LMFDB - Embedded newform 2250.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2250,2,Mod(1,2250)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2250, 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("2250.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2250 = 2 \cdot 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2250.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:	$17.9663404548$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 750)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	2250.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{7} -1.00000 q^{8} -1.38197 q^{11} -2.85410 q^{13} -1.23607 q^{14} +1.00000 q^{16} -4.85410 q^{17} +6.00000 q^{19} +1.38197 q^{22} +3.09017 q^{23} +2.85410 q^{26} +1.23607 q^{28} -9.32624 q^{29} +2.14590 q^{31} -1.00000 q^{32} +4.85410 q^{34} -7.85410 q^{37} -6.00000 q^{38} +3.23607 q^{41} -0.145898 q^{43} -1.38197 q^{44} -3.09017 q^{46} -10.8541 q^{47} -5.47214 q^{49} -2.85410 q^{52} -3.23607 q^{53} -1.23607 q^{56} +9.32624 q^{58} -6.38197 q^{59} +13.4164 q^{61} -2.14590 q^{62} +1.00000 q^{64} -2.38197 q^{67} -4.85410 q^{68} +8.94427 q^{71} -12.0000 q^{73} +7.85410 q^{74} +6.00000 q^{76} -1.70820 q^{77} -1.14590 q^{79} -3.23607 q^{82} +8.18034 q^{83} +0.145898 q^{86} +1.38197 q^{88} +9.23607 q^{89} -3.52786 q^{91} +3.09017 q^{92} +10.8541 q^{94} -18.1803 q^{97} +5.47214 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 5 q^{11} + q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} + 12 q^{19} + 5 q^{22} - 5 q^{23} - q^{26} - 2 q^{28} - 3 q^{29} + 11 q^{31} - 2 q^{32} + 3 q^{34} - 9 q^{37} - 12 q^{38} + 2 q^{41} - 7 q^{43} - 5 q^{44} + 5 q^{46} - 15 q^{47} - 2 q^{49} + q^{52} - 2 q^{53} + 2 q^{56} + 3 q^{58} - 15 q^{59} - 11 q^{62} + 2 q^{64} - 7 q^{67} - 3 q^{68} - 24 q^{73} + 9 q^{74} + 12 q^{76} + 10 q^{77} - 9 q^{79} - 2 q^{82} - 6 q^{83} + 7 q^{86} + 5 q^{88} + 14 q^{89} - 16 q^{91} - 5 q^{92} + 15 q^{94} - 14 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 5 * q^11 + q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 + 12 * q^19 + 5 * q^22 - 5 * q^23 - q^26 - 2 * q^28 - 3 * q^29 + 11 * q^31 - 2 * q^32 + 3 * q^34 - 9 * q^37 - 12 * q^38 + 2 * q^41 - 7 * q^43 - 5 * q^44 + 5 * q^46 - 15 * q^47 - 2 * q^49 + q^52 - 2 * q^53 + 2 * q^56 + 3 * q^58 - 15 * q^59 - 11 * q^62 + 2 * q^64 - 7 * q^67 - 3 * q^68 - 24 * q^73 + 9 * q^74 + 12 * q^76 + 10 * q^77 - 9 * q^79 - 2 * q^82 - 6 * q^83 + 7 * q^86 + 5 * q^88 + 14 * q^89 - 16 * q^91 - 5 * q^92 + 15 * q^94 - 14 * q^97 + 2 * 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
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−1.38197	−0.416678	−0.208339	−	0.978057i	$-0.566806\pi$
$11$	−1.38197	−0.416678	−0.208339	+	0.978057i	$0.566806\pi$
$12$	0	0
$13$	−2.85410	−0.791585	−0.395793	−	0.918340i	$-0.629530\pi$
$13$	−2.85410	−0.791585	−0.395793	+	0.918340i	$0.629530\pi$
$14$	−1.23607	−0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.85410	−1.17729	−0.588646	−	0.808391i	$-0.700339\pi$
$17$	−4.85410	−1.17729	−0.588646	+	0.808391i	$0.700339\pi$
$18$	0	0
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	0	0
$22$	1.38197	0.294636
$23$	3.09017	0.644345	0.322172	−	0.946681i	$-0.395587\pi$
$23$	3.09017	0.644345	0.322172	+	0.946681i	$0.395587\pi$
$24$	0	0
$25$	0	0
$26$	2.85410	0.559735
$27$	0	0
$28$	1.23607	0.233595
$29$	−9.32624	−1.73184	−0.865919	−	0.500183i	$-0.833266\pi$
$29$	−9.32624	−1.73184	−0.865919	+	0.500183i	$0.833266\pi$
$30$	0	0
$31$	2.14590	0.385415	0.192707	−	0.981256i	$-0.438273\pi$
$31$	2.14590	0.385415	0.192707	+	0.981256i	$0.438273\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.85410	0.832472
$35$	0	0
$36$	0	0
$37$	−7.85410	−1.29121	−0.645603	−	0.763673i	$-0.723394\pi$
$37$	−7.85410	−1.29121	−0.645603	+	0.763673i	$0.723394\pi$
$38$	−6.00000	−0.973329
$39$	0	0
$40$	0	0
$41$	3.23607	0.505389	0.252694	−	0.967546i	$-0.418683\pi$
$41$	3.23607	0.505389	0.252694	+	0.967546i	$0.418683\pi$
$42$	0	0
$43$	−0.145898	−0.0222492	−0.0111246	−	0.999938i	$-0.503541\pi$
$43$	−0.145898	−0.0222492	−0.0111246	+	0.999938i	$0.503541\pi$
$44$	−1.38197	−0.208339
$45$	0	0
$46$	−3.09017	−0.455621
$47$	−10.8541	−1.58323	−0.791617	−	0.611018i	$-0.790760\pi$
$47$	−10.8541	−1.58323	−0.791617	+	0.611018i	$0.790760\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	0	0
$52$	−2.85410	−0.395793
$53$	−3.23607	−0.444508	−0.222254	−	0.974989i	$-0.571341\pi$
$53$	−3.23607	−0.444508	−0.222254	+	0.974989i	$0.571341\pi$
$54$	0	0
$55$	0	0
$56$	−1.23607	−0.165177
$57$	0	0
$58$	9.32624	1.22460
$59$	−6.38197	−0.830861	−0.415431	−	0.909625i	$-0.636369\pi$
$59$	−6.38197	−0.830861	−0.415431	+	0.909625i	$0.636369\pi$
$60$	0	0
$61$	13.4164	1.71780	0.858898	−	0.512148i	$-0.171150\pi$
$61$	13.4164	1.71780	0.858898	+	0.512148i	$0.171150\pi$
$62$	−2.14590	−0.272529
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−2.38197	−0.291003	−0.145502	−	0.989358i	$-0.546480\pi$
$67$	−2.38197	−0.291003	−0.145502	+	0.989358i	$0.546480\pi$
$68$	−4.85410	−0.588646
$69$	0	0
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	7.85410	0.913021
$75$	0	0
$76$	6.00000	0.688247
$77$	−1.70820	−0.194668
$78$	0	0
$79$	−1.14590	−0.128924	−0.0644618	−	0.997920i	$-0.520533\pi$
$79$	−1.14590	−0.128924	−0.0644618	+	0.997920i	$0.520533\pi$
$80$	0	0
$81$	0	0
$82$	−3.23607	−0.357364
$83$	8.18034	0.897909	0.448954	−	0.893555i	$-0.351797\pi$
$83$	8.18034	0.897909	0.448954	+	0.893555i	$0.351797\pi$
$84$	0	0
$85$	0	0
$86$	0.145898	0.0157326
$87$	0	0
$88$	1.38197	0.147318
$89$	9.23607	0.979021	0.489511	−	0.871997i	$-0.337175\pi$
$89$	9.23607	0.979021	0.489511	+	0.871997i	$0.337175\pi$
$90$	0	0
$91$	−3.52786	−0.369821
$92$	3.09017	0.322172
$93$	0	0
$94$	10.8541	1.11952
$95$	0	0
$96$	0	0
$97$	−18.1803	−1.84593	−0.922967	−	0.384879i	$-0.874243\pi$
$97$	−18.1803	−1.84593	−0.922967	+	0.384879i	$0.874243\pi$
$98$	5.47214	0.552769
$99$	0	0
$100$	0	0
$101$	2.32624	0.231469	0.115735	−	0.993280i	$-0.463078\pi$
$101$	2.32624	0.231469	0.115735	+	0.993280i	$0.463078\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	2.85410	0.279868
$105$	0	0
$106$	3.23607	0.314315
$107$	−8.18034	−0.790823	−0.395412	−	0.918504i	$-0.629398\pi$
$107$	−8.18034	−0.790823	−0.395412	+	0.918504i	$0.629398\pi$
$108$	0	0
$109$	−13.4164	−1.28506	−0.642529	−	0.766261i	$-0.722115\pi$
$109$	−13.4164	−1.28506	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	0	0
$112$	1.23607	0.116797
$113$	12.3262	1.15955	0.579777	−	0.814775i	$-0.303139\pi$
$113$	12.3262	1.15955	0.579777	+	0.814775i	$0.303139\pi$
$114$	0	0
$115$	0	0
$116$	−9.32624	−0.865919
$117$	0	0
$118$	6.38197	0.587508
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−9.09017	−0.826379
$122$	−13.4164	−1.21466
$123$	0	0
$124$	2.14590	0.192707
$125$	0	0
$126$	0	0
$127$	13.4164	1.19051	0.595257	−	0.803535i	$-0.297050\pi$
$127$	13.4164	1.19051	0.595257	+	0.803535i	$0.297050\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−17.8885	−1.56293	−0.781465	−	0.623949i	$-0.785527\pi$
$131$	−17.8885	−1.56293	−0.781465	+	0.623949i	$0.785527\pi$
$132$	0	0
$133$	7.41641	0.643084
$134$	2.38197	0.205771
$135$	0	0
$136$	4.85410	0.416236
$137$	4.61803	0.394545	0.197273	−	0.980349i	$-0.436792\pi$
$137$	4.61803	0.394545	0.197273	+	0.980349i	$0.436792\pi$
$138$	0	0
$139$	−21.2361	−1.80122	−0.900610	−	0.434628i	$-0.856880\pi$
$139$	−21.2361	−1.80122	−0.900610	+	0.434628i	$0.856880\pi$
$140$	0	0
$141$	0	0
$142$	−8.94427	−0.750587
$143$	3.94427	0.329837
$144$	0	0
$145$	0	0
$146$	12.0000	0.993127
$147$	0	0
$148$	−7.85410	−0.645603
$149$	20.4721	1.67714	0.838571	−	0.544792i	$-0.183391\pi$
$149$	20.4721	1.67714	0.838571	+	0.544792i	$0.183391\pi$
$150$	0	0
$151$	−17.8541	−1.45295	−0.726473	−	0.687195i	$-0.758842\pi$
$151$	−17.8541	−1.45295	−0.726473	+	0.687195i	$0.758842\pi$
$152$	−6.00000	−0.486664
$153$	0	0
$154$	1.70820	0.137651
$155$	0	0
$156$	0	0
$157$	0.437694	0.0349318	0.0174659	−	0.999847i	$-0.494440\pi$
$157$	0.437694	0.0349318	0.0174659	+	0.999847i	$0.494440\pi$
$158$	1.14590	0.0911628
$159$	0	0
$160$	0	0
$161$	3.81966	0.301031
$162$	0	0
$163$	−20.2705	−1.58771	−0.793854	−	0.608108i	$-0.791929\pi$
$163$	−20.2705	−1.58771	−0.793854	+	0.608108i	$0.791929\pi$
$164$	3.23607	0.252694
$165$	0	0
$166$	−8.18034	−0.634918
$167$	−23.0902	−1.78677	−0.893385	−	0.449291i	$-0.851677\pi$
$167$	−23.0902	−1.78677	−0.893385	+	0.449291i	$0.851677\pi$
$168$	0	0
$169$	−4.85410	−0.373392
$170$	0	0
$171$	0	0
$172$	−0.145898	−0.0111246
$173$	−9.23607	−0.702205	−0.351103	−	0.936337i	$-0.614193\pi$
$173$	−9.23607	−0.702205	−0.351103	+	0.936337i	$0.614193\pi$
$174$	0	0
$175$	0	0
$176$	−1.38197	−0.104170
$177$	0	0
$178$	−9.23607	−0.692273
$179$	−7.41641	−0.554328	−0.277164	−	0.960823i	$-0.589395\pi$
$179$	−7.41641	−0.554328	−0.277164	+	0.960823i	$0.589395\pi$
$180$	0	0
$181$	7.41641	0.551257	0.275629	−	0.961264i	$-0.411114\pi$
$181$	7.41641	0.551257	0.275629	+	0.961264i	$0.411114\pi$
$182$	3.52786	0.261503
$183$	0	0
$184$	−3.09017	−0.227810
$185$	0	0
$186$	0	0
$187$	6.70820	0.490552
$188$	−10.8541	−0.791617
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	17.1246	1.23266	0.616328	−	0.787489i	$-0.288619\pi$
$193$	17.1246	1.23266	0.616328	+	0.787489i	$0.288619\pi$
$194$	18.1803	1.30527
$195$	0	0
$196$	−5.47214	−0.390867
$197$	−1.52786	−0.108856	−0.0544279	−	0.998518i	$-0.517334\pi$
$197$	−1.52786	−0.108856	−0.0544279	+	0.998518i	$0.517334\pi$
$198$	0	0
$199$	−15.6180	−1.10713	−0.553567	−	0.832805i	$-0.686734\pi$
$199$	−15.6180	−1.10713	−0.553567	+	0.832805i	$0.686734\pi$
$200$	0	0
$201$	0	0
$202$	−2.32624	−0.163674
$203$	−11.5279	−0.809097
$204$	0	0
$205$	0	0
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−2.85410	−0.197896
$209$	−8.29180	−0.573556
$210$	0	0
$211$	3.41641	0.235195	0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641	0.235195	0.117598	+	0.993061i	$0.462481\pi$
$212$	−3.23607	−0.222254
$213$	0	0
$214$	8.18034	0.559197
$215$	0	0
$216$	0	0
$217$	2.65248	0.180062
$218$	13.4164	0.908674
$219$	0	0
$220$	0	0
$221$	13.8541	0.931928
$222$	0	0
$223$	21.7082	1.45369	0.726844	−	0.686802i	$-0.240986\pi$
$223$	21.7082	1.45369	0.726844	+	0.686802i	$0.240986\pi$
$224$	−1.23607	−0.0825883
$225$	0	0
$226$	−12.3262	−0.819929
$227$	−16.4721	−1.09329	−0.546647	−	0.837363i	$-0.684096\pi$
$227$	−16.4721	−1.09329	−0.546647	+	0.837363i	$0.684096\pi$
$228$	0	0
$229$	−1.70820	−0.112881	−0.0564406	−	0.998406i	$-0.517975\pi$
$229$	−1.70820	−0.112881	−0.0564406	+	0.998406i	$0.517975\pi$
$230$	0	0
$231$	0	0
$232$	9.32624	0.612298
$233$	7.32624	0.479958	0.239979	−	0.970778i	$-0.422859\pi$
$233$	7.32624	0.479958	0.239979	+	0.970778i	$0.422859\pi$
$234$	0	0
$235$	0	0
$236$	−6.38197	−0.415431
$237$	0	0
$238$	6.00000	0.388922
$239$	26.6525	1.72401	0.862003	−	0.506904i	$-0.169210\pi$
$239$	26.6525	1.72401	0.862003	+	0.506904i	$0.169210\pi$
$240$	0	0
$241$	0.0901699	0.00580836	0.00290418	−	0.999996i	$-0.499076\pi$
$241$	0.0901699	0.00580836	0.00290418	+	0.999996i	$0.499076\pi$
$242$	9.09017	0.584338
$243$	0	0
$244$	13.4164	0.858898
$245$	0	0
$246$	0	0
$247$	−17.1246	−1.08961
$248$	−2.14590	−0.136265
$249$	0	0
$250$	0	0
$251$	−7.90983	−0.499264	−0.249632	−	0.968341i	$-0.580310\pi$
$251$	−7.90983	−0.499264	−0.249632	+	0.968341i	$0.580310\pi$
$252$	0	0
$253$	−4.27051	−0.268485
$254$	−13.4164	−0.841820
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.5066	1.46630	0.733150	−	0.680067i	$-0.238049\pi$
$257$	23.5066	1.46630	0.733150	+	0.680067i	$0.238049\pi$
$258$	0	0
$259$	−9.70820	−0.603238
$260$	0	0
$261$	0	0
$262$	17.8885	1.10516
$263$	−15.9787	−0.985290	−0.492645	−	0.870230i	$-0.663970\pi$
$263$	−15.9787	−0.985290	−0.492645	+	0.870230i	$0.663970\pi$
$264$	0	0
$265$	0	0
$266$	−7.41641	−0.454729
$267$	0	0
$268$	−2.38197	−0.145502
$269$	−22.6180	−1.37905	−0.689523	−	0.724264i	$-0.742180\pi$
$269$	−22.6180	−1.37905	−0.689523	+	0.724264i	$0.742180\pi$
$270$	0	0
$271$	−8.56231	−0.520123	−0.260062	−	0.965592i	$-0.583743\pi$
$271$	−8.56231	−0.520123	−0.260062	+	0.965592i	$0.583743\pi$
$272$	−4.85410	−0.294323
$273$	0	0
$274$	−4.61803	−0.278986
$275$	0	0
$276$	0	0
$277$	−14.3607	−0.862850	−0.431425	−	0.902149i	$-0.641989\pi$
$277$	−14.3607	−0.862850	−0.431425	+	0.902149i	$0.641989\pi$
$278$	21.2361	1.27365
$279$	0	0
$280$	0	0
$281$	−22.4721	−1.34058	−0.670288	−	0.742101i	$-0.733829\pi$
$281$	−22.4721	−1.34058	−0.670288	+	0.742101i	$0.733829\pi$
$282$	0	0
$283$	21.0902	1.25368	0.626840	−	0.779148i	$-0.284348\pi$
$283$	21.0902	1.25368	0.626840	+	0.779148i	$0.284348\pi$
$284$	8.94427	0.530745
$285$	0	0
$286$	−3.94427	−0.233230
$287$	4.00000	0.236113
$288$	0	0
$289$	6.56231	0.386018
$290$	0	0
$291$	0	0
$292$	−12.0000	−0.702247
$293$	−15.5967	−0.911172	−0.455586	−	0.890192i	$-0.650570\pi$
$293$	−15.5967	−0.911172	−0.455586	+	0.890192i	$0.650570\pi$
$294$	0	0
$295$	0	0
$296$	7.85410	0.456510
$297$	0	0
$298$	−20.4721	−1.18592
$299$	−8.81966	−0.510054
$300$	0	0
$301$	−0.180340	−0.0103946
$302$	17.8541	1.02739
$303$	0	0
$304$	6.00000	0.344124
$305$	0	0
$306$	0	0
$307$	33.0902	1.88856	0.944278	−	0.329149i	$-0.106762\pi$
$307$	33.0902	1.88856	0.944278	+	0.329149i	$0.106762\pi$
$308$	−1.70820	−0.0973340
$309$	0	0
$310$	0	0
$311$	4.47214	0.253592	0.126796	−	0.991929i	$-0.459531\pi$
$311$	4.47214	0.253592	0.126796	+	0.991929i	$0.459531\pi$
$312$	0	0
$313$	−30.9443	−1.74907	−0.874537	−	0.484959i	$-0.838834\pi$
$313$	−30.9443	−1.74907	−0.874537	+	0.484959i	$0.838834\pi$
$314$	−0.437694	−0.0247005
$315$	0	0
$316$	−1.14590	−0.0644618
$317$	9.70820	0.545267	0.272634	−	0.962118i	$-0.412105\pi$
$317$	9.70820	0.545267	0.272634	+	0.962118i	$0.412105\pi$
$318$	0	0
$319$	12.8885	0.721620
$320$	0	0
$321$	0	0
$322$	−3.81966	−0.212861
$323$	−29.1246	−1.62054
$324$	0	0
$325$	0	0
$326$	20.2705	1.12268
$327$	0	0
$328$	−3.23607	−0.178682
$329$	−13.4164	−0.739671
$330$	0	0
$331$	−21.4164	−1.17715	−0.588576	−	0.808442i	$-0.700311\pi$
$331$	−21.4164	−1.17715	−0.588576	+	0.808442i	$0.700311\pi$
$332$	8.18034	0.448954
$333$	0	0
$334$	23.0902	1.26344
$335$	0	0
$336$	0	0
$337$	12.9443	0.705119	0.352560	−	0.935789i	$-0.385311\pi$
$337$	12.9443	0.705119	0.352560	+	0.935789i	$0.385311\pi$
$338$	4.85410	0.264028
$339$	0	0
$340$	0	0
$341$	−2.96556	−0.160594
$342$	0	0
$343$	−15.4164	−0.832408
$344$	0.145898	0.00786629
$345$	0	0
$346$	9.23607	0.496534
$347$	24.3607	1.30775	0.653875	−	0.756603i	$-0.273142\pi$
$347$	24.3607	1.30775	0.653875	+	0.756603i	$0.273142\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	1.38197	0.0736590
$353$	1.20163	0.0639561	0.0319781	−	0.999489i	$-0.489819\pi$
$353$	1.20163	0.0639561	0.0319781	+	0.999489i	$0.489819\pi$
$354$	0	0
$355$	0	0
$356$	9.23607	0.489511
$357$	0	0
$358$	7.41641	0.391969
$359$	−23.8885	−1.26079	−0.630395	−	0.776275i	$-0.717107\pi$
$359$	−23.8885	−1.26079	−0.630395	+	0.776275i	$0.717107\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−7.41641	−0.389798
$363$	0	0
$364$	−3.52786	−0.184910
$365$	0	0
$366$	0	0
$367$	−10.2918	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$367$	−10.2918	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$368$	3.09017	0.161086
$369$	0	0
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	32.7426	1.69535	0.847675	−	0.530516i	$-0.178002\pi$
$373$	32.7426	1.69535	0.847675	+	0.530516i	$0.178002\pi$
$374$	−6.70820	−0.346873
$375$	0	0
$376$	10.8541	0.559758
$377$	26.6180	1.37090
$378$	0	0
$379$	−1.23607	−0.0634925	−0.0317463	−	0.999496i	$-0.510107\pi$
$379$	−1.23607	−0.0634925	−0.0317463	+	0.999496i	$0.510107\pi$
$380$	0	0
$381$	0	0
$382$	6.00000	0.306987
$383$	17.8885	0.914062	0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885	0.914062	0.457031	+	0.889451i	$0.348913\pi$
$384$	0	0
$385$	0	0
$386$	−17.1246	−0.871620
$387$	0	0
$388$	−18.1803	−0.922967
$389$	−18.2705	−0.926352	−0.463176	−	0.886266i	$-0.653290\pi$
$389$	−18.2705	−0.926352	−0.463176	+	0.886266i	$0.653290\pi$
$390$	0	0
$391$	−15.0000	−0.758583
$392$	5.47214	0.276385
$393$	0	0
$394$	1.52786	0.0769727
$395$	0	0
$396$	0	0
$397$	−32.4721	−1.62973	−0.814865	−	0.579651i	$-0.803189\pi$
$397$	−32.4721	−1.62973	−0.814865	+	0.579651i	$0.803189\pi$
$398$	15.6180	0.782861
$399$	0	0
$400$	0	0
$401$	10.4721	0.522954	0.261477	−	0.965210i	$-0.415791\pi$
$401$	10.4721	0.522954	0.261477	+	0.965210i	$0.415791\pi$
$402$	0	0
$403$	−6.12461	−0.305089
$404$	2.32624	0.115735
$405$	0	0
$406$	11.5279	0.572118
$407$	10.8541	0.538018
$408$	0	0
$409$	13.5623	0.670613	0.335306	−	0.942109i	$-0.391160\pi$
$409$	13.5623	0.670613	0.335306	+	0.942109i	$0.391160\pi$
$410$	0	0
$411$	0	0
$412$	4.00000	0.197066
$413$	−7.88854	−0.388170
$414$	0	0
$415$	0	0
$416$	2.85410	0.139934
$417$	0	0
$418$	8.29180	0.405565
$419$	8.94427	0.436956	0.218478	−	0.975842i	$-0.429891\pi$
$419$	8.94427	0.436956	0.218478	+	0.975842i	$0.429891\pi$
$420$	0	0
$421$	14.4721	0.705329	0.352664	−	0.935750i	$-0.385276\pi$
$421$	14.4721	0.705329	0.352664	+	0.935750i	$0.385276\pi$
$422$	−3.41641	−0.166308
$423$	0	0
$424$	3.23607	0.157157
$425$	0	0
$426$	0	0
$427$	16.5836	0.802536
$428$	−8.18034	−0.395412
$429$	0	0
$430$	0	0
$431$	21.7082	1.04565	0.522824	−	0.852441i	$-0.324879\pi$
$431$	21.7082	1.04565	0.522824	+	0.852441i	$0.324879\pi$
$432$	0	0
$433$	0.180340	0.00866658	0.00433329	−	0.999991i	$-0.498621\pi$
$433$	0.180340	0.00866658	0.00433329	+	0.999991i	$0.498621\pi$
$434$	−2.65248	−0.127323
$435$	0	0
$436$	−13.4164	−0.642529
$437$	18.5410	0.886937
$438$	0	0
$439$	28.3262	1.35194	0.675969	−	0.736930i	$-0.263725\pi$
$439$	28.3262	1.35194	0.675969	+	0.736930i	$0.263725\pi$
$440$	0	0
$441$	0	0
$442$	−13.8541	−0.658972
$443$	32.6525	1.55137	0.775683	−	0.631123i	$-0.217406\pi$
$443$	32.6525	1.55137	0.775683	+	0.631123i	$0.217406\pi$
$444$	0	0
$445$	0	0
$446$	−21.7082	−1.02791
$447$	0	0
$448$	1.23607	0.0583987
$449$	−2.94427	−0.138949	−0.0694744	−	0.997584i	$-0.522132\pi$
$449$	−2.94427	−0.138949	−0.0694744	+	0.997584i	$0.522132\pi$
$450$	0	0
$451$	−4.47214	−0.210585
$452$	12.3262	0.579777
$453$	0	0
$454$	16.4721	0.773076
$455$	0	0
$456$	0	0
$457$	22.9443	1.07329	0.536644	−	0.843809i	$-0.319692\pi$
$457$	22.9443	1.07329	0.536644	+	0.843809i	$0.319692\pi$
$458$	1.70820	0.0798191
$459$	0	0
$460$	0	0
$461$	15.9787	0.744203	0.372101	−	0.928192i	$-0.378637\pi$
$461$	15.9787	0.744203	0.372101	+	0.928192i	$0.378637\pi$
$462$	0	0
$463$	2.29180	0.106509	0.0532544	−	0.998581i	$-0.483041\pi$
$463$	2.29180	0.106509	0.0532544	+	0.998581i	$0.483041\pi$
$464$	−9.32624	−0.432960
$465$	0	0
$466$	−7.32624	−0.339381
$467$	2.29180	0.106052	0.0530258	−	0.998593i	$-0.483113\pi$
$467$	2.29180	0.106052	0.0530258	+	0.998593i	$0.483113\pi$
$468$	0	0
$469$	−2.94427	−0.135954
$470$	0	0
$471$	0	0
$472$	6.38197	0.293754
$473$	0.201626	0.00927078
$474$	0	0
$475$	0	0
$476$	−6.00000	−0.275010
$477$	0	0
$478$	−26.6525	−1.21906
$479$	−6.65248	−0.303959	−0.151980	−	0.988384i	$-0.548565\pi$
$479$	−6.65248	−0.303959	−0.151980	+	0.988384i	$0.548565\pi$
$480$	0	0
$481$	22.4164	1.02210
$482$	−0.0901699	−0.00410713
$483$	0	0
$484$	−9.09017	−0.413190
$485$	0	0
$486$	0	0
$487$	17.5967	0.797385	0.398692	−	0.917085i	$-0.369464\pi$
$487$	17.5967	0.797385	0.398692	+	0.917085i	$0.369464\pi$
$488$	−13.4164	−0.607332
$489$	0	0
$490$	0	0
$491$	−2.61803	−0.118150	−0.0590751	−	0.998254i	$-0.518815\pi$
$491$	−2.61803	−0.118150	−0.0590751	+	0.998254i	$0.518815\pi$
$492$	0	0
$493$	45.2705	2.03888
$494$	17.1246	0.770473
$495$	0	0
$496$	2.14590	0.0963537
$497$	11.0557	0.495917
$498$	0	0
$499$	13.2361	0.592528	0.296264	−	0.955106i	$-0.404259\pi$
$499$	13.2361	0.592528	0.296264	+	0.955106i	$0.404259\pi$
$500$	0	0
$501$	0	0
$502$	7.90983	0.353033
$503$	−10.4721	−0.466929	−0.233465	−	0.972365i	$-0.575006\pi$
$503$	−10.4721	−0.466929	−0.233465	+	0.972365i	$0.575006\pi$
$504$	0	0
$505$	0	0
$506$	4.27051	0.189847
$507$	0	0
$508$	13.4164	0.595257
$509$	−2.94427	−0.130503	−0.0652513	−	0.997869i	$-0.520785\pi$
$509$	−2.94427	−0.130503	−0.0652513	+	0.997869i	$0.520785\pi$
$510$	0	0
$511$	−14.8328	−0.656165
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−23.5066	−1.03683
$515$	0	0
$516$	0	0
$517$	15.0000	0.659699
$518$	9.70820	0.426554
$519$	0	0
$520$	0	0
$521$	8.18034	0.358387	0.179194	−	0.983814i	$-0.442651\pi$
$521$	8.18034	0.358387	0.179194	+	0.983814i	$0.442651\pi$
$522$	0	0
$523$	−11.9098	−0.520781	−0.260390	−	0.965503i	$-0.583851\pi$
$523$	−11.9098	−0.520781	−0.260390	+	0.965503i	$0.583851\pi$
$524$	−17.8885	−0.781465
$525$	0	0
$526$	15.9787	0.696705
$527$	−10.4164	−0.453746
$528$	0	0
$529$	−13.4508	−0.584820
$530$	0	0
$531$	0	0
$532$	7.41641	0.321542
$533$	−9.23607	−0.400059
$534$	0	0
$535$	0	0
$536$	2.38197	0.102885
$537$	0	0
$538$	22.6180	0.975133
$539$	7.56231	0.325732
$540$	0	0
$541$	11.7082	0.503375	0.251688	−	0.967809i	$-0.419014\pi$
$541$	11.7082	0.503375	0.251688	+	0.967809i	$0.419014\pi$
$542$	8.56231	0.367783
$543$	0	0
$544$	4.85410	0.208118
$545$	0	0
$546$	0	0
$547$	5.43769	0.232499	0.116250	−	0.993220i	$-0.462913\pi$
$547$	5.43769	0.232499	0.116250	+	0.993220i	$0.462913\pi$
$548$	4.61803	0.197273
$549$	0	0
$550$	0	0
$551$	−55.9574	−2.38387
$552$	0	0
$553$	−1.41641	−0.0602318
$554$	14.3607	0.610127
$555$	0	0
$556$	−21.2361	−0.900610
$557$	10.3607	0.438996	0.219498	−	0.975613i	$-0.429558\pi$
$557$	10.3607	0.438996	0.219498	+	0.975613i	$0.429558\pi$
$558$	0	0
$559$	0.416408	0.0176122
$560$	0	0
$561$	0	0
$562$	22.4721	0.947930
$563$	−12.6525	−0.533238	−0.266619	−	0.963802i	$-0.585907\pi$
$563$	−12.6525	−0.533238	−0.266619	+	0.963802i	$0.585907\pi$
$564$	0	0
$565$	0	0
$566$	−21.0902	−0.886486
$567$	0	0
$568$	−8.94427	−0.375293
$569$	13.0557	0.547325	0.273662	−	0.961826i	$-0.411765\pi$
$569$	13.0557	0.547325	0.273662	+	0.961826i	$0.411765\pi$
$570$	0	0
$571$	12.3607	0.517278	0.258639	−	0.965974i	$-0.416726\pi$
$571$	12.3607	0.517278	0.258639	+	0.965974i	$0.416726\pi$
$572$	3.94427	0.164918
$573$	0	0
$574$	−4.00000	−0.166957
$575$	0	0
$576$	0	0
$577$	−4.11146	−0.171162	−0.0855811	−	0.996331i	$-0.527275\pi$
$577$	−4.11146	−0.171162	−0.0855811	+	0.996331i	$0.527275\pi$
$578$	−6.56231	−0.272956
$579$	0	0
$580$	0	0
$581$	10.1115	0.419494
$582$	0	0
$583$	4.47214	0.185217
$584$	12.0000	0.496564
$585$	0	0
$586$	15.5967	0.644296
$587$	18.6525	0.769870	0.384935	−	0.922944i	$-0.374224\pi$
$587$	18.6525	0.769870	0.384935	+	0.922944i	$0.374224\pi$
$588$	0	0
$589$	12.8754	0.530521
$590$	0	0
$591$	0	0
$592$	−7.85410	−0.322802
$593$	34.7984	1.42900	0.714499	−	0.699636i	$-0.246655\pi$
$593$	34.7984	1.42900	0.714499	+	0.699636i	$0.246655\pi$
$594$	0	0
$595$	0	0
$596$	20.4721	0.838571
$597$	0	0
$598$	8.81966	0.360663
$599$	31.8885	1.30293	0.651465	−	0.758678i	$-0.274155\pi$
$599$	31.8885	1.30293	0.651465	+	0.758678i	$0.274155\pi$
$600$	0	0
$601$	40.6869	1.65965	0.829827	−	0.558021i	$-0.188439\pi$
$601$	40.6869	1.65965	0.829827	+	0.558021i	$0.188439\pi$
$602$	0.180340	0.00735011
$603$	0	0
$604$	−17.8541	−0.726473
$605$	0	0
$606$	0	0
$607$	47.4164	1.92457	0.962286	−	0.272039i	$-0.0876979\pi$
$607$	47.4164	1.92457	0.962286	+	0.272039i	$0.0876979\pi$
$608$	−6.00000	−0.243332
$609$	0	0
$610$	0	0
$611$	30.9787	1.25326
$612$	0	0
$613$	−37.7771	−1.52580	−0.762901	−	0.646515i	$-0.776226\pi$
$613$	−37.7771	−1.52580	−0.762901	+	0.646515i	$0.776226\pi$
$614$	−33.0902	−1.33541
$615$	0	0
$616$	1.70820	0.0688255
$617$	−31.3050	−1.26029	−0.630145	−	0.776478i	$-0.717005\pi$
$617$	−31.3050	−1.26029	−0.630145	+	0.776478i	$0.717005\pi$
$618$	0	0
$619$	−47.7082	−1.91755	−0.958777	−	0.284159i	$-0.908286\pi$
$619$	−47.7082	−1.91755	−0.958777	+	0.284159i	$0.908286\pi$
$620$	0	0
$621$	0	0
$622$	−4.47214	−0.179316
$623$	11.4164	0.457389
$624$	0	0
$625$	0	0
$626$	30.9443	1.23678
$627$	0	0
$628$	0.437694	0.0174659
$629$	38.1246	1.52013
$630$	0	0
$631$	−15.0344	−0.598512	−0.299256	−	0.954173i	$-0.596738\pi$
$631$	−15.0344	−0.598512	−0.299256	+	0.954173i	$0.596738\pi$
$632$	1.14590	0.0455814
$633$	0	0
$634$	−9.70820	−0.385562
$635$	0	0
$636$	0	0
$637$	15.6180	0.618809
$638$	−12.8885	−0.510262
$639$	0	0
$640$	0	0
$641$	30.6525	1.21070	0.605350	−	0.795959i	$-0.293033\pi$
$641$	30.6525	1.21070	0.605350	+	0.795959i	$0.293033\pi$
$642$	0	0
$643$	31.5623	1.24470	0.622348	−	0.782741i	$-0.286179\pi$
$643$	31.5623	1.24470	0.622348	+	0.782741i	$0.286179\pi$
$644$	3.81966	0.150516
$645$	0	0
$646$	29.1246	1.14589
$647$	12.6180	0.496066	0.248033	−	0.968752i	$-0.420216\pi$
$647$	12.6180	0.496066	0.248033	+	0.968752i	$0.420216\pi$
$648$	0	0
$649$	8.81966	0.346202
$650$	0	0
$651$	0	0
$652$	−20.2705	−0.793854
$653$	−9.88854	−0.386969	−0.193484	−	0.981103i	$-0.561979\pi$
$653$	−9.88854	−0.386969	−0.193484	+	0.981103i	$0.561979\pi$
$654$	0	0
$655$	0	0
$656$	3.23607	0.126347
$657$	0	0
$658$	13.4164	0.523026
$659$	−2.02129	−0.0787381	−0.0393691	−	0.999225i	$-0.512535\pi$
$659$	−2.02129	−0.0787381	−0.0393691	+	0.999225i	$0.512535\pi$
$660$	0	0
$661$	5.70820	0.222023	0.111012	−	0.993819i	$-0.464591\pi$
$661$	5.70820	0.222023	0.111012	+	0.993819i	$0.464591\pi$
$662$	21.4164	0.832372
$663$	0	0
$664$	−8.18034	−0.317459
$665$	0	0
$666$	0	0
$667$	−28.8197	−1.11590
$668$	−23.0902	−0.893385
$669$	0	0
$670$	0	0
$671$	−18.5410	−0.715768
$672$	0	0
$673$	−47.3050	−1.82347	−0.911736	−	0.410777i	$-0.865258\pi$
$673$	−47.3050	−1.82347	−0.911736	+	0.410777i	$0.865258\pi$
$674$	−12.9443	−0.498595
$675$	0	0
$676$	−4.85410	−0.186696
$677$	31.2361	1.20050	0.600250	−	0.799813i	$-0.295068\pi$
$677$	31.2361	1.20050	0.600250	+	0.799813i	$0.295068\pi$
$678$	0	0
$679$	−22.4721	−0.862401
$680$	0	0
$681$	0	0
$682$	2.96556	0.113557
$683$	10.7639	0.411870	0.205935	−	0.978566i	$-0.433976\pi$
$683$	10.7639	0.411870	0.205935	+	0.978566i	$0.433976\pi$
$684$	0	0
$685$	0	0
$686$	15.4164	0.588601
$687$	0	0
$688$	−0.145898	−0.00556231
$689$	9.23607	0.351866
$690$	0	0
$691$	26.7639	1.01815	0.509074	−	0.860723i	$-0.329988\pi$
$691$	26.7639	1.01815	0.509074	+	0.860723i	$0.329988\pi$
$692$	−9.23607	−0.351103
$693$	0	0
$694$	−24.3607	−0.924719
$695$	0	0
$696$	0	0
$697$	−15.7082	−0.594991
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12.2148	−0.461346	−0.230673	−	0.973031i	$-0.574093\pi$
$701$	−12.2148	−0.461346	−0.230673	+	0.973031i	$0.574093\pi$
$702$	0	0
$703$	−47.1246	−1.77734
$704$	−1.38197	−0.0520848
$705$	0	0
$706$	−1.20163	−0.0452238
$707$	2.87539	0.108140
$708$	0	0
$709$	−6.87539	−0.258211	−0.129105	−	0.991631i	$-0.541211\pi$
$709$	−6.87539	−0.258211	−0.129105	+	0.991631i	$0.541211\pi$
$710$	0	0
$711$	0	0
$712$	−9.23607	−0.346136
$713$	6.63119	0.248340
$714$	0	0
$715$	0	0
$716$	−7.41641	−0.277164
$717$	0	0
$718$	23.8885	0.891513
$719$	7.23607	0.269860	0.134930	−	0.990855i	$-0.456919\pi$
$719$	7.23607	0.269860	0.134930	+	0.990855i	$0.456919\pi$
$720$	0	0
$721$	4.94427	0.184134
$722$	−17.0000	−0.632674
$723$	0	0
$724$	7.41641	0.275629
$725$	0	0
$726$	0	0
$727$	7.34752	0.272505	0.136252	−	0.990674i	$-0.456494\pi$
$727$	7.34752	0.272505	0.136252	+	0.990674i	$0.456494\pi$
$728$	3.52786	0.130751
$729$	0	0
$730$	0	0
$731$	0.708204	0.0261939
$732$	0	0
$733$	34.6869	1.28119	0.640595	−	0.767879i	$-0.278688\pi$
$733$	34.6869	1.28119	0.640595	+	0.767879i	$0.278688\pi$
$734$	10.2918	0.379877
$735$	0	0
$736$	−3.09017	−0.113905
$737$	3.29180	0.121255
$738$	0	0
$739$	8.65248	0.318286	0.159143	−	0.987256i	$-0.449127\pi$
$739$	8.65248	0.318286	0.159143	+	0.987256i	$0.449127\pi$
$740$	0	0
$741$	0	0
$742$	4.00000	0.146845
$743$	19.6180	0.719716	0.359858	−	0.933007i	$-0.382825\pi$
$743$	19.6180	0.719716	0.359858	+	0.933007i	$0.382825\pi$
$744$	0	0
$745$	0	0
$746$	−32.7426	−1.19879
$747$	0	0
$748$	6.70820	0.245276
$749$	−10.1115	−0.369465
$750$	0	0
$751$	−43.4164	−1.58429	−0.792144	−	0.610335i	$-0.791035\pi$
$751$	−43.4164	−1.58429	−0.792144	+	0.610335i	$0.791035\pi$
$752$	−10.8541	−0.395808
$753$	0	0
$754$	−26.6180	−0.969372
$755$	0	0
$756$	0	0
$757$	4.83282	0.175652	0.0878258	−	0.996136i	$-0.472008\pi$
$757$	4.83282	0.175652	0.0878258	+	0.996136i	$0.472008\pi$
$758$	1.23607	0.0448960
$759$	0	0
$760$	0	0
$761$	32.6525	1.18365	0.591826	−	0.806066i	$-0.298407\pi$
$761$	32.6525	1.18365	0.591826	+	0.806066i	$0.298407\pi$
$762$	0	0
$763$	−16.5836	−0.600366
$764$	−6.00000	−0.217072
$765$	0	0
$766$	−17.8885	−0.646339
$767$	18.2148	0.657698
$768$	0	0
$769$	−22.9230	−0.826624	−0.413312	−	0.910589i	$-0.635628\pi$
$769$	−22.9230	−0.826624	−0.413312	+	0.910589i	$0.635628\pi$
$770$	0	0
$771$	0	0
$772$	17.1246	0.616328
$773$	6.47214	0.232787	0.116393	−	0.993203i	$-0.462867\pi$
$773$	6.47214	0.232787	0.116393	+	0.993203i	$0.462867\pi$
$774$	0	0
$775$	0	0
$776$	18.1803	0.652636
$777$	0	0
$778$	18.2705	0.655030
$779$	19.4164	0.695665
$780$	0	0
$781$	−12.3607	−0.442300
$782$	15.0000	0.536399
$783$	0	0
$784$	−5.47214	−0.195433
$785$	0	0
$786$	0	0
$787$	−38.5623	−1.37460	−0.687299	−	0.726375i	$-0.741204\pi$
$787$	−38.5623	−1.37460	−0.687299	+	0.726375i	$0.741204\pi$
$788$	−1.52786	−0.0544279
$789$	0	0
$790$	0	0
$791$	15.2361	0.541732
$792$	0	0
$793$	−38.2918	−1.35978
$794$	32.4721	1.15239
$795$	0	0
$796$	−15.6180	−0.553567
$797$	−34.4721	−1.22107	−0.610533	−	0.791991i	$-0.709045\pi$
$797$	−34.4721	−1.22107	−0.610533	+	0.791991i	$0.709045\pi$
$798$	0	0
$799$	52.6869	1.86393
$800$	0	0
$801$	0	0
$802$	−10.4721	−0.369784
$803$	16.5836	0.585222
$804$	0	0
$805$	0	0
$806$	6.12461	0.215730
$807$	0	0
$808$	−2.32624	−0.0818368
$809$	29.5279	1.03814	0.519072	−	0.854730i	$-0.326278\pi$
$809$	29.5279	1.03814	0.519072	+	0.854730i	$0.326278\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	−11.5279	−0.404549
$813$	0	0
$814$	−10.8541	−0.380436
$815$	0	0
$816$	0	0
$817$	−0.875388	−0.0306260
$818$	−13.5623	−0.474195
$819$	0	0
$820$	0	0
$821$	−36.2148	−1.26390	−0.631952	−	0.775007i	$-0.717746\pi$
$821$	−36.2148	−1.26390	−0.631952	+	0.775007i	$0.717746\pi$
$822$	0	0
$823$	15.1246	0.527211	0.263605	−	0.964631i	$-0.415088\pi$
$823$	15.1246	0.527211	0.263605	+	0.964631i	$0.415088\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	7.88854	0.274478
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	0	0
$829$	34.1803	1.18713	0.593566	−	0.804785i	$-0.297720\pi$
$829$	34.1803	1.18713	0.593566	+	0.804785i	$0.297720\pi$
$830$	0	0
$831$	0	0
$832$	−2.85410	−0.0989482
$833$	26.5623	0.920329
$834$	0	0
$835$	0	0
$836$	−8.29180	−0.286778
$837$	0	0
$838$	−8.94427	−0.308975
$839$	11.1246	0.384064	0.192032	−	0.981389i	$-0.438492\pi$
$839$	11.1246	0.384064	0.192032	+	0.981389i	$0.438492\pi$
$840$	0	0
$841$	57.9787	1.99927
$842$	−14.4721	−0.498743
$843$	0	0
$844$	3.41641	0.117598
$845$	0	0
$846$	0	0
$847$	−11.2361	−0.386076
$848$	−3.23607	−0.111127
$849$	0	0
$850$	0	0
$851$	−24.2705	−0.831982
$852$	0	0
$853$	29.0902	0.996028	0.498014	−	0.867169i	$-0.334063\pi$
$853$	29.0902	0.996028	0.498014	+	0.867169i	$0.334063\pi$
$854$	−16.5836	−0.567479
$855$	0	0
$856$	8.18034	0.279598
$857$	12.3262	0.421056	0.210528	−	0.977588i	$-0.432482\pi$
$857$	12.3262	0.421056	0.210528	+	0.977588i	$0.432482\pi$
$858$	0	0
$859$	−15.7082	−0.535957	−0.267979	−	0.963425i	$-0.586356\pi$
$859$	−15.7082	−0.535957	−0.267979	+	0.963425i	$0.586356\pi$
$860$	0	0
$861$	0	0
$862$	−21.7082	−0.739384
$863$	−8.56231	−0.291464	−0.145732	−	0.989324i	$-0.546554\pi$
$863$	−8.56231	−0.291464	−0.145732	+	0.989324i	$0.546554\pi$
$864$	0	0
$865$	0	0
$866$	−0.180340	−0.00612820
$867$	0	0
$868$	2.65248	0.0900309
$869$	1.58359	0.0537197
$870$	0	0
$871$	6.79837	0.230354
$872$	13.4164	0.454337
$873$	0	0
$874$	−18.5410	−0.627159
$875$	0	0
$876$	0	0
$877$	−26.3951	−0.891300	−0.445650	−	0.895207i	$-0.647027\pi$
$877$	−26.3951	−0.891300	−0.445650	+	0.895207i	$0.647027\pi$
$878$	−28.3262	−0.955964
$879$	0	0
$880$	0	0
$881$	5.05573	0.170332	0.0851659	−	0.996367i	$-0.472858\pi$
$881$	5.05573	0.170332	0.0851659	+	0.996367i	$0.472858\pi$
$882$	0	0
$883$	−46.9787	−1.58096	−0.790480	−	0.612488i	$-0.790169\pi$
$883$	−46.9787	−1.58096	−0.790480	+	0.612488i	$0.790169\pi$
$884$	13.8541	0.465964
$885$	0	0
$886$	−32.6525	−1.09698
$887$	54.4721	1.82900	0.914498	−	0.404591i	$-0.132586\pi$
$887$	54.4721	1.82900	0.914498	+	0.404591i	$0.132586\pi$
$888$	0	0
$889$	16.5836	0.556196
$890$	0	0
$891$	0	0
$892$	21.7082	0.726844
$893$	−65.1246	−2.17931
$894$	0	0
$895$	0	0
$896$	−1.23607	−0.0412941
$897$	0	0
$898$	2.94427	0.0982516
$899$	−20.0132	−0.667476
$900$	0	0
$901$	15.7082	0.523316
$902$	4.47214	0.148906
$903$	0	0
$904$	−12.3262	−0.409965
$905$	0	0
$906$	0	0
$907$	−23.1459	−0.768547	−0.384273	−	0.923219i	$-0.625548\pi$
$907$	−23.1459	−0.768547	−0.384273	+	0.923219i	$0.625548\pi$
$908$	−16.4721	−0.546647
$909$	0	0
$910$	0	0
$911$	−37.5279	−1.24335	−0.621677	−	0.783274i	$-0.713548\pi$
$911$	−37.5279	−1.24335	−0.621677	+	0.783274i	$0.713548\pi$
$912$	0	0
$913$	−11.3050	−0.374139
$914$	−22.9443	−0.758929
$915$	0	0
$916$	−1.70820	−0.0564406
$917$	−22.1115	−0.730185
$918$	0	0
$919$	21.8885	0.722036	0.361018	−	0.932559i	$-0.382429\pi$
$919$	21.8885	0.722036	0.361018	+	0.932559i	$0.382429\pi$
$920$	0	0
$921$	0	0
$922$	−15.9787	−0.526231
$923$	−25.5279	−0.840260
$924$	0	0
$925$	0	0
$926$	−2.29180	−0.0753131
$927$	0	0
$928$	9.32624	0.306149
$929$	24.5410	0.805165	0.402582	−	0.915384i	$-0.368113\pi$
$929$	24.5410	0.805165	0.402582	+	0.915384i	$0.368113\pi$
$930$	0	0
$931$	−32.8328	−1.07605
$932$	7.32624	0.239979
$933$	0	0
$934$	−2.29180	−0.0749899
$935$	0	0
$936$	0	0
$937$	27.4164	0.895655	0.447828	−	0.894120i	$-0.352198\pi$
$937$	27.4164	0.895655	0.447828	+	0.894120i	$0.352198\pi$
$938$	2.94427	0.0961339
$939$	0	0
$940$	0	0
$941$	22.4508	0.731877	0.365938	−	0.930639i	$-0.380748\pi$
$941$	22.4508	0.731877	0.365938	+	0.930639i	$0.380748\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	−6.38197	−0.207715
$945$	0	0
$946$	−0.201626	−0.00655543
$947$	−55.3050	−1.79717	−0.898585	−	0.438800i	$-0.855404\pi$
$947$	−55.3050	−1.79717	−0.898585	+	0.438800i	$0.855404\pi$
$948$	0	0
$949$	34.2492	1.11178
$950$	0	0
$951$	0	0
$952$	6.00000	0.194461
$953$	−1.63932	−0.0531028	−0.0265514	−	0.999647i	$-0.508453\pi$
$953$	−1.63932	−0.0531028	−0.0265514	+	0.999647i	$0.508453\pi$
$954$	0	0
$955$	0	0
$956$	26.6525	0.862003
$957$	0	0
$958$	6.65248	0.214932
$959$	5.70820	0.184328
$960$	0	0
$961$	−26.3951	−0.851456
$962$	−22.4164	−0.722734
$963$	0	0
$964$	0.0901699	0.00290418
$965$	0	0
$966$	0	0
$967$	−39.0132	−1.25458	−0.627289	−	0.778786i	$-0.715836\pi$
$967$	−39.0132	−1.25458	−0.627289	+	0.778786i	$0.715836\pi$
$968$	9.09017	0.292169
$969$	0	0
$970$	0	0
$971$	−52.0902	−1.67165	−0.835827	−	0.548994i	$-0.815011\pi$
$971$	−52.0902	−1.67165	−0.835827	+	0.548994i	$0.815011\pi$
$972$	0	0
$973$	−26.2492	−0.841511
$974$	−17.5967	−0.563836
$975$	0	0
$976$	13.4164	0.429449
$977$	−54.2837	−1.73669	−0.868344	−	0.495962i	$-0.834815\pi$
$977$	−54.2837	−1.73669	−0.868344	+	0.495962i	$0.834815\pi$
$978$	0	0
$979$	−12.7639	−0.407937
$980$	0	0
$981$	0	0
$982$	2.61803	0.0835448
$983$	−24.9787	−0.796697	−0.398349	−	0.917234i	$-0.630417\pi$
$983$	−24.9787	−0.796697	−0.398349	+	0.917234i	$0.630417\pi$
$984$	0	0
$985$	0	0
$986$	−45.2705	−1.44171
$987$	0	0
$988$	−17.1246	−0.544806
$989$	−0.450850	−0.0143362
$990$	0	0
$991$	−8.68692	−0.275949	−0.137975	−	0.990436i	$-0.544059\pi$
$991$	−8.68692	−0.275949	−0.137975	+	0.990436i	$0.544059\pi$
$992$	−2.14590	−0.0681323
$993$	0	0
$994$	−11.0557	−0.350666
$995$	0	0
$996$	0	0
$997$	38.2705	1.21204	0.606020	−	0.795450i	$-0.292765\pi$
$997$	38.2705	1.21204	0.606020	+	0.795450i	$0.292765\pi$
$998$	−13.2361	−0.418980
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2250.2.a.c.1.2		2
3.2	odd	2		750.2.a.f.1.2	yes	2
5.2	odd	4		2250.2.c.b.1999.2		4
5.3	odd	4		2250.2.c.b.1999.3		4
5.4	even	2		2250.2.a.n.1.1		2
12.11	even	2		6000.2.a.y.1.1		2
15.2	even	4		750.2.c.b.499.4		4
15.8	even	4		750.2.c.b.499.1		4
15.14	odd	2		750.2.a.c.1.1	✓	2
60.23	odd	4		6000.2.f.a.1249.4		4
60.47	odd	4		6000.2.f.a.1249.1		4
60.59	even	2		6000.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.2.a.c.1.1	✓	2	15.14	odd	2
750.2.a.f.1.2	yes	2	3.2	odd	2
750.2.c.b.499.1		4	15.8	even	4
750.2.c.b.499.4		4	15.2	even	4
2250.2.a.c.1.2		2	1.1	even	1	trivial
2250.2.a.n.1.1		2	5.4	even	2
2250.2.c.b.1999.2		4	5.2	odd	4
2250.2.c.b.1999.3		4	5.3	odd	4
6000.2.a.d.1.2		2	60.59	even	2
6000.2.a.y.1.1		2	12.11	even	2
6000.2.f.a.1249.1		4	60.47	odd	4
6000.2.f.a.1249.4		4	60.23	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:	\( N \)	\(=\)	\( 2250 = 2 \cdot 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2250.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.23607 q^{7} -1.00000 q^{8} -1.38197 q^{11} -2.85410 q^{13} -1.23607 q^{14} +1.00000 q^{16} -4.85410 q^{17} +6.00000 q^{19} +1.38197 q^{22} +3.09017 q^{23} +2.85410 q^{26} +1.23607 q^{28} -9.32624 q^{29} +2.14590 q^{31} -1.00000 q^{32} +4.85410 q^{34} -7.85410 q^{37} -6.00000 q^{38} +3.23607 q^{41} -0.145898 q^{43} -1.38197 q^{44} -3.09017 q^{46} -10.8541 q^{47} -5.47214 q^{49} -2.85410 q^{52} -3.23607 q^{53} -1.23607 q^{56} +9.32624 q^{58} -6.38197 q^{59} +13.4164 q^{61} -2.14590 q^{62} +1.00000 q^{64} -2.38197 q^{67} -4.85410 q^{68} +8.94427 q^{71} -12.0000 q^{73} +7.85410 q^{74} +6.00000 q^{76} -1.70820 q^{77} -1.14590 q^{79} -3.23607 q^{82} +8.18034 q^{83} +0.145898 q^{86} +1.38197 q^{88} +9.23607 q^{89} -3.52786 q^{91} +3.09017 q^{92} +10.8541 q^{94} -18.1803 q^{97} +5.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{7} - 2 q^{8} - 5 q^{11} + q^{13} + 2 q^{14} + 2 q^{16} - 3 q^{17} + 12 q^{19} + 5 q^{22} - 5 q^{23} - q^{26} - 2 q^{28} - 3 q^{29} + 11 q^{31} - 2 q^{32} + 3 q^{34} - 9 q^{37} - 12 q^{38} + 2 q^{41} - 7 q^{43} - 5 q^{44} + 5 q^{46} - 15 q^{47} - 2 q^{49} + q^{52} - 2 q^{53} + 2 q^{56} + 3 q^{58} - 15 q^{59} - 11 q^{62} + 2 q^{64} - 7 q^{67} - 3 q^{68} - 24 q^{73} + 9 q^{74} + 12 q^{76} + 10 q^{77} - 9 q^{79} - 2 q^{82} - 6 q^{83} + 7 q^{86} + 5 q^{88} + 14 q^{89} - 16 q^{91} - 5 q^{92} + 15 q^{94} - 14 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^7 - 2 * q^8 - 5 * q^11 + q^13 + 2 * q^14 + 2 * q^16 - 3 * q^17 + 12 * q^19 + 5 * q^22 - 5 * q^23 - q^26 - 2 * q^28 - 3 * q^29 + 11 * q^31 - 2 * q^32 + 3 * q^34 - 9 * q^37 - 12 * q^38 + 2 * q^41 - 7 * q^43 - 5 * q^44 + 5 * q^46 - 15 * q^47 - 2 * q^49 + q^52 - 2 * q^53 + 2 * q^56 + 3 * q^58 - 15 * q^59 - 11 * q^62 + 2 * q^64 - 7 * q^67 - 3 * q^68 - 24 * q^73 + 9 * q^74 + 12 * q^76 + 10 * q^77 - 9 * q^79 - 2 * q^82 - 6 * q^83 + 7 * q^86 + 5 * q^88 + 14 * q^89 - 16 * q^91 - 5 * q^92 + 15 * q^94 - 14 * q^97 + 2 * q^98

Introduction

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