LMFDB - Embedded newform 3750.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3750,2,Mod(1,3750)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3750, 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("3750.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3750 = 2 \cdot 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3750.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:	$29.9439007580$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 150)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	3750.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -5.23607 q^{11} +1.00000 q^{12} +4.85410 q^{13} -2.00000 q^{14} +1.00000 q^{16} -7.85410 q^{17} +1.00000 q^{18} -2.76393 q^{19} -2.00000 q^{21} -5.23607 q^{22} -6.00000 q^{23} +1.00000 q^{24} +4.85410 q^{26} +1.00000 q^{27} -2.00000 q^{28} -1.38197 q^{29} +3.70820 q^{31} +1.00000 q^{32} -5.23607 q^{33} -7.85410 q^{34} +1.00000 q^{36} +2.14590 q^{37} -2.76393 q^{38} +4.85410 q^{39} -6.09017 q^{41} -2.00000 q^{42} +1.23607 q^{43} -5.23607 q^{44} -6.00000 q^{46} -4.76393 q^{47} +1.00000 q^{48} -3.00000 q^{49} -7.85410 q^{51} +4.85410 q^{52} -8.56231 q^{53} +1.00000 q^{54} -2.00000 q^{56} -2.76393 q^{57} -1.38197 q^{58} +8.94427 q^{59} -8.85410 q^{61} +3.70820 q^{62} -2.00000 q^{63} +1.00000 q^{64} -5.23607 q^{66} +9.70820 q^{67} -7.85410 q^{68} -6.00000 q^{69} -14.1803 q^{71} +1.00000 q^{72} +3.14590 q^{73} +2.14590 q^{74} -2.76393 q^{76} +10.4721 q^{77} +4.85410 q^{78} +1.00000 q^{81} -6.09017 q^{82} -6.00000 q^{83} -2.00000 q^{84} +1.23607 q^{86} -1.38197 q^{87} -5.23607 q^{88} +1.38197 q^{89} -9.70820 q^{91} -6.00000 q^{92} +3.70820 q^{93} -4.76393 q^{94} +1.00000 q^{96} +13.8541 q^{97} -3.00000 q^{98} -5.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} - 6 q^{11} + 2 q^{12} + 3 q^{13} - 4 q^{14} + 2 q^{16} - 9 q^{17} + 2 q^{18} - 10 q^{19} - 4 q^{21} - 6 q^{22} - 12 q^{23} + 2 q^{24}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9 - 6 * q^11 + 2 * q^12 + 3 * q^13 - 4 * q^14 + 2 * q^16 - 9 * q^17 + 2 * q^18 - 10 * q^19 - 4 * q^21 - 6 * q^22 - 12 * q^23 + 2 * q^24 + 3 * q^26 + 2 * q^27 - 4 * q^28 - 5 * q^29 - 6 * q^31 + 2 * q^32 - 6 * q^33 - 9 * q^34 + 2 * q^36 + 11 * q^37 - 10 * q^38 + 3 * q^39 - q^41 - 4 * q^42 - 2 * q^43 - 6 * q^44 - 12 * q^46 - 14 * q^47 + 2 * q^48 - 6 * q^49 - 9 * q^51 + 3 * q^52 + 3 * q^53 + 2 * q^54 - 4 * q^56 - 10 * q^57 - 5 * q^58 - 11 * q^61 - 6 * q^62 - 4 * q^63 + 2 * q^64 - 6 * q^66 + 6 * q^67 - 9 * q^68 - 12 * q^69 - 6 * q^71 + 2 * q^72 + 13 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 + 3 * q^78 + 2 * q^81 - q^82 - 12 * q^83 - 4 * q^84 - 2 * q^86 - 5 * q^87 - 6 * q^88 + 5 * q^89 - 6 * q^91 - 12 * q^92 - 6 * q^93 - 14 * q^94 + 2 * q^96 + 21 * q^97 - 6 * q^98 - 6 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	1.00000	0.288675
$13$	4.85410	1.34629	0.673143	−	0.739512i	$-0.264944\pi$
$13$	4.85410	1.34629	0.673143	+	0.739512i	$0.264944\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.85410	−1.90490	−0.952450	−	0.304696i	$-0.901445\pi$
$17$	−7.85410	−1.90490	−0.952450	+	0.304696i	$0.901445\pi$
$18$	1.00000	0.235702
$19$	−2.76393	−0.634089	−0.317045	−	0.948411i	$-0.602691\pi$
$19$	−2.76393	−0.634089	−0.317045	+	0.948411i	$0.602691\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	−5.23607	−1.11633
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	4.85410	0.951968
$27$	1.00000	0.192450
$28$	−2.00000	−0.377964
$29$	−1.38197	−0.256625	−0.128312	−	0.991734i	$-0.540956\pi$
$29$	−1.38197	−0.256625	−0.128312	+	0.991734i	$0.540956\pi$
$30$	0	0
$31$	3.70820	0.666013	0.333007	−	0.942925i	$-0.391937\pi$
$31$	3.70820	0.666013	0.333007	+	0.942925i	$0.391937\pi$
$32$	1.00000	0.176777
$33$	−5.23607	−0.911482
$34$	−7.85410	−1.34697
$35$	0	0
$36$	1.00000	0.166667
$37$	2.14590	0.352783	0.176392	−	0.984320i	$-0.443557\pi$
$37$	2.14590	0.352783	0.176392	+	0.984320i	$0.443557\pi$
$38$	−2.76393	−0.448369
$39$	4.85410	0.777278
$40$	0	0
$41$	−6.09017	−0.951125	−0.475562	−	0.879682i	$-0.657755\pi$
$41$	−6.09017	−0.951125	−0.475562	+	0.879682i	$0.657755\pi$
$42$	−2.00000	−0.308607
$43$	1.23607	0.188499	0.0942493	−	0.995549i	$-0.469955\pi$
$43$	1.23607	0.188499	0.0942493	+	0.995549i	$0.469955\pi$
$44$	−5.23607	−0.789367
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−4.76393	−0.694891	−0.347445	−	0.937700i	$-0.612951\pi$
$47$	−4.76393	−0.694891	−0.347445	+	0.937700i	$0.612951\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−7.85410	−1.09979
$52$	4.85410	0.673143
$53$	−8.56231	−1.17612	−0.588062	−	0.808816i	$-0.700109\pi$
$53$	−8.56231	−1.17612	−0.588062	+	0.808816i	$0.700109\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	−2.76393	−0.366092
$58$	−1.38197	−0.181461
$59$	8.94427	1.16445	0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427	1.16445	0.582223	+	0.813029i	$0.302183\pi$
$60$	0	0
$61$	−8.85410	−1.13365	−0.566826	−	0.823838i	$-0.691829\pi$
$61$	−8.85410	−1.13365	−0.566826	+	0.823838i	$0.691829\pi$
$62$	3.70820	0.470942
$63$	−2.00000	−0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	−5.23607	−0.644515
$67$	9.70820	1.18605	0.593023	−	0.805186i	$-0.297934\pi$
$67$	9.70820	1.18605	0.593023	+	0.805186i	$0.297934\pi$
$68$	−7.85410	−0.952450
$69$	−6.00000	−0.722315
$70$	0	0
$71$	−14.1803	−1.68290	−0.841448	−	0.540338i	$-0.818297\pi$
$71$	−14.1803	−1.68290	−0.841448	+	0.540338i	$0.818297\pi$
$72$	1.00000	0.117851
$73$	3.14590	0.368200	0.184100	−	0.982908i	$-0.441063\pi$
$73$	3.14590	0.368200	0.184100	+	0.982908i	$0.441063\pi$
$74$	2.14590	0.249456
$75$	0	0
$76$	−2.76393	−0.317045
$77$	10.4721	1.19341
$78$	4.85410	0.549619
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.09017	−0.672547
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−2.00000	−0.218218
$85$	0	0
$86$	1.23607	0.133289
$87$	−1.38197	−0.148162
$88$	−5.23607	−0.558167
$89$	1.38197	0.146488	0.0732441	−	0.997314i	$-0.476665\pi$
$89$	1.38197	0.146488	0.0732441	+	0.997314i	$0.476665\pi$
$90$	0	0
$91$	−9.70820	−1.01770
$92$	−6.00000	−0.625543
$93$	3.70820	0.384523
$94$	−4.76393	−0.491362
$95$	0	0
$96$	1.00000	0.102062
$97$	13.8541	1.40667	0.703335	−	0.710858i	$-0.251693\pi$
$97$	13.8541	1.40667	0.703335	+	0.710858i	$0.251693\pi$
$98$	−3.00000	−0.303046
$99$	−5.23607	−0.526245
$100$	0	0
$101$	1.67376	0.166546	0.0832728	−	0.996527i	$-0.473463\pi$
$101$	1.67376	0.166546	0.0832728	+	0.996527i	$0.473463\pi$
$102$	−7.85410	−0.777672
$103$	2.29180	0.225817	0.112909	−	0.993605i	$-0.463983\pi$
$103$	2.29180	0.225817	0.112909	+	0.993605i	$0.463983\pi$
$104$	4.85410	0.475984
$105$	0	0
$106$	−8.56231	−0.831645
$107$	−10.9443	−1.05802	−0.529011	−	0.848615i	$-0.677437\pi$
$107$	−10.9443	−1.05802	−0.529011	+	0.848615i	$0.677437\pi$
$108$	1.00000	0.0962250
$109$	2.56231	0.245424	0.122712	−	0.992442i	$-0.460841\pi$
$109$	2.56231	0.245424	0.122712	+	0.992442i	$0.460841\pi$
$110$	0	0
$111$	2.14590	0.203680
$112$	−2.00000	−0.188982
$113$	11.5623	1.08769	0.543845	−	0.839186i	$-0.316968\pi$
$113$	11.5623	1.08769	0.543845	+	0.839186i	$0.316968\pi$
$114$	−2.76393	−0.258866
$115$	0	0
$116$	−1.38197	−0.128312
$117$	4.85410	0.448762
$118$	8.94427	0.823387
$119$	15.7082	1.43997
$120$	0	0
$121$	16.4164	1.49240
$122$	−8.85410	−0.801613
$123$	−6.09017	−0.549132
$124$	3.70820	0.333007
$125$	0	0
$126$	−2.00000	−0.178174
$127$	−0.291796	−0.0258927	−0.0129464	−	0.999916i	$-0.504121\pi$
$127$	−0.291796	−0.0258927	−0.0129464	+	0.999916i	$0.504121\pi$
$128$	1.00000	0.0883883
$129$	1.23607	0.108830
$130$	0	0
$131$	−0.763932	−0.0667451	−0.0333725	−	0.999443i	$-0.510625\pi$
$131$	−0.763932	−0.0667451	−0.0333725	+	0.999443i	$0.510625\pi$
$132$	−5.23607	−0.455741
$133$	5.52786	0.479327
$134$	9.70820	0.838661
$135$	0	0
$136$	−7.85410	−0.673484
$137$	15.0344	1.28448	0.642240	−	0.766504i	$-0.278005\pi$
$137$	15.0344	1.28448	0.642240	+	0.766504i	$0.278005\pi$
$138$	−6.00000	−0.510754
$139$	−13.4164	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$140$	0	0
$141$	−4.76393	−0.401195
$142$	−14.1803	−1.18999
$143$	−25.4164	−2.12543
$144$	1.00000	0.0833333
$145$	0	0
$146$	3.14590	0.260356
$147$	−3.00000	−0.247436
$148$	2.14590	0.176392
$149$	7.03444	0.576284	0.288142	−	0.957588i	$-0.406962\pi$
$149$	7.03444	0.576284	0.288142	+	0.957588i	$0.406962\pi$
$150$	0	0
$151$	−6.94427	−0.565117	−0.282558	−	0.959250i	$-0.591183\pi$
$151$	−6.94427	−0.565117	−0.282558	+	0.959250i	$0.591183\pi$
$152$	−2.76393	−0.224184
$153$	−7.85410	−0.634967
$154$	10.4721	0.843869
$155$	0	0
$156$	4.85410	0.388639
$157$	−17.8541	−1.42491	−0.712456	−	0.701717i	$-0.752417\pi$
$157$	−17.8541	−1.42491	−0.712456	+	0.701717i	$0.752417\pi$
$158$	0	0
$159$	−8.56231	−0.679035
$160$	0	0
$161$	12.0000	0.945732
$162$	1.00000	0.0785674
$163$	−20.4721	−1.60350	−0.801751	−	0.597659i	$-0.796098\pi$
$163$	−20.4721	−1.60350	−0.801751	+	0.597659i	$0.796098\pi$
$164$	−6.09017	−0.475562
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−6.47214	−0.500829	−0.250414	−	0.968139i	$-0.580567\pi$
$167$	−6.47214	−0.500829	−0.250414	+	0.968139i	$0.580567\pi$
$168$	−2.00000	−0.154303
$169$	10.5623	0.812485
$170$	0	0
$171$	−2.76393	−0.211363
$172$	1.23607	0.0942493
$173$	10.9098	0.829459	0.414730	−	0.909945i	$-0.363876\pi$
$173$	10.9098	0.829459	0.414730	+	0.909945i	$0.363876\pi$
$174$	−1.38197	−0.104767
$175$	0	0
$176$	−5.23607	−0.394683
$177$	8.94427	0.672293
$178$	1.38197	0.103583
$179$	−3.81966	−0.285495	−0.142747	−	0.989759i	$-0.545594\pi$
$179$	−3.81966	−0.285495	−0.142747	+	0.989759i	$0.545594\pi$
$180$	0	0
$181$	−20.0344	−1.48915	−0.744574	−	0.667540i	$-0.767347\pi$
$181$	−20.0344	−1.48915	−0.744574	+	0.667540i	$0.767347\pi$
$182$	−9.70820	−0.719620
$183$	−8.85410	−0.654514
$184$	−6.00000	−0.442326
$185$	0	0
$186$	3.70820	0.271899
$187$	41.1246	3.00733
$188$	−4.76393	−0.347445
$189$	−2.00000	−0.145479
$190$	0	0
$191$	−0.763932	−0.0552762	−0.0276381	−	0.999618i	$-0.508799\pi$
$191$	−0.763932	−0.0552762	−0.0276381	+	0.999618i	$0.508799\pi$
$192$	1.00000	0.0721688
$193$	−2.38197	−0.171458	−0.0857288	−	0.996319i	$-0.527322\pi$
$193$	−2.38197	−0.171458	−0.0857288	+	0.996319i	$0.527322\pi$
$194$	13.8541	0.994667
$195$	0	0
$196$	−3.00000	−0.214286
$197$	13.3262	0.949455	0.474728	−	0.880133i	$-0.342547\pi$
$197$	13.3262	0.949455	0.474728	+	0.880133i	$0.342547\pi$
$198$	−5.23607	−0.372111
$199$	−6.18034	−0.438113	−0.219056	−	0.975712i	$-0.570298\pi$
$199$	−6.18034	−0.438113	−0.219056	+	0.975712i	$0.570298\pi$
$200$	0	0
$201$	9.70820	0.684764
$202$	1.67376	0.117765
$203$	2.76393	0.193990
$204$	−7.85410	−0.549897
$205$	0	0
$206$	2.29180	0.159677
$207$	−6.00000	−0.417029
$208$	4.85410	0.336571
$209$	14.4721	1.00106
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−8.56231	−0.588062
$213$	−14.1803	−0.971621
$214$	−10.9443	−0.748135
$215$	0	0
$216$	1.00000	0.0680414
$217$	−7.41641	−0.503459
$218$	2.56231	0.173541
$219$	3.14590	0.212580
$220$	0	0
$221$	−38.1246	−2.56454
$222$	2.14590	0.144023
$223$	2.29180	0.153470	0.0767350	−	0.997052i	$-0.475550\pi$
$223$	2.29180	0.153470	0.0767350	+	0.997052i	$0.475550\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	11.5623	0.769113
$227$	16.9443	1.12463	0.562315	−	0.826923i	$-0.309911\pi$
$227$	16.9443	1.12463	0.562315	+	0.826923i	$0.309911\pi$
$228$	−2.76393	−0.183046
$229$	−24.2705	−1.60384	−0.801920	−	0.597431i	$-0.796188\pi$
$229$	−24.2705	−1.60384	−0.801920	+	0.597431i	$0.796188\pi$
$230$	0	0
$231$	10.4721	0.689016
$232$	−1.38197	−0.0907305
$233$	22.0902	1.44718	0.723588	−	0.690233i	$-0.242492\pi$
$233$	22.0902	1.44718	0.723588	+	0.690233i	$0.242492\pi$
$234$	4.85410	0.317323
$235$	0	0
$236$	8.94427	0.582223
$237$	0	0
$238$	15.7082	1.01821
$239$	26.8328	1.73567	0.867835	−	0.496852i	$-0.165511\pi$
$239$	26.8328	1.73567	0.867835	+	0.496852i	$0.165511\pi$
$240$	0	0
$241$	−13.8541	−0.892421	−0.446211	−	0.894928i	$-0.647227\pi$
$241$	−13.8541	−0.892421	−0.446211	+	0.894928i	$0.647227\pi$
$242$	16.4164	1.05529
$243$	1.00000	0.0641500
$244$	−8.85410	−0.566826
$245$	0	0
$246$	−6.09017	−0.388295
$247$	−13.4164	−0.853666
$248$	3.70820	0.235471
$249$	−6.00000	−0.380235
$250$	0	0
$251$	−12.4721	−0.787234	−0.393617	−	0.919274i	$-0.628776\pi$
$251$	−12.4721	−0.787234	−0.393617	+	0.919274i	$0.628776\pi$
$252$	−2.00000	−0.125988
$253$	31.4164	1.97513
$254$	−0.291796	−0.0183089
$255$	0	0
$256$	1.00000	0.0625000
$257$	11.6180	0.724713	0.362357	−	0.932040i	$-0.381972\pi$
$257$	11.6180	0.724713	0.362357	+	0.932040i	$0.381972\pi$
$258$	1.23607	0.0769542
$259$	−4.29180	−0.266679
$260$	0	0
$261$	−1.38197	−0.0855415
$262$	−0.763932	−0.0471959
$263$	−0.472136	−0.0291132	−0.0145566	−	0.999894i	$-0.504634\pi$
$263$	−0.472136	−0.0291132	−0.0145566	+	0.999894i	$0.504634\pi$
$264$	−5.23607	−0.322258
$265$	0	0
$266$	5.52786	0.338935
$267$	1.38197	0.0845749
$268$	9.70820	0.593023
$269$	−1.90983	−0.116444	−0.0582222	−	0.998304i	$-0.518543\pi$
$269$	−1.90983	−0.116444	−0.0582222	+	0.998304i	$0.518543\pi$
$270$	0	0
$271$	28.1803	1.71183	0.855917	−	0.517113i	$-0.172993\pi$
$271$	28.1803	1.71183	0.855917	+	0.517113i	$0.172993\pi$
$272$	−7.85410	−0.476225
$273$	−9.70820	−0.587567
$274$	15.0344	0.908264
$275$	0	0
$276$	−6.00000	−0.361158
$277$	22.1459	1.33062	0.665309	−	0.746568i	$-0.268300\pi$
$277$	22.1459	1.33062	0.665309	+	0.746568i	$0.268300\pi$
$278$	−13.4164	−0.804663
$279$	3.70820	0.222004
$280$	0	0
$281$	−6.09017	−0.363309	−0.181655	−	0.983362i	$-0.558145\pi$
$281$	−6.09017	−0.363309	−0.181655	+	0.983362i	$0.558145\pi$
$282$	−4.76393	−0.283688
$283$	−3.23607	−0.192364	−0.0961821	−	0.995364i	$-0.530663\pi$
$283$	−3.23607	−0.192364	−0.0961821	+	0.995364i	$0.530663\pi$
$284$	−14.1803	−0.841448
$285$	0	0
$286$	−25.4164	−1.50290
$287$	12.1803	0.718983
$288$	1.00000	0.0589256
$289$	44.6869	2.62864
$290$	0	0
$291$	13.8541	0.812142
$292$	3.14590	0.184100
$293$	28.7984	1.68242	0.841209	−	0.540709i	$-0.181844\pi$
$293$	28.7984	1.68242	0.841209	+	0.540709i	$0.181844\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	2.14590	0.124728
$297$	−5.23607	−0.303827
$298$	7.03444	0.407494
$299$	−29.1246	−1.68432
$300$	0	0
$301$	−2.47214	−0.142492
$302$	−6.94427	−0.399598
$303$	1.67376	0.0961551
$304$	−2.76393	−0.158522
$305$	0	0
$306$	−7.85410	−0.448989
$307$	16.2918	0.929822	0.464911	−	0.885357i	$-0.346086\pi$
$307$	16.2918	0.929822	0.464911	+	0.885357i	$0.346086\pi$
$308$	10.4721	0.596705
$309$	2.29180	0.130376
$310$	0	0
$311$	−18.6525	−1.05768	−0.528842	−	0.848720i	$-0.677374\pi$
$311$	−18.6525	−1.05768	−0.528842	+	0.848720i	$0.677374\pi$
$312$	4.85410	0.274809
$313$	26.3607	1.48999	0.744997	−	0.667068i	$-0.232451\pi$
$313$	26.3607	1.48999	0.744997	+	0.667068i	$0.232451\pi$
$314$	−17.8541	−1.00757
$315$	0	0
$316$	0	0
$317$	6.94427	0.390029	0.195015	−	0.980800i	$-0.437525\pi$
$317$	6.94427	0.390029	0.195015	+	0.980800i	$0.437525\pi$
$318$	−8.56231	−0.480150
$319$	7.23607	0.405142
$320$	0	0
$321$	−10.9443	−0.610850
$322$	12.0000	0.668734
$323$	21.7082	1.20788
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.4721	−1.13385
$327$	2.56231	0.141696
$328$	−6.09017	−0.336273
$329$	9.52786	0.525288
$330$	0	0
$331$	−32.4721	−1.78483	−0.892415	−	0.451216i	$-0.850991\pi$
$331$	−32.4721	−1.78483	−0.892415	+	0.451216i	$0.850991\pi$
$332$	−6.00000	−0.329293
$333$	2.14590	0.117594
$334$	−6.47214	−0.354140
$335$	0	0
$336$	−2.00000	−0.109109
$337$	−28.8328	−1.57062	−0.785312	−	0.619100i	$-0.787497\pi$
$337$	−28.8328	−1.57062	−0.785312	+	0.619100i	$0.787497\pi$
$338$	10.5623	0.574514
$339$	11.5623	0.627978
$340$	0	0
$341$	−19.4164	−1.05146
$342$	−2.76393	−0.149456
$343$	20.0000	1.07990
$344$	1.23607	0.0666443
$345$	0	0
$346$	10.9098	0.586516
$347$	5.23607	0.281087	0.140543	−	0.990075i	$-0.455115\pi$
$347$	5.23607	0.281087	0.140543	+	0.990075i	$0.455115\pi$
$348$	−1.38197	−0.0740812
$349$	14.7984	0.792139	0.396069	−	0.918221i	$-0.370374\pi$
$349$	14.7984	0.792139	0.396069	+	0.918221i	$0.370374\pi$
$350$	0	0
$351$	4.85410	0.259093
$352$	−5.23607	−0.279083
$353$	−20.4721	−1.08962	−0.544811	−	0.838559i	$-0.683399\pi$
$353$	−20.4721	−1.08962	−0.544811	+	0.838559i	$0.683399\pi$
$354$	8.94427	0.475383
$355$	0	0
$356$	1.38197	0.0732441
$357$	15.7082	0.831366
$358$	−3.81966	−0.201875
$359$	−27.8885	−1.47190	−0.735951	−	0.677035i	$-0.763264\pi$
$359$	−27.8885	−1.47190	−0.735951	+	0.677035i	$0.763264\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	−20.0344	−1.05299
$363$	16.4164	0.861638
$364$	−9.70820	−0.508848
$365$	0	0
$366$	−8.85410	−0.462811
$367$	12.4721	0.651040	0.325520	−	0.945535i	$-0.394461\pi$
$367$	12.4721	0.651040	0.325520	+	0.945535i	$0.394461\pi$
$368$	−6.00000	−0.312772
$369$	−6.09017	−0.317042
$370$	0	0
$371$	17.1246	0.889066
$372$	3.70820	0.192261
$373$	−2.58359	−0.133773	−0.0668867	−	0.997761i	$-0.521307\pi$
$373$	−2.58359	−0.133773	−0.0668867	+	0.997761i	$0.521307\pi$
$374$	41.1246	2.12650
$375$	0	0
$376$	−4.76393	−0.245681
$377$	−6.70820	−0.345490
$378$	−2.00000	−0.102869
$379$	−3.41641	−0.175489	−0.0877445	−	0.996143i	$-0.527966\pi$
$379$	−3.41641	−0.175489	−0.0877445	+	0.996143i	$0.527966\pi$
$380$	0	0
$381$	−0.291796	−0.0149492
$382$	−0.763932	−0.0390862
$383$	−20.4721	−1.04608	−0.523039	−	0.852309i	$-0.675202\pi$
$383$	−20.4721	−1.04608	−0.523039	+	0.852309i	$0.675202\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−2.38197	−0.121239
$387$	1.23607	0.0628329
$388$	13.8541	0.703335
$389$	15.9787	0.810153	0.405076	−	0.914283i	$-0.367245\pi$
$389$	15.9787	0.810153	0.405076	+	0.914283i	$0.367245\pi$
$390$	0	0
$391$	47.1246	2.38319
$392$	−3.00000	−0.151523
$393$	−0.763932	−0.0385353
$394$	13.3262	0.671366
$395$	0	0
$396$	−5.23607	−0.263122
$397$	−28.8328	−1.44708	−0.723539	−	0.690284i	$-0.757486\pi$
$397$	−28.8328	−1.44708	−0.723539	+	0.690284i	$0.757486\pi$
$398$	−6.18034	−0.309792
$399$	5.52786	0.276739
$400$	0	0
$401$	−26.0902	−1.30288	−0.651440	−	0.758700i	$-0.725835\pi$
$401$	−26.0902	−1.30288	−0.651440	+	0.758700i	$0.725835\pi$
$402$	9.70820	0.484201
$403$	18.0000	0.896644
$404$	1.67376	0.0832728
$405$	0	0
$406$	2.76393	0.137172
$407$	−11.2361	−0.556951
$408$	−7.85410	−0.388836
$409$	20.9787	1.03733	0.518665	−	0.854977i	$-0.326429\pi$
$409$	20.9787	1.03733	0.518665	+	0.854977i	$0.326429\pi$
$410$	0	0
$411$	15.0344	0.741594
$412$	2.29180	0.112909
$413$	−17.8885	−0.880238
$414$	−6.00000	−0.294884
$415$	0	0
$416$	4.85410	0.237992
$417$	−13.4164	−0.657004
$418$	14.4721	0.707855
$419$	−37.8885	−1.85098	−0.925488	−	0.378776i	$-0.876345\pi$
$419$	−37.8885	−1.85098	−0.925488	+	0.378776i	$0.876345\pi$
$420$	0	0
$421$	−11.0902	−0.540502	−0.270251	−	0.962790i	$-0.587107\pi$
$421$	−11.0902	−0.540502	−0.270251	+	0.962790i	$0.587107\pi$
$422$	−8.00000	−0.389434
$423$	−4.76393	−0.231630
$424$	−8.56231	−0.415822
$425$	0	0
$426$	−14.1803	−0.687040
$427$	17.7082	0.856960
$428$	−10.9443	−0.529011
$429$	−25.4164	−1.22712
$430$	0	0
$431$	3.70820	0.178618	0.0893089	−	0.996004i	$-0.471534\pi$
$431$	3.70820	0.178618	0.0893089	+	0.996004i	$0.471534\pi$
$432$	1.00000	0.0481125
$433$	23.1459	1.11232	0.556160	−	0.831075i	$-0.312274\pi$
$433$	23.1459	1.11232	0.556160	+	0.831075i	$0.312274\pi$
$434$	−7.41641	−0.355999
$435$	0	0
$436$	2.56231	0.122712
$437$	16.5836	0.793301
$438$	3.14590	0.150317
$439$	−25.1246	−1.19913	−0.599566	−	0.800325i	$-0.704660\pi$
$439$	−25.1246	−1.19913	−0.599566	+	0.800325i	$0.704660\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−38.1246	−1.81340
$443$	−30.0689	−1.42862	−0.714308	−	0.699832i	$-0.753258\pi$
$443$	−30.0689	−1.42862	−0.714308	+	0.699832i	$0.753258\pi$
$444$	2.14590	0.101840
$445$	0	0
$446$	2.29180	0.108520
$447$	7.03444	0.332718
$448$	−2.00000	−0.0944911
$449$	4.79837	0.226449	0.113225	−	0.993569i	$-0.463882\pi$
$449$	4.79837	0.226449	0.113225	+	0.993569i	$0.463882\pi$
$450$	0	0
$451$	31.8885	1.50157
$452$	11.5623	0.543845
$453$	−6.94427	−0.326270
$454$	16.9443	0.795234
$455$	0	0
$456$	−2.76393	−0.129433
$457$	12.4721	0.583422	0.291711	−	0.956507i	$-0.405775\pi$
$457$	12.4721	0.583422	0.291711	+	0.956507i	$0.405775\pi$
$458$	−24.2705	−1.13409
$459$	−7.85410	−0.366598
$460$	0	0
$461$	25.0902	1.16857	0.584283	−	0.811550i	$-0.301376\pi$
$461$	25.0902	1.16857	0.584283	+	0.811550i	$0.301376\pi$
$462$	10.4721	0.487208
$463$	12.2918	0.571248	0.285624	−	0.958342i	$-0.407799\pi$
$463$	12.2918	0.571248	0.285624	+	0.958342i	$0.407799\pi$
$464$	−1.38197	−0.0641562
$465$	0	0
$466$	22.0902	1.02331
$467$	−15.8197	−0.732047	−0.366023	−	0.930606i	$-0.619281\pi$
$467$	−15.8197	−0.732047	−0.366023	+	0.930606i	$0.619281\pi$
$468$	4.85410	0.224381
$469$	−19.4164	−0.896566
$470$	0	0
$471$	−17.8541	−0.822674
$472$	8.94427	0.411693
$473$	−6.47214	−0.297589
$474$	0	0
$475$	0	0
$476$	15.7082	0.719984
$477$	−8.56231	−0.392041
$478$	26.8328	1.22730
$479$	4.47214	0.204337	0.102169	−	0.994767i	$-0.467422\pi$
$479$	4.47214	0.204337	0.102169	+	0.994767i	$0.467422\pi$
$480$	0	0
$481$	10.4164	0.474947
$482$	−13.8541	−0.631037
$483$	12.0000	0.546019
$484$	16.4164	0.746200
$485$	0	0
$486$	1.00000	0.0453609
$487$	−10.2918	−0.466366	−0.233183	−	0.972433i	$-0.574914\pi$
$487$	−10.2918	−0.466366	−0.233183	+	0.972433i	$0.574914\pi$
$488$	−8.85410	−0.400806
$489$	−20.4721	−0.925782
$490$	0	0
$491$	29.8885	1.34885	0.674426	−	0.738343i	$-0.264391\pi$
$491$	29.8885	1.34885	0.674426	+	0.738343i	$0.264391\pi$
$492$	−6.09017	−0.274566
$493$	10.8541	0.488844
$494$	−13.4164	−0.603633
$495$	0	0
$496$	3.70820	0.166503
$497$	28.3607	1.27215
$498$	−6.00000	−0.268866
$499$	−33.4164	−1.49592	−0.747962	−	0.663742i	$-0.768967\pi$
$499$	−33.4164	−1.49592	−0.747962	+	0.663742i	$0.768967\pi$
$500$	0	0
$501$	−6.47214	−0.289154
$502$	−12.4721	−0.556659
$503$	−21.5279	−0.959880	−0.479940	−	0.877301i	$-0.659342\pi$
$503$	−21.5279	−0.959880	−0.479940	+	0.877301i	$0.659342\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	31.4164	1.39663
$507$	10.5623	0.469088
$508$	−0.291796	−0.0129464
$509$	−13.7426	−0.609132	−0.304566	−	0.952491i	$-0.598511\pi$
$509$	−13.7426	−0.609132	−0.304566	+	0.952491i	$0.598511\pi$
$510$	0	0
$511$	−6.29180	−0.278333
$512$	1.00000	0.0441942
$513$	−2.76393	−0.122031
$514$	11.6180	0.512450
$515$	0	0
$516$	1.23607	0.0544149
$517$	24.9443	1.09705
$518$	−4.29180	−0.188571
$519$	10.9098	0.478888
$520$	0	0
$521$	6.27051	0.274716	0.137358	−	0.990521i	$-0.456139\pi$
$521$	6.27051	0.274716	0.137358	+	0.990521i	$0.456139\pi$
$522$	−1.38197	−0.0604870
$523$	2.29180	0.100213	0.0501066	−	0.998744i	$-0.484044\pi$
$523$	2.29180	0.100213	0.0501066	+	0.998744i	$0.484044\pi$
$524$	−0.763932	−0.0333725
$525$	0	0
$526$	−0.472136	−0.0205861
$527$	−29.1246	−1.26869
$528$	−5.23607	−0.227871
$529$	13.0000	0.565217
$530$	0	0
$531$	8.94427	0.388148
$532$	5.52786	0.239663
$533$	−29.5623	−1.28049
$534$	1.38197	0.0598035
$535$	0	0
$536$	9.70820	0.419331
$537$	−3.81966	−0.164831
$538$	−1.90983	−0.0823386
$539$	15.7082	0.676600
$540$	0	0
$541$	21.2705	0.914491	0.457245	−	0.889341i	$-0.348836\pi$
$541$	21.2705	0.914491	0.457245	+	0.889341i	$0.348836\pi$
$542$	28.1803	1.21045
$543$	−20.0344	−0.859760
$544$	−7.85410	−0.336742
$545$	0	0
$546$	−9.70820	−0.415473
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	15.0344	0.642240
$549$	−8.85410	−0.377884
$550$	0	0
$551$	3.81966	0.162723
$552$	−6.00000	−0.255377
$553$	0	0
$554$	22.1459	0.940889
$555$	0	0
$556$	−13.4164	−0.568982
$557$	27.2705	1.15549	0.577744	−	0.816218i	$-0.303933\pi$
$557$	27.2705	1.15549	0.577744	+	0.816218i	$0.303933\pi$
$558$	3.70820	0.156981
$559$	6.00000	0.253773
$560$	0	0
$561$	41.1246	1.73628
$562$	−6.09017	−0.256898
$563$	−10.4721	−0.441348	−0.220674	−	0.975348i	$-0.570826\pi$
$563$	−10.4721	−0.441348	−0.220674	+	0.975348i	$0.570826\pi$
$564$	−4.76393	−0.200598
$565$	0	0
$566$	−3.23607	−0.136022
$567$	−2.00000	−0.0839921
$568$	−14.1803	−0.594994
$569$	21.3820	0.896379	0.448189	−	0.893939i	$-0.352069\pi$
$569$	21.3820	0.896379	0.448189	+	0.893939i	$0.352069\pi$
$570$	0	0
$571$	33.7082	1.41064	0.705322	−	0.708887i	$-0.250802\pi$
$571$	33.7082	1.41064	0.705322	+	0.708887i	$0.250802\pi$
$572$	−25.4164	−1.06271
$573$	−0.763932	−0.0319137
$574$	12.1803	0.508398
$575$	0	0
$576$	1.00000	0.0416667
$577$	−28.8328	−1.20033	−0.600163	−	0.799878i	$-0.704898\pi$
$577$	−28.8328	−1.20033	−0.600163	+	0.799878i	$0.704898\pi$
$578$	44.6869	1.85873
$579$	−2.38197	−0.0989911
$580$	0	0
$581$	12.0000	0.497844
$582$	13.8541	0.574271
$583$	44.8328	1.85679
$584$	3.14590	0.130178
$585$	0	0
$586$	28.7984	1.18965
$587$	15.8885	0.655790	0.327895	−	0.944714i	$-0.393661\pi$
$587$	15.8885	0.655790	0.327895	+	0.944714i	$0.393661\pi$
$588$	−3.00000	−0.123718
$589$	−10.2492	−0.422312
$590$	0	0
$591$	13.3262	0.548168
$592$	2.14590	0.0881959
$593$	6.03444	0.247805	0.123902	−	0.992294i	$-0.460459\pi$
$593$	6.03444	0.247805	0.123902	+	0.992294i	$0.460459\pi$
$594$	−5.23607	−0.214838
$595$	0	0
$596$	7.03444	0.288142
$597$	−6.18034	−0.252944
$598$	−29.1246	−1.19099
$599$	1.05573	0.0431359	0.0215679	−	0.999767i	$-0.493134\pi$
$599$	1.05573	0.0431359	0.0215679	+	0.999767i	$0.493134\pi$
$600$	0	0
$601$	7.32624	0.298843	0.149422	−	0.988774i	$-0.452259\pi$
$601$	7.32624	0.298843	0.149422	+	0.988774i	$0.452259\pi$
$602$	−2.47214	−0.100757
$603$	9.70820	0.395349
$604$	−6.94427	−0.282558
$605$	0	0
$606$	1.67376	0.0679919
$607$	1.81966	0.0738577	0.0369289	−	0.999318i	$-0.488243\pi$
$607$	1.81966	0.0738577	0.0369289	+	0.999318i	$0.488243\pi$
$608$	−2.76393	−0.112092
$609$	2.76393	0.112000
$610$	0	0
$611$	−23.1246	−0.935522
$612$	−7.85410	−0.317483
$613$	21.5623	0.870893	0.435447	−	0.900215i	$-0.356590\pi$
$613$	21.5623	0.870893	0.435447	+	0.900215i	$0.356590\pi$
$614$	16.2918	0.657483
$615$	0	0
$616$	10.4721	0.421934
$617$	−8.38197	−0.337445	−0.168723	−	0.985664i	$-0.553964\pi$
$617$	−8.38197	−0.337445	−0.168723	+	0.985664i	$0.553964\pi$
$618$	2.29180	0.0921896
$619$	3.41641	0.137317	0.0686585	−	0.997640i	$-0.478128\pi$
$619$	3.41641	0.137317	0.0686585	+	0.997640i	$0.478128\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	−18.6525	−0.747896
$623$	−2.76393	−0.110735
$624$	4.85410	0.194320
$625$	0	0
$626$	26.3607	1.05358
$627$	14.4721	0.577961
$628$	−17.8541	−0.712456
$629$	−16.8541	−0.672017
$630$	0	0
$631$	−43.1246	−1.71676	−0.858382	−	0.513011i	$-0.828530\pi$
$631$	−43.1246	−1.71676	−0.858382	+	0.513011i	$0.828530\pi$
$632$	0	0
$633$	−8.00000	−0.317971
$634$	6.94427	0.275792
$635$	0	0
$636$	−8.56231	−0.339518
$637$	−14.5623	−0.576980
$638$	7.23607	0.286479
$639$	−14.1803	−0.560966
$640$	0	0
$641$	−26.9443	−1.06423	−0.532117	−	0.846671i	$-0.678603\pi$
$641$	−26.9443	−1.06423	−0.532117	+	0.846671i	$0.678603\pi$
$642$	−10.9443	−0.431936
$643$	39.7771	1.56866	0.784328	−	0.620347i	$-0.213008\pi$
$643$	39.7771	1.56866	0.784328	+	0.620347i	$0.213008\pi$
$644$	12.0000	0.472866
$645$	0	0
$646$	21.7082	0.854098
$647$	35.8885	1.41092	0.705462	−	0.708748i	$-0.250739\pi$
$647$	35.8885	1.41092	0.705462	+	0.708748i	$0.250739\pi$
$648$	1.00000	0.0392837
$649$	−46.8328	−1.83835
$650$	0	0
$651$	−7.41641	−0.290672
$652$	−20.4721	−0.801751
$653$	32.2148	1.26066	0.630331	−	0.776327i	$-0.282919\pi$
$653$	32.2148	1.26066	0.630331	+	0.776327i	$0.282919\pi$
$654$	2.56231	0.100194
$655$	0	0
$656$	−6.09017	−0.237781
$657$	3.14590	0.122733
$658$	9.52786	0.371435
$659$	38.9443	1.51705	0.758527	−	0.651642i	$-0.225920\pi$
$659$	38.9443	1.51705	0.758527	+	0.651642i	$0.225920\pi$
$660$	0	0
$661$	−23.5279	−0.915128	−0.457564	−	0.889177i	$-0.651278\pi$
$661$	−23.5279	−0.915128	−0.457564	+	0.889177i	$0.651278\pi$
$662$	−32.4721	−1.26207
$663$	−38.1246	−1.48064
$664$	−6.00000	−0.232845
$665$	0	0
$666$	2.14590	0.0831519
$667$	8.29180	0.321060
$668$	−6.47214	−0.250414
$669$	2.29180	0.0886060
$670$	0	0
$671$	46.3607	1.78973
$672$	−2.00000	−0.0771517
$673$	18.7984	0.724624	0.362312	−	0.932057i	$-0.381987\pi$
$673$	18.7984	0.724624	0.362312	+	0.932057i	$0.381987\pi$
$674$	−28.8328	−1.11060
$675$	0	0
$676$	10.5623	0.406243
$677$	−36.4721	−1.40174	−0.700869	−	0.713290i	$-0.747204\pi$
$677$	−36.4721	−1.40174	−0.700869	+	0.713290i	$0.747204\pi$
$678$	11.5623	0.444048
$679$	−27.7082	−1.06334
$680$	0	0
$681$	16.9443	0.649306
$682$	−19.4164	−0.743493
$683$	−19.4164	−0.742948	−0.371474	−	0.928443i	$-0.621148\pi$
$683$	−19.4164	−0.742948	−0.371474	+	0.928443i	$0.621148\pi$
$684$	−2.76393	−0.105682
$685$	0	0
$686$	20.0000	0.763604
$687$	−24.2705	−0.925978
$688$	1.23607	0.0471246
$689$	−41.5623	−1.58340
$690$	0	0
$691$	−24.1803	−0.919863	−0.459932	−	0.887954i	$-0.652126\pi$
$691$	−24.1803	−0.919863	−0.459932	+	0.887954i	$0.652126\pi$
$692$	10.9098	0.414730
$693$	10.4721	0.397804
$694$	5.23607	0.198758
$695$	0	0
$696$	−1.38197	−0.0523833
$697$	47.8328	1.81180
$698$	14.7984	0.560127
$699$	22.0902	0.835527
$700$	0	0
$701$	−20.1591	−0.761397	−0.380698	−	0.924699i	$-0.624316\pi$
$701$	−20.1591	−0.761397	−0.380698	+	0.924699i	$0.624316\pi$
$702$	4.85410	0.183206
$703$	−5.93112	−0.223696
$704$	−5.23607	−0.197342
$705$	0	0
$706$	−20.4721	−0.770479
$707$	−3.34752	−0.125897
$708$	8.94427	0.336146
$709$	−29.7984	−1.11910	−0.559551	−	0.828796i	$-0.689026\pi$
$709$	−29.7984	−1.11910	−0.559551	+	0.828796i	$0.689026\pi$
$710$	0	0
$711$	0	0
$712$	1.38197	0.0517914
$713$	−22.2492	−0.833240
$714$	15.7082	0.587865
$715$	0	0
$716$	−3.81966	−0.142747
$717$	26.8328	1.00209
$718$	−27.8885	−1.04079
$719$	22.3607	0.833913	0.416956	−	0.908927i	$-0.363097\pi$
$719$	22.3607	0.833913	0.416956	+	0.908927i	$0.363097\pi$
$720$	0	0
$721$	−4.58359	−0.170702
$722$	−11.3607	−0.422801
$723$	−13.8541	−0.515240
$724$	−20.0344	−0.744574
$725$	0	0
$726$	16.4164	0.609270
$727$	−10.2918	−0.381702	−0.190851	−	0.981619i	$-0.561125\pi$
$727$	−10.2918	−0.381702	−0.190851	+	0.981619i	$0.561125\pi$
$728$	−9.70820	−0.359810
$729$	1.00000	0.0370370
$730$	0	0
$731$	−9.70820	−0.359071
$732$	−8.85410	−0.327257
$733$	6.36068	0.234937	0.117469	−	0.993077i	$-0.462522\pi$
$733$	6.36068	0.234937	0.117469	+	0.993077i	$0.462522\pi$
$734$	12.4721	0.460355
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−50.8328	−1.87245
$738$	−6.09017	−0.224182
$739$	8.29180	0.305019	0.152509	−	0.988302i	$-0.451265\pi$
$739$	8.29180	0.305019	0.152509	+	0.988302i	$0.451265\pi$
$740$	0	0
$741$	−13.4164	−0.492864
$742$	17.1246	0.628664
$743$	−6.65248	−0.244056	−0.122028	−	0.992527i	$-0.538940\pi$
$743$	−6.65248	−0.244056	−0.122028	+	0.992527i	$0.538940\pi$
$744$	3.70820	0.135949
$745$	0	0
$746$	−2.58359	−0.0945920
$747$	−6.00000	−0.219529
$748$	41.1246	1.50366
$749$	21.8885	0.799790
$750$	0	0
$751$	14.7639	0.538744	0.269372	−	0.963036i	$-0.413184\pi$
$751$	14.7639	0.538744	0.269372	+	0.963036i	$0.413184\pi$
$752$	−4.76393	−0.173723
$753$	−12.4721	−0.454510
$754$	−6.70820	−0.244298
$755$	0	0
$756$	−2.00000	−0.0727393
$757$	28.8541	1.04872	0.524360	−	0.851497i	$-0.324305\pi$
$757$	28.8541	1.04872	0.524360	+	0.851497i	$0.324305\pi$
$758$	−3.41641	−0.124090
$759$	31.4164	1.14034
$760$	0	0
$761$	24.1591	0.875765	0.437883	−	0.899032i	$-0.355729\pi$
$761$	24.1591	0.875765	0.437883	+	0.899032i	$0.355729\pi$
$762$	−0.291796	−0.0105707
$763$	−5.12461	−0.185523
$764$	−0.763932	−0.0276381
$765$	0	0
$766$	−20.4721	−0.739688
$767$	43.4164	1.56768
$768$	1.00000	0.0360844
$769$	−2.36068	−0.0851283	−0.0425641	−	0.999094i	$-0.513553\pi$
$769$	−2.36068	−0.0851283	−0.0425641	+	0.999094i	$0.513553\pi$
$770$	0	0
$771$	11.6180	0.418413
$772$	−2.38197	−0.0857288
$773$	−14.0902	−0.506788	−0.253394	−	0.967363i	$-0.581547\pi$
$773$	−14.0902	−0.506788	−0.253394	+	0.967363i	$0.581547\pi$
$774$	1.23607	0.0444295
$775$	0	0
$776$	13.8541	0.497333
$777$	−4.29180	−0.153967
$778$	15.9787	0.572865
$779$	16.8328	0.603098
$780$	0	0
$781$	74.2492	2.65685
$782$	47.1246	1.68517
$783$	−1.38197	−0.0493874
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−0.763932	−0.0272486
$787$	−30.5410	−1.08867	−0.544335	−	0.838868i	$-0.683218\pi$
$787$	−30.5410	−1.08867	−0.544335	+	0.838868i	$0.683218\pi$
$788$	13.3262	0.474728
$789$	−0.472136	−0.0168085
$790$	0	0
$791$	−23.1246	−0.822217
$792$	−5.23607	−0.186056
$793$	−42.9787	−1.52622
$794$	−28.8328	−1.02324
$795$	0	0
$796$	−6.18034	−0.219056
$797$	13.3262	0.472040	0.236020	−	0.971748i	$-0.424157\pi$
$797$	13.3262	0.472040	0.236020	+	0.971748i	$0.424157\pi$
$798$	5.52786	0.195684
$799$	37.4164	1.32370
$800$	0	0
$801$	1.38197	0.0488294
$802$	−26.0902	−0.921276
$803$	−16.4721	−0.581289
$804$	9.70820	0.342382
$805$	0	0
$806$	18.0000	0.634023
$807$	−1.90983	−0.0672292
$808$	1.67376	0.0588827
$809$	8.21478	0.288816	0.144408	−	0.989518i	$-0.453872\pi$
$809$	8.21478	0.288816	0.144408	+	0.989518i	$0.453872\pi$
$810$	0	0
$811$	10.2918	0.361394	0.180697	−	0.983539i	$-0.442165\pi$
$811$	10.2918	0.361394	0.180697	+	0.983539i	$0.442165\pi$
$812$	2.76393	0.0969950
$813$	28.1803	0.988328
$814$	−11.2361	−0.393824
$815$	0	0
$816$	−7.85410	−0.274949
$817$	−3.41641	−0.119525
$818$	20.9787	0.733504
$819$	−9.70820	−0.339232
$820$	0	0
$821$	−44.8328	−1.56468	−0.782338	−	0.622854i	$-0.785973\pi$
$821$	−44.8328	−1.56468	−0.782338	+	0.622854i	$0.785973\pi$
$822$	15.0344	0.524386
$823$	23.5967	0.822531	0.411265	−	0.911516i	$-0.365087\pi$
$823$	23.5967	0.822531	0.411265	+	0.911516i	$0.365087\pi$
$824$	2.29180	0.0798385
$825$	0	0
$826$	−17.8885	−0.622422
$827$	−14.7639	−0.513392	−0.256696	−	0.966492i	$-0.582634\pi$
$827$	−14.7639	−0.513392	−0.256696	+	0.966492i	$0.582634\pi$
$828$	−6.00000	−0.208514
$829$	39.1459	1.35959	0.679797	−	0.733401i	$-0.262068\pi$
$829$	39.1459	1.35959	0.679797	+	0.733401i	$0.262068\pi$
$830$	0	0
$831$	22.1459	0.768233
$832$	4.85410	0.168286
$833$	23.5623	0.816386
$834$	−13.4164	−0.464572
$835$	0	0
$836$	14.4721	0.500529
$837$	3.70820	0.128174
$838$	−37.8885	−1.30884
$839$	−5.12461	−0.176921	−0.0884606	−	0.996080i	$-0.528195\pi$
$839$	−5.12461	−0.176921	−0.0884606	+	0.996080i	$0.528195\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	−11.0902	−0.382192
$843$	−6.09017	−0.209757
$844$	−8.00000	−0.275371
$845$	0	0
$846$	−4.76393	−0.163787
$847$	−32.8328	−1.12815
$848$	−8.56231	−0.294031
$849$	−3.23607	−0.111062
$850$	0	0
$851$	−12.8754	−0.441363
$852$	−14.1803	−0.485810
$853$	−16.4508	−0.563266	−0.281633	−	0.959522i	$-0.590876\pi$
$853$	−16.4508	−0.563266	−0.281633	+	0.959522i	$0.590876\pi$
$854$	17.7082	0.605962
$855$	0	0
$856$	−10.9443	−0.374068
$857$	−23.3050	−0.796082	−0.398041	−	0.917368i	$-0.630310\pi$
$857$	−23.3050	−0.796082	−0.398041	+	0.917368i	$0.630310\pi$
$858$	−25.4164	−0.867702
$859$	45.5279	1.55339	0.776695	−	0.629876i	$-0.216894\pi$
$859$	45.5279	1.55339	0.776695	+	0.629876i	$0.216894\pi$
$860$	0	0
$861$	12.1803	0.415105
$862$	3.70820	0.126302
$863$	−12.1803	−0.414624	−0.207312	−	0.978275i	$-0.566471\pi$
$863$	−12.1803	−0.414624	−0.207312	+	0.978275i	$0.566471\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	23.1459	0.786530
$867$	44.6869	1.51765
$868$	−7.41641	−0.251729
$869$	0	0
$870$	0	0
$871$	47.1246	1.59676
$872$	2.56231	0.0867706
$873$	13.8541	0.468890
$874$	16.5836	0.560948
$875$	0	0
$876$	3.14590	0.106290
$877$	−17.9787	−0.607098	−0.303549	−	0.952816i	$-0.598172\pi$
$877$	−17.9787	−0.607098	−0.303549	+	0.952816i	$0.598172\pi$
$878$	−25.1246	−0.847915
$879$	28.7984	0.971345
$880$	0	0
$881$	46.7214	1.57408	0.787041	−	0.616900i	$-0.211612\pi$
$881$	46.7214	1.57408	0.787041	+	0.616900i	$0.211612\pi$
$882$	−3.00000	−0.101015
$883$	−42.1803	−1.41948	−0.709741	−	0.704463i	$-0.751188\pi$
$883$	−42.1803	−1.41948	−0.709741	+	0.704463i	$0.751188\pi$
$884$	−38.1246	−1.28227
$885$	0	0
$886$	−30.0689	−1.01018
$887$	−25.8197	−0.866939	−0.433470	−	0.901168i	$-0.642711\pi$
$887$	−25.8197	−0.866939	−0.433470	+	0.901168i	$0.642711\pi$
$888$	2.14590	0.0720116
$889$	0.583592	0.0195731
$890$	0	0
$891$	−5.23607	−0.175415
$892$	2.29180	0.0767350
$893$	13.1672	0.440623
$894$	7.03444	0.235267
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	−29.1246	−0.972442
$898$	4.79837	0.160124
$899$	−5.12461	−0.170915
$900$	0	0
$901$	67.2492	2.24040
$902$	31.8885	1.06177
$903$	−2.47214	−0.0822675
$904$	11.5623	0.384557
$905$	0	0
$906$	−6.94427	−0.230708
$907$	−20.2918	−0.673778	−0.336889	−	0.941544i	$-0.609375\pi$
$907$	−20.2918	−0.673778	−0.336889	+	0.941544i	$0.609375\pi$
$908$	16.9443	0.562315
$909$	1.67376	0.0555152
$910$	0	0
$911$	56.0689	1.85764	0.928822	−	0.370525i	$-0.120822\pi$
$911$	56.0689	1.85764	0.928822	+	0.370525i	$0.120822\pi$
$912$	−2.76393	−0.0915229
$913$	31.4164	1.03973
$914$	12.4721	0.412542
$915$	0	0
$916$	−24.2705	−0.801920
$917$	1.52786	0.0504545
$918$	−7.85410	−0.259224
$919$	−1.05573	−0.0348253	−0.0174126	−	0.999848i	$-0.505543\pi$
$919$	−1.05573	−0.0348253	−0.0174126	+	0.999848i	$0.505543\pi$
$920$	0	0
$921$	16.2918	0.536833
$922$	25.0902	0.826301
$923$	−68.8328	−2.26566
$924$	10.4721	0.344508
$925$	0	0
$926$	12.2918	0.403933
$927$	2.29180	0.0752725
$928$	−1.38197	−0.0453653
$929$	−30.9787	−1.01638	−0.508189	−	0.861245i	$-0.669685\pi$
$929$	−30.9787	−1.01638	−0.508189	+	0.861245i	$0.669685\pi$
$930$	0	0
$931$	8.29180	0.271753
$932$	22.0902	0.723588
$933$	−18.6525	−0.610655
$934$	−15.8197	−0.517635
$935$	0	0
$936$	4.85410	0.158661
$937$	−35.0902	−1.14635	−0.573173	−	0.819434i	$-0.694288\pi$
$937$	−35.0902	−1.14635	−0.573173	+	0.819434i	$0.694288\pi$
$938$	−19.4164	−0.633968
$939$	26.3607	0.860248
$940$	0	0
$941$	21.6738	0.706544	0.353272	−	0.935521i	$-0.385069\pi$
$941$	21.6738	0.706544	0.353272	+	0.935521i	$0.385069\pi$
$942$	−17.8541	−0.581718
$943$	36.5410	1.18994
$944$	8.94427	0.291111
$945$	0	0
$946$	−6.47214	−0.210427
$947$	−10.9443	−0.355641	−0.177821	−	0.984063i	$-0.556905\pi$
$947$	−10.9443	−0.355641	−0.177821	+	0.984063i	$0.556905\pi$
$948$	0	0
$949$	15.2705	0.495702
$950$	0	0
$951$	6.94427	0.225183
$952$	15.7082	0.509106
$953$	32.8673	1.06467	0.532337	−	0.846532i	$-0.321314\pi$
$953$	32.8673	1.06467	0.532337	+	0.846532i	$0.321314\pi$
$954$	−8.56231	−0.277215
$955$	0	0
$956$	26.8328	0.867835
$957$	7.23607	0.233909
$958$	4.47214	0.144488
$959$	−30.0689	−0.970975
$960$	0	0
$961$	−17.2492	−0.556427
$962$	10.4164	0.335838
$963$	−10.9443	−0.352674
$964$	−13.8541	−0.446211
$965$	0	0
$966$	12.0000	0.386094
$967$	−40.9443	−1.31668	−0.658340	−	0.752721i	$-0.728741\pi$
$967$	−40.9443	−1.31668	−0.658340	+	0.752721i	$0.728741\pi$
$968$	16.4164	0.527643
$969$	21.7082	0.697368
$970$	0	0
$971$	−33.7771	−1.08396	−0.541979	−	0.840392i	$-0.682325\pi$
$971$	−33.7771	−1.08396	−0.541979	+	0.840392i	$0.682325\pi$
$972$	1.00000	0.0320750
$973$	26.8328	0.860221
$974$	−10.2918	−0.329770
$975$	0	0
$976$	−8.85410	−0.283413
$977$	20.5623	0.657846	0.328923	−	0.944357i	$-0.393314\pi$
$977$	20.5623	0.657846	0.328923	+	0.944357i	$0.393314\pi$
$978$	−20.4721	−0.654627
$979$	−7.23607	−0.231266
$980$	0	0
$981$	2.56231	0.0818081
$982$	29.8885	0.953782
$983$	42.5410	1.35685	0.678424	−	0.734671i	$-0.262663\pi$
$983$	42.5410	1.35685	0.678424	+	0.734671i	$0.262663\pi$
$984$	−6.09017	−0.194148
$985$	0	0
$986$	10.8541	0.345665
$987$	9.52786	0.303275
$988$	−13.4164	−0.426833
$989$	−7.41641	−0.235828
$990$	0	0
$991$	60.5410	1.92315	0.961574	−	0.274544i	$-0.0885270\pi$
$991$	60.5410	1.92315	0.961574	+	0.274544i	$0.0885270\pi$
$992$	3.70820	0.117736
$993$	−32.4721	−1.03047
$994$	28.3607	0.899546
$995$	0	0
$996$	−6.00000	−0.190117
$997$	1.41641	0.0448581	0.0224290	−	0.999748i	$-0.492860\pi$
$997$	1.41641	0.0448581	0.0224290	+	0.999748i	$0.492860\pi$
$998$	−33.4164	−1.05778
$999$	2.14590	0.0678932

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3750.2.a.g.1.1		2
5.2	odd	4		3750.2.c.c.1249.3		4
5.3	odd	4		3750.2.c.c.1249.1		4
5.4	even	2		3750.2.a.b.1.1		2
25.2	odd	20		750.2.h.a.649.2		8
25.9	even	10		150.2.g.b.31.1	✓	4
25.11	even	5		750.2.g.a.601.1		4
25.12	odd	20		750.2.h.a.349.1		8
25.13	odd	20		750.2.h.a.349.2		8
25.14	even	10		150.2.g.b.121.1	yes	4
25.16	even	5		750.2.g.a.151.1		4
25.23	odd	20		750.2.h.a.649.1		8
75.14	odd	10		450.2.h.b.271.1		4
75.59	odd	10		450.2.h.b.181.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
150.2.g.b.31.1	✓	4	25.9	even	10
150.2.g.b.121.1	yes	4	25.14	even	10
450.2.h.b.181.1		4	75.59	odd	10
450.2.h.b.271.1		4	75.14	odd	10
750.2.g.a.151.1		4	25.16	even	5
750.2.g.a.601.1		4	25.11	even	5
750.2.h.a.349.1		8	25.12	odd	20
750.2.h.a.349.2		8	25.13	odd	20
750.2.h.a.649.1		8	25.23	odd	20
750.2.h.a.649.2		8	25.2	odd	20
3750.2.a.b.1.1		2	5.4	even	2
3750.2.a.g.1.1		2	1.1	even	1	trivial
3750.2.c.c.1249.1		4	5.3	odd	4
3750.2.c.c.1249.3		4	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -5.23607 q^{11} +1.00000 q^{12} +4.85410 q^{13} -2.00000 q^{14} +1.00000 q^{16} -7.85410 q^{17} +1.00000 q^{18} -2.76393 q^{19} -2.00000 q^{21} -5.23607 q^{22} -6.00000 q^{23} +1.00000 q^{24} +4.85410 q^{26} +1.00000 q^{27} -2.00000 q^{28} -1.38197 q^{29} +3.70820 q^{31} +1.00000 q^{32} -5.23607 q^{33} -7.85410 q^{34} +1.00000 q^{36} +2.14590 q^{37} -2.76393 q^{38} +4.85410 q^{39} -6.09017 q^{41} -2.00000 q^{42} +1.23607 q^{43} -5.23607 q^{44} -6.00000 q^{46} -4.76393 q^{47} +1.00000 q^{48} -3.00000 q^{49} -7.85410 q^{51} +4.85410 q^{52} -8.56231 q^{53} +1.00000 q^{54} -2.00000 q^{56} -2.76393 q^{57} -1.38197 q^{58} +8.94427 q^{59} -8.85410 q^{61} +3.70820 q^{62} -2.00000 q^{63} +1.00000 q^{64} -5.23607 q^{66} +9.70820 q^{67} -7.85410 q^{68} -6.00000 q^{69} -14.1803 q^{71} +1.00000 q^{72} +3.14590 q^{73} +2.14590 q^{74} -2.76393 q^{76} +10.4721 q^{77} +4.85410 q^{78} +1.00000 q^{81} -6.09017 q^{82} -6.00000 q^{83} -2.00000 q^{84} +1.23607 q^{86} -1.38197 q^{87} -5.23607 q^{88} +1.38197 q^{89} -9.70820 q^{91} -6.00000 q^{92} +3.70820 q^{93} -4.76393 q^{94} +1.00000 q^{96} +13.8541 q^{97} -3.00000 q^{98} -5.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 4 q^{7} + 2 q^{8} + 2 q^{9} - 6 q^{11} + 2 q^{12} + 3 q^{13} - 4 q^{14} + 2 q^{16} - 9 q^{17} + 2 q^{18} - 10 q^{19} - 4 q^{21} - 6 q^{22} - 12 q^{23} + 2 q^{24}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 4 * q^7 + 2 * q^8 + 2 * q^9 - 6 * q^11 + 2 * q^12 + 3 * q^13 - 4 * q^14 + 2 * q^16 - 9 * q^17 + 2 * q^18 - 10 * q^19 - 4 * q^21 - 6 * q^22 - 12 * q^23 + 2 * q^24 + 3 * q^26 + 2 * q^27 - 4 * q^28 - 5 * q^29 - 6 * q^31 + 2 * q^32 - 6 * q^33 - 9 * q^34 + 2 * q^36 + 11 * q^37 - 10 * q^38 + 3 * q^39 - q^41 - 4 * q^42 - 2 * q^43 - 6 * q^44 - 12 * q^46 - 14 * q^47 + 2 * q^48 - 6 * q^49 - 9 * q^51 + 3 * q^52 + 3 * q^53 + 2 * q^54 - 4 * q^56 - 10 * q^57 - 5 * q^58 - 11 * q^61 - 6 * q^62 - 4 * q^63 + 2 * q^64 - 6 * q^66 + 6 * q^67 - 9 * q^68 - 12 * q^69 - 6 * q^71 + 2 * q^72 + 13 * q^73 + 11 * q^74 - 10 * q^76 + 12 * q^77 + 3 * q^78 + 2 * q^81 - q^82 - 12 * q^83 - 4 * q^84 - 2 * q^86 - 5 * q^87 - 6 * q^88 + 5 * q^89 - 6 * q^91 - 12 * q^92 - 6 * q^93 - 14 * q^94 + 2 * q^96 + 21 * q^97 - 6 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more