LMFDB - Embedded newform 375.2.b.a.124.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(124,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("375.124");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 375 = 3 \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	375.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:	$2.99439007580$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			124.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	375.124
Dual form			375.2.b.a.124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.61803i q^{2} +1.00000i q^{3} -4.85410 q^{4} -2.61803 q^{6} +1.38197i q^{7} -7.47214i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.61803i q^{2} +1.00000i q^{3} -4.85410 q^{4} -2.61803 q^{6} +1.38197i q^{7} -7.47214i q^{8} -1.00000 q^{9} -6.23607 q^{11} -4.85410i q^{12} -3.00000i q^{13} -3.61803 q^{14} +9.85410 q^{16} +3.38197i q^{17} -2.61803i q^{18} +1.00000 q^{19} -1.38197 q^{21} -16.3262i q^{22} +6.70820i q^{23} +7.47214 q^{24} +7.85410 q^{26} -1.00000i q^{27} -6.70820i q^{28} +4.23607 q^{29} -3.09017 q^{31} +10.8541i q^{32} -6.23607i q^{33} -8.85410 q^{34} +4.85410 q^{36} +5.00000i q^{37} +2.61803i q^{38} +3.00000 q^{39} -4.14590 q^{41} -3.61803i q^{42} +2.38197i q^{43} +30.2705 q^{44} -17.5623 q^{46} +9.18034i q^{47} +9.85410i q^{48} +5.09017 q^{49} -3.38197 q^{51} +14.5623i q^{52} +3.61803i q^{53} +2.61803 q^{54} +10.3262 q^{56} +1.00000i q^{57} +11.0902i q^{58} -10.7984 q^{59} -5.09017 q^{61} -8.09017i q^{62} -1.38197i q^{63} -8.70820 q^{64} +16.3262 q^{66} -8.00000i q^{67} -16.4164i q^{68} -6.70820 q^{69} +1.14590 q^{71} +7.47214i q^{72} -9.14590i q^{73} -13.0902 q^{74} -4.85410 q^{76} -8.61803i q^{77} +7.85410i q^{78} -2.76393 q^{79} +1.00000 q^{81} -10.8541i q^{82} +8.32624i q^{83} +6.70820 q^{84} -6.23607 q^{86} +4.23607i q^{87} +46.5967i q^{88} +5.47214 q^{89} +4.14590 q^{91} -32.5623i q^{92} -3.09017i q^{93} -24.0344 q^{94} -10.8541 q^{96} +9.56231i q^{97} +13.3262i q^{98} +6.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) $ 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9	$ 4 q - 6 q^{4} - 6 q^{6} - 4 q^{9} - 16 q^{11} - 10 q^{14} + 26 q^{16} + 4 q^{19} - 10 q^{21} + 12 q^{24} + 18 q^{26} + 8 q^{29} + 10 q^{31} - 22 q^{34} + 6 q^{36} + 12 q^{39} - 30 q^{41} + 54 q^{44} - 30 q^{46} - 2 q^{49} - 18 q^{51} + 6 q^{54} + 10 q^{56} + 6 q^{59} + 2 q^{61} - 8 q^{64} + 34 q^{66} + 18 q^{71} - 30 q^{74} - 6 q^{76} - 20 q^{79} + 4 q^{81} - 16 q^{86} + 4 q^{89} + 30 q^{91} - 38 q^{94} - 30 q^{96} + 16 q^{99}+O(q^{100}) $ 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9 - 16 * q^11 - 10 * q^14 + 26 * q^16 + 4 * q^19 - 10 * q^21 + 12 * q^24 + 18 * q^26 + 8 * q^29 + 10 * q^31 - 22 * q^34 + 6 * q^36 + 12 * q^39 - 30 * q^41 + 54 * q^44 - 30 * q^46 - 2 * q^49 - 18 * q^51 + 6 * q^54 + 10 * q^56 + 6 * q^59 + 2 * q^61 - 8 * q^64 + 34 * q^66 + 18 * q^71 - 30 * q^74 - 6 * q^76 - 20 * q^79 + 4 * q^81 - 16 * q^86 + 4 * q^89 + 30 * q^91 - 38 * q^94 - 30 * q^96 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$.

$n$	$127$	$251$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.61803i	1.85123i	0.378467	+	0.925615i	$0.376451\pi$
$2$	2.61803i	1.85123i	−0.378467	+	0.925615i	$0.623549\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−4.85410	−2.42705
$5$	0	0
$5$	0	0
$6$	−2.61803	−1.06881
$7$	1.38197i	0.522334i	0.965294	+	0.261167i	$0.0841073\pi$
$7$	1.38197i	0.522334i	−0.965294	+	0.261167i	$0.915893\pi$
$8$	− 7.47214i	− 2.64180i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−6.23607	−1.88025	−0.940123	−	0.340836i	$-0.889290\pi$
$11$	−6.23607	−1.88025	−0.940123	+	0.340836i	$0.889290\pi$
$12$	− 4.85410i	− 1.40126i
$13$	− 3.00000i	− 0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$	− 3.00000i	− 0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$	−3.61803	−0.966960
$15$	0	0
$16$	9.85410	2.46353
$17$	3.38197i	0.820247i	0.912030	+	0.410124i	$0.134514\pi$
$17$	3.38197i	0.820247i	−0.912030	+	0.410124i	$0.865486\pi$
$18$	− 2.61803i	− 0.617077i
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−1.38197	−0.301570
$22$	− 16.3262i	− 3.48077i
$23$	6.70820i	1.39876i	0.714751	+	0.699379i	$0.246540\pi$
$23$	6.70820i	1.39876i	−0.714751	+	0.699379i	$0.753460\pi$
$24$	7.47214	1.52524
$25$	0	0
$26$	7.85410	1.54032
$27$	− 1.00000i	− 0.192450i
$28$	− 6.70820i	− 1.26773i
$29$	4.23607	0.786618	0.393309	−	0.919406i	$-0.371330\pi$
$29$	4.23607	0.786618	0.393309	+	0.919406i	$0.371330\pi$
$30$	0	0
$31$	−3.09017	−0.555011	−0.277505	−	0.960724i	$-0.589508\pi$
$31$	−3.09017	−0.555011	−0.277505	+	0.960724i	$0.589508\pi$
$32$	10.8541i	1.91875i
$33$	− 6.23607i	− 1.08556i
$34$	−8.85410	−1.51847
$35$	0	0
$36$	4.85410	0.809017
$37$	5.00000i	0.821995i	0.911636	+	0.410997i	$0.134819\pi$
$37$	5.00000i	0.821995i	−0.911636	+	0.410997i	$0.865181\pi$
$38$	2.61803i	0.424701i
$39$	3.00000	0.480384
$40$	0	0
$41$	−4.14590	−0.647480	−0.323740	−	0.946146i	$-0.604940\pi$
$41$	−4.14590	−0.647480	−0.323740	+	0.946146i	$0.604940\pi$
$42$	− 3.61803i	− 0.558275i
$43$	2.38197i	0.363246i	0.983368	+	0.181623i	$0.0581351\pi$
$43$	2.38197i	0.363246i	−0.983368	+	0.181623i	$0.941865\pi$
$44$	30.2705	4.56345
$45$	0	0
$46$	−17.5623	−2.58942
$47$	9.18034i	1.33909i	0.742771	+	0.669545i	$0.233511\pi$
$47$	9.18034i	1.33909i	−0.742771	+	0.669545i	$0.766489\pi$
$48$	9.85410i	1.42232i
$49$	5.09017	0.727167
$50$	0	0
$51$	−3.38197	−0.473570
$52$	14.5623i	2.01943i
$53$	3.61803i	0.496975i	0.968635	+	0.248488i	$0.0799335\pi$
$53$	3.61803i	0.496975i	−0.968635	+	0.248488i	$0.920066\pi$
$54$	2.61803	0.356269
$55$	0	0
$56$	10.3262	1.37990
$57$	1.00000i	0.132453i
$58$	11.0902i	1.45621i
$59$	−10.7984	−1.40583	−0.702914	−	0.711275i	$-0.748118\pi$
$59$	−10.7984	−1.40583	−0.702914	+	0.711275i	$0.748118\pi$
$60$	0	0
$61$	−5.09017	−0.651729	−0.325865	−	0.945416i	$-0.605655\pi$
$61$	−5.09017	−0.651729	−0.325865	+	0.945416i	$0.605655\pi$
$62$	− 8.09017i	− 1.02745i
$63$	− 1.38197i	− 0.174111i
$64$	−8.70820	−1.08853
$65$	0	0
$66$	16.3262	2.00962
$67$	− 8.00000i	− 0.977356i	−0.872464	−	0.488678i	$-0.837479\pi$
$67$	− 8.00000i	− 0.977356i	0.872464	−	0.488678i	$-0.162521\pi$
$68$	− 16.4164i	− 1.99078i
$69$	−6.70820	−0.807573
$70$	0	0
$71$	1.14590	0.135993	0.0679965	−	0.997686i	$-0.478339\pi$
$71$	1.14590	0.135993	0.0679965	+	0.997686i	$0.478339\pi$
$72$	7.47214i	0.880600i
$73$	− 9.14590i	− 1.07045i	−0.844711	−	0.535223i	$-0.820228\pi$
$73$	− 9.14590i	− 1.07045i	0.844711	−	0.535223i	$-0.179772\pi$
$74$	−13.0902	−1.52170
$75$	0	0
$76$	−4.85410	−0.556804
$77$	− 8.61803i	− 0.982116i
$78$	7.85410i	0.889302i
$79$	−2.76393	−0.310967	−0.155483	−	0.987839i	$-0.549693\pi$
$79$	−2.76393	−0.310967	−0.155483	+	0.987839i	$0.549693\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 10.8541i	− 1.19864i
$83$	8.32624i	0.913923i	0.889486	+	0.456962i	$0.151062\pi$
$83$	8.32624i	0.913923i	−0.889486	+	0.456962i	$0.848938\pi$
$84$	6.70820	0.731925
$85$	0	0
$86$	−6.23607	−0.672453
$87$	4.23607i	0.454154i
$88$	46.5967i	4.96723i
$89$	5.47214	0.580045	0.290023	−	0.957020i	$-0.406337\pi$
$89$	5.47214	0.580045	0.290023	+	0.957020i	$0.406337\pi$
$90$	0	0
$91$	4.14590	0.434608
$92$	− 32.5623i	− 3.39486i
$93$	− 3.09017i	− 0.320436i
$94$	−24.0344	−2.47896
$95$	0	0
$96$	−10.8541	−1.10779
$97$	9.56231i	0.970905i	0.874263	+	0.485453i	$0.161345\pi$
$97$	9.56231i	0.970905i	−0.874263	+	0.485453i	$0.838655\pi$
$98$	13.3262i	1.34615i
$99$	6.23607	0.626748
$100$	0	0
$101$	10.4721	1.04202	0.521008	−	0.853552i	$-0.325556\pi$
$101$	10.4721	1.04202	0.521008	+	0.853552i	$0.325556\pi$
$102$	− 8.85410i	− 0.876687i
$103$	− 12.0902i	− 1.19128i	−0.803252	−	0.595640i	$-0.796899\pi$
$103$	− 12.0902i	− 1.19128i	0.803252	−	0.595640i	$-0.203101\pi$
$104$	−22.4164	−2.19811
$105$	0	0
$106$	−9.47214	−0.920015
$107$	1.61803i	0.156421i	0.996937	+	0.0782106i	$0.0249207\pi$
$107$	1.61803i	0.156421i	−0.996937	+	0.0782106i	$0.975079\pi$
$108$	4.85410i	0.467086i
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0	0
$111$	−5.00000	−0.474579
$112$	13.6180i	1.28678i
$113$	5.23607i	0.492568i	0.969198	+	0.246284i	$0.0792096\pi$
$113$	5.23607i	0.492568i	−0.969198	+	0.246284i	$0.920790\pi$
$114$	−2.61803	−0.245201
$115$	0	0
$116$	−20.5623	−1.90916
$117$	3.00000i	0.277350i
$118$	− 28.2705i	− 2.60251i
$119$	−4.67376	−0.428443
$120$	0	0
$121$	27.8885	2.53532
$122$	− 13.3262i	− 1.20650i
$123$	− 4.14590i	− 0.373823i
$124$	15.0000	1.34704
$125$	0	0
$126$	3.61803	0.322320
$127$	20.4164i	1.81166i	0.423638	+	0.905832i	$0.360753\pi$
$127$	20.4164i	1.81166i	−0.423638	+	0.905832i	$0.639247\pi$
$128$	− 1.09017i	− 0.0963583i
$129$	−2.38197	−0.209720
$130$	0	0
$131$	1.09017	0.0952486	0.0476243	−	0.998865i	$-0.484835\pi$
$131$	1.09017	0.0952486	0.0476243	+	0.998865i	$0.484835\pi$
$132$	30.2705i	2.63471i
$133$	1.38197i	0.119832i
$134$	20.9443	1.80931
$135$	0	0
$136$	25.2705	2.16693
$137$	11.7639i	1.00506i	0.864560	+	0.502530i	$0.167597\pi$
$137$	11.7639i	1.00506i	−0.864560	+	0.502530i	$0.832403\pi$
$138$	− 17.5623i	− 1.49500i
$139$	6.23607	0.528936	0.264468	−	0.964394i	$-0.414804\pi$
$139$	6.23607	0.528936	0.264468	+	0.964394i	$0.414804\pi$
$140$	0	0
$141$	−9.18034	−0.773124
$142$	3.00000i	0.251754i
$143$	18.7082i	1.56446i
$144$	−9.85410	−0.821175
$145$	0	0
$146$	23.9443	1.98164
$147$	5.09017i	0.419830i
$148$	− 24.2705i	− 1.99502i
$149$	−5.94427	−0.486974	−0.243487	−	0.969904i	$-0.578291\pi$
$149$	−5.94427	−0.486974	−0.243487	+	0.969904i	$0.578291\pi$
$150$	0	0
$151$	18.1803	1.47950	0.739748	−	0.672885i	$-0.234945\pi$
$151$	18.1803	1.47950	0.739748	+	0.672885i	$0.234945\pi$
$152$	− 7.47214i	− 0.606070i
$153$	− 3.38197i	− 0.273416i
$154$	22.5623	1.81812
$155$	0	0
$156$	−14.5623	−1.16592
$157$	− 3.56231i	− 0.284303i	−0.989845	−	0.142151i	$-0.954598\pi$
$157$	− 3.56231i	− 0.284303i	0.989845	−	0.142151i	$-0.0454020\pi$
$158$	− 7.23607i	− 0.575671i
$159$	−3.61803	−0.286929
$160$	0	0
$161$	−9.27051	−0.730619
$162$	2.61803i	0.205692i
$163$	9.00000i	0.704934i	0.935824	+	0.352467i	$0.114657\pi$
$163$	9.00000i	0.704934i	−0.935824	+	0.352467i	$0.885343\pi$
$164$	20.1246	1.57147
$165$	0	0
$166$	−21.7984	−1.69188
$167$	− 4.47214i	− 0.346064i	−0.984916	−	0.173032i	$-0.944644\pi$
$167$	− 4.47214i	− 0.346064i	0.984916	−	0.173032i	$-0.0553564\pi$
$168$	10.3262i	0.796687i
$169$	4.00000	0.307692
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	− 11.5623i	− 0.881618i
$173$	− 12.3262i	− 0.937147i	−0.883425	−	0.468573i	$-0.844768\pi$
$173$	− 12.3262i	− 0.937147i	0.883425	−	0.468573i	$-0.155232\pi$
$174$	−11.0902	−0.840744
$175$	0	0
$176$	−61.4508	−4.63203
$177$	− 10.7984i	− 0.811655i
$178$	14.3262i	1.07380i
$179$	22.2361	1.66200	0.831001	−	0.556271i	$-0.187768\pi$
$179$	22.2361	1.66200	0.831001	+	0.556271i	$0.187768\pi$
$180$	0	0
$181$	13.4164	0.997234	0.498617	−	0.866822i	$-0.333841\pi$
$181$	13.4164	0.997234	0.498617	+	0.866822i	$0.333841\pi$
$182$	10.8541i	0.804560i
$183$	− 5.09017i	− 0.376276i
$184$	50.1246	3.69524
$185$	0	0
$186$	8.09017	0.593200
$187$	− 21.0902i	− 1.54227i
$188$	− 44.5623i	− 3.25004i
$189$	1.38197	0.100523
$190$	0	0
$191$	−0.0557281	−0.00403234	−0.00201617	−	0.999998i	$-0.500642\pi$
$191$	−0.0557281	−0.00403234	−0.00201617	+	0.999998i	$0.500642\pi$
$192$	− 8.70820i	− 0.628460i
$193$	4.70820i	0.338904i	0.985538	+	0.169452i	$0.0541998\pi$
$193$	4.70820i	0.338904i	−0.985538	+	0.169452i	$0.945800\pi$
$194$	−25.0344	−1.79737
$195$	0	0
$196$	−24.7082	−1.76487
$197$	− 15.1803i	− 1.08155i	−0.841166	−	0.540777i	$-0.818130\pi$
$197$	− 15.1803i	− 1.08155i	0.841166	−	0.540777i	$-0.181870\pi$
$198$	16.3262i	1.16026i
$199$	−18.7082	−1.32619	−0.663095	−	0.748536i	$-0.730757\pi$
$199$	−18.7082	−1.32619	−0.663095	+	0.748536i	$0.730757\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	27.4164i	1.92901i
$203$	5.85410i	0.410877i
$204$	16.4164	1.14938
$205$	0	0
$206$	31.6525	2.20533
$207$	− 6.70820i	− 0.466252i
$208$	− 29.5623i	− 2.04978i
$209$	−6.23607	−0.431358
$210$	0	0
$211$	3.41641	0.235195	0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641	0.235195	0.117598	+	0.993061i	$0.462481\pi$
$212$	− 17.5623i	− 1.20618i
$213$	1.14590i	0.0785156i
$214$	−4.23607	−0.289572
$215$	0	0
$216$	−7.47214	−0.508414
$217$	− 4.27051i	− 0.289901i
$218$	− 23.5623i	− 1.59584i
$219$	9.14590	0.618023
$220$	0	0
$221$	10.1459	0.682487
$222$	− 13.0902i	− 0.878555i
$223$	24.8885i	1.66666i	0.552776	+	0.833330i	$0.313569\pi$
$223$	24.8885i	1.66666i	−0.552776	+	0.833330i	$0.686431\pi$
$224$	−15.0000	−1.00223
$225$	0	0
$226$	−13.7082	−0.911856
$227$	− 16.1803i	− 1.07393i	−0.843606	−	0.536963i	$-0.819571\pi$
$227$	− 16.1803i	− 1.07393i	0.843606	−	0.536963i	$-0.180429\pi$
$228$	− 4.85410i	− 0.321471i
$229$	19.1246	1.26379	0.631895	−	0.775054i	$-0.282277\pi$
$229$	19.1246	1.26379	0.631895	+	0.775054i	$0.282277\pi$
$230$	0	0
$231$	8.61803	0.567025
$232$	− 31.6525i	− 2.07809i
$233$	− 25.5066i	− 1.67099i	−0.549497	−	0.835496i	$-0.685181\pi$
$233$	− 25.5066i	− 1.67099i	0.549497	−	0.835496i	$-0.314819\pi$
$234$	−7.85410	−0.513439
$235$	0	0
$236$	52.4164	3.41202
$237$	− 2.76393i	− 0.179537i
$238$	− 12.2361i	− 0.793146i
$239$	12.9098	0.835067	0.417534	−	0.908661i	$-0.362895\pi$
$239$	12.9098	0.835067	0.417534	+	0.908661i	$0.362895\pi$
$240$	0	0
$241$	−2.29180	−0.147628	−0.0738138	−	0.997272i	$-0.523517\pi$
$241$	−2.29180	−0.147628	−0.0738138	+	0.997272i	$0.523517\pi$
$242$	73.0132i	4.69346i
$243$	1.00000i	0.0641500i
$244$	24.7082	1.58178
$245$	0	0
$246$	10.8541	0.692032
$247$	− 3.00000i	− 0.190885i
$248$	23.0902i	1.46623i
$249$	−8.32624	−0.527654
$250$	0	0
$251$	−13.3262	−0.841145	−0.420572	−	0.907259i	$-0.638171\pi$
$251$	−13.3262	−0.841145	−0.420572	+	0.907259i	$0.638171\pi$
$252$	6.70820i	0.422577i
$253$	− 41.8328i	− 2.63001i
$254$	−53.4508	−3.35380
$255$	0	0
$256$	−14.5623	−0.910144
$257$	19.1803i	1.19644i	0.801333	+	0.598218i	$0.204124\pi$
$257$	19.1803i	1.19644i	−0.801333	+	0.598218i	$0.795876\pi$
$258$	− 6.23607i	− 0.388241i
$259$	−6.90983	−0.429356
$260$	0	0
$261$	−4.23607	−0.262206
$262$	2.85410i	0.176327i
$263$	19.8541i	1.22426i	0.790759	+	0.612128i	$0.209686\pi$
$263$	19.8541i	1.22426i	−0.790759	+	0.612128i	$0.790314\pi$
$264$	−46.5967	−2.86783
$265$	0	0
$266$	−3.61803	−0.221836
$267$	5.47214i	0.334889i
$268$	38.8328i	2.37209i
$269$	−22.6180	−1.37905	−0.689523	−	0.724264i	$-0.742180\pi$
$269$	−22.6180	−1.37905	−0.689523	+	0.724264i	$0.742180\pi$
$270$	0	0
$271$	−4.85410	−0.294866	−0.147433	−	0.989072i	$-0.547101\pi$
$271$	−4.85410	−0.294866	−0.147433	+	0.989072i	$0.547101\pi$
$272$	33.3262i	2.02070i
$273$	4.14590i	0.250921i
$274$	−30.7984	−1.86060
$275$	0	0
$276$	32.5623	1.96002
$277$	− 18.2705i	− 1.09777i	−0.835898	−	0.548884i	$-0.815053\pi$
$277$	− 18.2705i	− 1.09777i	0.835898	−	0.548884i	$-0.184947\pi$
$278$	16.3262i	0.979183i
$279$	3.09017	0.185004
$280$	0	0
$281$	−30.2705	−1.80579	−0.902894	−	0.429864i	$-0.858561\pi$
$281$	−30.2705	−1.80579	−0.902894	+	0.429864i	$0.858561\pi$
$282$	− 24.0344i	− 1.43123i
$283$	12.5279i	0.744704i	0.928092	+	0.372352i	$0.121449\pi$
$283$	12.5279i	0.744704i	−0.928092	+	0.372352i	$0.878551\pi$
$284$	−5.56231	−0.330062
$285$	0	0
$286$	−48.9787	−2.89617
$287$	− 5.72949i	− 0.338201i
$288$	− 10.8541i	− 0.639584i
$289$	5.56231	0.327194
$290$	0	0
$291$	−9.56231	−0.560552
$292$	44.3951i	2.59803i
$293$	7.41641i	0.433271i	0.976253	+	0.216636i	$0.0695083\pi$
$293$	7.41641i	0.433271i	−0.976253	+	0.216636i	$0.930492\pi$
$294$	−13.3262	−0.777202
$295$	0	0
$296$	37.3607	2.17155
$297$	6.23607i	0.361853i
$298$	− 15.5623i	− 0.901500i
$299$	20.1246	1.16384
$300$	0	0
$301$	−3.29180	−0.189736
$302$	47.5967i	2.73889i
$303$	10.4721i	0.601608i
$304$	9.85410	0.565172
$305$	0	0
$306$	8.85410	0.506155
$307$	18.1246i	1.03443i	0.855857	+	0.517213i	$0.173031\pi$
$307$	18.1246i	1.03443i	−0.855857	+	0.517213i	$0.826969\pi$
$308$	41.8328i	2.38365i
$309$	12.0902	0.687786
$310$	0	0
$311$	7.90983	0.448525	0.224263	−	0.974529i	$-0.428003\pi$
$311$	7.90983	0.448525	0.224263	+	0.974529i	$0.428003\pi$
$312$	− 22.4164i	− 1.26908i
$313$	− 26.9787i	− 1.52493i	−0.647031	−	0.762464i	$-0.723990\pi$
$313$	− 26.9787i	− 1.52493i	0.647031	−	0.762464i	$-0.276010\pi$
$314$	9.32624	0.526310
$315$	0	0
$316$	13.4164	0.754732
$317$	− 10.4164i	− 0.585044i	−0.956259	−	0.292522i	$-0.905506\pi$
$317$	− 10.4164i	− 0.585044i	0.956259	−	0.292522i	$-0.0944944\pi$
$318$	− 9.47214i	− 0.531171i
$319$	−26.4164	−1.47903
$320$	0	0
$321$	−1.61803	−0.0903099
$322$	− 24.2705i	− 1.35254i
$323$	3.38197i	0.188178i
$324$	−4.85410	−0.269672
$325$	0	0
$326$	−23.5623	−1.30500
$327$	− 9.00000i	− 0.497701i
$328$	30.9787i	1.71051i
$329$	−12.6869	−0.699452
$330$	0	0
$331$	12.8541	0.706525	0.353263	−	0.935524i	$-0.385072\pi$
$331$	12.8541	0.706525	0.353263	+	0.935524i	$0.385072\pi$
$332$	− 40.4164i	− 2.21814i
$333$	− 5.00000i	− 0.273998i
$334$	11.7082	0.640644
$335$	0	0
$336$	−13.6180	−0.742925
$337$	− 9.52786i	− 0.519016i	−0.965741	−	0.259508i	$-0.916440\pi$
$337$	− 9.52786i	− 0.519016i	0.965741	−	0.259508i	$-0.0835604\pi$
$338$	10.4721i	0.569609i
$339$	−5.23607	−0.284384
$340$	0	0
$341$	19.2705	1.04356
$342$	− 2.61803i	− 0.141567i
$343$	16.7082i	0.902158i
$344$	17.7984	0.959624
$345$	0	0
$346$	32.2705	1.73487
$347$	23.5066i	1.26190i	0.775824	+	0.630950i	$0.217335\pi$
$347$	23.5066i	1.26190i	−0.775824	+	0.630950i	$0.782665\pi$
$348$	− 20.5623i	− 1.10226i
$349$	30.0902	1.61069	0.805345	−	0.592806i	$-0.201980\pi$
$349$	30.0902	1.61069	0.805345	+	0.592806i	$0.201980\pi$
$350$	0	0
$351$	−3.00000	−0.160128
$352$	− 67.6869i	− 3.60772i
$353$	− 17.6738i	− 0.940679i	−0.882485	−	0.470340i	$-0.844131\pi$
$353$	− 17.6738i	− 0.940679i	0.882485	−	0.470340i	$-0.155869\pi$
$354$	28.2705	1.50256
$355$	0	0
$356$	−26.5623	−1.40780
$357$	− 4.67376i	− 0.247362i
$358$	58.2148i	3.07675i
$359$	−20.4721	−1.08048	−0.540239	−	0.841512i	$-0.681666\pi$
$359$	−20.4721	−1.08048	−0.540239	+	0.841512i	$0.681666\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	35.1246i	1.84611i
$363$	27.8885i	1.46377i
$364$	−20.1246	−1.05482
$365$	0	0
$366$	13.3262	0.696574
$367$	− 32.0902i	− 1.67509i	−0.546366	−	0.837547i	$-0.683989\pi$
$367$	− 32.0902i	− 1.67509i	0.546366	−	0.837547i	$-0.316011\pi$
$368$	66.1033i	3.44587i
$369$	4.14590	0.215827
$370$	0	0
$371$	−5.00000	−0.259587
$372$	15.0000i	0.777714i
$373$	1.72949i	0.0895496i	0.998997	+	0.0447748i	$0.0142570\pi$
$373$	1.72949i	0.0895496i	−0.998997	+	0.0447748i	$0.985743\pi$
$374$	55.2148	2.85509
$375$	0	0
$376$	68.5967	3.53761
$377$	− 12.7082i	− 0.654506i
$378$	3.61803i	0.186092i
$379$	−17.5623	−0.902115	−0.451058	−	0.892495i	$-0.648953\pi$
$379$	−17.5623	−0.902115	−0.451058	+	0.892495i	$0.648953\pi$
$380$	0	0
$381$	−20.4164	−1.04596
$382$	− 0.145898i	− 0.00746479i
$383$	− 36.9787i	− 1.88952i	−0.327757	−	0.944762i	$-0.606293\pi$
$383$	− 36.9787i	− 1.88952i	0.327757	−	0.944762i	$-0.393707\pi$
$384$	1.09017	0.0556325
$385$	0	0
$386$	−12.3262	−0.627389
$387$	− 2.38197i	− 0.121082i
$388$	− 46.4164i	− 2.35644i
$389$	11.0344	0.559468	0.279734	−	0.960077i	$-0.409754\pi$
$389$	11.0344	0.559468	0.279734	+	0.960077i	$0.409754\pi$
$390$	0	0
$391$	−22.6869	−1.14733
$392$	− 38.0344i	− 1.92103i
$393$	1.09017i	0.0549918i
$394$	39.7426	2.00221
$395$	0	0
$396$	−30.2705	−1.52115
$397$	− 0.944272i	− 0.0473916i	−0.999719	−	0.0236958i	$-0.992457\pi$
$397$	− 0.944272i	− 0.0473916i	0.999719	−	0.0236958i	$-0.00754332\pi$
$398$	− 48.9787i	− 2.45508i
$399$	−1.38197	−0.0691848
$400$	0	0
$401$	−17.6525	−0.881523	−0.440761	−	0.897624i	$-0.645291\pi$
$401$	−17.6525	−0.881523	−0.440761	+	0.897624i	$0.645291\pi$
$402$	20.9443i	1.04461i
$403$	9.27051i	0.461797i
$404$	−50.8328	−2.52903
$405$	0	0
$406$	−15.3262	−0.760628
$407$	− 31.1803i	− 1.54555i
$408$	25.2705i	1.25108i
$409$	21.2361	1.05006	0.525028	−	0.851085i	$-0.324055\pi$
$409$	21.2361	1.05006	0.525028	+	0.851085i	$0.324055\pi$
$410$	0	0
$411$	−11.7639	−0.580272
$412$	58.6869i	2.89130i
$413$	− 14.9230i	− 0.734312i
$414$	17.5623	0.863140
$415$	0	0
$416$	32.5623	1.59650
$417$	6.23607i	0.305382i
$418$	− 16.3262i	− 0.798542i
$419$	30.2705	1.47881	0.739406	−	0.673260i	$-0.235107\pi$
$419$	30.2705	1.47881	0.739406	+	0.673260i	$0.235107\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	8.94427i	0.435400i
$423$	− 9.18034i	− 0.446363i
$424$	27.0344	1.31291
$425$	0	0
$426$	−3.00000	−0.145350
$427$	− 7.03444i	− 0.340421i
$428$	− 7.85410i	− 0.379642i
$429$	−18.7082	−0.903241
$430$	0	0
$431$	−12.2361	−0.589391	−0.294695	−	0.955591i	$-0.595218\pi$
$431$	−12.2361	−0.589391	−0.294695	+	0.955591i	$0.595218\pi$
$432$	− 9.85410i	− 0.474106i
$433$	35.3050i	1.69665i	0.529478	+	0.848324i	$0.322388\pi$
$433$	35.3050i	1.69665i	−0.529478	+	0.848324i	$0.677612\pi$
$434$	11.1803	0.536673
$435$	0	0
$436$	43.6869	2.09222
$437$	6.70820i	0.320897i
$438$	23.9443i	1.14410i
$439$	−3.61803	−0.172679	−0.0863397	−	0.996266i	$-0.527517\pi$
$439$	−3.61803	−0.172679	−0.0863397	+	0.996266i	$0.527517\pi$
$440$	0	0
$441$	−5.09017	−0.242389
$442$	26.5623i	1.26344i
$443$	− 33.4508i	− 1.58930i	−0.607069	−	0.794649i	$-0.707655\pi$
$443$	− 33.4508i	− 1.58930i	0.607069	−	0.794649i	$-0.292345\pi$
$444$	24.2705	1.15183
$445$	0	0
$446$	−65.1591	−3.08537
$447$	− 5.94427i	− 0.281154i
$448$	− 12.0344i	− 0.568574i
$449$	−33.5066	−1.58127	−0.790637	−	0.612286i	$-0.790250\pi$
$449$	−33.5066	−1.58127	−0.790637	+	0.612286i	$0.790250\pi$
$450$	0	0
$451$	25.8541	1.21742
$452$	− 25.4164i	− 1.19549i
$453$	18.1803i	0.854187i
$454$	42.3607	1.98809
$455$	0	0
$456$	7.47214	0.349915
$457$	42.6869i	1.99681i	0.0564594	+	0.998405i	$0.482019\pi$
$457$	42.6869i	1.99681i	−0.0564594	+	0.998405i	$0.517981\pi$
$458$	50.0689i	2.33957i
$459$	3.38197	0.157857
$460$	0	0
$461$	−8.52786	−0.397182	−0.198591	−	0.980082i	$-0.563637\pi$
$461$	−8.52786	−0.397182	−0.198591	+	0.980082i	$0.563637\pi$
$462$	22.5623i	1.04969i
$463$	16.3262i	0.758745i	0.925244	+	0.379372i	$0.123860\pi$
$463$	16.3262i	0.758745i	−0.925244	+	0.379372i	$0.876140\pi$
$464$	41.7426	1.93785
$465$	0	0
$466$	66.7771	3.09339
$467$	16.0902i	0.744564i	0.928120	+	0.372282i	$0.121425\pi$
$467$	16.0902i	0.744564i	−0.928120	+	0.372282i	$0.878575\pi$
$468$	− 14.5623i	− 0.673143i
$469$	11.0557	0.510506
$470$	0	0
$471$	3.56231	0.164142
$472$	80.6869i	3.71392i
$473$	− 14.8541i	− 0.682992i
$474$	7.23607	0.332364
$475$	0	0
$476$	22.6869	1.03985
$477$	− 3.61803i	− 0.165658i
$478$	33.7984i	1.54590i
$479$	29.5967	1.35231	0.676155	−	0.736759i	$-0.263645\pi$
$479$	29.5967	1.35231	0.676155	+	0.736759i	$0.263645\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	− 6.00000i	− 0.273293i
$483$	− 9.27051i	− 0.421823i
$484$	−135.374	−6.15336
$485$	0	0
$486$	−2.61803	−0.118756
$487$	10.9443i	0.495932i	0.968769	+	0.247966i	$0.0797622\pi$
$487$	10.9443i	0.495932i	−0.968769	+	0.247966i	$0.920238\pi$
$488$	38.0344i	1.72174i
$489$	−9.00000	−0.406994
$490$	0	0
$491$	7.79837	0.351936	0.175968	−	0.984396i	$-0.443695\pi$
$491$	7.79837	0.351936	0.175968	+	0.984396i	$0.443695\pi$
$492$	20.1246i	0.907288i
$493$	14.3262i	0.645221i
$494$	7.85410	0.353373
$495$	0	0
$496$	−30.4508	−1.36728
$497$	1.58359i	0.0710338i
$498$	− 21.7984i	− 0.976808i
$499$	−30.7082	−1.37469	−0.687344	−	0.726332i	$-0.741223\pi$
$499$	−30.7082	−1.37469	−0.687344	+	0.726332i	$0.741223\pi$
$500$	0	0
$501$	4.47214	0.199800
$502$	− 34.8885i	− 1.55715i
$503$	− 7.14590i	− 0.318620i	−0.987229	−	0.159310i	$-0.949073\pi$
$503$	− 7.14590i	− 0.318620i	0.987229	−	0.159310i	$-0.0509269\pi$
$504$	−10.3262	−0.459967
$505$	0	0
$506$	109.520	4.86875
$507$	4.00000i	0.177646i
$508$	− 99.1033i	− 4.39700i
$509$	3.34752	0.148376	0.0741882	−	0.997244i	$-0.476363\pi$
$509$	3.34752	0.148376	0.0741882	+	0.997244i	$0.476363\pi$
$510$	0	0
$511$	12.6393	0.559131
$512$	− 40.3050i	− 1.78124i
$513$	− 1.00000i	− 0.0441511i
$514$	−50.2148	−2.21488
$515$	0	0
$516$	11.5623	0.509002
$517$	− 57.2492i	− 2.51782i
$518$	− 18.0902i	− 0.794836i
$519$	12.3262	0.541062
$520$	0	0
$521$	−15.3262	−0.671455	−0.335727	−	0.941959i	$-0.608982\pi$
$521$	−15.3262	−0.671455	−0.335727	+	0.941959i	$0.608982\pi$
$522$	− 11.0902i	− 0.485404i
$523$	− 4.00000i	− 0.174908i	−0.996169	−	0.0874539i	$-0.972127\pi$
$523$	− 4.00000i	− 0.174908i	0.996169	−	0.0874539i	$-0.0278730\pi$
$524$	−5.29180	−0.231173
$525$	0	0
$526$	−51.9787	−2.26638
$527$	− 10.4508i	− 0.455246i
$528$	− 61.4508i	− 2.67430i
$529$	−22.0000	−0.956522
$530$	0	0
$531$	10.7984	0.468610
$532$	− 6.70820i	− 0.290838i
$533$	12.4377i	0.538736i
$534$	−14.3262	−0.619957
$535$	0	0
$536$	−59.7771	−2.58198
$537$	22.2361i	0.959557i
$538$	− 59.2148i	− 2.55293i
$539$	−31.7426	−1.36725
$540$	0	0
$541$	−12.0344	−0.517401	−0.258701	−	0.965958i	$-0.583294\pi$
$541$	−12.0344	−0.517401	−0.258701	+	0.965958i	$0.583294\pi$
$542$	− 12.7082i	− 0.545864i
$543$	13.4164i	0.575753i
$544$	−36.7082	−1.57385
$545$	0	0
$546$	−10.8541	−0.464513
$547$	36.5066i	1.56091i	0.625213	+	0.780454i	$0.285012\pi$
$547$	36.5066i	1.56091i	−0.625213	+	0.780454i	$0.714988\pi$
$548$	− 57.1033i	− 2.43933i
$549$	5.09017	0.217243
$550$	0	0
$551$	4.23607	0.180463
$552$	50.1246i	2.13345i
$553$	− 3.81966i	− 0.162428i
$554$	47.8328	2.03222
$555$	0	0
$556$	−30.2705	−1.28376
$557$	− 31.2148i	− 1.32261i	−0.750116	−	0.661306i	$-0.770002\pi$
$557$	− 31.2148i	− 1.32261i	0.750116	−	0.661306i	$-0.229998\pi$
$558$	8.09017i	0.342484i
$559$	7.14590	0.302239
$560$	0	0
$561$	21.0902	0.890428
$562$	− 79.2492i	− 3.34293i
$563$	21.2148i	0.894096i	0.894510	+	0.447048i	$0.147525\pi$
$563$	21.2148i	0.894096i	−0.894510	+	0.447048i	$0.852475\pi$
$564$	44.5623	1.87641
$565$	0	0
$566$	−32.7984	−1.37862
$567$	1.38197i	0.0580371i
$568$	− 8.56231i	− 0.359266i
$569$	1.65248	0.0692754	0.0346377	−	0.999400i	$-0.488972\pi$
$569$	1.65248	0.0692754	0.0346377	+	0.999400i	$0.488972\pi$
$570$	0	0
$571$	−24.4508	−1.02324	−0.511618	−	0.859213i	$-0.670954\pi$
$571$	−24.4508	−1.02324	−0.511618	+	0.859213i	$0.670954\pi$
$572$	− 90.8115i	− 3.79702i
$573$	− 0.0557281i	− 0.00232807i
$574$	15.0000	0.626088
$575$	0	0
$576$	8.70820	0.362842
$577$	43.8328i	1.82478i	0.409318	+	0.912392i	$0.365767\pi$
$577$	43.8328i	1.82478i	−0.409318	+	0.912392i	$0.634233\pi$
$578$	14.5623i	0.605712i
$579$	−4.70820	−0.195666
$580$	0	0
$581$	−11.5066	−0.477373
$582$	− 25.0344i	− 1.03771i
$583$	− 22.5623i	− 0.934435i
$584$	−68.3394	−2.82790
$585$	0	0
$586$	−19.4164	−0.802084
$587$	− 8.05573i	− 0.332495i	−0.986084	−	0.166248i	$-0.946835\pi$
$587$	− 8.05573i	− 0.332495i	0.986084	−	0.166248i	$-0.0531651\pi$
$588$	− 24.7082i	− 1.01895i
$589$	−3.09017	−0.127328
$590$	0	0
$591$	15.1803	0.624436
$592$	49.2705i	2.02501i
$593$	13.3607i	0.548657i	0.961636	+	0.274329i	$0.0884556\pi$
$593$	13.3607i	0.548657i	−0.961636	+	0.274329i	$0.911544\pi$
$594$	−16.3262	−0.669874
$595$	0	0
$596$	28.8541	1.18191
$597$	− 18.7082i	− 0.765676i
$598$	52.6869i	2.15453i
$599$	−17.4508	−0.713022	−0.356511	−	0.934291i	$-0.616034\pi$
$599$	−17.4508	−0.713022	−0.356511	+	0.934291i	$0.616034\pi$
$600$	0	0
$601$	−9.00000	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$602$	− 8.61803i	− 0.351245i
$603$	8.00000i	0.325785i
$604$	−88.2492	−3.59081
$605$	0	0
$606$	−27.4164	−1.11372
$607$	− 20.3607i	− 0.826414i	−0.910637	−	0.413207i	$-0.864409\pi$
$607$	− 20.3607i	− 0.826414i	0.910637	−	0.413207i	$-0.135591\pi$
$608$	10.8541i	0.440192i
$609$	−5.85410	−0.237220
$610$	0	0
$611$	27.5410	1.11419
$612$	16.4164i	0.663594i
$613$	− 16.5967i	− 0.670336i	−0.942158	−	0.335168i	$-0.891207\pi$
$613$	− 16.5967i	− 0.670336i	0.942158	−	0.335168i	$-0.108793\pi$
$614$	−47.4508	−1.91496
$615$	0	0
$616$	−64.3951	−2.59455
$617$	1.76393i	0.0710132i	0.999369	+	0.0355066i	$0.0113045\pi$
$617$	1.76393i	0.0710132i	−0.999369	+	0.0355066i	$0.988696\pi$
$618$	31.6525i	1.27325i
$619$	7.41641	0.298091	0.149045	−	0.988830i	$-0.452380\pi$
$619$	7.41641	0.298091	0.149045	+	0.988830i	$0.452380\pi$
$620$	0	0
$621$	6.70820	0.269191
$622$	20.7082i	0.830323i
$623$	7.56231i	0.302977i
$624$	29.5623	1.18344
$625$	0	0
$626$	70.6312	2.82299
$627$	− 6.23607i	− 0.249045i
$628$	17.2918i	0.690018i
$629$	−16.9098	−0.674239
$630$	0	0
$631$	−14.7082	−0.585524	−0.292762	−	0.956185i	$-0.594574\pi$
$631$	−14.7082	−0.585524	−0.292762	+	0.956185i	$0.594574\pi$
$632$	20.6525i	0.821511i
$633$	3.41641i	0.135790i
$634$	27.2705	1.08305
$635$	0	0
$636$	17.5623	0.696391
$637$	− 15.2705i	− 0.605040i
$638$	− 69.1591i	− 2.73803i
$639$	−1.14590	−0.0453310
$640$	0	0
$641$	39.5967	1.56398	0.781989	−	0.623293i	$-0.214205\pi$
$641$	39.5967	1.56398	0.781989	+	0.623293i	$0.214205\pi$
$642$	− 4.23607i	− 0.167184i
$643$	− 5.05573i	− 0.199378i	−0.995019	−	0.0996892i	$-0.968215\pi$
$643$	− 5.05573i	− 0.199378i	0.995019	−	0.0996892i	$-0.0317849\pi$
$644$	45.0000	1.77325
$645$	0	0
$646$	−8.85410	−0.348360
$647$	16.1459i	0.634761i	0.948298	+	0.317380i	$0.102803\pi$
$647$	16.1459i	0.634761i	−0.948298	+	0.317380i	$0.897197\pi$
$648$	− 7.47214i	− 0.293533i
$649$	67.3394	2.64330
$650$	0	0
$651$	4.27051	0.167374
$652$	− 43.6869i	− 1.71091i
$653$	− 39.0132i	− 1.52670i	−0.645983	−	0.763351i	$-0.723553\pi$
$653$	− 39.0132i	− 1.52670i	0.645983	−	0.763351i	$-0.276447\pi$
$654$	23.5623	0.921359
$655$	0	0
$656$	−40.8541	−1.59508
$657$	9.14590i	0.356815i
$658$	− 33.2148i	− 1.29485i
$659$	29.0689	1.13236	0.566181	−	0.824281i	$-0.308420\pi$
$659$	29.0689	1.13236	0.566181	+	0.824281i	$0.308420\pi$
$660$	0	0
$661$	32.9787	1.28272	0.641362	−	0.767239i	$-0.278370\pi$
$661$	32.9787	1.28272	0.641362	+	0.767239i	$0.278370\pi$
$662$	33.6525i	1.30794i
$663$	10.1459i	0.394034i
$664$	62.2148	2.41440
$665$	0	0
$666$	13.0902	0.507234
$667$	28.4164i	1.10029i
$668$	21.7082i	0.839916i
$669$	−24.8885	−0.962247
$670$	0	0
$671$	31.7426	1.22541
$672$	− 15.0000i	− 0.578638i
$673$	43.4164i	1.67358i	0.547524	+	0.836790i	$0.315570\pi$
$673$	43.4164i	1.67358i	−0.547524	+	0.836790i	$0.684430\pi$
$674$	24.9443	0.960817
$675$	0	0
$676$	−19.4164	−0.746785
$677$	20.0689i	0.771310i	0.922643	+	0.385655i	$0.126024\pi$
$677$	20.0689i	0.771310i	−0.922643	+	0.385655i	$0.873976\pi$
$678$	− 13.7082i	− 0.526460i
$679$	−13.2148	−0.507137
$680$	0	0
$681$	16.1803	0.620032
$682$	50.4508i	1.93186i
$683$	− 21.0000i	− 0.803543i	−0.915740	−	0.401771i	$-0.868395\pi$
$683$	− 21.0000i	− 0.803543i	0.915740	−	0.401771i	$-0.131605\pi$
$684$	4.85410	0.185601
$685$	0	0
$686$	−43.7426	−1.67010
$687$	19.1246i	0.729649i
$688$	23.4721i	0.894867i
$689$	10.8541	0.413508
$690$	0	0
$691$	37.1803	1.41441	0.707203	−	0.707010i	$-0.249957\pi$
$691$	37.1803	1.41441	0.707203	+	0.707010i	$0.249957\pi$
$692$	59.8328i	2.27450i
$693$	8.61803i	0.327372i
$694$	−61.5410	−2.33607
$695$	0	0
$696$	31.6525	1.19978
$697$	− 14.0213i	− 0.531094i
$698$	78.7771i	2.98176i
$699$	25.5066	0.964747
$700$	0	0
$701$	11.8197	0.446422	0.223211	−	0.974770i	$-0.428346\pi$
$701$	11.8197	0.446422	0.223211	+	0.974770i	$0.428346\pi$
$702$	− 7.85410i	− 0.296434i
$703$	5.00000i	0.188579i
$704$	54.3050	2.04669
$705$	0	0
$706$	46.2705	1.74141
$707$	14.4721i	0.544281i
$708$	52.4164i	1.96993i
$709$	12.7082	0.477267	0.238633	−	0.971110i	$-0.423301\pi$
$709$	12.7082	0.477267	0.238633	+	0.971110i	$0.423301\pi$
$710$	0	0
$711$	2.76393	0.103656
$712$	− 40.8885i	− 1.53236i
$713$	− 20.7295i	− 0.776326i
$714$	12.2361	0.457923
$715$	0	0
$716$	−107.936	−4.03376
$717$	12.9098i	0.482126i
$718$	− 53.5967i	− 2.00021i
$719$	7.47214	0.278664	0.139332	−	0.990246i	$-0.455505\pi$
$719$	7.47214	0.278664	0.139332	+	0.990246i	$0.455505\pi$
$720$	0	0
$721$	16.7082	0.622246
$722$	− 47.1246i	− 1.75380i
$723$	− 2.29180i	− 0.0852328i
$724$	−65.1246	−2.42034
$725$	0	0
$726$	−73.0132	−2.70977
$727$	22.4164i	0.831379i	0.909507	+	0.415689i	$0.136460\pi$
$727$	22.4164i	0.831379i	−0.909507	+	0.415689i	$0.863540\pi$
$728$	− 30.9787i	− 1.14815i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−8.05573	−0.297952
$732$	24.7082i	0.913241i
$733$	24.9443i	0.921338i	0.887572	+	0.460669i	$0.152390\pi$
$733$	24.9443i	0.921338i	−0.887572	+	0.460669i	$0.847610\pi$
$734$	84.0132	3.10098
$735$	0	0
$736$	−72.8115	−2.68387
$737$	49.8885i	1.83767i
$738$	10.8541i	0.399545i
$739$	−6.72949	−0.247548	−0.123774	−	0.992310i	$-0.539500\pi$
$739$	−6.72949	−0.247548	−0.123774	+	0.992310i	$0.539500\pi$
$740$	0	0
$741$	3.00000	0.110208
$742$	− 13.0902i	− 0.480555i
$743$	− 18.0689i	− 0.662883i	−0.943476	−	0.331442i	$-0.892465\pi$
$743$	− 18.0689i	− 0.662883i	0.943476	−	0.331442i	$-0.107535\pi$
$744$	−23.0902	−0.846527
$745$	0	0
$746$	−4.52786	−0.165777
$747$	− 8.32624i	− 0.304641i
$748$	102.374i	3.74316i
$749$	−2.23607	−0.0817041
$750$	0	0
$751$	−52.2148	−1.90534	−0.952672	−	0.303999i	$-0.901678\pi$
$751$	−52.2148	−1.90534	−0.952672	+	0.303999i	$0.901678\pi$
$752$	90.4640i	3.29888i
$753$	− 13.3262i	− 0.485635i
$754$	33.2705	1.21164
$755$	0	0
$756$	−6.70820	−0.243975
$757$	19.6525i	0.714281i	0.934051	+	0.357141i	$0.116248\pi$
$757$	19.6525i	0.714281i	−0.934051	+	0.357141i	$0.883752\pi$
$758$	− 45.9787i	− 1.67002i
$759$	41.8328	1.51844
$760$	0	0
$761$	23.0689	0.836246	0.418123	−	0.908390i	$-0.362688\pi$
$761$	23.0689	0.836246	0.418123	+	0.908390i	$0.362688\pi$
$762$	− 53.4508i	− 1.93632i
$763$	− 12.4377i	− 0.450275i
$764$	0.270510	0.00978670
$765$	0	0
$766$	96.8115	3.49794
$767$	32.3951i	1.16972i
$768$	− 14.5623i	− 0.525472i
$769$	−0.944272	−0.0340513	−0.0170257	−	0.999855i	$-0.505420\pi$
$769$	−0.944272	−0.0340513	−0.0170257	+	0.999855i	$0.505420\pi$
$770$	0	0
$771$	−19.1803	−0.690763
$772$	− 22.8541i	− 0.822537i
$773$	− 21.0000i	− 0.755318i	−0.925945	−	0.377659i	$-0.876729\pi$
$773$	− 21.0000i	− 0.755318i	0.925945	−	0.377659i	$-0.123271\pi$
$774$	6.23607	0.224151
$775$	0	0
$776$	71.4508	2.56494
$777$	− 6.90983i	− 0.247889i
$778$	28.8885i	1.03570i
$779$	−4.14590	−0.148542
$780$	0	0
$781$	−7.14590	−0.255700
$782$	− 59.3951i	− 2.12397i
$783$	− 4.23607i	− 0.151385i
$784$	50.1591	1.79139
$785$	0	0
$786$	−2.85410	−0.101802
$787$	− 34.5623i	− 1.23201i	−0.787741	−	0.616007i	$-0.788749\pi$
$787$	− 34.5623i	− 1.23201i	0.787741	−	0.616007i	$-0.211251\pi$
$788$	73.6869i	2.62499i
$789$	−19.8541	−0.706825
$790$	0	0
$791$	−7.23607	−0.257285
$792$	− 46.5967i	− 1.65574i
$793$	15.2705i	0.542272i
$794$	2.47214	0.0877328
$795$	0	0
$796$	90.8115	3.21873
$797$	− 16.6738i	− 0.590615i	−0.955402	−	0.295307i	$-0.904578\pi$
$797$	− 16.6738i	− 0.590615i	0.955402	−	0.295307i	$-0.0954221\pi$
$798$	− 3.61803i	− 0.128077i
$799$	−31.0476	−1.09839
$800$	0	0
$801$	−5.47214	−0.193348
$802$	− 46.2148i	− 1.63190i
$803$	57.0344i	2.01270i
$804$	−38.8328	−1.36953
$805$	0	0
$806$	−24.2705	−0.854892
$807$	− 22.6180i	− 0.796193i
$808$	− 78.2492i	− 2.75280i
$809$	4.79837	0.168702	0.0843509	−	0.996436i	$-0.473118\pi$
$809$	4.79837	0.168702	0.0843509	+	0.996436i	$0.473118\pi$
$810$	0	0
$811$	29.0344	1.01954	0.509769	−	0.860312i	$-0.329731\pi$
$811$	29.0344	1.01954	0.509769	+	0.860312i	$0.329731\pi$
$812$	− 28.4164i	− 0.997220i
$813$	− 4.85410i	− 0.170241i
$814$	81.6312	2.86117
$815$	0	0
$816$	−33.3262	−1.16665
$817$	2.38197i	0.0833344i
$818$	55.5967i	1.94389i
$819$	−4.14590	−0.144869
$820$	0	0
$821$	−3.76393	−0.131362	−0.0656811	−	0.997841i	$-0.520922\pi$
$821$	−3.76393	−0.131362	−0.0656811	+	0.997841i	$0.520922\pi$
$822$	− 30.7984i	− 1.07422i
$823$	− 9.63932i	− 0.336006i	−0.985787	−	0.168003i	$-0.946268\pi$
$823$	− 9.63932i	− 0.336006i	0.985787	−	0.168003i	$-0.0537318\pi$
$824$	−90.3394	−3.14712
$825$	0	0
$826$	39.0689	1.35938
$827$	1.67376i	0.0582024i	0.999576	+	0.0291012i	$0.00926451\pi$
$827$	1.67376i	0.0582024i	−0.999576	+	0.0291012i	$0.990735\pi$
$828$	32.5623i	1.13162i
$829$	4.87539	0.169329	0.0846646	−	0.996410i	$-0.473018\pi$
$829$	4.87539	0.169329	0.0846646	+	0.996410i	$0.473018\pi$
$830$	0	0
$831$	18.2705	0.633797
$832$	26.1246i	0.905708i
$833$	17.2148i	0.596457i
$834$	−16.3262	−0.565331
$835$	0	0
$836$	30.2705	1.04693
$837$	3.09017i	0.106812i
$838$	79.2492i	2.73762i
$839$	−9.47214	−0.327014	−0.163507	−	0.986542i	$-0.552281\pi$
$839$	−9.47214	−0.327014	−0.163507	+	0.986542i	$0.552281\pi$
$840$	0	0
$841$	−11.0557	−0.381232
$842$	− 68.0689i	− 2.34581i
$843$	− 30.2705i	− 1.04257i
$844$	−16.5836	−0.570831
$845$	0	0
$846$	24.0344	0.826321
$847$	38.5410i	1.32429i
$848$	35.6525i	1.22431i
$849$	−12.5279	−0.429955
$850$	0	0
$851$	−33.5410	−1.14977
$852$	− 5.56231i	− 0.190561i
$853$	− 22.7082i	− 0.777514i	−0.921340	−	0.388757i	$-0.872905\pi$
$853$	− 22.7082i	− 0.777514i	0.921340	−	0.388757i	$-0.127095\pi$
$854$	18.4164	0.630197
$855$	0	0
$856$	12.0902	0.413234
$857$	41.2148i	1.40787i	0.710264	+	0.703935i	$0.248575\pi$
$857$	41.2148i	1.40787i	−0.710264	+	0.703935i	$0.751425\pi$
$858$	− 48.9787i	− 1.67211i
$859$	31.1803	1.06386	0.531930	−	0.846788i	$-0.321467\pi$
$859$	31.1803	1.06386	0.531930	+	0.846788i	$0.321467\pi$
$860$	0	0
$861$	5.72949	0.195261
$862$	− 32.0344i	− 1.09110i
$863$	19.2148i	0.654079i	0.945011	+	0.327039i	$0.106051\pi$
$863$	19.2148i	0.654079i	−0.945011	+	0.327039i	$0.893949\pi$
$864$	10.8541	0.369264
$865$	0	0
$866$	−92.4296	−3.14088
$867$	5.56231i	0.188906i
$868$	20.7295i	0.703605i
$869$	17.2361	0.584694
$870$	0	0
$871$	−24.0000	−0.813209
$872$	67.2492i	2.27735i
$873$	− 9.56231i	− 0.323635i
$874$	−17.5623	−0.594054
$875$	0	0
$876$	−44.3951	−1.49997
$877$	− 16.0557i	− 0.542163i	−0.962556	−	0.271082i	$-0.912619\pi$
$877$	− 16.0557i	− 0.542163i	0.962556	−	0.271082i	$-0.0873814\pi$
$878$	− 9.47214i	− 0.319669i
$879$	−7.41641	−0.250149
$880$	0	0
$881$	51.2361	1.72619	0.863093	−	0.505044i	$-0.168524\pi$
$881$	51.2361	1.72619	0.863093	+	0.505044i	$0.168524\pi$
$882$	− 13.3262i	− 0.448718i
$883$	9.94427i	0.334651i	0.985902	+	0.167326i	$0.0535131\pi$
$883$	9.94427i	0.334651i	−0.985902	+	0.167326i	$0.946487\pi$
$884$	−49.2492	−1.65643
$885$	0	0
$886$	87.5755	2.94216
$887$	− 8.30495i	− 0.278853i	−0.990232	−	0.139426i	$-0.955474\pi$
$887$	− 8.30495i	− 0.278853i	0.990232	−	0.139426i	$-0.0445259\pi$
$888$	37.3607i	1.25374i
$889$	−28.2148	−0.946293
$890$	0	0
$891$	−6.23607	−0.208916
$892$	− 120.812i	− 4.04507i
$893$	9.18034i	0.307208i
$894$	15.5623	0.520481
$895$	0	0
$896$	1.50658	0.0503312
$897$	20.1246i	0.671941i
$898$	− 87.7214i	− 2.92730i
$899$	−13.0902	−0.436582
$900$	0	0
$901$	−12.2361	−0.407643
$902$	67.6869i	2.25373i
$903$	− 3.29180i	− 0.109544i
$904$	39.1246	1.30127
$905$	0	0
$906$	−47.5967	−1.58130
$907$	− 15.5410i	− 0.516031i	−0.966141	−	0.258016i	$-0.916931\pi$
$907$	− 15.5410i	− 0.516031i	0.966141	−	0.258016i	$-0.0830686\pi$
$908$	78.5410i	2.60648i
$909$	−10.4721	−0.347339
$910$	0	0
$911$	21.2705	0.704723	0.352362	−	0.935864i	$-0.385379\pi$
$911$	21.2705	0.704723	0.352362	+	0.935864i	$0.385379\pi$
$912$	9.85410i	0.326302i
$913$	− 51.9230i	− 1.71840i
$914$	−111.756	−3.69655
$915$	0	0
$916$	−92.8328	−3.06728
$917$	1.50658i	0.0497516i
$918$	8.85410i	0.292229i
$919$	5.72949	0.188998	0.0944992	−	0.995525i	$-0.469875\pi$
$919$	5.72949	0.188998	0.0944992	+	0.995525i	$0.469875\pi$
$920$	0	0
$921$	−18.1246	−0.597226
$922$	− 22.3262i	− 0.735275i
$923$	− 3.43769i	− 0.113153i
$924$	−41.8328	−1.37620
$925$	0	0
$926$	−42.7426	−1.40461
$927$	12.0902i	0.397093i
$928$	45.9787i	1.50933i
$929$	35.9230	1.17859	0.589297	−	0.807916i	$-0.299405\pi$
$929$	35.9230	1.17859	0.589297	+	0.807916i	$0.299405\pi$
$930$	0	0
$931$	5.09017	0.166824
$932$	123.812i	4.05558i
$933$	7.90983i	0.258956i
$934$	−42.1246	−1.37836
$935$	0	0
$936$	22.4164	0.732703
$937$	52.1803i	1.70466i	0.523006	+	0.852329i	$0.324810\pi$
$937$	52.1803i	1.70466i	−0.523006	+	0.852329i	$0.675190\pi$
$938$	28.9443i	0.945064i
$939$	26.9787	0.880417
$940$	0	0
$941$	12.4377	0.405457	0.202729	−	0.979235i	$-0.435019\pi$
$941$	12.4377	0.405457	0.202729	+	0.979235i	$0.435019\pi$
$942$	9.32624i	0.303865i
$943$	− 27.8115i	− 0.905668i
$944$	−106.408	−3.46329
$945$	0	0
$946$	38.8885	1.26438
$947$	− 3.11146i	− 0.101109i	−0.998721	−	0.0505544i	$-0.983901\pi$
$947$	− 3.11146i	− 0.101109i	0.998721	−	0.0505544i	$-0.0160988\pi$
$948$	13.4164i	0.435745i
$949$	−27.4377	−0.890665
$950$	0	0
$951$	10.4164	0.337775
$952$	34.9230i	1.13186i
$953$	56.9230i	1.84392i	0.387289	+	0.921958i	$0.373411\pi$
$953$	56.9230i	1.84392i	−0.387289	+	0.921958i	$0.626589\pi$
$954$	9.47214	0.306672
$955$	0	0
$956$	−62.6656	−2.02675
$957$	− 26.4164i	− 0.853921i
$958$	77.4853i	2.50344i
$959$	−16.2574	−0.524977
$960$	0	0
$961$	−21.4508	−0.691963
$962$	39.2705i	1.26613i
$963$	− 1.61803i	− 0.0521404i
$964$	11.1246	0.358300
$965$	0	0
$966$	24.2705	0.780891
$967$	− 6.41641i	− 0.206338i	−0.994664	−	0.103169i	$-0.967102\pi$
$967$	− 6.41641i	− 0.206338i	0.994664	−	0.103169i	$-0.0328982\pi$
$968$	− 208.387i	− 6.69781i
$969$	−3.38197	−0.108644
$970$	0	0
$971$	−9.59675	−0.307974	−0.153987	−	0.988073i	$-0.549211\pi$
$971$	−9.59675	−0.307974	−0.153987	+	0.988073i	$0.549211\pi$
$972$	− 4.85410i	− 0.155695i
$973$	8.61803i	0.276281i
$974$	−28.6525	−0.918085
$975$	0	0
$976$	−50.1591	−1.60555
$977$	− 37.1459i	− 1.18840i	−0.804316	−	0.594201i	$-0.797468\pi$
$977$	− 37.1459i	− 1.18840i	0.804316	−	0.594201i	$-0.202532\pi$
$978$	− 23.5623i	− 0.753439i
$979$	−34.1246	−1.09063
$980$	0	0
$981$	9.00000	0.287348
$982$	20.4164i	0.651514i
$983$	− 27.0344i	− 0.862265i	−0.902289	−	0.431132i	$-0.858114\pi$
$983$	− 27.0344i	− 0.862265i	0.902289	−	0.431132i	$-0.141886\pi$
$984$	−30.9787	−0.987565
$985$	0	0
$986$	−37.5066	−1.19445
$987$	− 12.6869i	− 0.403829i
$988$	14.5623i	0.463289i
$989$	−15.9787	−0.508094
$990$	0	0
$991$	−38.8541	−1.23424	−0.617121	−	0.786869i	$-0.711701\pi$
$991$	−38.8541	−1.23424	−0.617121	+	0.786869i	$0.711701\pi$
$992$	− 33.5410i	− 1.06493i
$993$	12.8541i	0.407913i
$994$	−4.14590	−0.131500
$995$	0	0
$996$	40.4164	1.28064
$997$	− 45.7214i	− 1.44801i	−0.689795	−	0.724005i	$-0.742299\pi$
$997$	− 45.7214i	− 1.44801i	0.689795	−	0.724005i	$-0.257701\pi$
$998$	− 80.3951i	− 2.54486i
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	375.2.b.a.124.4		4
3.2	odd	2		1125.2.b.b.874.1		4
4.3	odd	2		6000.2.f.n.1249.2		4
5.2	odd	4		375.2.a.a.1.1	✓	2
5.3	odd	4		375.2.a.d.1.2	yes	2
5.4	even	2	inner	375.2.b.a.124.1		4
15.2	even	4		1125.2.a.f.1.2		2
15.8	even	4		1125.2.a.a.1.1		2
15.14	odd	2		1125.2.b.b.874.4		4
20.3	even	4		6000.2.a.q.1.2		2
20.7	even	4		6000.2.a.m.1.1		2
20.19	odd	2		6000.2.f.n.1249.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
375.2.a.a.1.1	✓	2	5.2	odd	4
375.2.a.d.1.2	yes	2	5.3	odd	4
375.2.b.a.124.1		4	5.4	even	2	inner
375.2.b.a.124.4		4	1.1	even	1	trivial
1125.2.a.a.1.1		2	15.8	even	4
1125.2.a.f.1.2		2	15.2	even	4
1125.2.b.b.874.1		4	3.2	odd	2
1125.2.b.b.874.4		4	15.14	odd	2
6000.2.a.m.1.1		2	20.7	even	4
6000.2.a.q.1.2		2	20.3	even	4
6000.2.f.n.1249.2		4	4.3	odd	2
6000.2.f.n.1249.3		4	20.19	odd	2

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:	\( N \)	\(=\)	\( 375 = 3 \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	375.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.61803i q^{2} +1.00000i q^{3} -4.85410 q^{4} -2.61803 q^{6} +1.38197i q^{7} -7.47214i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.61803i q^{2} +1.00000i q^{3} -4.85410 q^{4} -2.61803 q^{6} +1.38197i q^{7} -7.47214i q^{8} -1.00000 q^{9} -6.23607 q^{11} -4.85410i q^{12} -3.00000i q^{13} -3.61803 q^{14} +9.85410 q^{16} +3.38197i q^{17} -2.61803i q^{18} +1.00000 q^{19} -1.38197 q^{21} -16.3262i q^{22} +6.70820i q^{23} +7.47214 q^{24} +7.85410 q^{26} -1.00000i q^{27} -6.70820i q^{28} +4.23607 q^{29} -3.09017 q^{31} +10.8541i q^{32} -6.23607i q^{33} -8.85410 q^{34} +4.85410 q^{36} +5.00000i q^{37} +2.61803i q^{38} +3.00000 q^{39} -4.14590 q^{41} -3.61803i q^{42} +2.38197i q^{43} +30.2705 q^{44} -17.5623 q^{46} +9.18034i q^{47} +9.85410i q^{48} +5.09017 q^{49} -3.38197 q^{51} +14.5623i q^{52} +3.61803i q^{53} +2.61803 q^{54} +10.3262 q^{56} +1.00000i q^{57} +11.0902i q^{58} -10.7984 q^{59} -5.09017 q^{61} -8.09017i q^{62} -1.38197i q^{63} -8.70820 q^{64} +16.3262 q^{66} -8.00000i q^{67} -16.4164i q^{68} -6.70820 q^{69} +1.14590 q^{71} +7.47214i q^{72} -9.14590i q^{73} -13.0902 q^{74} -4.85410 q^{76} -8.61803i q^{77} +7.85410i q^{78} -2.76393 q^{79} +1.00000 q^{81} -10.8541i q^{82} +8.32624i q^{83} +6.70820 q^{84} -6.23607 q^{86} +4.23607i q^{87} +46.5967i q^{88} +5.47214 q^{89} +4.14590 q^{91} -32.5623i q^{92} -3.09017i q^{93} -24.0344 q^{94} -10.8541 q^{96} +9.56231i q^{97} +13.3262i q^{98} +6.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) \) 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9	\( 4 q - 6 q^{4} - 6 q^{6} - 4 q^{9} - 16 q^{11} - 10 q^{14} + 26 q^{16} + 4 q^{19} - 10 q^{21} + 12 q^{24} + 18 q^{26} + 8 q^{29} + 10 q^{31} - 22 q^{34} + 6 q^{36} + 12 q^{39} - 30 q^{41} + 54 q^{44} - 30 q^{46} - 2 q^{49} - 18 q^{51} + 6 q^{54} + 10 q^{56} + 6 q^{59} + 2 q^{61} - 8 q^{64} + 34 q^{66} + 18 q^{71} - 30 q^{74} - 6 q^{76} - 20 q^{79} + 4 q^{81} - 16 q^{86} + 4 q^{89} + 30 q^{91} - 38 q^{94} - 30 q^{96} + 16 q^{99}+O(q^{100}) \) 4 * q - 6 * q^4 - 6 * q^6 - 4 * q^9 - 16 * q^11 - 10 * q^14 + 26 * q^16 + 4 * q^19 - 10 * q^21 + 12 * q^24 + 18 * q^26 + 8 * q^29 + 10 * q^31 - 22 * q^34 + 6 * q^36 + 12 * q^39 - 30 * q^41 + 54 * q^44 - 30 * q^46 - 2 * q^49 - 18 * q^51 + 6 * q^54 + 10 * q^56 + 6 * q^59 + 2 * q^61 - 8 * q^64 + 34 * q^66 + 18 * q^71 - 30 * q^74 - 6 * q^76 - 20 * q^79 + 4 * q^81 - 16 * q^86 + 4 * q^89 + 30 * q^91 - 38 * q^94 - 30 * q^96 + 16 * q^99

Introduction

L-functions

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