LMFDB - Embedded newform 725.2.b.f.349.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [725,2,Mod(349,725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("725.349"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-8,0,-2,0,0,-22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.59416223908864.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 14x^{8} + 71x^{6} + 159x^{4} + 151x^{2} + 49 $ x^10 + 14x^8 + 71x^6 + 159x^4 + 151x^2 + 49
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.2
Root			$-1.88481i$ of defining polynomial
Character	$\chi$	$=$	725.349
Dual form			725.2.b.f.349.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.88481i q^{2} -1.10085i q^{3} -1.55251 q^{4} -2.07489 q^{6} +1.70908i q^{7} -0.843434i q^{8} +1.78814 q^{9} -2.85130 q^{11} +1.70908i q^{12} -6.22128i q^{13} +3.22128 q^{14} -4.69473 q^{16} -1.40381i q^{17} -3.37030i q^{18} -6.74028 q^{19} +1.88143 q^{21} +5.37416i q^{22} -1.42153i q^{23} -0.928492 q^{24} -11.7259 q^{26} -5.27100i q^{27} -2.65336i q^{28} -1.00000 q^{29} -1.29510 q^{31} +7.16181i q^{32} +3.13884i q^{33} -2.64591 q^{34} -2.77610 q^{36} -0.989830i q^{37} +12.7042i q^{38} -6.84868 q^{39} +6.32213 q^{41} -3.54614i q^{42} -6.03720i q^{43} +4.42667 q^{44} -2.67932 q^{46} +8.97549i q^{47} +5.16818i q^{48} +4.07906 q^{49} -1.54538 q^{51} +9.65860i q^{52} -2.11288i q^{53} -9.93484 q^{54} +1.44149 q^{56} +7.42002i q^{57} +1.88481i q^{58} +10.0504 q^{59} +5.73236 q^{61} +2.44101i q^{62} +3.05606i q^{63} +4.10919 q^{64} +5.91612 q^{66} -2.45274i q^{67} +2.17943i q^{68} -1.56489 q^{69} +5.04282 q^{71} -1.50817i q^{72} -5.55668i q^{73} -1.86564 q^{74} +10.4644 q^{76} -4.87309i q^{77} +12.9085i q^{78} -9.37892 q^{79} -0.438162 q^{81} -11.9160i q^{82} -5.06744i q^{83} -2.92094 q^{84} -11.3790 q^{86} +1.10085i q^{87} +2.40488i q^{88} +1.75693 q^{89} +10.6326 q^{91} +2.20694i q^{92} +1.42570i q^{93} +16.9171 q^{94} +7.88406 q^{96} +14.6037i q^{97} -7.68825i q^{98} -5.09851 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 8 q^{4} - 2 q^{6} - 22 q^{9} - 4 q^{11} - 22 q^{14} - 20 q^{16} - 4 q^{19} - 2 q^{21} - 26 q^{24} - 32 q^{26} - 10 q^{29} - 2 q^{31} - 6 q^{34} - 26 q^{36} + 12 q^{39} + 10 q^{41} - 14 q^{44} - 40 q^{46}+ \cdots + 52 q^{99}+O(q^{100}) $ 10 * q - 8 * q^4 - 2 * q^6 - 22 * q^9 - 4 * q^11 - 22 * q^14 - 20 * q^16 - 4 * q^19 - 2 * q^21 - 26 * q^24 - 32 * q^26 - 10 * q^29 - 2 * q^31 - 6 * q^34 - 26 * q^36 + 12 * q^39 + 10 * q^41 - 14 * q^44 - 40 * q^46 + 6 * q^49 + 10 * q^51 - 20 * q^54 - 32 * q^56 + 22 * q^59 - 10 * q^61 + 10 * q^64 - 54 * q^66 + 52 * q^69 - 10 * q^71 - 16 * q^74 + 32 * q^76 + 20 * q^79 + 98 * q^81 - 64 * q^84 - 26 * q^86 + 36 * q^89 + 2 * q^91 + 26 * q^94 - 76 * q^96 + 52 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$.

$n$	$176$	$552$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 1.88481i	− 1.33276i	−0.745611	−	0.666381i	$-0.767842\pi$
$2$	− 1.88481i	− 1.33276i	0.745611	−	0.666381i	$-0.232158\pi$
$3$	− 1.10085i	− 0.635574i	−0.948162	−	0.317787i	$-0.897060\pi$
$3$	− 1.10085i	− 0.635574i	0.948162	−	0.317787i	$-0.102940\pi$
$4$	−1.55251	−0.776255
$5$	0	0
$5$	0	0
$6$	−2.07489	−0.847069
$7$	1.70908i	0.645970i	0.946404	+	0.322985i	$0.104686\pi$
$7$	1.70908i	0.645970i	−0.946404	+	0.322985i	$0.895314\pi$
$8$	− 0.843434i	− 0.298199i
$9$	1.78814	0.596045
$10$	0	0
$11$	−2.85130	−0.859699	−0.429849	−	0.902901i	$-0.641433\pi$
$11$	−2.85130	−0.859699	−0.429849	+	0.902901i	$0.641433\pi$
$12$	1.70908i	0.493368i
$13$	− 6.22128i	− 1.72547i	−0.505653	−	0.862737i	$-0.668748\pi$
$13$	− 6.22128i	− 1.72547i	0.505653	−	0.862737i	$-0.331252\pi$
$14$	3.22128	0.860924
$15$	0	0
$16$	−4.69473	−1.17368
$17$	− 1.40381i	− 0.340474i	−0.985403	−	0.170237i	$-0.945547\pi$
$17$	− 1.40381i	− 0.340474i	0.985403	−	0.170237i	$-0.0544533\pi$
$18$	− 3.37030i	− 0.794387i
$19$	−6.74028	−1.54633	−0.773163	−	0.634207i	$-0.781327\pi$
$19$	−6.74028	−1.54633	−0.773163	+	0.634207i	$0.781327\pi$
$20$	0	0
$21$	1.88143	0.410562
$22$	5.37416i	1.14577i
$23$	− 1.42153i	− 0.296410i	−0.988957	−	0.148205i	$-0.952651\pi$
$23$	− 1.42153i	− 0.296410i	0.988957	−	0.148205i	$-0.0473495\pi$
$24$	−0.928492	−0.189528
$25$	0	0
$26$	−11.7259	−2.29965
$27$	− 5.27100i	− 1.01441i
$28$	− 2.65336i	− 0.501437i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−1.29510	−0.232606	−0.116303	−	0.993214i	$-0.537104\pi$
$31$	−1.29510	−0.232606	−0.116303	+	0.993214i	$0.537104\pi$
$32$	7.16181i	1.26604i
$33$	3.13884i	0.546403i
$34$	−2.64591	−0.453770
$35$	0	0
$36$	−2.77610	−0.462683
$37$	− 0.989830i	− 0.162727i	−0.996684	−	0.0813635i	$-0.974073\pi$
$37$	− 0.989830i	− 0.162727i	0.996684	−	0.0813635i	$-0.0259275\pi$
$38$	12.7042i	2.06089i
$39$	−6.84868	−1.09667
$40$	0	0
$41$	6.32213	0.987351	0.493675	−	0.869646i	$-0.335653\pi$
$41$	6.32213	0.987351	0.493675	+	0.869646i	$0.335653\pi$
$42$	− 3.54614i	− 0.547181i
$43$	− 6.03720i	− 0.920665i	−0.887747	−	0.460332i	$-0.847730\pi$
$43$	− 6.03720i	− 0.920665i	0.887747	−	0.460332i	$-0.152270\pi$
$44$	4.42667	0.667346
$45$	0	0
$46$	−2.67932	−0.395044
$47$	8.97549i	1.30921i	0.755971	+	0.654605i	$0.227165\pi$
$47$	8.97549i	1.30921i	−0.755971	+	0.654605i	$0.772835\pi$
$48$	5.16818i	0.745963i
$49$	4.07906	0.582723
$50$	0	0
$51$	−1.54538	−0.216396
$52$	9.65860i	1.33941i
$53$	− 2.11288i	− 0.290227i	−0.989415	−	0.145113i	$-0.953645\pi$
$53$	− 2.11288i	− 0.290227i	0.989415	−	0.145113i	$-0.0463547\pi$
$54$	−9.93484	−1.35196
$55$	0	0
$56$	1.44149	0.192628
$57$	7.42002i	0.982805i
$58$	1.88481i	0.247488i
$59$	10.0504	1.30845	0.654224	−	0.756301i	$-0.272995\pi$
$59$	10.0504	1.30845	0.654224	+	0.756301i	$0.272995\pi$
$60$	0	0
$61$	5.73236	0.733953	0.366977	−	0.930230i	$-0.380393\pi$
$61$	5.73236	0.733953	0.366977	+	0.930230i	$0.380393\pi$
$62$	2.44101i	0.310009i
$63$	3.05606i	0.385027i
$64$	4.10919	0.513649
$65$	0	0
$66$	5.91612	0.728225
$67$	− 2.45274i	− 0.299650i	−0.988713	−	0.149825i	$-0.952129\pi$
$67$	− 2.45274i	− 0.299650i	0.988713	−	0.149825i	$-0.0478710\pi$
$68$	2.17943i	0.264294i
$69$	−1.56489	−0.188390
$70$	0	0
$71$	5.04282	0.598473	0.299236	−	0.954179i	$-0.403268\pi$
$71$	5.04282	0.598473	0.299236	+	0.954179i	$0.403268\pi$
$72$	− 1.50817i	− 0.177740i
$73$	− 5.55668i	− 0.650360i	−0.945652	−	0.325180i	$-0.894575\pi$
$73$	− 5.55668i	− 0.650360i	0.945652	−	0.325180i	$-0.105425\pi$
$74$	−1.86564	−0.216876
$75$	0	0
$76$	10.4644	1.20034
$77$	− 4.87309i	− 0.555340i
$78$	12.9085i	1.46160i
$79$	−9.37892	−1.05521	−0.527606	−	0.849489i	$-0.676910\pi$
$79$	−9.37892	−1.05521	−0.527606	+	0.849489i	$0.676910\pi$
$80$	0	0
$81$	−0.438162	−0.0486846
$82$	− 11.9160i	− 1.31590i
$83$	− 5.06744i	− 0.556224i	−0.960549	−	0.278112i	$-0.910291\pi$
$83$	− 5.06744i	− 0.556224i	0.960549	−	0.278112i	$-0.0897087\pi$
$84$	−2.92094	−0.318701
$85$	0	0
$86$	−11.3790	−1.22703
$87$	1.10085i	0.118023i
$88$	2.40488i	0.256361i
$89$	1.75693	0.186234	0.0931171	−	0.995655i	$-0.470317\pi$
$89$	1.75693	0.186234	0.0931171	+	0.995655i	$0.470317\pi$
$90$	0	0
$91$	10.6326	1.11460
$92$	2.20694i	0.230089i
$93$	1.42570i	0.147839i
$94$	16.9171	1.74486
$95$	0	0
$96$	7.88406	0.804663
$97$	14.6037i	1.48278i	0.671075	+	0.741389i	$0.265833\pi$
$97$	14.6037i	1.48278i	−0.671075	+	0.741389i	$0.734167\pi$
$98$	− 7.68825i	− 0.776631i
$99$	−5.09851	−0.512420
$100$	0	0
$101$	15.8868	1.58079	0.790397	−	0.612595i	$-0.209874\pi$
$101$	15.8868	1.58079	0.790397	+	0.612595i	$0.209874\pi$
$102$	2.91275i	0.288405i
$103$	− 11.6322i	− 1.14616i	−0.819500	−	0.573079i	$-0.805749\pi$
$103$	− 11.6322i	− 1.14616i	0.819500	−	0.573079i	$-0.194251\pi$
$104$	−5.24724	−0.514534
$105$	0	0
$106$	−3.98239	−0.386804
$107$	13.0887i	1.26533i	0.774426	+	0.632664i	$0.218039\pi$
$107$	13.0887i	1.26533i	−0.774426	+	0.632664i	$0.781961\pi$
$108$	8.18329i	0.787437i
$109$	−10.5634	−1.01179	−0.505894	−	0.862596i	$-0.668837\pi$
$109$	−10.5634	−1.01179	−0.505894	+	0.862596i	$0.668837\pi$
$110$	0	0
$111$	−1.08965	−0.103425
$112$	− 8.02365i	− 0.758164i
$113$	18.4952i	1.73988i	0.493157	+	0.869940i	$0.335843\pi$
$113$	18.4952i	1.73988i	−0.493157	+	0.869940i	$0.664157\pi$
$114$	13.9853	1.30985
$115$	0	0
$116$	1.55251	0.144147
$117$	− 11.1245i	− 1.02846i
$118$	− 18.9430i	− 1.74385i
$119$	2.39922	0.219936
$120$	0	0
$121$	−2.87010	−0.260918
$122$	− 10.8044i	− 0.978185i
$123$	− 6.95970i	− 0.627535i
$124$	2.01065	0.180562
$125$	0	0
$126$	5.76009	0.513150
$127$	− 2.95697i	− 0.262389i	−0.991357	−	0.131194i	$-0.958119\pi$
$127$	− 2.95697i	− 0.262389i	0.991357	−	0.131194i	$-0.0418812\pi$
$128$	6.57858i	0.581470i
$129$	−6.64604	−0.585151
$130$	0	0
$131$	−20.9785	−1.83290	−0.916450	−	0.400150i	$-0.868958\pi$
$131$	−20.9785	−1.83290	−0.916450	+	0.400150i	$0.868958\pi$
$132$	− 4.87309i	− 0.424148i
$133$	− 11.5197i	− 0.998880i
$134$	−4.62294	−0.399362
$135$	0	0
$136$	−1.18402	−0.101529
$137$	− 19.5953i	− 1.67414i	−0.547094	−	0.837071i	$-0.684266\pi$
$137$	− 19.5953i	− 1.67414i	0.547094	−	0.837071i	$-0.315734\pi$
$138$	2.94952i	0.251080i
$139$	21.2859	1.80545	0.902725	−	0.430218i	$-0.141563\pi$
$139$	21.2859	1.80545	0.902725	+	0.430218i	$0.141563\pi$
$140$	0	0
$141$	9.88064	0.832100
$142$	− 9.50476i	− 0.797622i
$143$	17.7387i	1.48339i
$144$	−8.39482	−0.699568
$145$	0	0
$146$	−10.4733	−0.866776
$147$	− 4.49042i	− 0.370364i
$148$	1.53672i	0.126318i
$149$	16.6123	1.36093	0.680466	−	0.732779i	$-0.261777\pi$
$149$	16.6123	1.36093	0.680466	+	0.732779i	$0.261777\pi$
$150$	0	0
$151$	8.74718	0.711835	0.355918	−	0.934517i	$-0.384168\pi$
$151$	8.74718	0.711835	0.355918	+	0.934517i	$0.384168\pi$
$152$	5.68498i	0.461113i
$153$	− 2.51020i	− 0.202938i
$154$	−9.18484	−0.740136
$155$	0	0
$156$	10.6326	0.851293
$157$	− 20.0971i	− 1.60392i	−0.597378	−	0.801960i	$-0.703791\pi$
$157$	− 20.0971i	− 1.60392i	0.597378	−	0.801960i	$-0.296209\pi$
$158$	17.6775i	1.40635i
$159$	−2.32596	−0.184461
$160$	0	0
$161$	2.42950	0.191472
$162$	0.825852i	0.0648850i
$163$	3.21100i	0.251505i	0.992062	+	0.125753i	$0.0401346\pi$
$163$	3.21100i	0.251505i	−0.992062	+	0.125753i	$0.959865\pi$
$164$	−9.81517	−0.766436
$165$	0	0
$166$	−9.55117	−0.741315
$167$	6.83823i	0.529158i	0.964364	+	0.264579i	$0.0852330\pi$
$167$	6.83823i	0.529158i	−0.964364	+	0.264579i	$0.914767\pi$
$168$	− 1.58686i	− 0.122429i
$169$	−25.7044	−1.97726
$170$	0	0
$171$	−12.0525	−0.921681
$172$	9.37282i	0.714671i
$173$	− 3.47907i	− 0.264509i	−0.991216	−	0.132254i	$-0.957778\pi$
$173$	− 3.47907i	− 0.264509i	0.991216	−	0.132254i	$-0.0422216\pi$
$174$	2.07489	0.157297
$175$	0	0
$176$	13.3861	1.00901
$177$	− 11.0639i	− 0.831615i
$178$	− 3.31148i	− 0.248206i
$179$	23.3359	1.74421	0.872105	−	0.489319i	$-0.162755\pi$
$179$	23.3359	1.74421	0.872105	+	0.489319i	$0.162755\pi$
$180$	0	0
$181$	−3.34240	−0.248439	−0.124219	−	0.992255i	$-0.539643\pi$
$181$	−3.34240	−0.248439	−0.124219	+	0.992255i	$0.539643\pi$
$182$	− 20.0405i	− 1.48550i
$183$	− 6.31045i	− 0.466482i
$184$	−1.19897	−0.0883891
$185$	0	0
$186$	2.68718	0.197034
$187$	4.00268i	0.292705i
$188$	− 13.9345i	− 1.01628i
$189$	9.00855	0.655275
$190$	0	0
$191$	4.61979	0.334276	0.167138	−	0.985934i	$-0.446547\pi$
$191$	4.61979	0.334276	0.167138	+	0.985934i	$0.446547\pi$
$192$	− 4.52359i	− 0.326462i
$193$	− 24.9023i	− 1.79251i	−0.443542	−	0.896254i	$-0.646278\pi$
$193$	− 24.9023i	− 1.79251i	0.443542	−	0.896254i	$-0.353722\pi$
$194$	27.5252	1.97619
$195$	0	0
$196$	−6.33278	−0.452342
$197$	− 5.96322i	− 0.424862i	−0.977176	−	0.212431i	$-0.931862\pi$
$197$	− 5.96322i	− 0.424862i	0.977176	−	0.212431i	$-0.0681380\pi$
$198$	9.60972i	0.682933i
$199$	19.0033	1.34711	0.673555	−	0.739137i	$-0.264767\pi$
$199$	19.0033	1.34711	0.673555	+	0.739137i	$0.264767\pi$
$200$	0	0
$201$	−2.70009	−0.190450
$202$	− 29.9436i	− 2.10682i
$203$	− 1.70908i	− 0.119954i
$204$	2.39922	0.167979
$205$	0	0
$206$	−21.9245	−1.52755
$207$	− 2.54189i	− 0.176674i
$208$	29.2073i	2.02516i
$209$	19.2186	1.32938
$210$	0	0
$211$	−8.60278	−0.592240	−0.296120	−	0.955151i	$-0.595693\pi$
$211$	−8.60278	−0.592240	−0.296120	+	0.955151i	$0.595693\pi$
$212$	3.28027i	0.225290i
$213$	− 5.55138i	− 0.380374i
$214$	24.6696	1.68638
$215$	0	0
$216$	−4.44575	−0.302495
$217$	− 2.21342i	− 0.150257i
$218$	19.9100i	1.34847i
$219$	−6.11706	−0.413352
$220$	0	0
$221$	−8.73349	−0.587478
$222$	2.05379i	0.137841i
$223$	7.77513i	0.520661i	0.965520	+	0.260331i	$0.0838316\pi$
$223$	7.77513i	0.520661i	−0.965520	+	0.260331i	$0.916168\pi$
$224$	−12.2401	−0.817825
$225$	0	0
$226$	34.8599	2.31885
$227$	− 11.4310i	− 0.758699i	−0.925253	−	0.379349i	$-0.876148\pi$
$227$	− 11.4310i	− 0.758699i	0.925253	−	0.379349i	$-0.123852\pi$
$228$	− 11.5197i	− 0.762908i
$229$	6.14881	0.406325	0.203162	−	0.979145i	$-0.434878\pi$
$229$	6.14881	0.406325	0.203162	+	0.979145i	$0.434878\pi$
$230$	0	0
$231$	−5.36452	−0.352960
$232$	0.843434i	0.0553742i
$233$	0.829745i	0.0543584i	0.999631	+	0.0271792i	$0.00865247\pi$
$233$	0.829745i	0.0543584i	−0.999631	+	0.0271792i	$0.991348\pi$
$234$	−20.9676	−1.37069
$235$	0	0
$236$	−15.6033	−1.01569
$237$	10.3248i	0.670665i
$238$	− 4.52207i	− 0.293122i
$239$	0.473622	0.0306361	0.0153180	−	0.999883i	$-0.495124\pi$
$239$	0.473622	0.0306361	0.0153180	+	0.999883i	$0.495124\pi$
$240$	0	0
$241$	−22.0803	−1.42232	−0.711160	−	0.703030i	$-0.751830\pi$
$241$	−22.0803	−1.42232	−0.711160	+	0.703030i	$0.751830\pi$
$242$	5.40958i	0.347741i
$243$	− 15.3307i	− 0.983463i
$244$	−8.89954	−0.569735
$245$	0	0
$246$	−13.1177	−0.836355
$247$	41.9332i	2.66815i
$248$	1.09233i	0.0693630i
$249$	−5.57848	−0.353522
$250$	0	0
$251$	6.98079	0.440624	0.220312	−	0.975429i	$-0.429292\pi$
$251$	6.98079	0.440624	0.220312	+	0.975429i	$0.429292\pi$
$252$	− 4.74456i	− 0.298879i
$253$	4.05321i	0.254823i
$254$	−5.57333	−0.349702
$255$	0	0
$256$	20.6178	1.28861
$257$	5.35099i	0.333785i	0.985975	+	0.166893i	$0.0533734\pi$
$257$	5.35099i	0.333785i	−0.985975	+	0.166893i	$0.946627\pi$
$258$	12.5265i	0.779867i
$259$	1.69169	0.105117
$260$	0	0
$261$	−1.78814	−0.110683
$262$	39.5405i	2.44282i
$263$	− 8.56128i	− 0.527911i	−0.964535	−	0.263955i	$-0.914973\pi$
$263$	− 8.56128i	− 0.527911i	0.964535	−	0.263955i	$-0.0850272\pi$
$264$	2.64741	0.162937
$265$	0	0
$266$	−21.7124	−1.33127
$267$	− 1.93411i	− 0.118366i
$268$	3.80790i	0.232604i
$269$	7.36998	0.449356	0.224678	−	0.974433i	$-0.427867\pi$
$269$	7.36998	0.449356	0.224678	+	0.974433i	$0.427867\pi$
$270$	0	0
$271$	−5.49008	−0.333499	−0.166749	−	0.985999i	$-0.553327\pi$
$271$	−5.49008	−0.333499	−0.166749	+	0.985999i	$0.553327\pi$
$272$	6.59051i	0.399608i
$273$	− 11.7049i	− 0.708414i
$274$	−36.9335	−2.23123
$275$	0	0
$276$	2.42950	0.146239
$277$	− 24.0013i	− 1.44210i	−0.692882	−	0.721051i	$-0.743659\pi$
$277$	− 24.0013i	− 1.44210i	0.692882	−	0.721051i	$-0.256341\pi$
$278$	− 40.1200i	− 2.40624i
$279$	−2.31581	−0.138644
$280$	0	0
$281$	10.0364	0.598722	0.299361	−	0.954140i	$-0.403227\pi$
$281$	10.0364	0.598722	0.299361	+	0.954140i	$0.403227\pi$
$282$	− 18.6231i	− 1.10899i
$283$	8.41013i	0.499930i	0.968255	+	0.249965i	$0.0804192\pi$
$283$	8.41013i	0.499930i	−0.968255	+	0.249965i	$0.919581\pi$
$284$	−7.82903	−0.464568
$285$	0	0
$286$	33.4342	1.97700
$287$	10.8050i	0.637799i
$288$	12.8063i	0.754618i
$289$	15.0293	0.884078
$290$	0	0
$291$	16.0764	0.942416
$292$	8.62680i	0.504846i
$293$	6.38571i	0.373057i	0.982450	+	0.186529i	$0.0597237\pi$
$293$	6.38571i	0.373057i	−0.982450	+	0.186529i	$0.940276\pi$
$294$	−8.46359	−0.493607
$295$	0	0
$296$	−0.834856	−0.0485250
$297$	15.0292i	0.872083i
$298$	− 31.3110i	− 1.81380i
$299$	−8.84375	−0.511447
$300$	0	0
$301$	10.3180	0.594722
$302$	− 16.4868i	− 0.948707i
$303$	− 17.4889i	− 1.00471i
$304$	31.6438	1.81490
$305$	0	0
$306$	−4.73125	−0.270468
$307$	28.7573i	1.64126i	0.571458	+	0.820632i	$0.306378\pi$
$307$	28.7573i	1.64126i	−0.571458	+	0.820632i	$0.693622\pi$
$308$	7.56551i	0.431085i
$309$	−12.8053	−0.728468
$310$	0	0
$311$	4.20464	0.238423	0.119211	−	0.992869i	$-0.461963\pi$
$311$	4.20464	0.238423	0.119211	+	0.992869i	$0.461963\pi$
$312$	5.77641i	0.327025i
$313$	18.2272i	1.03026i	0.857111	+	0.515132i	$0.172257\pi$
$313$	18.2272i	1.03026i	−0.857111	+	0.515132i	$0.827743\pi$
$314$	−37.8791	−2.13764
$315$	0	0
$316$	14.5609	0.819113
$317$	13.4593i	0.755951i	0.925816	+	0.377975i	$0.123380\pi$
$317$	13.4593i	0.755951i	−0.925816	+	0.377975i	$0.876620\pi$
$318$	4.38400i	0.245842i
$319$	2.85130	0.159642
$320$	0	0
$321$	14.4086	0.804210
$322$	− 4.57915i	− 0.255186i
$323$	9.46207i	0.526483i
$324$	0.680250	0.0377917
$325$	0	0
$326$	6.05214	0.335197
$327$	11.6287i	0.643066i
$328$	− 5.33230i	− 0.294427i
$329$	−15.3398	−0.845710
$330$	0	0
$331$	15.2501	0.838223	0.419112	−	0.907935i	$-0.362342\pi$
$331$	15.2501	0.838223	0.419112	+	0.907935i	$0.362342\pi$
$332$	7.86726i	0.431772i
$333$	− 1.76995i	− 0.0969927i
$334$	12.8888	0.705242
$335$	0	0
$336$	−8.83281	−0.481870
$337$	− 5.07452i	− 0.276426i	−0.990402	−	0.138213i	$-0.955864\pi$
$337$	− 5.07452i	− 0.276426i	0.990402	−	0.138213i	$-0.0441359\pi$
$338$	48.4479i	2.63522i
$339$	20.3604	1.10582
$340$	0	0
$341$	3.69271	0.199971
$342$	22.7168i	1.22838i
$343$	18.9350i	1.02239i
$344$	−5.09198	−0.274541
$345$	0	0
$346$	−6.55738	−0.352527
$347$	28.0386i	1.50519i	0.658482	+	0.752596i	$0.271199\pi$
$347$	28.0386i	1.50519i	−0.658482	+	0.752596i	$0.728801\pi$
$348$	− 1.70908i	− 0.0916161i
$349$	27.7759	1.48681	0.743406	−	0.668840i	$-0.233209\pi$
$349$	27.7759	1.48681	0.743406	+	0.668840i	$0.233209\pi$
$350$	0	0
$351$	−32.7924	−1.75033
$352$	− 20.4205i	− 1.08841i
$353$	31.4223i	1.67244i	0.548392	+	0.836221i	$0.315240\pi$
$353$	31.4223i	1.67244i	−0.548392	+	0.836221i	$0.684760\pi$
$354$	−20.8534	−1.10835
$355$	0	0
$356$	−2.72765	−0.144565
$357$	− 2.64117i	− 0.139785i
$358$	− 43.9838i	− 2.32462i
$359$	−27.0194	−1.42603	−0.713014	−	0.701150i	$-0.752671\pi$
$359$	−27.0194	−1.42603	−0.713014	+	0.701150i	$0.752671\pi$
$360$	0	0
$361$	26.4314	1.39113
$362$	6.29980i	0.331110i
$363$	3.15954i	0.165833i
$364$	−16.5073	−0.865217
$365$	0	0
$366$	−11.8940	−0.621709
$367$	− 20.3243i	− 1.06092i	−0.847710	−	0.530460i	$-0.822019\pi$
$367$	− 20.3243i	− 1.06092i	0.847710	−	0.530460i	$-0.177981\pi$
$368$	6.67371i	0.347891i
$369$	11.3048	0.588506
$370$	0	0
$371$	3.61108	0.187478
$372$	− 2.21342i	− 0.114760i
$373$	− 15.9724i	− 0.827021i	−0.910499	−	0.413511i	$-0.864302\pi$
$373$	− 15.9724i	− 0.827021i	0.910499	−	0.413511i	$-0.135698\pi$
$374$	7.54429	0.390106
$375$	0	0
$376$	7.57023	0.390405
$377$	6.22128i	0.320412i
$378$	− 16.9794i	− 0.873326i
$379$	−29.4840	−1.51449	−0.757247	−	0.653129i	$-0.773456\pi$
$379$	−29.4840	−1.51449	−0.757247	+	0.653129i	$0.773456\pi$
$380$	0	0
$381$	−3.25517	−0.166768
$382$	− 8.70742i	− 0.445510i
$383$	35.1376i	1.79545i	0.440560	+	0.897723i	$0.354780\pi$
$383$	35.1376i	1.79545i	−0.440560	+	0.897723i	$0.645220\pi$
$384$	7.24201	0.369567
$385$	0	0
$386$	−46.9361	−2.38899
$387$	− 10.7953i	− 0.548758i
$388$	− 22.6724i	− 1.15101i
$389$	−28.3776	−1.43880	−0.719401	−	0.694595i	$-0.755583\pi$
$389$	−28.3776	−1.43880	−0.719401	+	0.694595i	$0.755583\pi$
$390$	0	0
$391$	−1.99556	−0.100920
$392$	− 3.44042i	− 0.173767i
$393$	23.0941i	1.16494i
$394$	−11.2395	−0.566240
$395$	0	0
$396$	7.91549	0.397768
$397$	24.5435i	1.23180i	0.787823	+	0.615901i	$0.211208\pi$
$397$	24.5435i	1.23180i	−0.787823	+	0.615901i	$0.788792\pi$
$398$	− 35.8176i	− 1.79538i
$399$	−12.6814	−0.634863
$400$	0	0
$401$	4.95811	0.247596	0.123798	−	0.992307i	$-0.460493\pi$
$401$	4.95811	0.247596	0.123798	+	0.992307i	$0.460493\pi$
$402$	5.08915i	0.253824i
$403$	8.05717i	0.401356i
$404$	−24.6644	−1.22710
$405$	0	0
$406$	−3.22128	−0.159870
$407$	2.82230i	0.139896i
$408$	1.30342i	0.0645291i
$409$	29.6350	1.46535	0.732677	−	0.680576i	$-0.238271\pi$
$409$	29.6350	1.46535	0.732677	+	0.680576i	$0.238271\pi$
$410$	0	0
$411$	−21.5715	−1.06404
$412$	18.0591i	0.889710i
$413$	17.1769i	0.845218i
$414$	−4.79098	−0.235464
$415$	0	0
$416$	44.5557	2.18452
$417$	− 23.4326i	− 1.14750i
$418$	− 36.2233i	− 1.77174i
$419$	−27.1403	−1.32589	−0.662945	−	0.748668i	$-0.730694\pi$
$419$	−27.1403	−1.32589	−0.662945	+	0.748668i	$0.730694\pi$
$420$	0	0
$421$	−27.0310	−1.31741	−0.658705	−	0.752401i	$-0.728896\pi$
$421$	−27.0310	−1.31741	−0.658705	+	0.752401i	$0.728896\pi$
$422$	16.2146i	0.789314i
$423$	16.0494i	0.780348i
$424$	−1.78208	−0.0865454
$425$	0	0
$426$	−10.4633	−0.506948
$427$	9.79703i	0.474112i
$428$	− 20.3203i	− 0.982217i
$429$	19.5276	0.942803
$430$	0	0
$431$	−27.7496	−1.33665	−0.668327	−	0.743868i	$-0.732989\pi$
$431$	−27.7496	−1.33665	−0.668327	+	0.743868i	$0.732989\pi$
$432$	24.7460i	1.19059i
$433$	14.3507i	0.689649i	0.938667	+	0.344825i	$0.112062\pi$
$433$	14.3507i	0.689649i	−0.938667	+	0.344825i	$0.887938\pi$
$434$	−4.17187	−0.200256
$435$	0	0
$436$	16.3997	0.785405
$437$	9.58152i	0.458346i
$438$	11.5295i	0.550900i
$439$	−19.0540	−0.909398	−0.454699	−	0.890645i	$-0.650253\pi$
$439$	−19.0540	−0.909398	−0.454699	+	0.890645i	$0.650253\pi$
$440$	0	0
$441$	7.29391	0.347329
$442$	16.4610i	0.782969i
$443$	− 36.2772i	− 1.72358i	−0.507263	−	0.861791i	$-0.669343\pi$
$443$	− 36.2772i	− 1.72358i	0.507263	−	0.861791i	$-0.330657\pi$
$444$	1.69169	0.0802843
$445$	0	0
$446$	14.6546	0.693918
$447$	− 18.2876i	− 0.864974i
$448$	7.02292i	0.331802i
$449$	−3.04282	−0.143600	−0.0717998	−	0.997419i	$-0.522874\pi$
$449$	−3.04282	−0.143600	−0.0717998	+	0.997419i	$0.522874\pi$
$450$	0	0
$451$	−18.0263	−0.848825
$452$	− 28.7140i	− 1.35059i
$453$	− 9.62931i	− 0.452424i
$454$	−21.5452	−1.01117
$455$	0	0
$456$	6.25830	0.293072
$457$	− 24.8217i	− 1.16111i	−0.814220	−	0.580556i	$-0.802835\pi$
$457$	− 24.8217i	− 1.16111i	0.814220	−	0.580556i	$-0.197165\pi$
$458$	− 11.5893i	− 0.541534i
$459$	−7.39948	−0.345378
$460$	0	0
$461$	−14.1662	−0.659786	−0.329893	−	0.944018i	$-0.607013\pi$
$461$	−14.1662	−0.659786	−0.329893	+	0.944018i	$0.607013\pi$
$462$	10.1111i	0.470411i
$463$	− 9.03538i	− 0.419910i	−0.977711	−	0.209955i	$-0.932668\pi$
$463$	− 9.03538i	− 0.419910i	0.977711	−	0.209955i	$-0.0673317\pi$
$464$	4.69473	0.217948
$465$	0	0
$466$	1.56391	0.0724468
$467$	− 5.28374i	− 0.244503i	−0.992499	−	0.122251i	$-0.960989\pi$
$467$	− 5.28374i	− 0.244503i	0.992499	−	0.122251i	$-0.0390114\pi$
$468$	17.2709i	0.798347i
$469$	4.19191	0.193565
$470$	0	0
$471$	−22.1238	−1.01941
$472$	− 8.47683i	− 0.390178i
$473$	17.2139i	0.791495i
$474$	19.4602	0.893837
$475$	0	0
$476$	−3.72481	−0.170726
$477$	− 3.77812i	− 0.172988i
$478$	− 0.892688i	− 0.0408306i
$479$	16.3846	0.748634	0.374317	−	0.927301i	$-0.377877\pi$
$479$	16.3846	0.748634	0.374317	+	0.927301i	$0.377877\pi$
$480$	0	0
$481$	−6.15801	−0.280781
$482$	41.6173i	1.89561i
$483$	− 2.67451i	− 0.121695i
$484$	4.45585	0.202539
$485$	0	0
$486$	−28.8954	−1.31072
$487$	2.74404i	0.124344i	0.998065	+	0.0621722i	$0.0198028\pi$
$487$	2.74404i	0.124344i	−0.998065	+	0.0621722i	$0.980197\pi$
$488$	− 4.83487i	− 0.218864i
$489$	3.53483	0.159850
$490$	0	0
$491$	9.46362	0.427087	0.213544	−	0.976934i	$-0.431499\pi$
$491$	9.46362	0.427087	0.213544	+	0.976934i	$0.431499\pi$
$492$	10.8050i	0.487127i
$493$	1.40381i	0.0632244i
$494$	79.0361	3.55600
$495$	0	0
$496$	6.08013	0.273006
$497$	8.61857i	0.386595i
$498$	10.5144i	0.471160i
$499$	36.0076	1.61192	0.805960	−	0.591970i	$-0.201650\pi$
$499$	36.0076	1.61192	0.805960	+	0.591970i	$0.201650\pi$
$500$	0	0
$501$	7.52784	0.336319
$502$	− 13.1575i	− 0.587247i
$503$	− 21.6537i	− 0.965490i	−0.875761	−	0.482745i	$-0.839640\pi$
$503$	− 21.6537i	− 0.965490i	0.875761	−	0.482745i	$-0.160360\pi$
$504$	2.57759	0.114815
$505$	0	0
$506$	7.63953	0.339619
$507$	28.2966i	1.25669i
$508$	4.59073i	0.203681i
$509$	−16.5219	−0.732321	−0.366160	−	0.930552i	$-0.619328\pi$
$509$	−16.5219	−0.732321	−0.366160	+	0.930552i	$0.619328\pi$
$510$	0	0
$511$	9.49679	0.420113
$512$	− 25.7034i	− 1.13594i
$513$	35.5281i	1.56860i
$514$	10.0856	0.444857
$515$	0	0
$516$	10.3180	0.454226
$517$	− 25.5918i	− 1.12553i
$518$	− 3.18852i	− 0.140096i
$519$	−3.82992	−0.168115
$520$	0	0
$521$	4.77373	0.209141	0.104571	−	0.994517i	$-0.466653\pi$
$521$	4.77373	0.209141	0.104571	+	0.994517i	$0.466653\pi$
$522$	3.37030i	0.147514i
$523$	− 18.4589i	− 0.807150i	−0.914947	−	0.403575i	$-0.867767\pi$
$523$	− 18.4589i	− 0.807150i	0.914947	−	0.403575i	$-0.132233\pi$
$524$	32.5693	1.42280
$525$	0	0
$526$	−16.1364	−0.703580
$527$	1.81807i	0.0791963i
$528$	− 14.7360i	− 0.641303i
$529$	20.9792	0.912141
$530$	0	0
$531$	17.9714	0.779894
$532$	17.8844i	0.775386i
$533$	− 39.3318i	− 1.70365i
$534$	−3.64543	−0.157753
$535$	0	0
$536$	−2.06872	−0.0893552
$537$	− 25.6893i	− 1.10857i
$538$	− 13.8910i	− 0.598885i
$539$	−11.6306	−0.500966
$540$	0	0
$541$	−43.5067	−1.87050	−0.935250	−	0.353989i	$-0.884825\pi$
$541$	−43.5067	−1.87050	−0.935250	+	0.353989i	$0.884825\pi$
$542$	10.3478i	0.444474i
$543$	3.67948i	0.157901i
$544$	10.0538	0.431054
$545$	0	0
$546$	−22.0615	−0.944147
$547$	5.97641i	0.255533i	0.991804	+	0.127766i	$0.0407808\pi$
$547$	5.97641i	0.255533i	−0.991804	+	0.127766i	$0.959219\pi$
$548$	30.4219i	1.29956i
$549$	10.2502	0.437469
$550$	0	0
$551$	6.74028	0.287146
$552$	1.31988i	0.0561778i
$553$	− 16.0293i	− 0.681635i
$554$	−45.2380	−1.92198
$555$	0	0
$556$	−33.0466	−1.40149
$557$	− 30.0628i	− 1.27380i	−0.770946	−	0.636900i	$-0.780216\pi$
$557$	− 30.0628i	− 1.27380i	0.770946	−	0.636900i	$-0.219784\pi$
$558$	4.36486i	0.184779i
$559$	−37.5592	−1.58858
$560$	0	0
$561$	4.40634	0.186036
$562$	− 18.9167i	− 0.797954i
$563$	− 18.5651i	− 0.782425i	−0.920300	−	0.391212i	$-0.872056\pi$
$563$	− 18.5651i	− 0.782425i	0.920300	−	0.391212i	$-0.127944\pi$
$564$	−15.3398	−0.645922
$565$	0	0
$566$	15.8515	0.666288
$567$	− 0.748851i	− 0.0314488i
$568$	− 4.25329i	− 0.178464i
$569$	30.6606	1.28536	0.642679	−	0.766135i	$-0.277823\pi$
$569$	30.6606	1.28536	0.642679	+	0.766135i	$0.277823\pi$
$570$	0	0
$571$	−6.78472	−0.283932	−0.141966	−	0.989872i	$-0.545342\pi$
$571$	−6.78472	−0.283932	−0.141966	+	0.989872i	$0.545342\pi$
$572$	− 27.5396i	− 1.15149i
$573$	− 5.08568i	− 0.212457i
$574$	20.3654	0.850034
$575$	0	0
$576$	7.34779	0.306158
$577$	27.7326i	1.15453i	0.816559	+	0.577263i	$0.195879\pi$
$577$	27.7326i	1.15453i	−0.816559	+	0.577263i	$0.804121\pi$
$578$	− 28.3274i	− 1.17827i
$579$	−27.4136	−1.13927
$580$	0	0
$581$	8.66064	0.359304
$582$	− 30.3010i	− 1.25602i
$583$	6.02446i	0.249508i
$584$	−4.68670	−0.193937
$585$	0	0
$586$	12.0359	0.497197
$587$	− 16.6049i	− 0.685359i	−0.939452	−	0.342679i	$-0.888666\pi$
$587$	− 16.6049i	− 0.685359i	0.939452	−	0.342679i	$-0.111334\pi$
$588$	6.97142i	0.287497i
$589$	8.72932	0.359685
$590$	0	0
$591$	−6.56459	−0.270031
$592$	4.64699i	0.190990i
$593$	20.5406i	0.843499i	0.906712	+	0.421750i	$0.138584\pi$
$593$	20.5406i	0.843499i	−0.906712	+	0.421750i	$0.861416\pi$
$594$	28.3272	1.16228
$595$	0	0
$596$	−25.7908	−1.05643
$597$	− 20.9197i	− 0.856188i
$598$	16.6688i	0.681637i
$599$	−31.3560	−1.28117	−0.640586	−	0.767886i	$-0.721309\pi$
$599$	−31.3560	−1.28117	−0.640586	+	0.767886i	$0.721309\pi$
$600$	0	0
$601$	14.3674	0.586057	0.293029	−	0.956104i	$-0.405337\pi$
$601$	14.3674	0.586057	0.293029	+	0.956104i	$0.405337\pi$
$602$	− 19.4475i	− 0.792623i
$603$	− 4.38583i	− 0.178605i
$604$	−13.5801	−0.552566
$605$	0	0
$606$	−32.9633	−1.33904
$607$	25.1167i	1.01945i	0.860336	+	0.509727i	$0.170254\pi$
$607$	25.1167i	1.01945i	−0.860336	+	0.509727i	$0.829746\pi$
$608$	− 48.2726i	− 1.95771i
$609$	−1.88143	−0.0762394
$610$	0	0
$611$	55.8390	2.25901
$612$	3.89711i	0.157531i
$613$	36.4069i	1.47046i	0.677818	+	0.735230i	$0.262926\pi$
$613$	36.4069i	1.47046i	−0.677818	+	0.735230i	$0.737074\pi$
$614$	54.2020	2.18741
$615$	0	0
$616$	−4.11013	−0.165602
$617$	− 28.9915i	− 1.16715i	−0.812058	−	0.583576i	$-0.801653\pi$
$617$	− 28.9915i	− 1.16715i	0.812058	−	0.583576i	$-0.198347\pi$
$618$	24.1356i	0.970874i
$619$	38.6097	1.55185	0.775927	−	0.630823i	$-0.217283\pi$
$619$	38.6097	1.55185	0.775927	+	0.630823i	$0.217283\pi$
$620$	0	0
$621$	−7.49290	−0.300680
$622$	− 7.92494i	− 0.317761i
$623$	3.00273i	0.120302i
$624$	32.1527	1.28714
$625$	0	0
$626$	34.3549	1.37310
$627$	− 21.1567i	− 0.844917i
$628$	31.2009i	1.24505i
$629$	−1.38953	−0.0554043
$630$	0	0
$631$	12.4946	0.497401	0.248700	−	0.968580i	$-0.419997\pi$
$631$	12.4946	0.497401	0.248700	+	0.968580i	$0.419997\pi$
$632$	7.91050i	0.314663i
$633$	9.47034i	0.376412i
$634$	25.3683	1.00750
$635$	0	0
$636$	3.61108	0.143189
$637$	− 25.3770i	− 1.00547i
$638$	− 5.37416i	− 0.212765i
$639$	9.01725	0.356717
$640$	0	0
$641$	18.6523	0.736723	0.368361	−	0.929683i	$-0.379919\pi$
$641$	18.6523	0.736723	0.368361	+	0.929683i	$0.379919\pi$
$642$	− 27.1575i	− 1.07182i
$643$	− 9.99161i	− 0.394030i	−0.980400	−	0.197015i	$-0.936875\pi$
$643$	− 9.99161i	− 0.394030i	0.980400	−	0.197015i	$-0.0631248\pi$
$644$	−3.77183	−0.148631
$645$	0	0
$646$	17.8342	0.701677
$647$	− 6.59170i	− 0.259146i	−0.991570	−	0.129573i	$-0.958639\pi$
$647$	− 6.59170i	− 0.259146i	0.991570	−	0.129573i	$-0.0413607\pi$
$648$	0.369561i	0.0145177i
$649$	−28.6566	−1.12487
$650$	0	0
$651$	−2.43664	−0.0954993
$652$	− 4.98512i	− 0.195232i
$653$	47.5008i	1.85885i	0.369011	+	0.929425i	$0.379696\pi$
$653$	47.5008i	1.85885i	−0.369011	+	0.929425i	$0.620304\pi$
$654$	21.9178	0.857054
$655$	0	0
$656$	−29.6807	−1.15884
$657$	− 9.93610i	− 0.387644i
$658$	28.9126i	1.12713i
$659$	−32.7065	−1.27407	−0.637033	−	0.770837i	$-0.719838\pi$
$659$	−32.7065	−1.27407	−0.637033	+	0.770837i	$0.719838\pi$
$660$	0	0
$661$	−37.6003	−1.46248	−0.731242	−	0.682119i	$-0.761059\pi$
$661$	−37.6003	−1.46248	−0.731242	+	0.682119i	$0.761059\pi$
$662$	− 28.7436i	− 1.11715i
$663$	9.61424i	0.373386i
$664$	−4.27405	−0.165866
$665$	0	0
$666$	−3.33602	−0.129268
$667$	1.42153i	0.0550419i
$668$	− 10.6164i	− 0.410762i
$669$	8.55923	0.330919
$670$	0	0
$671$	−16.3447	−0.630979
$672$	13.4745i	0.519788i
$673$	− 47.2395i	− 1.82095i	−0.413564	−	0.910475i	$-0.635716\pi$
$673$	− 47.2395i	− 1.82095i	0.413564	−	0.910475i	$-0.364284\pi$
$674$	−9.56450	−0.368411
$675$	0	0
$676$	39.9063	1.53486
$677$	17.3323i	0.666134i	0.942903	+	0.333067i	$0.108083\pi$
$677$	17.3323i	0.666134i	−0.942903	+	0.333067i	$0.891917\pi$
$678$	− 38.3754i	− 1.47380i
$679$	−24.9588	−0.957830
$680$	0	0
$681$	−12.5837	−0.482209
$682$	− 6.96005i	− 0.266514i
$683$	− 4.70774i	− 0.180137i	−0.995936	−	0.0900683i	$-0.971291\pi$
$683$	− 4.70774i	− 0.180137i	0.995936	−	0.0900683i	$-0.0287085\pi$
$684$	18.7117	0.715459
$685$	0	0
$686$	35.6888	1.36260
$687$	− 6.76890i	− 0.258250i
$688$	28.3431i	1.08057i
$689$	−13.1449	−0.500779
$690$	0	0
$691$	−2.88811	−0.109869	−0.0549345	−	0.998490i	$-0.517495\pi$
$691$	−2.88811	−0.109869	−0.0549345	+	0.998490i	$0.517495\pi$
$692$	5.40129i	0.205326i
$693$	− 8.71374i	− 0.331008i
$694$	52.8475	2.00606
$695$	0	0
$696$	0.928492	0.0351944
$697$	− 8.87506i	− 0.336167i
$698$	− 52.3524i	− 1.98157i
$699$	0.913422	0.0345488
$700$	0	0
$701$	39.1914	1.48024	0.740120	−	0.672475i	$-0.234769\pi$
$701$	39.1914	1.48024	0.740120	+	0.672475i	$0.234769\pi$
$702$	61.8075i	2.33277i
$703$	6.67173i	0.251629i
$704$	−11.7165	−0.441584
$705$	0	0
$706$	59.2251	2.22897
$707$	27.1517i	1.02115i
$708$	17.1769i	0.645546i
$709$	−35.0367	−1.31583	−0.657915	−	0.753092i	$-0.728561\pi$
$709$	−35.0367	−1.31583	−0.657915	+	0.753092i	$0.728561\pi$
$710$	0	0
$711$	−16.7708	−0.628954
$712$	− 1.48185i	− 0.0555349i
$713$	1.84102i	0.0689468i
$714$	−4.97810	−0.186301
$715$	0	0
$716$	−36.2293	−1.35395
$717$	− 0.521385i	− 0.0194715i
$718$	50.9264i	1.90056i
$719$	−9.93705	−0.370590	−0.185295	−	0.982683i	$-0.559324\pi$
$719$	−9.93705	−0.370590	−0.185295	+	0.982683i	$0.559324\pi$
$720$	0	0
$721$	19.8804	0.740383
$722$	− 49.8182i	− 1.85404i
$723$	24.3071i	0.903990i
$724$	5.18912	0.192852
$725$	0	0
$726$	5.95513	0.221015
$727$	2.37679i	0.0881504i	0.999028	+	0.0440752i	$0.0140341\pi$
$727$	2.37679i	0.0881504i	−0.999028	+	0.0440752i	$0.985966\pi$
$728$	− 8.96793i	− 0.332374i
$729$	−18.1912	−0.673748
$730$	0	0
$731$	−8.47508	−0.313462
$732$	9.79703i	0.362109i
$733$	32.6214i	1.20490i	0.798157	+	0.602450i	$0.205809\pi$
$733$	32.6214i	1.20490i	−0.798157	+	0.602450i	$0.794191\pi$
$734$	−38.3075	−1.41396
$735$	0	0
$736$	10.1807	0.375267
$737$	6.99349i	0.257608i
$738$	− 21.3075i	− 0.784338i
$739$	−16.6544	−0.612642	−0.306321	−	0.951928i	$-0.599098\pi$
$739$	−16.6544	−0.612642	−0.306321	+	0.951928i	$0.599098\pi$
$740$	0	0
$741$	46.1620	1.69580
$742$	− 6.80620i	− 0.249863i
$743$	35.2240i	1.29224i	0.763235	+	0.646121i	$0.223610\pi$
$743$	35.2240i	1.29224i	−0.763235	+	0.646121i	$0.776390\pi$
$744$	1.20249	0.0440853
$745$	0	0
$746$	−30.1050	−1.10222
$747$	− 9.06128i	− 0.331535i
$748$	− 6.21420i	− 0.227214i
$749$	−22.3695	−0.817364
$750$	0	0
$751$	−46.9948	−1.71487	−0.857433	−	0.514596i	$-0.827942\pi$
$751$	−46.9948	−1.71487	−0.857433	+	0.514596i	$0.827942\pi$
$752$	− 42.1375i	− 1.53660i
$753$	− 7.68479i	− 0.280049i
$754$	11.7259	0.427034
$755$	0	0
$756$	−13.9859	−0.508661
$757$	16.5681i	0.602179i	0.953596	+	0.301089i	$0.0973502\pi$
$757$	16.5681i	0.602179i	−0.953596	+	0.301089i	$0.902650\pi$
$758$	55.5718i	2.01846i
$759$	4.46196	0.161959
$760$	0	0
$761$	43.1593	1.56453	0.782263	−	0.622949i	$-0.214066\pi$
$761$	43.1593	1.56453	0.782263	+	0.622949i	$0.214066\pi$
$762$	6.13538i	0.222262i
$763$	− 18.0536i	− 0.653585i
$764$	−7.17226	−0.259483
$765$	0	0
$766$	66.2277	2.39290
$767$	− 62.5262i	− 2.25769i
$768$	− 22.6970i	− 0.819007i
$769$	16.7337	0.603434	0.301717	−	0.953398i	$-0.402440\pi$
$769$	16.7337	0.603434	0.301717	+	0.953398i	$0.402440\pi$
$770$	0	0
$771$	5.89062	0.212145
$772$	38.6611i	1.39144i
$773$	− 8.41758i	− 0.302759i	−0.988476	−	0.151380i	$-0.951628\pi$
$773$	− 8.41758i	− 0.302759i	0.988476	−	0.151380i	$-0.0483716\pi$
$774$	−20.3472	−0.731364
$775$	0	0
$776$	12.3172	0.442163
$777$	− 1.86230i	− 0.0668095i
$778$	53.4864i	1.91758i
$779$	−42.6129	−1.52677
$780$	0	0
$781$	−14.3786	−0.514507
$782$	3.76125i	0.134502i
$783$	5.27100i	0.188370i
$784$	−19.1501	−0.683932
$785$	0	0
$786$	43.5280	1.55259
$787$	50.6781i	1.80648i	0.429136	+	0.903240i	$0.358818\pi$
$787$	50.6781i	1.80648i	−0.429136	+	0.903240i	$0.641182\pi$
$788$	9.25795i	0.329801i
$789$	−9.42465	−0.335527
$790$	0	0
$791$	−31.6097	−1.12391
$792$	4.30026i	0.152803i
$793$	− 35.6626i	− 1.26642i
$794$	46.2598	1.64170
$795$	0	0
$796$	−29.5028	−1.04570
$797$	− 21.8932i	− 0.775498i	−0.921765	−	0.387749i	$-0.873253\pi$
$797$	− 21.8932i	− 0.775498i	0.921765	−	0.387749i	$-0.126747\pi$
$798$	23.9020i	0.846121i
$799$	12.5999	0.445751
$800$	0	0
$801$	3.14163	0.111004
$802$	− 9.34509i	− 0.329987i
$803$	15.8438i	0.559114i
$804$	4.19191	0.147837
$805$	0	0
$806$	15.1862	0.534912
$807$	− 8.11323i	− 0.285599i
$808$	− 13.3995i	− 0.471391i
$809$	8.70213	0.305951	0.152975	−	0.988230i	$-0.451115\pi$
$809$	8.70213	0.305951	0.152975	+	0.988230i	$0.451115\pi$
$810$	0	0
$811$	39.0632	1.37170	0.685848	−	0.727745i	$-0.259431\pi$
$811$	39.0632	1.37170	0.685848	+	0.727745i	$0.259431\pi$
$812$	2.65336i	0.0931146i
$813$	6.04374i	0.211963i
$814$	5.31950	0.186448
$815$	0	0
$816$	7.25514	0.253981
$817$	40.6925i	1.42365i
$818$	− 55.8563i	− 1.95297i
$819$	19.0126	0.664354
$820$	0	0
$821$	16.7509	0.584612	0.292306	−	0.956325i	$-0.405577\pi$
$821$	16.7509	0.584612	0.292306	+	0.956325i	$0.405577\pi$
$822$	40.6581i	1.41811i
$823$	− 38.3755i	− 1.33768i	−0.743405	−	0.668842i	$-0.766790\pi$
$823$	− 38.3755i	− 1.33768i	0.743405	−	0.668842i	$-0.233210\pi$
$824$	−9.81101	−0.341783
$825$	0	0
$826$	32.3751	1.12647
$827$	6.70137i	0.233029i	0.993189	+	0.116515i	$0.0371722\pi$
$827$	6.70137i	0.233029i	−0.993189	+	0.116515i	$0.962828\pi$
$828$	3.94631i	0.137144i
$829$	−7.98795	−0.277433	−0.138716	−	0.990332i	$-0.544298\pi$
$829$	−7.98795	−0.277433	−0.138716	+	0.990332i	$0.544298\pi$
$830$	0	0
$831$	−26.4218	−0.916562
$832$	− 25.5644i	− 0.886288i
$833$	− 5.72622i	− 0.198402i
$834$	−44.1659	−1.52934
$835$	0	0
$836$	−29.8370	−1.03193
$837$	6.82646i	0.235957i
$838$	51.1543i	1.76710i
$839$	13.9407	0.481286	0.240643	−	0.970614i	$-0.422642\pi$
$839$	13.9407	0.481286	0.240643	+	0.970614i	$0.422642\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	50.9483i	1.75580i
$843$	− 11.0485i	− 0.380532i
$844$	13.3559	0.459729
$845$	0	0
$846$	30.2501	1.04002
$847$	− 4.90521i	− 0.168545i
$848$	9.91943i	0.340635i
$849$	9.25827	0.317743
$850$	0	0
$851$	−1.40707	−0.0482339
$852$	8.61857i	0.295267i
$853$	5.78046i	0.197919i	0.995091	+	0.0989597i	$0.0315515\pi$
$853$	5.78046i	0.197919i	−0.995091	+	0.0989597i	$0.968449\pi$
$854$	18.4655	0.631878
$855$	0	0
$856$	11.0394	0.377319
$857$	− 33.2410i	− 1.13549i	−0.823205	−	0.567745i	$-0.807816\pi$
$857$	− 33.2410i	− 1.13549i	0.823205	−	0.567745i	$-0.192184\pi$
$858$	− 36.8059i	− 1.25653i
$859$	−1.19425	−0.0407473	−0.0203736	−	0.999792i	$-0.506486\pi$
$859$	−1.19425	−0.0407473	−0.0203736	+	0.999792i	$0.506486\pi$
$860$	0	0
$861$	11.8947	0.405369
$862$	52.3028i	1.78144i
$863$	42.0216i	1.43043i	0.698904	+	0.715215i	$0.253671\pi$
$863$	42.0216i	1.43043i	−0.698904	+	0.715215i	$0.746329\pi$
$864$	37.7499	1.28428
$865$	0	0
$866$	27.0483	0.919138
$867$	− 16.5450i	− 0.561897i
$868$	3.43635i	0.116637i
$869$	26.7421	0.907164
$870$	0	0
$871$	−15.2592	−0.517037
$872$	8.90951i	0.301714i
$873$	26.1134i	0.883803i
$874$	18.0593	0.610866
$875$	0	0
$876$	9.49679	0.320867
$877$	− 22.8893i	− 0.772917i	−0.922307	−	0.386458i	$-0.873698\pi$
$877$	− 22.8893i	− 0.772917i	0.922307	−	0.386458i	$-0.126302\pi$
$878$	35.9132i	1.21201i
$879$	7.02969	0.237106
$880$	0	0
$881$	13.6751	0.460727	0.230363	−	0.973105i	$-0.426009\pi$
$881$	13.6751	0.460727	0.230363	+	0.973105i	$0.426009\pi$
$882$	− 13.7476i	− 0.462907i
$883$	− 1.49072i	− 0.0501668i	−0.999685	−	0.0250834i	$-0.992015\pi$
$883$	− 1.49072i	− 0.0501668i	0.999685	−	0.0250834i	$-0.00798513\pi$
$884$	13.5588	0.456033
$885$	0	0
$886$	−68.3757	−2.29713
$887$	− 2.58705i	− 0.0868646i	−0.999056	−	0.0434323i	$-0.986171\pi$
$887$	− 2.58705i	− 0.0868646i	0.999056	−	0.0434323i	$-0.0138293\pi$
$888$	0.919049i	0.0308413i
$889$	5.05369	0.169495
$890$	0	0
$891$	1.24933	0.0418541
$892$	− 12.0710i	− 0.404166i
$893$	− 60.4973i	− 2.02447i
$894$	−34.4687	−1.15280
$895$	0	0
$896$	−11.2433	−0.375612
$897$	9.73561i	0.325063i
$898$	5.73514i	0.191384i
$899$	1.29510	0.0431939
$900$	0	0
$901$	−2.96609	−0.0988146
$902$	33.9761i	1.13128i
$903$	− 11.3586i	− 0.377990i
$904$	15.5995	0.518831
$905$	0	0
$906$	−18.1494	−0.602974
$907$	18.3747i	0.610121i	0.952333	+	0.305060i	$0.0986767\pi$
$907$	18.3747i	0.610121i	−0.952333	+	0.305060i	$0.901323\pi$
$908$	17.7467i	0.588944i
$909$	28.4077	0.942225
$910$	0	0
$911$	32.8875	1.08961	0.544806	−	0.838562i	$-0.316603\pi$
$911$	32.8875	1.08961	0.544806	+	0.838562i	$0.316603\pi$
$912$	− 34.8350i	− 1.15350i
$913$	14.4488i	0.478185i
$914$	−46.7843	−1.54749
$915$	0	0
$916$	−9.54608	−0.315412
$917$	− 35.8538i	− 1.18400i
$918$	13.9466i	0.460307i
$919$	−28.2361	−0.931422	−0.465711	−	0.884937i	$-0.654201\pi$
$919$	−28.2361	−0.931422	−0.465711	+	0.884937i	$0.654201\pi$
$920$	0	0
$921$	31.6573	1.04314
$922$	26.7006i	0.879337i
$923$	− 31.3728i	− 1.03265i
$924$	8.32847	0.273987
$925$	0	0
$926$	−17.0300	−0.559640
$927$	− 20.8000i	− 0.683162i
$928$	− 7.16181i	− 0.235098i
$929$	44.5794	1.46260	0.731301	−	0.682055i	$-0.238913\pi$
$929$	44.5794	1.46260	0.731301	+	0.682055i	$0.238913\pi$
$930$	0	0
$931$	−27.4940	−0.901080
$932$	− 1.28819i	− 0.0421960i
$933$	− 4.62866i	− 0.151536i
$934$	−9.95886	−0.325864
$935$	0	0
$936$	−9.38278	−0.306686
$937$	− 47.0602i	− 1.53739i	−0.639615	−	0.768695i	$-0.720906\pi$
$937$	− 47.0602i	− 1.53739i	0.639615	−	0.768695i	$-0.279094\pi$
$938$	− 7.90096i	− 0.257976i
$939$	20.0654	0.654809
$940$	0	0
$941$	−45.5100	−1.48358	−0.741792	−	0.670630i	$-0.766024\pi$
$941$	−45.5100	−1.48358	−0.741792	+	0.670630i	$0.766024\pi$
$942$	41.6991i	1.35863i
$943$	− 8.98710i	− 0.292660i
$944$	−47.1838	−1.53570
$945$	0	0
$946$	32.4449	1.05487
$947$	9.26597i	0.301104i	0.988602	+	0.150552i	$0.0481050\pi$
$947$	9.26597i	0.301104i	−0.988602	+	0.150552i	$0.951895\pi$
$948$	− 16.0293i	− 0.520607i
$949$	−34.5697	−1.12218
$950$	0	0
$951$	14.8167	0.480463
$952$	− 2.02358i	− 0.0655846i
$953$	17.7750i	0.575788i	0.957662	+	0.287894i	$0.0929550\pi$
$953$	17.7750i	0.575788i	−0.957662	+	0.287894i	$0.907045\pi$
$954$	−7.12105	−0.230552
$955$	0	0
$956$	−0.735303	−0.0237814
$957$	− 3.13884i	− 0.101464i
$958$	− 30.8820i	− 0.997751i
$959$	33.4899	1.08145
$960$	0	0
$961$	−29.3227	−0.945894
$962$	11.6067i	0.374215i
$963$	23.4043i	0.754193i
$964$	34.2800	1.10408
$965$	0	0
$966$	−5.04095	−0.162190
$967$	− 45.7913i	− 1.47255i	−0.676683	−	0.736275i	$-0.736583\pi$
$967$	− 45.7913i	− 1.47255i	0.676683	−	0.736275i	$-0.263417\pi$
$968$	2.42074i	0.0778054i
$969$	10.4163	0.334619
$970$	0	0
$971$	−14.0095	−0.449587	−0.224794	−	0.974406i	$-0.572171\pi$
$971$	−14.0095	−0.449587	−0.224794	+	0.974406i	$0.572171\pi$
$972$	23.8010i	0.763418i
$973$	36.3793i	1.16627i
$974$	5.17199	0.165721
$975$	0	0
$976$	−26.9119	−0.861428
$977$	51.0534i	1.63334i	0.577103	+	0.816671i	$0.304183\pi$
$977$	51.0534i	1.63334i	−0.577103	+	0.816671i	$0.695817\pi$
$978$	− 6.66247i	− 0.213042i
$979$	−5.00953	−0.160105
$980$	0	0
$981$	−18.8888	−0.603071
$982$	− 17.8371i	− 0.569206i
$983$	− 4.02331i	− 0.128324i	−0.997940	−	0.0641618i	$-0.979563\pi$
$983$	− 4.02331i	− 0.128324i	0.997940	−	0.0641618i	$-0.0204374\pi$
$984$	−5.87005	−0.187130
$985$	0	0
$986$	2.64591	0.0842630
$987$	16.8868i	0.537511i
$988$	− 65.1017i	− 2.07116i
$989$	−8.58207	−0.272894
$990$	0	0
$991$	−49.5829	−1.57505	−0.787526	−	0.616282i	$-0.788638\pi$
$991$	−49.5829	−1.57505	−0.787526	+	0.616282i	$0.788638\pi$
$992$	− 9.27524i	− 0.294489i
$993$	− 16.7881i	− 0.532753i
$994$	16.2444	0.515240
$995$	0	0
$996$	8.66064	0.274423
$997$	34.8740i	1.10447i	0.833688	+	0.552235i	$0.186225\pi$
$997$	34.8740i	1.10447i	−0.833688	+	0.552235i	$0.813775\pi$
$998$	− 67.8674i	− 2.14831i
$999$	−5.21740	−0.165071

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.b.f.349.2		10
5.2	odd	4		725.2.a.h.1.5	✓	5
5.3	odd	4		725.2.a.k.1.1	yes	5
5.4	even	2	inner	725.2.b.f.349.9		10
15.2	even	4		6525.2.a.bq.1.1		5
15.8	even	4		6525.2.a.bm.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
725.2.a.h.1.5	✓	5	5.2	odd	4
725.2.a.k.1.1	yes	5	5.3	odd	4
725.2.b.f.349.2		10	1.1	even	1	trivial
725.2.b.f.349.9		10	5.4	even	2	inner
6525.2.a.bm.1.5		5	15.8	even	4
6525.2.a.bq.1.1		5	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.88481i q^{2} -1.10085i q^{3} -1.55251 q^{4} -2.07489 q^{6} +1.70908i q^{7} -0.843434i q^{8} +1.78814 q^{9} -2.85130 q^{11} +1.70908i q^{12} -6.22128i q^{13} +3.22128 q^{14} -4.69473 q^{16} -1.40381i q^{17} -3.37030i q^{18} -6.74028 q^{19} +1.88143 q^{21} +5.37416i q^{22} -1.42153i q^{23} -0.928492 q^{24} -11.7259 q^{26} -5.27100i q^{27} -2.65336i q^{28} -1.00000 q^{29} -1.29510 q^{31} +7.16181i q^{32} +3.13884i q^{33} -2.64591 q^{34} -2.77610 q^{36} -0.989830i q^{37} +12.7042i q^{38} -6.84868 q^{39} +6.32213 q^{41} -3.54614i q^{42} -6.03720i q^{43} +4.42667 q^{44} -2.67932 q^{46} +8.97549i q^{47} +5.16818i q^{48} +4.07906 q^{49} -1.54538 q^{51} +9.65860i q^{52} -2.11288i q^{53} -9.93484 q^{54} +1.44149 q^{56} +7.42002i q^{57} +1.88481i q^{58} +10.0504 q^{59} +5.73236 q^{61} +2.44101i q^{62} +3.05606i q^{63} +4.10919 q^{64} +5.91612 q^{66} -2.45274i q^{67} +2.17943i q^{68} -1.56489 q^{69} +5.04282 q^{71} -1.50817i q^{72} -5.55668i q^{73} -1.86564 q^{74} +10.4644 q^{76} -4.87309i q^{77} +12.9085i q^{78} -9.37892 q^{79} -0.438162 q^{81} -11.9160i q^{82} -5.06744i q^{83} -2.92094 q^{84} -11.3790 q^{86} +1.10085i q^{87} +2.40488i q^{88} +1.75693 q^{89} +10.6326 q^{91} +2.20694i q^{92} +1.42570i q^{93} +16.9171 q^{94} +7.88406 q^{96} +14.6037i q^{97} -7.68825i q^{98} -5.09851 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{4} - 2 q^{6} - 22 q^{9} - 4 q^{11} - 22 q^{14} - 20 q^{16} - 4 q^{19} - 2 q^{21} - 26 q^{24} - 32 q^{26} - 10 q^{29} - 2 q^{31} - 6 q^{34} - 26 q^{36} + 12 q^{39} + 10 q^{41} - 14 q^{44} - 40 q^{46}+ \cdots + 52 q^{99}+O(q^{100}) \) 10 * q - 8 * q^4 - 2 * q^6 - 22 * q^9 - 4 * q^11 - 22 * q^14 - 20 * q^16 - 4 * q^19 - 2 * q^21 - 26 * q^24 - 32 * q^26 - 10 * q^29 - 2 * q^31 - 6 * q^34 - 26 * q^36 + 12 * q^39 + 10 * q^41 - 14 * q^44 - 40 * q^46 + 6 * q^49 + 10 * q^51 - 20 * q^54 - 32 * q^56 + 22 * q^59 - 10 * q^61 + 10 * q^64 - 54 * q^66 + 52 * q^69 - 10 * q^71 - 16 * q^74 + 32 * q^76 + 20 * q^79 + 98 * q^81 - 64 * q^84 - 26 * q^86 + 36 * q^89 + 2 * q^91 + 26 * q^94 - 76 * q^96 + 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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