$690$	0	0
$691$	−31.2713	−1.18962	−0.594808	−	0.803868i	$-0.702772\pi$
$691$	−31.2713	−1.18962	−0.594808	+	0.803868i	$0.702772\pi$
$692$	− 87.0080i	− 3.30755i
$693$	5.14637i	0.195494i
$694$	−38.0722	−1.44520
$695$	0	0
$696$	28.0147	1.06189
$697$	15.1751i	0.574799i
$698$	54.3320i	2.05650i
$699$	14.3931	0.544398
$700$	0	0
$701$	−6.02456	−0.227544	−0.113772	−	0.993507i	$-0.536293\pi$
$701$	−6.02456	−0.227544	−0.113772	+	0.993507i	$0.536293\pi$
$702$	− 2.00000i	− 0.0754851i
$703$	− 9.53948i	− 0.359788i
$704$	−13.9572	−0.526030
$705$	0	0
$706$	17.5296	0.659736
$707$	52.4941i	1.97424i
$708$	52.3650i	1.96800i
$709$	−37.6974	−1.41576	−0.707878	−	0.706335i	$-0.750347\pi$
$709$	−37.6974	−1.41576	−0.707878	+	0.706335i	$0.750347\pi$
$710$	0	0
$711$	0.585462	0.0219565
$712$	0.350269i	0.0131269i
$713$	− 2.19656i	− 0.0822617i
$714$	14.1249	0.528613
$715$	0	0
$716$	−23.3288	−0.871840
$717$	− 13.1077i	− 0.489517i
$718$	4.39312i	0.163950i
$719$	−19.8708	−0.741058	−0.370529	−	0.928821i	$-0.620824\pi$
$719$	−19.8708	−0.741058	−0.370529	+	0.928821i	$0.620824\pi$
$720$	0	0
$721$	−88.5376	−3.29731
$722$	10.1077i	0.376171i
$723$	− 12.5181i	− 0.465552i
$724$	71.8799	2.67139
$725$	0	0
$726$	22.6932	0.842223
$727$	15.8757i	0.588796i	0.955683	+	0.294398i	$0.0951191\pi$
$727$	15.8757i	0.588796i	−0.955683	+	0.294398i	$0.904881\pi$
$728$	13.3717i	0.495588i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	14.3503	0.530764
$732$	− 20.3503i	− 0.752168i
$733$	− 37.1751i	− 1.37309i	−0.727085	−	0.686547i	$-0.759125\pi$
$733$	− 37.1751i	− 1.37309i	0.727085	−	0.686547i	$-0.240875\pi$
$734$	−10.9687	−0.404863
$735$	0	0
$736$	4.17513	0.153898
$737$	13.2614i	0.488492i
$738$	26.4752i	0.974566i
$739$	−44.0189	−1.61926	−0.809631	−	0.586940i	$-0.800333\pi$
$739$	−44.0189	−1.61926	−0.809631	+	0.586940i	$0.800333\pi$
$740$	0	0
$741$	−3.27131	−0.120175
$742$	− 42.3748i	− 1.55563i
$743$	− 51.5296i	− 1.89044i	−0.326437	−	0.945219i	$-0.605848\pi$
$743$	− 51.5296i	− 1.89044i	0.326437	−	0.945219i	$-0.394152\pi$
$744$	7.66442	0.280991
$745$	0	0
$746$	−43.6791	−1.59921
$747$	5.63565i	0.206198i
$748$	5.37169i	0.196409i
$749$	−24.4366	−0.892893
$750$	0	0
$751$	−3.91790	−0.142966	−0.0714832	−	0.997442i	$-0.522773\pi$
$751$	−3.91790	−0.142966	−0.0714832	+	0.997442i	$0.522773\pi$
$752$	1.84208i	0.0671736i
$753$	26.3503i	0.960257i
$754$	−16.0575	−0.584781
$755$	0	0
$756$	15.6644	0.569710
$757$	25.7606i	0.936285i	0.883653	+	0.468142i	$0.155077\pi$
$757$	25.7606i	0.936285i	−0.883653	+	0.468142i	$0.844923\pi$
$758$	69.9865i	2.54203i
$759$	1.14637	0.0416104
$760$	0	0
$761$	−15.7507	−0.570964	−0.285482	−	0.958384i	$-0.592154\pi$
$761$	−15.7507	−0.570964	−0.285482	+	0.958384i	$0.592154\pi$
$762$	− 12.8438i	− 0.465281i
$763$	56.1972i	2.03447i
$764$	−17.2860	−0.625386
$765$	0	0
$766$	−48.1972	−1.74143
$767$	− 12.8108i	− 0.462571i
$768$	22.9185i	0.827001i
$769$	19.0031	0.685271	0.342635	−	0.939468i	$-0.388680\pi$
$769$	19.0031	0.685271	0.342635	+	0.939468i	$0.388680\pi$
$770$	0	0
$771$	−27.0116	−0.972797
$772$	21.1365i	0.760719i
$773$	41.7795i	1.50270i	0.659902	+	0.751352i	$0.270598\pi$
$773$	41.7795i	1.50270i	−0.659902	+	0.751352i	$0.729402\pi$
$774$	25.0361	0.899905
$775$	0	0
$776$	−40.3503	−1.44849
$777$	− 11.1751i	− 0.400906i
$778$	− 5.47208i	− 0.196183i
$779$	43.3043	1.55154
$780$	0	0
$781$	−4.95402	−0.177269
$782$	− 3.14637i	− 0.112514i
$783$	8.02877i	0.286925i
$784$	−15.7392	−0.562113
$785$	0	0
$786$	0	0
$787$	− 21.8610i	− 0.779260i	−0.920972	−	0.389630i	$-0.872603\pi$
$787$	− 21.8610i	− 0.779260i	0.920972	−	0.389630i	$-0.127397\pi$
$788$	85.1659i	3.03391i
$789$	−21.8855	−0.779146
$790$	0	0
$791$	−63.4298	−2.25531
$792$	4.00000i	0.142134i
$793$	4.97858i	0.176794i
$794$	68.2646	2.42262
$795$	0	0
$796$	−13.4574	−0.476984
$797$	− 27.8280i	− 0.985718i	−0.870109	−	0.492859i	$-0.835952\pi$
$797$	− 27.8280i	− 0.985718i	0.870109	−	0.492859i	$-0.164048\pi$
$798$	− 40.3074i	− 1.42687i
$799$	−2.06740	−0.0731395
$800$	0	0
$801$	−0.100384	−0.00354690
$802$	34.7581i	1.22735i
$803$	− 15.0214i	− 0.530095i
$804$	40.3650	1.42356
$805$	0	0
$806$	−4.39312	−0.154741
$807$	3.69319i	0.130007i
$808$	40.8009i	1.43537i
$809$	21.0565	0.740306	0.370153	−	0.928971i	$-0.379305\pi$
$809$	21.0565	0.740306	0.370153	+	0.928971i	$0.379305\pi$
$810$	0	0
$811$	−16.8249	−0.590801	−0.295400	−	0.955374i	$-0.595453\pi$
$811$	−16.8249	−0.590801	−0.295400	+	0.955374i	$0.595453\pi$
$812$	− 125.766i	− 4.41352i
$813$	− 0.288520i	− 0.0101188i
$814$	6.68585	0.234339
$815$	0	0
$816$	1.60688	0.0562522
$817$	− 40.9504i	− 1.43267i
$818$	− 89.0754i	− 3.11445i
$819$	−3.83221	−0.133908
$820$	0	0
$821$	−2.59388	−0.0905272	−0.0452636	−	0.998975i	$-0.514413\pi$
$821$	−2.59388	−0.0905272	−0.0452636	+	0.998975i	$0.514413\pi$
$822$	− 27.5640i	− 0.961406i
$823$	− 0.786230i	− 0.0274063i	−0.999906	−	0.0137031i	$-0.995638\pi$
$823$	− 0.786230i	− 0.0274063i	0.999906	−	0.0137031i	$-0.00436198\pi$
$824$	−68.8156	−2.39731
$825$	0	0
$826$	157.848	5.49224
$827$	10.2211i	0.355423i	0.984083	+	0.177712i	$0.0568694\pi$
$827$	10.2211i	0.355423i	−0.984083	+	0.177712i	$0.943131\pi$
$828$	− 3.48929i	− 0.121261i
$829$	8.97437	0.311693	0.155846	−	0.987781i	$-0.450190\pi$
$829$	8.97437	0.311693	0.155846	+	0.987781i	$0.450190\pi$
$830$	0	0
$831$	21.3288	0.739889
$832$	− 10.3931i	− 0.360316i
$833$	− 17.6644i	− 0.612036i
$834$	19.1035	0.661501
$835$	0	0
$836$	15.3288	0.530159
$837$	2.19656i	0.0759241i
$838$	51.1856i	1.76818i
$839$	−19.6069	−0.676905	−0.338452	−	0.940984i	$-0.609903\pi$
$839$	−19.6069	−0.676905	−0.338452	+	0.940984i	$0.609903\pi$
$840$	0	0
$841$	35.4611	1.22280
$842$	− 50.2302i	− 1.73105i
$843$	26.6676i	0.918480i
$844$	2.87819	0.0990715
$845$	0	0
$846$	−3.60688	−0.124007
$847$	− 43.4826i	− 1.49408i
$848$	− 4.82065i	− 0.165542i
$849$	−25.9185	−0.889521
$850$	0	0
$851$	−2.48929	−0.0853317
$852$	15.0790i	0.516597i
$853$	14.3650i	0.491847i	0.969289	+	0.245923i	$0.0790912\pi$
$853$	14.3650i	0.491847i	−0.969289	+	0.245923i	$0.920909\pi$
$854$	−61.3435	−2.09913
$855$	0	0
$856$	−18.9933	−0.649177
$857$	22.0000i	0.751506i	0.926720	+	0.375753i	$0.122616\pi$
$857$	22.0000i	0.751506i	−0.926720	+	0.375753i	$0.877384\pi$
$858$	− 2.29273i	− 0.0782725i
$859$	4.85425	0.165625	0.0828125	−	0.996565i	$-0.473610\pi$
$859$	4.85425	0.165625	0.0828125	+	0.996565i	$0.473610\pi$
$860$	0	0
$861$	50.7293	1.72885
$862$	− 69.8715i	− 2.37983i
$863$	− 41.8139i	− 1.42336i	−0.702503	−	0.711681i	$-0.747934\pi$
$863$	− 41.8139i	− 1.42336i	0.702503	−	0.711681i	$-0.252066\pi$
$864$	−4.17513	−0.142041
$865$	0	0
$866$	2.16779	0.0736644
$867$	− 15.1966i	− 0.516102i
$868$	− 34.4078i	− 1.16788i
$869$	0.671153	0.0227673
$870$	0	0
$871$	−9.87506	−0.334604
$872$	43.6791i	1.47916i
$873$	− 11.5640i	− 0.391383i
$874$	−8.97858	−0.303705
$875$	0	0
$876$	−45.7220	−1.54480
$877$	− 1.27973i	− 0.0432134i	−0.999767	−	0.0216067i	$-0.993122\pi$
$877$	− 1.27973i	− 0.0432134i	0.999767	−	0.0216067i	$-0.00687816\pi$
$878$	− 50.7925i	− 1.71416i
$879$	−13.6503	−0.460415
$880$	0	0
$881$	52.9687	1.78456	0.892281	−	0.451481i	$-0.149104\pi$
$881$	52.9687	1.78456	0.892281	+	0.451481i	$0.149104\pi$
$882$	− 30.8181i	− 1.03770i
$883$	17.5479i	0.590534i	0.955415	+	0.295267i	$0.0954086\pi$
$883$	17.5479i	0.590534i	−0.955415	+	0.295267i	$0.904591\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−12.1923	−0.409610
$887$	− 11.2713i	− 0.378453i	−0.981933	−	0.189227i	$-0.939402\pi$
$887$	− 11.2713i	− 0.378453i	0.981933	−	0.189227i	$-0.0605981\pi$
$888$	− 8.68585i	− 0.291478i
$889$	−24.6100	−0.825394
$890$	0	0
$891$	−1.14637	−0.0384047
$892$	19.4145i	0.650047i
$893$	5.89962i	0.197423i
$894$	−8.74338	−0.292423
$895$	0	0
$896$	90.5720	3.02580
$897$	0.853635i	0.0285020i
$898$	− 25.4391i	− 0.848914i
$899$	17.6357	0.588182
$900$	0	0
$901$	5.41033	0.180244
$902$	30.3503i	1.01055i
$903$	− 47.9718i	− 1.59640i
$904$	−49.3007	−1.63972
$905$	0	0
$906$	−38.9933	−1.29546
$907$	27.2045i	0.903311i	0.892192	+	0.451656i	$0.149166\pi$
$907$	27.2045i	0.903311i	−0.892192	+	0.451656i	$0.850834\pi$
$908$	− 57.8715i	− 1.92053i
$909$	−11.6932	−0.387839
$910$	0	0
$911$	−3.41454	−0.113129	−0.0565643	−	0.998399i	$-0.518015\pi$
$911$	−3.41454	−0.113129	−0.0565643	+	0.998399i	$0.518015\pi$
$912$	− 4.58546i	− 0.151840i
$913$	6.46052i	0.213812i
$914$	63.9044	2.11377
$915$	0	0
$916$	23.1793	0.765867
$917$	0	0
$918$	3.14637i	0.103846i
$919$	−20.4998	−0.676225	−0.338113	−	0.941106i	$-0.609788\pi$
$919$	−20.4998	−0.676225	−0.338113	+	0.941106i	$0.609788\pi$
$920$	0	0
$921$	−9.78937	−0.322571
$922$	− 47.3435i	− 1.55918i
$923$	− 3.68898i	− 0.121424i
$924$	17.9572	0.590747
$925$	0	0
$926$	−18.8207	−0.618485
$927$	− 19.7220i	− 0.647754i
$928$	33.5212i	1.10039i
$929$	−38.2640	−1.25540	−0.627700	−	0.778455i	$-0.716004\pi$
$929$	−38.2640	−1.25540	−0.627700	+	0.778455i	$0.716004\pi$
$930$	0	0
$931$	−50.4078	−1.65205
$932$	− 50.2217i	− 1.64507i
$933$	3.32885i	0.108982i
$934$	−45.5886	−1.49170
$935$	0	0
$936$	−2.97858	−0.0973578
$937$	35.6791i	1.16559i	0.812621	+	0.582793i	$0.198040\pi$
$937$	35.6791i	1.16559i	−0.812621	+	0.582793i	$0.801960\pi$
$938$	− 121.676i	− 3.97285i
$939$	18.8824	0.616204
$940$	0	0
$941$	15.2321	0.496551	0.248275	−	0.968689i	$-0.420136\pi$
$941$	15.2321	0.496551	0.248275	+	0.968689i	$0.420136\pi$
$942$	31.0031i	1.01014i
$943$	− 11.3001i	− 0.367981i
$944$	17.9572	0.584456
$945$	0	0
$946$	28.7005	0.933135
$947$	6.29273i	0.204486i	0.994759	+	0.102243i	$0.0326020\pi$
$947$	6.29273i	0.204486i	−0.994759	+	0.102243i	$0.967398\pi$
$948$	− 2.04285i	− 0.0663485i
$949$	11.1856	0.363100
$950$	0	0
$951$	−8.06740	−0.261604
$952$	− 21.0361i	− 0.681784i
$953$	− 20.3418i	− 0.658937i	−0.944167	−	0.329469i	$-0.893130\pi$
$953$	− 20.3418i	− 0.658937i	0.944167	−	0.329469i	$-0.106870\pi$
$954$	9.43910	0.305602
$955$	0	0
$956$	−45.7367	−1.47923
$957$	9.20390i	0.297520i
$958$	67.8370i	2.19172i
$959$	−52.8156	−1.70551
$960$	0	0
$961$	−26.1751	−0.844359
$962$	4.97858i	0.160516i
$963$	− 5.44331i	− 0.175408i
$964$	−43.6791	−1.40681
$965$	0	0
$966$	−10.5181	−0.338413
$967$	− 37.6461i	− 1.21062i	−0.795991	−	0.605309i	$-0.793050\pi$
$967$	− 37.6461i	− 1.21062i	0.795991	−	0.605309i	$-0.206950\pi$
$968$	− 33.7967i	− 1.08627i
$969$	5.14637	0.165325
$970$	0	0
$971$	−29.6216	−0.950602	−0.475301	−	0.879823i	$-0.657661\pi$
$971$	−29.6216	−0.950602	−0.475301	+	0.879823i	$0.657661\pi$
$972$	3.48929i	0.111919i
$973$	− 36.6044i	− 1.17348i
$974$	48.7581	1.56231
$975$	0	0
$976$	−6.97858	−0.223379
$977$	21.0158i	0.672354i	0.941799	+	0.336177i	$0.109134\pi$
$977$	21.0158i	0.672354i	−0.941799	+	0.336177i	$0.890866\pi$
$978$	38.4078i	1.22815i
$979$	−0.115077	−0.00367788
$980$	0	0
$981$	−12.5181	−0.399671
$982$	− 82.2694i	− 2.62532i
$983$	27.5647i	0.879176i	0.898200	+	0.439588i	$0.144876\pi$
$983$	27.5647i	0.879176i	−0.898200	+	0.439588i	$0.855124\pi$
$984$	39.4292	1.25696
$985$	0	0
$986$	25.2614	0.804488
$987$	6.91117i	0.219985i
$988$	11.4145i	0.363145i
$989$	−10.6858	−0.339790
$990$	0	0
$991$	−20.1537	−0.640204	−0.320102	−	0.947383i	$-0.603717\pi$
$991$	−20.1537	−0.640204	−0.320102	+	0.947383i	$0.603717\pi$
$992$	9.17092i	0.291177i
$993$	26.6388i	0.845356i
$994$	45.4538	1.44171
$995$	0	0
$996$	19.6644	0.623091
$997$	14.9295i	0.472821i	0.971653	+	0.236410i	$0.0759710\pi$
$997$	14.9295i	0.472821i	−0.971653	+	0.236410i	$0.924029\pi$
$998$	− 17.1464i	− 0.542759i
$999$	2.48929	0.0787576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.2.b.u.1174.1		6
5.2	odd	4		1725.2.a.bi.1.3		3
5.3	odd	4		345.2.a.j.1.1	✓	3
5.4	even	2	inner	1725.2.b.u.1174.6		6
15.2	even	4		5175.2.a.br.1.1		3
15.8	even	4		1035.2.a.n.1.3		3
20.3	even	4		5520.2.a.by.1.1		3
115.68	even	4		7935.2.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.j.1.1	✓	3	5.3	odd	4
1035.2.a.n.1.3		3	15.8	even	4
1725.2.a.bi.1.3		3	5.2	odd	4
1725.2.b.u.1174.1		6	1.1	even	1	trivial
1725.2.b.u.1174.6		6	5.4	even	2	inner
5175.2.a.br.1.1		3	15.2	even	4
5520.2.a.by.1.1		3	20.3	even	4
7935.2.a.u.1.1		3	115.68	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:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1725.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.34292i q^{2} -1.00000i q^{3} -3.48929 q^{4} -2.34292 q^{6} +4.48929i q^{7} +3.48929i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-2.34292i q^{2} -1.00000i q^{3} -3.48929 q^{4} -2.34292 q^{6} +4.48929i q^{7} +3.48929i q^{8} -1.00000 q^{9} -1.14637 q^{11} +3.48929i q^{12} -0.853635i q^{13} +10.5181 q^{14} +1.19656 q^{16} +1.34292i q^{17} +2.34292i q^{18} +3.83221 q^{19} +4.48929 q^{21} +2.68585i q^{22} -1.00000i q^{23} +3.48929 q^{24} -2.00000 q^{26} +1.00000i q^{27} -15.6644i q^{28} +8.02877 q^{29} +2.19656 q^{31} +4.17513i q^{32} +1.14637i q^{33} +3.14637 q^{34} +3.48929 q^{36} -2.48929i q^{37} -8.97858i q^{38} -0.853635 q^{39} +11.3001 q^{41} -10.5181i q^{42} -10.6858i q^{43} +4.00000 q^{44} -2.34292 q^{46} +1.53948i q^{47} -1.19656i q^{48} -13.1537 q^{49} +1.34292 q^{51} +2.97858i q^{52} -4.02877i q^{53} +2.34292 q^{54} -15.6644 q^{56} -3.83221i q^{57} -18.8108i q^{58} +15.0073 q^{59} -5.83221 q^{61} -5.14637i q^{62} -4.48929i q^{63} +12.1751 q^{64} +2.68585 q^{66} -11.5682i q^{67} -4.68585i q^{68} -1.00000 q^{69} +4.32150 q^{71} -3.48929i q^{72} +13.1035i q^{73} -5.83221 q^{74} -13.3717 q^{76} -5.14637i q^{77} +2.00000i q^{78} -0.585462 q^{79} +1.00000 q^{81} -26.4752i q^{82} -5.63565i q^{83} -15.6644 q^{84} -25.0361 q^{86} -8.02877i q^{87} -4.00000i q^{88} +0.100384 q^{89} +3.83221 q^{91} +3.48929i q^{92} -2.19656i q^{93} +3.60688 q^{94} +4.17513 q^{96} +11.5640i q^{97} +30.8181i q^{98} +1.14637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^4 - 2 * q^6 - 6 * q^9	\( 6 q - 6 q^{4} - 2 q^{6} - 6 q^{9} - 4 q^{11} + 12 q^{14} - 2 q^{16} - 4 q^{19} + 12 q^{21} + 6 q^{24} - 12 q^{26} + 12 q^{29} + 4 q^{31} + 16 q^{34} + 6 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} - 2 q^{46} - 10 q^{49} - 4 q^{51} + 2 q^{54} - 40 q^{56} + 24 q^{59} - 8 q^{61} + 34 q^{64} - 8 q^{66} - 6 q^{69} - 16 q^{71} - 8 q^{74} - 32 q^{76} + 8 q^{79} + 6 q^{81} - 40 q^{84} - 48 q^{86} - 12 q^{89} - 4 q^{91} + 40 q^{94} - 14 q^{96} + 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^4 - 2 * q^6 - 6 * q^9 - 4 * q^11 + 12 * q^14 - 2 * q^16 - 4 * q^19 + 12 * q^21 + 6 * q^24 - 12 * q^26 + 12 * q^29 + 4 * q^31 + 16 * q^34 + 6 * q^36 - 8 * q^39 - 4 * q^41 + 24 * q^44 - 2 * q^46 - 10 * q^49 - 4 * q^51 + 2 * q^54 - 40 * q^56 + 24 * q^59 - 8 * q^61 + 34 * q^64 - 8 * q^66 - 6 * q^69 - 16 * q^71 - 8 * q^74 - 32 * q^76 + 8 * q^79 + 6 * q^81 - 40 * q^84 - 48 * q^86 - 12 * q^89 - 4 * q^91 + 40 * q^94 - 14 * q^96 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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