LMFDB - Embedded newform 4598.2.a.by.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.a (trivial)

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:	yes
Analytic conductor:	$36.7152148494$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 $ x^8 - 16x^6 - 4x^5 + 75x^4 + 32x^3 - 90x^2 - 28x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.02984$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.55594 q^{3} +1.00000 q^{4} +2.24033 q^{5} -1.55594 q^{6} +2.27752 q^{7} -1.00000 q^{8} -0.579065 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.55594 q^{3} +1.00000 q^{4} +2.24033 q^{5} -1.55594 q^{6} +2.27752 q^{7} -1.00000 q^{8} -0.579065 q^{9} -2.24033 q^{10} +1.55594 q^{12} -2.75611 q^{13} -2.27752 q^{14} +3.48581 q^{15} +1.00000 q^{16} +0.571514 q^{17} +0.579065 q^{18} +1.00000 q^{19} +2.24033 q^{20} +3.54367 q^{21} +9.47493 q^{23} -1.55594 q^{24} +0.0190711 q^{25} +2.75611 q^{26} -5.56879 q^{27} +2.27752 q^{28} -1.99533 q^{29} -3.48581 q^{30} +8.92659 q^{31} -1.00000 q^{32} -0.571514 q^{34} +5.10239 q^{35} -0.579065 q^{36} +7.48207 q^{37} -1.00000 q^{38} -4.28832 q^{39} -2.24033 q^{40} +0.0862369 q^{41} -3.54367 q^{42} -5.53342 q^{43} -1.29730 q^{45} -9.47493 q^{46} -9.68313 q^{47} +1.55594 q^{48} -1.81292 q^{49} -0.0190711 q^{50} +0.889238 q^{51} -2.75611 q^{52} +4.40072 q^{53} +5.56879 q^{54} -2.27752 q^{56} +1.55594 q^{57} +1.99533 q^{58} +8.02502 q^{59} +3.48581 q^{60} -3.73883 q^{61} -8.92659 q^{62} -1.31883 q^{63} +1.00000 q^{64} -6.17458 q^{65} +6.91634 q^{67} +0.571514 q^{68} +14.7424 q^{69} -5.10239 q^{70} +4.91454 q^{71} +0.579065 q^{72} +10.7414 q^{73} -7.48207 q^{74} +0.0296734 q^{75} +1.00000 q^{76} +4.28832 q^{78} +5.92179 q^{79} +2.24033 q^{80} -6.92749 q^{81} -0.0862369 q^{82} +16.8470 q^{83} +3.54367 q^{84} +1.28038 q^{85} +5.53342 q^{86} -3.10460 q^{87} +5.14468 q^{89} +1.29730 q^{90} -6.27708 q^{91} +9.47493 q^{92} +13.8892 q^{93} +9.68313 q^{94} +2.24033 q^{95} -1.55594 q^{96} +3.98358 q^{97} +1.81292 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9	$ 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^14 + 4 * q^15 + 8 * q^16 + 4 * q^17 - 22 * q^18 + 8 * q^19 + 20 * q^21 + 14 * q^23 - 8 * q^24 + 36 * q^25 - 12 * q^26 + 32 * q^27 - 4 * q^28 + 2 * q^29 - 4 * q^30 - 8 * q^32 - 4 * q^34 - 36 * q^35 + 22 * q^36 + 24 * q^37 - 8 * q^38 - 16 * q^39 - 8 * q^41 - 20 * q^42 - 8 * q^43 + 16 * q^45 - 14 * q^46 - 16 * q^47 + 8 * q^48 + 34 * q^49 - 36 * q^50 - 18 * q^51 + 12 * q^52 + 36 * q^53 - 32 * q^54 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 24 * q^59 + 4 * q^60 - 12 * q^61 - 24 * q^63 + 8 * q^64 - 16 * q^65 + 16 * q^67 + 4 * q^68 + 4 * q^69 + 36 * q^70 + 4 * q^71 - 22 * q^72 + 20 * q^73 - 24 * q^74 + 40 * q^75 + 8 * q^76 + 16 * q^78 + 12 * q^79 + 40 * q^81 + 8 * q^82 - 20 * q^83 + 20 * q^84 - 12 * q^85 + 8 * q^86 + 36 * q^87 + 8 * q^89 - 16 * q^90 - 24 * q^91 + 14 * q^92 + 12 * q^93 + 16 * q^94 - 8 * q^96 + 4 * q^97 - 34 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.55594	0.898320	0.449160	−	0.893451i	$-0.351723\pi$
$3$	1.55594	0.898320	0.449160	+	0.893451i	$0.351723\pi$
$4$	1.00000	0.500000
$5$	2.24033	1.00191	0.500953	−	0.865475i	$-0.332983\pi$
$5$	2.24033	1.00191	0.500953	+	0.865475i	$0.332983\pi$
$6$	−1.55594	−0.635208
$7$	2.27752	0.860821	0.430410	−	0.902633i	$-0.358369\pi$
$7$	2.27752	0.860821	0.430410	+	0.902633i	$0.358369\pi$
$8$	−1.00000	−0.353553
$9$	−0.579065	−0.193022
$10$	−2.24033	−0.708454
$11$	0	0
$11$	0	0
$12$	1.55594	0.449160
$13$	−2.75611	−0.764406	−0.382203	−	0.924078i	$-0.624835\pi$
$13$	−2.75611	−0.764406	−0.382203	+	0.924078i	$0.624835\pi$
$14$	−2.27752	−0.608692
$15$	3.48581	0.900031
$16$	1.00000	0.250000
$17$	0.571514	0.138612	0.0693062	−	0.997595i	$-0.477921\pi$
$17$	0.571514	0.138612	0.0693062	+	0.997595i	$0.477921\pi$
$18$	0.579065	0.136487
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.24033	0.500953
$21$	3.54367	0.773292
$22$	0	0
$23$	9.47493	1.97566	0.987830	−	0.155537i	$-0.0497107\pi$
$23$	9.47493	1.97566	0.987830	+	0.155537i	$0.0497107\pi$
$24$	−1.55594	−0.317604
$25$	0.0190711	0.00381423
$26$	2.75611	0.540517
$27$	−5.56879	−1.07171
$28$	2.27752	0.430410
$29$	−1.99533	−0.370523	−0.185261	−	0.982689i	$-0.559313\pi$
$29$	−1.99533	−0.370523	−0.185261	+	0.982689i	$0.559313\pi$
$30$	−3.48581	−0.636418
$31$	8.92659	1.60326	0.801631	−	0.597819i	$-0.203966\pi$
$31$	8.92659	1.60326	0.801631	+	0.597819i	$0.203966\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.571514	−0.0980138
$35$	5.10239	0.862461
$36$	−0.579065	−0.0965108
$37$	7.48207	1.23004	0.615022	−	0.788510i	$-0.289147\pi$
$37$	7.48207	1.23004	0.615022	+	0.788510i	$0.289147\pi$
$38$	−1.00000	−0.162221
$39$	−4.28832	−0.686681
$40$	−2.24033	−0.354227
$41$	0.0862369	0.0134679	0.00673397	−	0.999977i	$-0.497856\pi$
$41$	0.0862369	0.0134679	0.00673397	+	0.999977i	$0.497856\pi$
$42$	−3.54367	−0.546800
$43$	−5.53342	−0.843839	−0.421920	−	0.906633i	$-0.638644\pi$
$43$	−5.53342	−0.843839	−0.421920	+	0.906633i	$0.638644\pi$
$44$	0	0
$45$	−1.29730	−0.193389
$46$	−9.47493	−1.39700
$47$	−9.68313	−1.41243	−0.706215	−	0.707997i	$-0.749599\pi$
$47$	−9.68313	−1.41243	−0.706215	+	0.707997i	$0.749599\pi$
$48$	1.55594	0.224580
$49$	−1.81292	−0.258988
$50$	−0.0190711	−0.00269706
$51$	0.889238	0.124518
$52$	−2.75611	−0.382203
$53$	4.40072	0.604485	0.302243	−	0.953231i	$-0.402265\pi$
$53$	4.40072	0.604485	0.302243	+	0.953231i	$0.402265\pi$
$54$	5.56879	0.757817
$55$	0	0
$56$	−2.27752	−0.304346
$57$	1.55594	0.206089
$58$	1.99533	0.261999
$59$	8.02502	1.04477	0.522384	−	0.852710i	$-0.325043\pi$
$59$	8.02502	1.04477	0.522384	+	0.852710i	$0.325043\pi$
$60$	3.48581	0.450016
$61$	−3.73883	−0.478709	−0.239354	−	0.970932i	$-0.576936\pi$
$61$	−3.73883	−0.478709	−0.239354	+	0.970932i	$0.576936\pi$
$62$	−8.92659	−1.13368
$63$	−1.31883	−0.166157
$64$	1.00000	0.125000
$65$	−6.17458	−0.765863
$66$	0	0
$67$	6.91634	0.844965	0.422483	−	0.906371i	$-0.361159\pi$
$67$	6.91634	0.844965	0.422483	+	0.906371i	$0.361159\pi$
$68$	0.571514	0.0693062
$69$	14.7424	1.77477
$70$	−5.10239	−0.609852
$71$	4.91454	0.583248	0.291624	−	0.956533i	$-0.405804\pi$
$71$	4.91454	0.583248	0.291624	+	0.956533i	$0.405804\pi$
$72$	0.579065	0.0682435
$73$	10.7414	1.25719	0.628593	−	0.777734i	$-0.283631\pi$
$73$	10.7414	1.25719	0.628593	+	0.777734i	$0.283631\pi$
$74$	−7.48207	−0.869773
$75$	0.0296734	0.00342639
$76$	1.00000	0.114708
$77$	0	0
$78$	4.28832	0.485557
$79$	5.92179	0.666254	0.333127	−	0.942882i	$-0.391896\pi$
$79$	5.92179	0.666254	0.333127	+	0.942882i	$0.391896\pi$
$80$	2.24033	0.250476
$81$	−6.92749	−0.769721
$82$	−0.0862369	−0.00952328
$83$	16.8470	1.84920	0.924601	−	0.380938i	$-0.124399\pi$
$83$	16.8470	1.84920	0.924601	+	0.380938i	$0.124399\pi$
$84$	3.54367	0.386646
$85$	1.28038	0.138877
$86$	5.53342	0.596684
$87$	−3.10460	−0.332848
$88$	0	0
$89$	5.14468	0.545335	0.272668	−	0.962108i	$-0.412094\pi$
$89$	5.14468	0.545335	0.272668	+	0.962108i	$0.412094\pi$
$90$	1.29730	0.136747
$91$	−6.27708	−0.658017
$92$	9.47493	0.987830
$93$	13.8892	1.44024
$94$	9.68313	0.998739
$95$	2.24033	0.229853
$96$	−1.55594	−0.158802
$97$	3.98358	0.404471	0.202236	−	0.979337i	$-0.435179\pi$
$97$	3.98358	0.404471	0.202236	+	0.979337i	$0.435179\pi$
$98$	1.81292	0.183132
$99$	0	0
$100$	0.0190711	0.00190711
$101$	−1.22629	−0.122021	−0.0610104	−	0.998137i	$-0.519432\pi$
$101$	−1.22629	−0.122021	−0.0610104	+	0.998137i	$0.519432\pi$
$102$	−0.889238	−0.0880477
$103$	6.31633	0.622367	0.311183	−	0.950350i	$-0.399275\pi$
$103$	6.31633	0.622367	0.311183	+	0.950350i	$0.399275\pi$
$104$	2.75611	0.270258
$105$	7.93898	0.774765
$106$	−4.40072	−0.427436
$107$	−15.9287	−1.53988	−0.769940	−	0.638116i	$-0.779714\pi$
$107$	−15.9287	−1.53988	−0.769940	+	0.638116i	$0.779714\pi$
$108$	−5.56879	−0.535857
$109$	−8.67279	−0.830703	−0.415352	−	0.909661i	$-0.636341\pi$
$109$	−8.67279	−0.830703	−0.415352	+	0.909661i	$0.636341\pi$
$110$	0	0
$111$	11.6416	1.10497
$112$	2.27752	0.215205
$113$	−0.297807	−0.0280153	−0.0140077	−	0.999902i	$-0.504459\pi$
$113$	−0.297807	−0.0280153	−0.0140077	+	0.999902i	$0.504459\pi$
$114$	−1.55594	−0.145727
$115$	21.2270	1.97942
$116$	−1.99533	−0.185261
$117$	1.59596	0.147547
$118$	−8.02502	−0.738763
$119$	1.30163	0.119320
$120$	−3.48581	−0.318209
$121$	0	0
$122$	3.73883	0.338498
$123$	0.134179	0.0120985
$124$	8.92659	0.801631
$125$	−11.1589	−0.998084
$126$	1.31883	0.117491
$127$	0.637264	0.0565480	0.0282740	−	0.999600i	$-0.490999\pi$
$127$	0.637264	0.0565480	0.0282740	+	0.999600i	$0.490999\pi$
$128$	−1.00000	−0.0883883
$129$	−8.60965	−0.758037
$130$	6.17458	0.541547
$131$	21.8110	1.90564	0.952818	−	0.303542i	$-0.0981693\pi$
$131$	21.8110	1.90564	0.952818	+	0.303542i	$0.0981693\pi$
$132$	0	0
$133$	2.27752	0.197486
$134$	−6.91634	−0.597481
$135$	−12.4759	−1.07376
$136$	−0.571514	−0.0490069
$137$	−1.60089	−0.136773	−0.0683866	−	0.997659i	$-0.521785\pi$
$137$	−1.60089	−0.136773	−0.0683866	+	0.997659i	$0.521785\pi$
$138$	−14.7424	−1.25496
$139$	15.9213	1.35043	0.675214	−	0.737621i	$-0.264051\pi$
$139$	15.9213	1.35043	0.675214	+	0.737621i	$0.264051\pi$
$140$	5.10239	0.431230
$141$	−15.0663	−1.26881
$142$	−4.91454	−0.412419
$143$	0	0
$144$	−0.579065	−0.0482554
$145$	−4.47019	−0.371229
$146$	−10.7414	−0.888965
$147$	−2.82078	−0.232654
$148$	7.48207	0.615022
$149$	−4.19884	−0.343982	−0.171991	−	0.985098i	$-0.555020\pi$
$149$	−4.19884	−0.343982	−0.171991	+	0.985098i	$0.555020\pi$
$150$	−0.0296734	−0.00242283
$151$	0.918503	0.0747467	0.0373733	−	0.999301i	$-0.488101\pi$
$151$	0.918503	0.0747467	0.0373733	+	0.999301i	$0.488101\pi$
$152$	−1.00000	−0.0811107
$153$	−0.330944	−0.0267552
$154$	0	0
$155$	19.9985	1.60632
$156$	−4.28832	−0.343341
$157$	−11.8346	−0.944505	−0.472252	−	0.881463i	$-0.656559\pi$
$157$	−11.8346	−0.944505	−0.472252	+	0.881463i	$0.656559\pi$
$158$	−5.92179	−0.471113
$159$	6.84723	0.543021
$160$	−2.24033	−0.177114
$161$	21.5793	1.70069
$162$	6.92749	0.544275
$163$	−7.54426	−0.590912	−0.295456	−	0.955356i	$-0.595472\pi$
$163$	−7.54426	−0.590912	−0.295456	+	0.955356i	$0.595472\pi$
$164$	0.0862369	0.00673397
$165$	0	0
$166$	−16.8470	−1.30758
$167$	9.75174	0.754612	0.377306	−	0.926089i	$-0.376850\pi$
$167$	9.75174	0.754612	0.377306	+	0.926089i	$0.376850\pi$
$168$	−3.54367	−0.273400
$169$	−5.40388	−0.415683
$170$	−1.28038	−0.0982005
$171$	−0.579065	−0.0442822
$172$	−5.53342	−0.421920
$173$	−9.90137	−0.752787	−0.376393	−	0.926460i	$-0.622836\pi$
$173$	−9.90137	−0.752787	−0.376393	+	0.926460i	$0.622836\pi$
$174$	3.10460	0.235359
$175$	0.0434348	0.00328336
$176$	0	0
$177$	12.4864	0.938536
$178$	−5.14468	−0.385610
$179$	15.8720	1.18633	0.593164	−	0.805081i	$-0.297878\pi$
$179$	15.8720	1.18633	0.593164	+	0.805081i	$0.297878\pi$
$180$	−1.29730	−0.0966947
$181$	−17.7160	−1.31682	−0.658410	−	0.752660i	$-0.728771\pi$
$181$	−17.7160	−1.31682	−0.658410	+	0.752660i	$0.728771\pi$
$182$	6.27708	0.465288
$183$	−5.81738	−0.430033
$184$	−9.47493	−0.698501
$185$	16.7623	1.23239
$186$	−13.8892	−1.01841
$187$	0	0
$188$	−9.68313	−0.706215
$189$	−12.6830	−0.922554
$190$	−2.24033	−0.162531
$191$	−17.5478	−1.26971	−0.634857	−	0.772629i	$-0.718941\pi$
$191$	−17.5478	−1.26971	−0.634857	+	0.772629i	$0.718941\pi$
$192$	1.55594	0.112290
$193$	13.1066	0.943437	0.471719	−	0.881749i	$-0.343634\pi$
$193$	13.1066	0.943437	0.471719	+	0.881749i	$0.343634\pi$
$194$	−3.98358	−0.286004
$195$	−9.60725	−0.687990
$196$	−1.81292	−0.129494
$197$	−15.2492	−1.08646	−0.543230	−	0.839584i	$-0.682799\pi$
$197$	−15.2492	−1.08646	−0.543230	+	0.839584i	$0.682799\pi$
$198$	0	0
$199$	13.4862	0.956009	0.478005	−	0.878357i	$-0.341360\pi$
$199$	13.4862	0.956009	0.478005	+	0.878357i	$0.341360\pi$
$200$	−0.0190711	−0.00134853
$201$	10.7614	0.759049
$202$	1.22629	0.0862818
$203$	−4.54439	−0.318954
$204$	0.889238	0.0622591
$205$	0.193199	0.0134936
$206$	−6.31633	−0.440080
$207$	−5.48660	−0.381345
$208$	−2.75611	−0.191102
$209$	0	0
$210$	−7.93898	−0.547842
$211$	−2.15813	−0.148572	−0.0742858	−	0.997237i	$-0.523668\pi$
$211$	−2.15813	−0.148572	−0.0742858	+	0.997237i	$0.523668\pi$
$212$	4.40072	0.302243
$213$	7.64670	0.523943
$214$	15.9287	1.08886
$215$	−12.3967	−0.845447
$216$	5.56879	0.378908
$217$	20.3305	1.38012
$218$	8.67279	0.587396
$219$	16.7129	1.12936
$220$	0	0
$221$	−1.57515	−0.105956
$222$	−11.6416	−0.781334
$223$	6.76418	0.452963	0.226482	−	0.974015i	$-0.427278\pi$
$223$	6.76418	0.452963	0.226482	+	0.974015i	$0.427278\pi$
$224$	−2.27752	−0.152173
$225$	−0.0110434	−0.000736228 0
$226$	0.297807	0.0198098
$227$	6.68912	0.443973	0.221986	−	0.975050i	$-0.428746\pi$
$227$	6.68912	0.443973	0.221986	+	0.975050i	$0.428746\pi$
$228$	1.55594	0.103044
$229$	0.645439	0.0426518	0.0213259	−	0.999773i	$-0.493211\pi$
$229$	0.645439	0.0426518	0.0213259	+	0.999773i	$0.493211\pi$
$230$	−21.2270	−1.39966
$231$	0	0
$232$	1.99533	0.131000
$233$	−5.52617	−0.362031	−0.181016	−	0.983480i	$-0.557938\pi$
$233$	−5.52617	−0.362031	−0.181016	+	0.983480i	$0.557938\pi$
$234$	−1.59596	−0.104331
$235$	−21.6934	−1.41512
$236$	8.02502	0.522384
$237$	9.21393	0.598509
$238$	−1.30163	−0.0843723
$239$	−24.0112	−1.55316	−0.776578	−	0.630021i	$-0.783046\pi$
$239$	−24.0112	−1.55316	−0.776578	+	0.630021i	$0.783046\pi$
$240$	3.48581	0.225008
$241$	12.5817	0.810461	0.405231	−	0.914214i	$-0.367191\pi$
$241$	12.5817	0.810461	0.405231	+	0.914214i	$0.367191\pi$
$242$	0	0
$243$	5.92766	0.380259
$244$	−3.73883	−0.239354
$245$	−4.06153	−0.259481
$246$	−0.134179	−0.00855495
$247$	−2.75611	−0.175367
$248$	−8.92659	−0.566839
$249$	26.2129	1.66117
$250$	11.1589	0.705752
$251$	−27.2335	−1.71896	−0.859480	−	0.511169i	$-0.829213\pi$
$251$	−27.2335	−1.71896	−0.859480	+	0.511169i	$0.829213\pi$
$252$	−1.31883	−0.0830785
$253$	0	0
$254$	−0.637264	−0.0399855
$255$	1.99219	0.124756
$256$	1.00000	0.0625000
$257$	−27.8288	−1.73591	−0.867956	−	0.496641i	$-0.834567\pi$
$257$	−27.8288	−1.73591	−0.867956	+	0.496641i	$0.834567\pi$
$258$	8.60965	0.536013
$259$	17.0405	1.05885
$260$	−6.17458	−0.382931
$261$	1.15542	0.0715189
$262$	−21.8110	−1.34749
$263$	−29.9986	−1.84979	−0.924895	−	0.380222i	$-0.875848\pi$
$263$	−29.9986	−1.84979	−0.924895	+	0.380222i	$0.875848\pi$
$264$	0	0
$265$	9.85905	0.605637
$266$	−2.27752	−0.139644
$267$	8.00479	0.489885
$268$	6.91634	0.422483
$269$	−1.89146	−0.115325	−0.0576623	−	0.998336i	$-0.518365\pi$
$269$	−1.89146	−0.115325	−0.0576623	+	0.998336i	$0.518365\pi$
$270$	12.4759	0.759261
$271$	−18.2983	−1.11154	−0.555772	−	0.831335i	$-0.687577\pi$
$271$	−18.2983	−1.11154	−0.555772	+	0.831335i	$0.687577\pi$
$272$	0.571514	0.0346531
$273$	−9.76673	−0.591109
$274$	1.60089	0.0967132
$275$	0	0
$276$	14.7424	0.887387
$277$	3.69005	0.221714	0.110857	−	0.993836i	$-0.464640\pi$
$277$	3.69005	0.221714	0.110857	+	0.993836i	$0.464640\pi$
$278$	−15.9213	−0.954897
$279$	−5.16908	−0.309464
$280$	−5.10239	−0.304926
$281$	23.5460	1.40463	0.702317	−	0.711864i	$-0.252149\pi$
$281$	23.5460	1.40463	0.702317	+	0.711864i	$0.252149\pi$
$282$	15.0663	0.897187
$283$	15.4242	0.916871	0.458435	−	0.888728i	$-0.348410\pi$
$283$	15.4242	0.916871	0.458435	+	0.888728i	$0.348410\pi$
$284$	4.91454	0.291624
$285$	3.48581	0.206481
$286$	0	0
$287$	0.196406	0.0115935
$288$	0.579065	0.0341217
$289$	−16.6734	−0.980787
$290$	4.47019	0.262498
$291$	6.19819	0.363344
$292$	10.7414	0.628593
$293$	15.3414	0.896251	0.448126	−	0.893971i	$-0.352092\pi$
$293$	15.3414	0.896251	0.448126	+	0.893971i	$0.352092\pi$
$294$	2.82078	0.164511
$295$	17.9787	1.04676
$296$	−7.48207	−0.434887
$297$	0	0
$298$	4.19884	0.243232
$299$	−26.1139	−1.51021
$300$	0.0296734	0.00171320
$301$	−12.6025	−0.726394
$302$	−0.918503	−0.0528539
$303$	−1.90803	−0.109614
$304$	1.00000	0.0573539
$305$	−8.37622	−0.479621
$306$	0.330944	0.0189188
$307$	23.5750	1.34550	0.672749	−	0.739870i	$-0.265113\pi$
$307$	23.5750	1.34550	0.672749	+	0.739870i	$0.265113\pi$
$308$	0	0
$309$	9.82780	0.559084
$310$	−19.9985	−1.13584
$311$	−4.17781	−0.236902	−0.118451	−	0.992960i	$-0.537793\pi$
$311$	−4.17781	−0.236902	−0.118451	+	0.992960i	$0.537793\pi$
$312$	4.28832	0.242779
$313$	−30.2227	−1.70829	−0.854144	−	0.520036i	$-0.825919\pi$
$313$	−30.2227	−1.70829	−0.854144	+	0.520036i	$0.825919\pi$
$314$	11.8346	0.667866
$315$	−2.95461	−0.166474
$316$	5.92179	0.333127
$317$	17.3175	0.972647	0.486324	−	0.873779i	$-0.338338\pi$
$317$	17.3175	0.972647	0.486324	+	0.873779i	$0.338338\pi$
$318$	−6.84723	−0.383974
$319$	0	0
$320$	2.24033	0.125238
$321$	−24.7840	−1.38331
$322$	−21.5793	−1.20257
$323$	0.571514	0.0317999
$324$	−6.92749	−0.384860
$325$	−0.0525621	−0.00291562
$326$	7.54426	0.417838
$327$	−13.4943	−0.746237
$328$	−0.0862369	−0.00476164
$329$	−22.0535	−1.21585
$330$	0	0
$331$	−9.13686	−0.502207	−0.251104	−	0.967960i	$-0.580793\pi$
$331$	−9.13686	−0.502207	−0.251104	+	0.967960i	$0.580793\pi$
$332$	16.8470	0.924601
$333$	−4.33261	−0.237425
$334$	−9.75174	−0.533591
$335$	15.4949	0.846575
$336$	3.54367	0.193323
$337$	−9.49487	−0.517219	−0.258609	−	0.965982i	$-0.583264\pi$
$337$	−9.49487	−0.517219	−0.258609	+	0.965982i	$0.583264\pi$
$338$	5.40388	0.293932
$339$	−0.463368	−0.0251667
$340$	1.28038	0.0694383
$341$	0	0
$342$	0.579065	0.0313122
$343$	−20.0716	−1.08376
$344$	5.53342	0.298342
$345$	33.0278	1.77816
$346$	9.90137	0.532301
$347$	12.4664	0.669234	0.334617	−	0.942354i	$-0.391393\pi$
$347$	12.4664	0.669234	0.334617	+	0.942354i	$0.391393\pi$
$348$	−3.10460	−0.166424
$349$	29.2430	1.56534	0.782672	−	0.622434i	$-0.213856\pi$
$349$	29.2430	1.56534	0.782672	+	0.622434i	$0.213856\pi$
$350$	−0.0434348	−0.00232169
$351$	15.3482	0.819226
$352$	0	0
$353$	−9.95957	−0.530095	−0.265047	−	0.964235i	$-0.585388\pi$
$353$	−9.95957	−0.530095	−0.265047	+	0.964235i	$0.585388\pi$
$354$	−12.4864	−0.663646
$355$	11.0102	0.584359
$356$	5.14468	0.272668
$357$	2.02526	0.107188
$358$	−15.8720	−0.838861
$359$	9.89422	0.522197	0.261099	−	0.965312i	$-0.415915\pi$
$359$	9.89422	0.522197	0.261099	+	0.965312i	$0.415915\pi$
$360$	1.29730	0.0683735
$361$	1.00000	0.0526316
$362$	17.7160	0.931132
$363$	0	0
$364$	−6.27708	−0.329008
$365$	24.0643	1.25958
$366$	5.81738	0.304080
$367$	−1.78588	−0.0932222	−0.0466111	−	0.998913i	$-0.514842\pi$
$367$	−1.78588	−0.0932222	−0.0466111	+	0.998913i	$0.514842\pi$
$368$	9.47493	0.493915
$369$	−0.0499368	−0.00259961
$370$	−16.7623	−0.871430
$371$	10.0227	0.520353
$372$	13.8892	0.720121
$373$	11.8730	0.614762	0.307381	−	0.951586i	$-0.400547\pi$
$373$	11.8730	0.614762	0.307381	+	0.951586i	$0.400547\pi$
$374$	0	0
$375$	−17.3626	−0.896598
$376$	9.68313	0.499369
$377$	5.49933	0.283230
$378$	12.6830	0.652344
$379$	14.5469	0.747226	0.373613	−	0.927585i	$-0.378119\pi$
$379$	14.5469	0.747226	0.373613	+	0.927585i	$0.378119\pi$
$380$	2.24033	0.114926
$381$	0.991542	0.0507982
$382$	17.5478	0.897824
$383$	10.9005	0.556988	0.278494	−	0.960438i	$-0.410165\pi$
$383$	10.9005	0.556988	0.278494	+	0.960438i	$0.410165\pi$
$384$	−1.55594	−0.0794010
$385$	0	0
$386$	−13.1066	−0.667111
$387$	3.20421	0.162879
$388$	3.98358	0.202236
$389$	2.85935	0.144975	0.0724874	−	0.997369i	$-0.476906\pi$
$389$	2.85935	0.144975	0.0724874	+	0.997369i	$0.476906\pi$
$390$	9.60725	0.486482
$391$	5.41505	0.273851
$392$	1.81292	0.0915661
$393$	33.9365	1.71187
$394$	15.2492	0.768244
$395$	13.2668	0.667523
$396$	0	0
$397$	−1.34242	−0.0673741	−0.0336870	−	0.999432i	$-0.510725\pi$
$397$	−1.34242	−0.0673741	−0.0336870	+	0.999432i	$0.510725\pi$
$398$	−13.4862	−0.676001
$399$	3.54367	0.177405
$400$	0.0190711	0.000953556 0
$401$	−25.2632	−1.26159	−0.630793	−	0.775951i	$-0.717270\pi$
$401$	−25.2632	−1.26159	−0.630793	+	0.775951i	$0.717270\pi$
$402$	−10.7614	−0.536729
$403$	−24.6026	−1.22554
$404$	−1.22629	−0.0610104
$405$	−15.5198	−0.771188
$406$	4.54439	0.225534
$407$	0	0
$408$	−0.889238	−0.0440239
$409$	−10.9069	−0.539312	−0.269656	−	0.962957i	$-0.586910\pi$
$409$	−10.9069	−0.539312	−0.269656	+	0.962957i	$0.586910\pi$
$410$	−0.193199	−0.00954142
$411$	−2.49088	−0.122866
$412$	6.31633	0.311183
$413$	18.2771	0.899358
$414$	5.48660	0.269652
$415$	37.7429	1.85272
$416$	2.75611	0.135129
$417$	24.7725	1.21312
$418$	0	0
$419$	9.08955	0.444054	0.222027	−	0.975041i	$-0.428733\pi$
$419$	9.08955	0.444054	0.222027	+	0.975041i	$0.428733\pi$
$420$	7.93898	0.387383
$421$	34.5218	1.68249	0.841244	−	0.540655i	$-0.181824\pi$
$421$	34.5218	1.68249	0.841244	+	0.540655i	$0.181824\pi$
$422$	2.15813	0.105056
$423$	5.60716	0.272630
$424$	−4.40072	−0.213718
$425$	0.0108994	0.000528699 0
$426$	−7.64670	−0.370484
$427$	−8.51526	−0.412082
$428$	−15.9287	−0.769940
$429$	0	0
$430$	12.3967	0.597821
$431$	3.67160	0.176855	0.0884273	−	0.996083i	$-0.471816\pi$
$431$	3.67160	0.176855	0.0884273	+	0.996083i	$0.471816\pi$
$432$	−5.56879	−0.267929
$433$	−9.59701	−0.461203	−0.230602	−	0.973048i	$-0.574069\pi$
$433$	−9.59701	−0.461203	−0.230602	+	0.973048i	$0.574069\pi$
$434$	−20.3305	−0.975893
$435$	−6.95532	−0.333482
$436$	−8.67279	−0.415352
$437$	9.47493	0.453248
$438$	−16.7129	−0.798575
$439$	5.20790	0.248560	0.124280	−	0.992247i	$-0.460338\pi$
$439$	5.20790	0.248560	0.124280	+	0.992247i	$0.460338\pi$
$440$	0	0
$441$	1.04980	0.0499903
$442$	1.57515	0.0749224
$443$	−11.5968	−0.550981	−0.275490	−	0.961304i	$-0.588840\pi$
$443$	−11.5968	−0.550981	−0.275490	+	0.961304i	$0.588840\pi$
$444$	11.6416	0.552487
$445$	11.5258	0.546374
$446$	−6.76418	−0.320293
$447$	−6.53312	−0.309006
$448$	2.27752	0.107603
$449$	−19.1493	−0.903711	−0.451856	−	0.892091i	$-0.649238\pi$
$449$	−19.1493	−0.903711	−0.451856	+	0.892091i	$0.649238\pi$
$450$	0.0110434	0.000520592 0
$451$	0	0
$452$	−0.297807	−0.0140077
$453$	1.42913	0.0671464
$454$	−6.68912	−0.313936
$455$	−14.0627	−0.659270
$456$	−1.55594	−0.0728634
$457$	−15.2474	−0.713243	−0.356622	−	0.934249i	$-0.616071\pi$
$457$	−15.2474	−0.713243	−0.356622	+	0.934249i	$0.616071\pi$
$458$	−0.645439	−0.0301594
$459$	−3.18264	−0.148553
$460$	21.2270	0.989712
$461$	−13.1903	−0.614336	−0.307168	−	0.951655i	$-0.599381\pi$
$461$	−13.1903	−0.614336	−0.307168	+	0.951655i	$0.599381\pi$
$462$	0	0
$463$	−15.8312	−0.735740	−0.367870	−	0.929877i	$-0.619913\pi$
$463$	−15.8312	−0.735740	−0.367870	+	0.929877i	$0.619913\pi$
$464$	−1.99533	−0.0926307
$465$	31.1164	1.44299
$466$	5.52617	0.255995
$467$	−14.8060	−0.685140	−0.342570	−	0.939492i	$-0.611297\pi$
$467$	−14.8060	−0.685140	−0.342570	+	0.939492i	$0.611297\pi$
$468$	1.59596	0.0737735
$469$	15.7521	0.727363
$470$	21.6934	1.00064
$471$	−18.4139	−0.848467
$472$	−8.02502	−0.369382
$473$	0	0
$474$	−9.21393	−0.423210
$475$	0.0190711	0.000875043 0
$476$	1.30163	0.0596602
$477$	−2.54830	−0.116679
$478$	24.0112	1.09825
$479$	−21.4058	−0.978055	−0.489028	−	0.872268i	$-0.662648\pi$
$479$	−21.4058	−0.978055	−0.489028	+	0.872268i	$0.662648\pi$
$480$	−3.48581	−0.159105
$481$	−20.6214	−0.940254
$482$	−12.5817	−0.573083
$483$	33.5760	1.52776
$484$	0	0
$485$	8.92453	0.405242
$486$	−5.92766	−0.268884
$487$	−18.0587	−0.818320	−0.409160	−	0.912463i	$-0.634178\pi$
$487$	−18.0587	−0.818320	−0.409160	+	0.912463i	$0.634178\pi$
$488$	3.73883	0.169249
$489$	−11.7384	−0.530828
$490$	4.06153	0.183481
$491$	−26.9808	−1.21763	−0.608813	−	0.793314i	$-0.708354\pi$
$491$	−26.9808	−1.21763	−0.608813	+	0.793314i	$0.708354\pi$
$492$	0.134179	0.00604926
$493$	−1.14036	−0.0513590
$494$	2.75611	0.124003
$495$	0	0
$496$	8.92659	0.400816
$497$	11.1929	0.502072
$498$	−26.2129	−1.17463
$499$	41.4218	1.85429	0.927147	−	0.374699i	$-0.122254\pi$
$499$	41.4218	1.85429	0.927147	+	0.374699i	$0.122254\pi$
$500$	−11.1589	−0.499042
$501$	15.1731	0.677883
$502$	27.2335	1.21549
$503$	−36.6381	−1.63361	−0.816806	−	0.576912i	$-0.804258\pi$
$503$	−36.6381	−1.63361	−0.816806	+	0.576912i	$0.804258\pi$
$504$	1.31883	0.0587454
$505$	−2.74730	−0.122253
$506$	0	0
$507$	−8.40808	−0.373416
$508$	0.637264	0.0282740
$509$	−1.94309	−0.0861260	−0.0430630	−	0.999072i	$-0.513712\pi$
$509$	−1.94309	−0.0861260	−0.0430630	+	0.999072i	$0.513712\pi$
$510$	−1.99219	−0.0882155
$511$	24.4637	1.08221
$512$	−1.00000	−0.0441942
$513$	−5.56879	−0.245868
$514$	27.8288	1.22748
$515$	14.1507	0.623552
$516$	−8.60965	−0.379019
$517$	0	0
$518$	−17.0405	−0.748719
$519$	−15.4059	−0.676243
$520$	6.17458	0.270773
$521$	−42.3465	−1.85523	−0.927616	−	0.373535i	$-0.878145\pi$
$521$	−42.3465	−1.85523	−0.927616	+	0.373535i	$0.878145\pi$
$522$	−1.15542	−0.0505715
$523$	−40.4000	−1.76657	−0.883284	−	0.468838i	$-0.844673\pi$
$523$	−40.4000	−1.76657	−0.883284	+	0.468838i	$0.844673\pi$
$524$	21.8110	0.952818
$525$	0.0675818	0.00294951
$526$	29.9986	1.30800
$527$	5.10167	0.222232
$528$	0	0
$529$	66.7744	2.90323
$530$	−9.85905	−0.428250
$531$	−4.64701	−0.201663
$532$	2.27752	0.0987429
$533$	−0.237678	−0.0102950
$534$	−8.00479	−0.346401
$535$	−35.6854	−1.54281
$536$	−6.91634	−0.298740
$537$	24.6958	1.06570
$538$	1.89146	0.0815468
$539$	0	0
$540$	−12.4759	−0.536878
$541$	−4.72409	−0.203104	−0.101552	−	0.994830i	$-0.532381\pi$
$541$	−4.72409	−0.203104	−0.101552	+	0.994830i	$0.532381\pi$
$542$	18.2983	0.785981
$543$	−27.5649	−1.18293
$544$	−0.571514	−0.0245034
$545$	−19.4299	−0.832286
$546$	9.76673	0.417978
$547$	−26.5069	−1.13335	−0.566677	−	0.823940i	$-0.691771\pi$
$547$	−26.5069	−1.13335	−0.566677	+	0.823940i	$0.691771\pi$
$548$	−1.60089	−0.0683866
$549$	2.16503	0.0924011
$550$	0	0
$551$	−1.99533	−0.0850037
$552$	−14.7424	−0.627478
$553$	13.4870	0.573525
$554$	−3.69005	−0.156775
$555$	26.0811	1.10708
$556$	15.9213	0.675214
$557$	19.2433	0.815364	0.407682	−	0.913124i	$-0.366337\pi$
$557$	19.2433	0.815364	0.407682	+	0.913124i	$0.366337\pi$
$558$	5.16908	0.218824
$559$	15.2507	0.645036
$560$	5.10239	0.215615
$561$	0	0
$562$	−23.5460	−0.993226
$563$	−3.50880	−0.147878	−0.0739391	−	0.997263i	$-0.523557\pi$
$563$	−3.50880	−0.147878	−0.0739391	+	0.997263i	$0.523557\pi$
$564$	−15.0663	−0.634407
$565$	−0.667185	−0.0280687
$566$	−15.4242	−0.648326
$567$	−15.7775	−0.662592
$568$	−4.91454	−0.206209
$569$	−34.8083	−1.45924	−0.729620	−	0.683852i	$-0.760303\pi$
$569$	−34.8083	−1.45924	−0.729620	+	0.683852i	$0.760303\pi$
$570$	−3.48581	−0.146004
$571$	8.86605	0.371033	0.185516	−	0.982641i	$-0.440604\pi$
$571$	8.86605	0.371033	0.185516	+	0.982641i	$0.440604\pi$
$572$	0	0
$573$	−27.3033	−1.14061
$574$	−0.196406	−0.00819783
$575$	0.180698	0.00753561
$576$	−0.579065	−0.0241277
$577$	15.5873	0.648910	0.324455	−	0.945901i	$-0.394819\pi$
$577$	15.5873	0.648910	0.324455	+	0.945901i	$0.394819\pi$
$578$	16.6734	0.693521
$579$	20.3931	0.847508
$580$	−4.47019	−0.185614
$581$	38.3694	1.59183
$582$	−6.19819	−0.256923
$583$	0	0
$584$	−10.7414	−0.444483
$585$	3.57549	0.147828
$586$	−15.3414	−0.633745
$587$	26.9304	1.11154	0.555769	−	0.831337i	$-0.312424\pi$
$587$	26.9304	1.11154	0.555769	+	0.831337i	$0.312424\pi$
$588$	−2.82078	−0.116327
$589$	8.92659	0.367814
$590$	−17.9787	−0.740171
$591$	−23.7268	−0.975989
$592$	7.48207	0.307511
$593$	23.7386	0.974826	0.487413	−	0.873172i	$-0.337941\pi$
$593$	23.7386	0.974826	0.487413	+	0.873172i	$0.337941\pi$
$594$	0	0
$595$	2.91608	0.119548
$596$	−4.19884	−0.171991
$597$	20.9836	0.858802
$598$	26.1139	1.06788
$599$	21.9522	0.896944	0.448472	−	0.893797i	$-0.351968\pi$
$599$	21.9522	0.896944	0.448472	+	0.893797i	$0.351968\pi$
$600$	−0.0296734	−0.00121141
$601$	9.40878	0.383792	0.191896	−	0.981415i	$-0.438536\pi$
$601$	9.40878	0.383792	0.191896	+	0.981415i	$0.438536\pi$
$602$	12.6025	0.513638
$603$	−4.00501	−0.163097
$604$	0.918503	0.0373733
$605$	0	0
$606$	1.90803	0.0775086
$607$	29.5786	1.20056	0.600278	−	0.799791i	$-0.295056\pi$
$607$	29.5786	1.20056	0.600278	+	0.799791i	$0.295056\pi$
$608$	−1.00000	−0.0405554
$609$	−7.07078	−0.286522
$610$	8.37622	0.339143
$611$	26.6877	1.07967
$612$	−0.330944	−0.0133776
$613$	43.1876	1.74433	0.872166	−	0.489211i	$-0.162715\pi$
$613$	43.1876	1.74433	0.872166	+	0.489211i	$0.162715\pi$
$614$	−23.5750	−0.951411
$615$	0.300605	0.0121216
$616$	0	0
$617$	−29.5678	−1.19036	−0.595178	−	0.803594i	$-0.702919\pi$
$617$	−29.5678	−1.19036	−0.595178	+	0.803594i	$0.702919\pi$
$618$	−9.82780	−0.395332
$619$	6.02976	0.242357	0.121178	−	0.992631i	$-0.461333\pi$
$619$	6.02976	0.242357	0.121178	+	0.992631i	$0.461333\pi$
$620$	19.9985	0.803159
$621$	−52.7640	−2.11734
$622$	4.17781	0.167515
$623$	11.7171	0.469436
$624$	−4.28832	−0.171670
$625$	−25.0950	−1.00380
$626$	30.2227	1.20794
$627$	0	0
$628$	−11.8346	−0.472252
$629$	4.27611	0.170500
$630$	2.95461	0.117715
$631$	−41.8326	−1.66533	−0.832665	−	0.553777i	$-0.813186\pi$
$631$	−41.8326	−1.66533	−0.832665	+	0.553777i	$0.813186\pi$
$632$	−5.92179	−0.235556
$633$	−3.35791	−0.133465
$634$	−17.3175	−0.687766
$635$	1.42768	0.0566558
$636$	6.84723	0.271510
$637$	4.99659	0.197972
$638$	0	0
$639$	−2.84584	−0.112580
$640$	−2.24033	−0.0885568
$641$	46.1632	1.82334	0.911668	−	0.410928i	$-0.134795\pi$
$641$	46.1632	1.82334	0.911668	+	0.410928i	$0.134795\pi$
$642$	24.7840	0.978145
$643$	15.9119	0.627504	0.313752	−	0.949505i	$-0.398414\pi$
$643$	15.9119	0.627504	0.313752	+	0.949505i	$0.398414\pi$
$644$	21.5793	0.850344
$645$	−19.2884	−0.759482
$646$	−0.571514	−0.0224859
$647$	4.91972	0.193414	0.0967071	−	0.995313i	$-0.469169\pi$
$647$	4.91972	0.193414	0.0967071	+	0.995313i	$0.469169\pi$
$648$	6.92749	0.272137
$649$	0	0
$650$	0.0525621	0.00206165
$651$	31.6329	1.23979
$652$	−7.54426	−0.295456
$653$	8.25465	0.323029	0.161515	−	0.986870i	$-0.448362\pi$
$653$	8.25465	0.323029	0.161515	+	0.986870i	$0.448362\pi$
$654$	13.4943	0.527669
$655$	48.8638	1.90927
$656$	0.0862369	0.00336699
$657$	−6.21997	−0.242664
$658$	22.0535	0.859735
$659$	−49.5250	−1.92922	−0.964611	−	0.263677i	$-0.915065\pi$
$659$	−49.5250	−1.92922	−0.964611	+	0.263677i	$0.915065\pi$
$660$	0	0
$661$	49.1275	1.91084	0.955420	−	0.295252i	$-0.0954035\pi$
$661$	49.1275	1.91084	0.955420	+	0.295252i	$0.0954035\pi$
$662$	9.13686	0.355114
$663$	−2.45084	−0.0951826
$664$	−16.8470	−0.653791
$665$	5.10239	0.197862
$666$	4.33261	0.167885
$667$	−18.9056	−0.732027
$668$	9.75174	0.377306
$669$	10.5246	0.406906
$670$	−15.4949	−0.598619
$671$	0	0
$672$	−3.54367	−0.136700
$673$	39.5853	1.52590	0.762951	−	0.646457i	$-0.223750\pi$
$673$	39.5853	1.52590	0.762951	+	0.646457i	$0.223750\pi$
$674$	9.49487	0.365729
$675$	−0.106203	−0.00408776
$676$	−5.40388	−0.207841
$677$	−44.3011	−1.70263	−0.851315	−	0.524655i	$-0.824194\pi$
$677$	−44.3011	−1.70263	−0.851315	+	0.524655i	$0.824194\pi$
$678$	0.463368	0.0177955
$679$	9.07267	0.348177
$680$	−1.28038	−0.0491003
$681$	10.4078	0.398829
$682$	0	0
$683$	4.18585	0.160167	0.0800835	−	0.996788i	$-0.474481\pi$
$683$	4.18585	0.160167	0.0800835	+	0.996788i	$0.474481\pi$
$684$	−0.579065	−0.0221411
$685$	−3.58652	−0.137034
$686$	20.0716	0.766336
$687$	1.00426	0.0383149
$688$	−5.53342	−0.210960
$689$	−12.1288	−0.462072
$690$	−33.0278	−1.25735
$691$	−17.3471	−0.659916	−0.329958	−	0.943996i	$-0.607035\pi$
$691$	−17.3471	−0.659916	−0.329958	+	0.943996i	$0.607035\pi$
$692$	−9.90137	−0.376393
$693$	0	0
$694$	−12.4664	−0.473220
$695$	35.6690	1.35300
$696$	3.10460	0.117679
$697$	0.0492856	0.00186682
$698$	−29.2430	−1.10687
$699$	−8.59836	−0.325220
$700$	0.0434348	0.00164168
$701$	30.9365	1.16845	0.584227	−	0.811590i	$-0.301398\pi$
$701$	30.9365	1.16845	0.584227	+	0.811590i	$0.301398\pi$
$702$	−15.3482	−0.579280
$703$	7.48207	0.282192
$704$	0	0
$705$	−33.7535	−1.27123
$706$	9.95957	0.374833
$707$	−2.79291	−0.105038
$708$	12.4864	0.469268
$709$	−17.4009	−0.653505	−0.326752	−	0.945110i	$-0.605954\pi$
$709$	−17.4009	−0.653505	−0.326752	+	0.945110i	$0.605954\pi$
$710$	−11.0102	−0.413205
$711$	−3.42910	−0.128601
$712$	−5.14468	−0.192805
$713$	84.5788	3.16750
$714$	−2.02526	−0.0757933
$715$	0	0
$716$	15.8720	0.593164
$717$	−37.3599	−1.39523
$718$	−9.89422	−0.369249
$719$	−15.1001	−0.563138	−0.281569	−	0.959541i	$-0.590855\pi$
$719$	−15.1001	−0.563138	−0.281569	+	0.959541i	$0.590855\pi$
$720$	−1.29730	−0.0483474
$721$	14.3856	0.535746
$722$	−1.00000	−0.0372161
$723$	19.5764	0.728053
$724$	−17.7160	−0.658410
$725$	−0.0380531	−0.00141326
$726$	0	0
$727$	12.5584	0.465764	0.232882	−	0.972505i	$-0.425184\pi$
$727$	12.5584	0.465764	0.232882	+	0.972505i	$0.425184\pi$
$728$	6.27708	0.232644
$729$	30.0055	1.11132
$730$	−24.0643	−0.890659
$731$	−3.16243	−0.116967
$732$	−5.81738	−0.215017
$733$	−8.63342	−0.318883	−0.159441	−	0.987207i	$-0.550969\pi$
$733$	−8.63342	−0.318883	−0.159441	+	0.987207i	$0.550969\pi$
$734$	1.78588	0.0659180
$735$	−6.31947	−0.233097
$736$	−9.47493	−0.349251
$737$	0	0
$738$	0.0499368	0.00183820
$739$	25.8140	0.949584	0.474792	−	0.880098i	$-0.342523\pi$
$739$	25.8140	0.949584	0.474792	+	0.880098i	$0.342523\pi$
$740$	16.7623	0.616194
$741$	−4.28832	−0.157536
$742$	−10.0227	−0.367945
$743$	−27.8252	−1.02081	−0.510404	−	0.859935i	$-0.670504\pi$
$743$	−27.8252	−1.02081	−0.510404	+	0.859935i	$0.670504\pi$
$744$	−13.8892	−0.509203
$745$	−9.40678	−0.344638
$746$	−11.8730	−0.434703
$747$	−9.75552	−0.356936
$748$	0	0
$749$	−36.2778	−1.32556
$750$	17.3626	0.633991
$751$	16.8060	0.613259	0.306630	−	0.951829i	$-0.400799\pi$
$751$	16.8060	0.613259	0.306630	+	0.951829i	$0.400799\pi$
$752$	−9.68313	−0.353107
$753$	−42.3735	−1.54418
$754$	−5.49933	−0.200274
$755$	2.05775	0.0748891
$756$	−12.6830	−0.461277
$757$	−41.6137	−1.51248	−0.756238	−	0.654296i	$-0.772965\pi$
$757$	−41.6137	−1.51248	−0.756238	+	0.654296i	$0.772965\pi$
$758$	−14.5469	−0.528369
$759$	0	0
$760$	−2.24033	−0.0812653
$761$	17.3354	0.628409	0.314204	−	0.949355i	$-0.398262\pi$
$761$	17.3354	0.628409	0.314204	+	0.949355i	$0.398262\pi$
$762$	−0.991542	−0.0359198
$763$	−19.7524	−0.715086
$764$	−17.5478	−0.634857
$765$	−0.741422	−0.0268062
$766$	−10.9005	−0.393850
$767$	−22.1178	−0.798628
$768$	1.55594	0.0561450
$769$	6.63601	0.239301	0.119650	−	0.992816i	$-0.461823\pi$
$769$	6.63601	0.239301	0.119650	+	0.992816i	$0.461823\pi$
$770$	0	0
$771$	−43.2998	−1.55940
$772$	13.1066	0.471719
$773$	−23.1698	−0.833360	−0.416680	−	0.909053i	$-0.636807\pi$
$773$	−23.1698	−0.833360	−0.416680	+	0.909053i	$0.636807\pi$
$774$	−3.20421	−0.115173
$775$	0.170240	0.00611521
$776$	−3.98358	−0.143002
$777$	26.5140	0.951184
$778$	−2.85935	−0.102513
$779$	0.0862369	0.00308976
$780$	−9.60725	−0.343995
$781$	0	0
$782$	−5.41505	−0.193642
$783$	11.1116	0.397095
$784$	−1.81292	−0.0647470
$785$	−26.5134	−0.946305
$786$	−33.9365	−1.21048
$787$	−35.8057	−1.27634	−0.638168	−	0.769897i	$-0.720307\pi$
$787$	−35.8057	−1.27634	−0.638168	+	0.769897i	$0.720307\pi$
$788$	−15.2492	−0.543230
$789$	−46.6758	−1.66170
$790$	−13.2668	−0.472010
$791$	−0.678260	−0.0241162
$792$	0	0
$793$	10.3046	0.365928
$794$	1.34242	0.0476407
$795$	15.3400	0.544056
$796$	13.4862	0.478005
$797$	12.9247	0.457816	0.228908	−	0.973448i	$-0.426485\pi$
$797$	12.9247	0.457816	0.228908	+	0.973448i	$0.426485\pi$
$798$	−3.54367	−0.125445
$799$	−5.53404	−0.195780
$800$	−0.0190711	−0.000674266 0
$801$	−2.97910	−0.105261
$802$	25.2632	0.892076
$803$	0	0
$804$	10.7614	0.379524
$805$	48.3448	1.70393
$806$	24.6026	0.866591
$807$	−2.94300	−0.103598
$808$	1.22629	0.0431409
$809$	1.69221	0.0594950	0.0297475	−	0.999557i	$-0.490530\pi$
$809$	1.69221	0.0594950	0.0297475	+	0.999557i	$0.490530\pi$
$810$	15.5198	0.545312
$811$	14.2119	0.499048	0.249524	−	0.968369i	$-0.419726\pi$
$811$	14.2119	0.499048	0.249524	+	0.968369i	$0.419726\pi$
$812$	−4.54439	−0.159477
$813$	−28.4710	−0.998523
$814$	0	0
$815$	−16.9016	−0.592038
$816$	0.889238	0.0311296
$817$	−5.53342	−0.193590
$818$	10.9069	0.381351
$819$	3.63484	0.127011
$820$	0.193199	0.00674680
$821$	−36.7897	−1.28397	−0.641985	−	0.766717i	$-0.721889\pi$
$821$	−36.7897	−1.28397	−0.641985	+	0.766717i	$0.721889\pi$
$822$	2.49088	0.0868794
$823$	16.9368	0.590379	0.295189	−	0.955439i	$-0.404617\pi$
$823$	16.9368	0.590379	0.295189	+	0.955439i	$0.404617\pi$
$824$	−6.31633	−0.220040
$825$	0	0
$826$	−18.2771	−0.635942
$827$	28.9883	1.00802	0.504012	−	0.863697i	$-0.331857\pi$
$827$	28.9883	1.00802	0.504012	+	0.863697i	$0.331857\pi$
$828$	−5.48660	−0.190673
$829$	38.3557	1.33215	0.666074	−	0.745886i	$-0.267973\pi$
$829$	38.3557	1.33215	0.666074	+	0.745886i	$0.267973\pi$
$830$	−37.7429	−1.31007
$831$	5.74148	0.199170
$832$	−2.75611	−0.0955508
$833$	−1.03611	−0.0358990
$834$	−24.7725	−0.857803
$835$	21.8471	0.756050
$836$	0	0
$837$	−49.7103	−1.71824
$838$	−9.08955	−0.313993
$839$	13.1347	0.453462	0.226731	−	0.973957i	$-0.427196\pi$
$839$	13.1347	0.453462	0.226731	+	0.973957i	$0.427196\pi$
$840$	−7.93898	−0.273921
$841$	−25.0187	−0.862713
$842$	−34.5218	−1.18970
$843$	36.6360	1.26181
$844$	−2.15813	−0.0742858
$845$	−12.1065	−0.416475
$846$	−5.60716	−0.192778
$847$	0	0
$848$	4.40072	0.151121
$849$	23.9990	0.823643
$850$	−0.0108994	−0.000373847 0
$851$	70.8921	2.43015
$852$	7.64670	0.261972
$853$	47.1433	1.61415	0.807077	−	0.590446i	$-0.201048\pi$
$853$	47.1433	1.61415	0.807077	+	0.590446i	$0.201048\pi$
$854$	8.51526	0.291386
$855$	−1.29730	−0.0443666
$856$	15.9287	0.544430
$857$	−4.63462	−0.158316	−0.0791579	−	0.996862i	$-0.525223\pi$
$857$	−4.63462	−0.158316	−0.0791579	+	0.996862i	$0.525223\pi$
$858$	0	0
$859$	−28.6906	−0.978911	−0.489456	−	0.872028i	$-0.662805\pi$
$859$	−28.6906	−0.978911	−0.489456	+	0.872028i	$0.662805\pi$
$860$	−12.3967	−0.422723
$861$	0.305595	0.0104147
$862$	−3.67160	−0.125055
$863$	−29.7927	−1.01416	−0.507078	−	0.861900i	$-0.669275\pi$
$863$	−29.7927	−1.01416	−0.507078	+	0.861900i	$0.669275\pi$
$864$	5.56879	0.189454
$865$	−22.1823	−0.754221
$866$	9.59701	0.326120
$867$	−25.9427	−0.881060
$868$	20.3305	0.690061
$869$	0	0
$870$	6.95532	0.235807
$871$	−19.0622	−0.645897
$872$	8.67279	0.293698
$873$	−2.30675	−0.0780717
$874$	−9.47493	−0.320494
$875$	−25.4146	−0.859171
$876$	16.7129	0.564678
$877$	−35.2763	−1.19120	−0.595598	−	0.803283i	$-0.703085\pi$
$877$	−35.2763	−1.19120	−0.595598	+	0.803283i	$0.703085\pi$
$878$	−5.20790	−0.175758
$879$	23.8702	0.805120
$880$	0	0
$881$	−0.536228	−0.0180660	−0.00903300	−	0.999959i	$-0.502875\pi$
$881$	−0.536228	−0.0180660	−0.00903300	+	0.999959i	$0.502875\pi$
$882$	−1.04980	−0.0353485
$883$	9.43044	0.317360	0.158680	−	0.987330i	$-0.449276\pi$
$883$	9.43044	0.317360	0.158680	+	0.987330i	$0.449276\pi$
$884$	−1.57515	−0.0529781
$885$	27.9737	0.940325
$886$	11.5968	0.389602
$887$	15.7866	0.530062	0.265031	−	0.964240i	$-0.414618\pi$
$887$	15.7866	0.530062	0.265031	+	0.964240i	$0.414618\pi$
$888$	−11.6416	−0.390667
$889$	1.45138	0.0486777
$890$	−11.5258	−0.386345
$891$	0	0
$892$	6.76418	0.226482
$893$	−9.68313	−0.324034
$894$	6.53312	0.218500
$895$	35.5585	1.18859
$896$	−2.27752	−0.0760865
$897$	−40.6316	−1.35665
$898$	19.1493	0.639020
$899$	−17.8115	−0.594045
$900$	−0.0110434	−0.000368114 0
$901$	2.51507	0.0837891
$902$	0	0
$903$	−19.6086	−0.652534
$904$	0.297807	0.00990491
$905$	−39.6896	−1.31933
$906$	−1.42913	−0.0474797
$907$	−44.7761	−1.48677	−0.743383	−	0.668866i	$-0.766780\pi$
$907$	−44.7761	−1.48677	−0.743383	+	0.668866i	$0.766780\pi$
$908$	6.68912	0.221986
$909$	0.710104	0.0235527
$910$	14.0627	0.466175
$911$	3.05935	0.101361	0.0506804	−	0.998715i	$-0.483861\pi$
$911$	3.05935	0.101361	0.0506804	+	0.998715i	$0.483861\pi$
$912$	1.55594	0.0515222
$913$	0	0
$914$	15.2474	0.504339
$915$	−13.0329	−0.430853
$916$	0.645439	0.0213259
$917$	49.6749	1.64041
$918$	3.18264	0.105043
$919$	55.2774	1.82343	0.911717	−	0.410818i	$-0.134757\pi$
$919$	55.2774	1.82343	0.911717	+	0.410818i	$0.134757\pi$
$920$	−21.2270	−0.699832
$921$	36.6812	1.20869
$922$	13.1903	0.434401
$923$	−13.5450	−0.445839
$924$	0	0
$925$	0.142692	0.00469167
$926$	15.8312	0.520247
$927$	−3.65757	−0.120130
$928$	1.99533	0.0654998
$929$	−29.1942	−0.957832	−0.478916	−	0.877861i	$-0.658970\pi$
$929$	−29.1942	−0.957832	−0.478916	+	0.877861i	$0.658970\pi$
$930$	−31.1164	−1.02035
$931$	−1.81292	−0.0594159
$932$	−5.52617	−0.181016
$933$	−6.50041	−0.212814
$934$	14.8060	0.484467
$935$	0	0
$936$	−1.59596	−0.0521657
$937$	18.6969	0.610801	0.305401	−	0.952224i	$-0.401210\pi$
$937$	18.6969	0.610801	0.305401	+	0.952224i	$0.401210\pi$
$938$	−15.7521	−0.514324
$939$	−47.0246	−1.53459
$940$	−21.6934	−0.707561
$941$	−4.47168	−0.145773	−0.0728863	−	0.997340i	$-0.523221\pi$
$941$	−4.47168	−0.145773	−0.0728863	+	0.997340i	$0.523221\pi$
$942$	18.4139	0.599957
$943$	0.817089	0.0266081
$944$	8.02502	0.261192
$945$	−28.4141	−0.924312
$946$	0	0
$947$	8.78679	0.285532	0.142766	−	0.989756i	$-0.454400\pi$
$947$	8.78679	0.285532	0.142766	+	0.989756i	$0.454400\pi$
$948$	9.21393	0.299254
$949$	−29.6045	−0.961002
$950$	−0.0190711	−0.000618749 0
$951$	26.9449	0.873748
$952$	−1.30163	−0.0421861
$953$	33.1114	1.07258	0.536292	−	0.844033i	$-0.319825\pi$
$953$	33.1114	1.07258	0.536292	+	0.844033i	$0.319825\pi$
$954$	2.54830	0.0825043
$955$	−39.3129	−1.27213
$956$	−24.0112	−0.776578
$957$	0	0
$958$	21.4058	0.691590
$959$	−3.64605	−0.117737
$960$	3.48581	0.112504
$961$	48.6840	1.57045
$962$	20.6214	0.664860
$963$	9.22372	0.297230
$964$	12.5817	0.405231
$965$	29.3632	0.945235
$966$	−33.5760	−1.08029
$967$	39.9996	1.28630	0.643150	−	0.765740i	$-0.277627\pi$
$967$	39.9996	1.28630	0.643150	+	0.765740i	$0.277627\pi$
$968$	0	0
$969$	0.889238	0.0285665
$970$	−8.92453	−0.286549
$971$	4.06604	0.130486	0.0652428	−	0.997869i	$-0.479218\pi$
$971$	4.06604	0.130486	0.0652428	+	0.997869i	$0.479218\pi$
$972$	5.92766	0.190130
$973$	36.2611	1.16248
$974$	18.0587	0.578639
$975$	−0.0817832	−0.00261916
$976$	−3.73883	−0.119677
$977$	−37.4002	−1.19654	−0.598270	−	0.801295i	$-0.704145\pi$
$977$	−37.4002	−1.19654	−0.598270	+	0.801295i	$0.704145\pi$
$978$	11.7384	0.375352
$979$	0	0
$980$	−4.06153	−0.129741
$981$	5.02211	0.160344
$982$	26.9808	0.860992
$983$	−52.9556	−1.68902	−0.844511	−	0.535538i	$-0.820109\pi$
$983$	−52.9556	−1.68902	−0.844511	+	0.535538i	$0.820109\pi$
$984$	−0.134179	−0.00427747
$985$	−34.1632	−1.08853
$986$	1.14036	0.0363163
$987$	−34.3138	−1.09222
$988$	−2.75611	−0.0876834
$989$	−52.4288	−1.66714
$990$	0	0
$991$	−3.98829	−0.126692	−0.0633461	−	0.997992i	$-0.520177\pi$
$991$	−3.98829	−0.126692	−0.0633461	+	0.997992i	$0.520177\pi$
$992$	−8.92659	−0.283420
$993$	−14.2164	−0.451143
$994$	−11.1929	−0.355019
$995$	30.2135	0.957831
$996$	26.2129	0.830587
$997$	−32.7194	−1.03623	−0.518117	−	0.855309i	$-0.673367\pi$
$997$	−32.7194	−1.03623	−0.518117	+	0.855309i	$0.673367\pi$
$998$	−41.4218	−1.31118
$999$	−41.6661	−1.31826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.by.1.4	✓	8
11.10	odd	2		4598.2.a.cb.1.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4598.2.a.by.1.4	✓	8	1.1	even	1	trivial
4598.2.a.cb.1.4	yes	8	11.10	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:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.55594 q^{3} +1.00000 q^{4} +2.24033 q^{5} -1.55594 q^{6} +2.27752 q^{7} -1.00000 q^{8} -0.579065 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.55594 q^{3} +1.00000 q^{4} +2.24033 q^{5} -1.55594 q^{6} +2.27752 q^{7} -1.00000 q^{8} -0.579065 q^{9} -2.24033 q^{10} +1.55594 q^{12} -2.75611 q^{13} -2.27752 q^{14} +3.48581 q^{15} +1.00000 q^{16} +0.571514 q^{17} +0.579065 q^{18} +1.00000 q^{19} +2.24033 q^{20} +3.54367 q^{21} +9.47493 q^{23} -1.55594 q^{24} +0.0190711 q^{25} +2.75611 q^{26} -5.56879 q^{27} +2.27752 q^{28} -1.99533 q^{29} -3.48581 q^{30} +8.92659 q^{31} -1.00000 q^{32} -0.571514 q^{34} +5.10239 q^{35} -0.579065 q^{36} +7.48207 q^{37} -1.00000 q^{38} -4.28832 q^{39} -2.24033 q^{40} +0.0862369 q^{41} -3.54367 q^{42} -5.53342 q^{43} -1.29730 q^{45} -9.47493 q^{46} -9.68313 q^{47} +1.55594 q^{48} -1.81292 q^{49} -0.0190711 q^{50} +0.889238 q^{51} -2.75611 q^{52} +4.40072 q^{53} +5.56879 q^{54} -2.27752 q^{56} +1.55594 q^{57} +1.99533 q^{58} +8.02502 q^{59} +3.48581 q^{60} -3.73883 q^{61} -8.92659 q^{62} -1.31883 q^{63} +1.00000 q^{64} -6.17458 q^{65} +6.91634 q^{67} +0.571514 q^{68} +14.7424 q^{69} -5.10239 q^{70} +4.91454 q^{71} +0.579065 q^{72} +10.7414 q^{73} -7.48207 q^{74} +0.0296734 q^{75} +1.00000 q^{76} +4.28832 q^{78} +5.92179 q^{79} +2.24033 q^{80} -6.92749 q^{81} -0.0862369 q^{82} +16.8470 q^{83} +3.54367 q^{84} +1.28038 q^{85} +5.53342 q^{86} -3.10460 q^{87} +5.14468 q^{89} +1.29730 q^{90} -6.27708 q^{91} +9.47493 q^{92} +13.8892 q^{93} +9.68313 q^{94} +2.24033 q^{95} -1.55594 q^{96} +3.98358 q^{97} +1.81292 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9	\( 8 q - 8 q^{2} + 8 q^{3} + 8 q^{4} - 8 q^{6} - 4 q^{7} - 8 q^{8} + 22 q^{9} + 8 q^{12} + 12 q^{13} + 4 q^{14} + 4 q^{15} + 8 q^{16} + 4 q^{17} - 22 q^{18} + 8 q^{19} + 20 q^{21} + 14 q^{23} - 8 q^{24} + 36 q^{25} - 12 q^{26} + 32 q^{27} - 4 q^{28} + 2 q^{29} - 4 q^{30} - 8 q^{32} - 4 q^{34} - 36 q^{35} + 22 q^{36} + 24 q^{37} - 8 q^{38} - 16 q^{39} - 8 q^{41} - 20 q^{42} - 8 q^{43} + 16 q^{45} - 14 q^{46} - 16 q^{47} + 8 q^{48} + 34 q^{49} - 36 q^{50} - 18 q^{51} + 12 q^{52} + 36 q^{53} - 32 q^{54} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 24 q^{59} + 4 q^{60} - 12 q^{61} - 24 q^{63} + 8 q^{64} - 16 q^{65} + 16 q^{67} + 4 q^{68} + 4 q^{69} + 36 q^{70} + 4 q^{71} - 22 q^{72} + 20 q^{73} - 24 q^{74} + 40 q^{75} + 8 q^{76} + 16 q^{78} + 12 q^{79} + 40 q^{81} + 8 q^{82} - 20 q^{83} + 20 q^{84} - 12 q^{85} + 8 q^{86} + 36 q^{87} + 8 q^{89} - 16 q^{90} - 24 q^{91} + 14 q^{92} + 12 q^{93} + 16 q^{94} - 8 q^{96} + 4 q^{97} - 34 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^3 + 8 * q^4 - 8 * q^6 - 4 * q^7 - 8 * q^8 + 22 * q^9 + 8 * q^12 + 12 * q^13 + 4 * q^14 + 4 * q^15 + 8 * q^16 + 4 * q^17 - 22 * q^18 + 8 * q^19 + 20 * q^21 + 14 * q^23 - 8 * q^24 + 36 * q^25 - 12 * q^26 + 32 * q^27 - 4 * q^28 + 2 * q^29 - 4 * q^30 - 8 * q^32 - 4 * q^34 - 36 * q^35 + 22 * q^36 + 24 * q^37 - 8 * q^38 - 16 * q^39 - 8 * q^41 - 20 * q^42 - 8 * q^43 + 16 * q^45 - 14 * q^46 - 16 * q^47 + 8 * q^48 + 34 * q^49 - 36 * q^50 - 18 * q^51 + 12 * q^52 + 36 * q^53 - 32 * q^54 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 24 * q^59 + 4 * q^60 - 12 * q^61 - 24 * q^63 + 8 * q^64 - 16 * q^65 + 16 * q^67 + 4 * q^68 + 4 * q^69 + 36 * q^70 + 4 * q^71 - 22 * q^72 + 20 * q^73 - 24 * q^74 + 40 * q^75 + 8 * q^76 + 16 * q^78 + 12 * q^79 + 40 * q^81 + 8 * q^82 - 20 * q^83 + 20 * q^84 - 12 * q^85 + 8 * q^86 + 36 * q^87 + 8 * q^89 - 16 * q^90 - 24 * q^91 + 14 * q^92 + 12 * q^93 + 16 * q^94 - 8 * q^96 + 4 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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