LMFDB - Embedded newform 1225.2.b.k.99.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(99,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1225.99");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$9.78167424761$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 175)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			99.4
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1225.99
Dual form			1225.2.b.k.99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803i q^{2} +1.23607i q^{3} -0.618034 q^{4} -2.00000 q^{6} +2.23607i q^{8} +1.47214 q^{9} +O(q^{10})$	\(q+1.61803i q^{2} +1.23607i q^{3} -0.618034 q^{4} -2.00000 q^{6} +2.23607i q^{8} +1.47214 q^{9} +4.23607 q^{11} -0.763932i q^{12} -3.23607i q^{13} -4.85410 q^{16} +6.47214i q^{17} +2.38197i q^{18} +4.47214 q^{19} +6.85410i q^{22} -1.76393i q^{23} -2.76393 q^{24} +5.23607 q^{26} +5.52786i q^{27} -5.00000 q^{29} +9.70820 q^{31} -3.38197i q^{32} +5.23607i q^{33} -10.4721 q^{34} -0.909830 q^{36} +3.00000i q^{37} +7.23607i q^{38} +4.00000 q^{39} -9.23607 q^{41} -6.23607i q^{43} -2.61803 q^{44} +2.85410 q^{46} +2.00000i q^{47} -6.00000i q^{48} -8.00000 q^{51} +2.00000i q^{52} +0.472136i q^{53} -8.94427 q^{54} +5.52786i q^{57} -8.09017i q^{58} -1.70820 q^{59} -3.70820 q^{61} +15.7082i q^{62} -4.23607 q^{64} -8.47214 q^{66} +0.236068i q^{67} -4.00000i q^{68} +2.18034 q^{69} -4.70820 q^{71} +3.29180i q^{72} -13.2361i q^{73} -4.85410 q^{74} -2.76393 q^{76} +6.47214i q^{78} -11.1803 q^{79} -2.41641 q^{81} -14.9443i q^{82} +5.70820i q^{83} +10.0902 q^{86} -6.18034i q^{87} +9.47214i q^{88} +12.7639 q^{89} +1.09017i q^{92} +12.0000i q^{93} -3.23607 q^{94} +4.18034 q^{96} -0.763932i q^{97} +6.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 8 q^{6} - 12 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 8 * q^6 - 12 * q^9	$ 4 q + 2 q^{4} - 8 q^{6} - 12 q^{9} + 8 q^{11} - 6 q^{16} - 20 q^{24} + 12 q^{26} - 20 q^{29} + 12 q^{31} - 24 q^{34} - 26 q^{36} + 16 q^{39} - 28 q^{41} - 6 q^{44} - 2 q^{46} - 32 q^{51} + 20 q^{59} + 12 q^{61} - 8 q^{64} - 16 q^{66} - 36 q^{69} + 8 q^{71} - 6 q^{74} - 20 q^{76} + 44 q^{81} + 18 q^{86} + 60 q^{89} - 4 q^{94} - 28 q^{96} + 16 q^{99}+O(q^{100}) $ 4 * q + 2 * q^4 - 8 * q^6 - 12 * q^9 + 8 * q^11 - 6 * q^16 - 20 * q^24 + 12 * q^26 - 20 * q^29 + 12 * q^31 - 24 * q^34 - 26 * q^36 + 16 * q^39 - 28 * q^41 - 6 * q^44 - 2 * q^46 - 32 * q^51 + 20 * q^59 + 12 * q^61 - 8 * q^64 - 16 * q^66 - 36 * q^69 + 8 * q^71 - 6 * q^74 - 20 * q^76 + 44 * q^81 + 18 * q^86 + 60 * q^89 - 4 * q^94 - 28 * q^96 + 16 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times$.

$n$	$101$	$1177$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.61803i	1.14412i	0.820211	+	0.572061i	$0.193856\pi$
$2$	1.61803i	1.14412i	−0.820211	+	0.572061i	$0.806144\pi$
$3$	1.23607i	0.713644i	0.934172	+	0.356822i	$0.116140\pi$
$3$	1.23607i	0.713644i	−0.934172	+	0.356822i	$0.883860\pi$
$4$	−0.618034	−0.309017
$5$	0	0
$5$	0	0
$6$	−2.00000	−0.816497
$7$	0	0
$7$	0	0
$8$	2.23607i	0.790569i
$9$	1.47214	0.490712
$10$	0	0
$11$	4.23607	1.27722	0.638611	−	0.769529i	$-0.279509\pi$
$11$	4.23607	1.27722	0.638611	+	0.769529i	$0.279509\pi$
$12$	− 0.763932i	− 0.220528i
$13$	− 3.23607i	− 0.897524i	−0.893651	−	0.448762i	$-0.851865\pi$
$13$	− 3.23607i	− 0.897524i	0.893651	−	0.448762i	$-0.148135\pi$
$14$	0	0
$15$	0	0
$16$	−4.85410	−1.21353
$17$	6.47214i	1.56972i	0.619671	+	0.784862i	$0.287266\pi$
$17$	6.47214i	1.56972i	−0.619671	+	0.784862i	$0.712734\pi$
$18$	2.38197i	0.561435i
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	0	0
$21$	0	0
$22$	6.85410i	1.46130i
$23$	− 1.76393i	− 0.367805i	−0.982944	−	0.183903i	$-0.941127\pi$
$23$	− 1.76393i	− 0.367805i	0.982944	−	0.183903i	$-0.0588731\pi$
$24$	−2.76393	−0.564185
$25$	0	0
$26$	5.23607	1.02688
$27$	5.52786i	1.06384i
$28$	0	0
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	9.70820	1.74364	0.871822	−	0.489822i	$-0.162938\pi$
$31$	9.70820	1.74364	0.871822	+	0.489822i	$0.162938\pi$
$32$	− 3.38197i	− 0.597853i
$33$	5.23607i	0.911482i
$34$	−10.4721	−1.79596
$35$	0	0
$36$	−0.909830	−0.151638
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	7.23607i	1.17385i
$39$	4.00000	0.640513
$40$	0	0
$41$	−9.23607	−1.44243	−0.721216	−	0.692711i	$-0.756416\pi$
$41$	−9.23607	−1.44243	−0.721216	+	0.692711i	$0.756416\pi$
$42$	0	0
$43$	− 6.23607i	− 0.950991i	−0.879718	−	0.475496i	$-0.842269\pi$
$43$	− 6.23607i	− 0.950991i	0.879718	−	0.475496i	$-0.157731\pi$
$44$	−2.61803	−0.394683
$45$	0	0
$46$	2.85410	0.420814
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	− 6.00000i	− 0.866025i
$49$	0	0
$50$	0	0
$51$	−8.00000	−1.12022
$52$	2.00000i	0.277350i
$53$	0.472136i	0.0648529i	0.999474	+	0.0324264i	$0.0103235\pi$
$53$	0.472136i	0.0648529i	−0.999474	+	0.0324264i	$0.989677\pi$
$54$	−8.94427	−1.21716
$55$	0	0
$56$	0	0
$57$	5.52786i	0.732183i
$58$	− 8.09017i	− 1.06229i
$59$	−1.70820	−0.222389	−0.111195	−	0.993799i	$-0.535468\pi$
$59$	−1.70820	−0.222389	−0.111195	+	0.993799i	$0.535468\pi$
$60$	0	0
$61$	−3.70820	−0.474787	−0.237393	−	0.971414i	$-0.576293\pi$
$61$	−3.70820	−0.474787	−0.237393	+	0.971414i	$0.576293\pi$
$62$	15.7082i	1.99494i
$63$	0	0
$64$	−4.23607	−0.529508
$65$	0	0
$66$	−8.47214	−1.04285
$67$	0.236068i	0.0288403i	0.999896	+	0.0144201i	$0.00459023\pi$
$67$	0.236068i	0.0288403i	−0.999896	+	0.0144201i	$0.995410\pi$
$68$	− 4.00000i	− 0.485071i
$69$	2.18034	0.262482
$70$	0	0
$71$	−4.70820	−0.558761	−0.279381	−	0.960180i	$-0.590129\pi$
$71$	−4.70820	−0.558761	−0.279381	+	0.960180i	$0.590129\pi$
$72$	3.29180i	0.387942i
$73$	− 13.2361i	− 1.54916i	−0.632473	−	0.774582i	$-0.717960\pi$
$73$	− 13.2361i	− 1.54916i	0.632473	−	0.774582i	$-0.282040\pi$
$74$	−4.85410	−0.564278
$75$	0	0
$76$	−2.76393	−0.317045
$77$	0	0
$78$	6.47214i	0.732825i
$79$	−11.1803	−1.25789	−0.628943	−	0.777451i	$-0.716512\pi$
$79$	−11.1803	−1.25789	−0.628943	+	0.777451i	$0.716512\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	− 14.9443i	− 1.65032i
$83$	5.70820i	0.626557i	0.949661	+	0.313278i	$0.101427\pi$
$83$	5.70820i	0.626557i	−0.949661	+	0.313278i	$0.898573\pi$
$84$	0	0
$85$	0	0
$86$	10.0902	1.08805
$87$	− 6.18034i	− 0.662602i
$88$	9.47214i	1.00973i
$89$	12.7639	1.35297	0.676487	−	0.736455i	$-0.263501\pi$
$89$	12.7639	1.35297	0.676487	+	0.736455i	$0.263501\pi$
$90$	0	0
$91$	0	0
$92$	1.09017i	0.113658i
$93$	12.0000i	1.24434i
$94$	−3.23607	−0.333775
$95$	0	0
$96$	4.18034	0.426654
$97$	− 0.763932i	− 0.0775655i	−0.999248	−	0.0387828i	$-0.987652\pi$
$97$	− 0.763932i	− 0.0775655i	0.999248	−	0.0387828i	$-0.0123480\pi$
$98$	0	0
$99$	6.23607	0.626748
$100$	0	0
$101$	−9.23607	−0.919023	−0.459512	−	0.888172i	$-0.651976\pi$
$101$	−9.23607	−0.919023	−0.459512	+	0.888172i	$0.651976\pi$
$102$	− 12.9443i	− 1.28167i
$103$	− 0.472136i	− 0.0465209i	−0.999729	−	0.0232605i	$-0.992595\pi$
$103$	− 0.472136i	− 0.0465209i	0.999729	−	0.0232605i	$-0.00740471\pi$
$104$	7.23607	0.709555
$105$	0	0
$106$	−0.763932	−0.0741996
$107$	8.00000i	0.773389i	0.922208	+	0.386695i	$0.126383\pi$
$107$	8.00000i	0.773389i	−0.922208	+	0.386695i	$0.873617\pi$
$108$	− 3.41641i	− 0.328744i
$109$	18.4164	1.76397	0.881986	−	0.471276i	$-0.156206\pi$
$109$	18.4164	1.76397	0.881986	+	0.471276i	$0.156206\pi$
$110$	0	0
$111$	−3.70820	−0.351967
$112$	0	0
$113$	− 12.4164i	− 1.16804i	−0.811740	−	0.584019i	$-0.801479\pi$
$113$	− 12.4164i	− 1.16804i	0.811740	−	0.584019i	$-0.198521\pi$
$114$	−8.94427	−0.837708
$115$	0	0
$116$	3.09017	0.286915
$117$	− 4.76393i	− 0.440426i
$118$	− 2.76393i	− 0.254441i
$119$	0	0
$120$	0	0
$121$	6.94427	0.631297
$122$	− 6.00000i	− 0.543214i
$123$	− 11.4164i	− 1.02938i
$124$	−6.00000	−0.538816
$125$	0	0
$126$	0	0
$127$	− 17.6525i	− 1.56640i	−0.621767	−	0.783202i	$-0.713585\pi$
$127$	− 17.6525i	− 1.56640i	0.621767	−	0.783202i	$-0.286415\pi$
$128$	− 13.6180i	− 1.20368i
$129$	7.70820	0.678670
$130$	0	0
$131$	−0.944272	−0.0825014	−0.0412507	−	0.999149i	$-0.513134\pi$
$131$	−0.944272	−0.0825014	−0.0412507	+	0.999149i	$0.513134\pi$
$132$	− 3.23607i	− 0.281664i
$133$	0	0
$134$	−0.381966	−0.0329968
$135$	0	0
$136$	−14.4721	−1.24098
$137$	6.94427i	0.593289i	0.954988	+	0.296645i	$0.0958677\pi$
$137$	6.94427i	0.593289i	−0.954988	+	0.296645i	$0.904132\pi$
$138$	3.52786i	0.300312i
$139$	−20.6525	−1.75172	−0.875860	−	0.482565i	$-0.839705\pi$
$139$	−20.6525	−1.75172	−0.875860	+	0.482565i	$0.839705\pi$
$140$	0	0
$141$	−2.47214	−0.208191
$142$	− 7.61803i	− 0.639291i
$143$	− 13.7082i	− 1.14634i
$144$	−7.14590	−0.595492
$145$	0	0
$146$	21.4164	1.77243
$147$	0	0
$148$	− 1.85410i	− 0.152406i
$149$	13.9443	1.14236	0.571180	−	0.820825i	$-0.306486\pi$
$149$	13.9443	1.14236	0.571180	+	0.820825i	$0.306486\pi$
$150$	0	0
$151$	−15.7639	−1.28285	−0.641425	−	0.767185i	$-0.721657\pi$
$151$	−15.7639	−1.28285	−0.641425	+	0.767185i	$0.721657\pi$
$152$	10.0000i	0.811107i
$153$	9.52786i	0.770282i
$154$	0	0
$155$	0	0
$156$	−2.47214	−0.197929
$157$	− 5.23607i	− 0.417884i	−0.977928	−	0.208942i	$-0.932998\pi$
$157$	− 5.23607i	− 0.417884i	0.977928	−	0.208942i	$-0.0670019\pi$
$158$	− 18.0902i	− 1.43918i
$159$	−0.583592	−0.0462819
$160$	0	0
$161$	0	0
$162$	− 3.90983i	− 0.307185i
$163$	10.4721i	0.820241i	0.912031	+	0.410120i	$0.134513\pi$
$163$	10.4721i	0.820241i	−0.912031	+	0.410120i	$0.865487\pi$
$164$	5.70820	0.445736
$165$	0	0
$166$	−9.23607	−0.716858
$167$	− 0.763932i	− 0.0591148i	−0.999563	−	0.0295574i	$-0.990590\pi$
$167$	− 0.763932i	− 0.0591148i	0.999563	−	0.0295574i	$-0.00940979\pi$
$168$	0	0
$169$	2.52786	0.194451
$170$	0	0
$171$	6.58359	0.503460
$172$	3.85410i	0.293873i
$173$	− 20.4721i	− 1.55647i	−0.627975	−	0.778234i	$-0.716116\pi$
$173$	− 20.4721i	− 1.55647i	0.627975	−	0.778234i	$-0.283884\pi$
$174$	10.0000	0.758098
$175$	0	0
$176$	−20.5623	−1.54994
$177$	− 2.11146i	− 0.158707i
$178$	20.6525i	1.54797i
$179$	−3.41641	−0.255354	−0.127677	−	0.991816i	$-0.540752\pi$
$179$	−3.41641	−0.255354	−0.127677	+	0.991816i	$0.540752\pi$
$180$	0	0
$181$	14.1803	1.05402	0.527008	−	0.849860i	$-0.323314\pi$
$181$	14.1803	1.05402	0.527008	+	0.849860i	$0.323314\pi$
$182$	0	0
$183$	− 4.58359i	− 0.338829i
$184$	3.94427	0.290776
$185$	0	0
$186$	−19.4164	−1.42368
$187$	27.4164i	2.00489i
$188$	− 1.23607i	− 0.0901495i
$189$	0	0
$190$	0	0
$191$	−2.47214	−0.178877	−0.0894387	−	0.995992i	$-0.528507\pi$
$191$	−2.47214	−0.178877	−0.0894387	+	0.995992i	$0.528507\pi$
$192$	− 5.23607i	− 0.377881i
$193$	14.4164i	1.03772i	0.854861	+	0.518858i	$0.173643\pi$
$193$	14.4164i	1.03772i	−0.854861	+	0.518858i	$0.826357\pi$
$194$	1.23607	0.0887445
$195$	0	0
$196$	0	0
$197$	7.47214i	0.532368i	0.963922	+	0.266184i	$0.0857628\pi$
$197$	7.47214i	0.532368i	−0.963922	+	0.266184i	$0.914237\pi$
$198$	10.0902i	0.717077i
$199$	2.76393	0.195930	0.0979650	−	0.995190i	$-0.468767\pi$
$199$	2.76393	0.195930	0.0979650	+	0.995190i	$0.468767\pi$
$200$	0	0
$201$	−0.291796	−0.0205817
$202$	− 14.9443i	− 1.05148i
$203$	0	0
$204$	4.94427	0.346168
$205$	0	0
$206$	0.763932	0.0532257
$207$	− 2.59675i	− 0.180486i
$208$	15.7082i	1.08917i
$209$	18.9443	1.31040
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	− 0.291796i	− 0.0200406i
$213$	− 5.81966i	− 0.398757i
$214$	−12.9443	−0.884852
$215$	0	0
$216$	−12.3607	−0.841038
$217$	0	0
$218$	29.7984i	2.01820i
$219$	16.3607	1.10555
$220$	0	0
$221$	20.9443	1.40886
$222$	− 6.00000i	− 0.402694i
$223$	− 2.18034i	− 0.146006i	−0.997332	−	0.0730032i	$-0.976742\pi$
$223$	− 2.18034i	− 0.146006i	0.997332	−	0.0730032i	$-0.0232583\pi$
$224$	0	0
$225$	0	0
$226$	20.0902	1.33638
$227$	5.41641i	0.359500i	0.983712	+	0.179750i	$0.0575288\pi$
$227$	5.41641i	0.359500i	−0.983712	+	0.179750i	$0.942471\pi$
$228$	− 3.41641i	− 0.226257i
$229$	4.47214	0.295527	0.147764	−	0.989023i	$-0.452793\pi$
$229$	4.47214	0.295527	0.147764	+	0.989023i	$0.452793\pi$
$230$	0	0
$231$	0	0
$232$	− 11.1803i	− 0.734025i
$233$	9.94427i	0.651471i	0.945461	+	0.325735i	$0.105612\pi$
$233$	9.94427i	0.651471i	−0.945461	+	0.325735i	$0.894388\pi$
$234$	7.70820	0.503901
$235$	0	0
$236$	1.05573	0.0687220
$237$	− 13.8197i	− 0.897683i
$238$	0	0
$239$	14.4721	0.936125	0.468062	−	0.883695i	$-0.344952\pi$
$239$	14.4721	0.936125	0.468062	+	0.883695i	$0.344952\pi$
$240$	0	0
$241$	12.4721	0.803401	0.401700	−	0.915771i	$-0.368419\pi$
$241$	12.4721	0.803401	0.401700	+	0.915771i	$0.368419\pi$
$242$	11.2361i	0.722282i
$243$	13.5967i	0.872232i
$244$	2.29180	0.146717
$245$	0	0
$246$	18.4721	1.17774
$247$	− 14.4721i	− 0.920840i
$248$	21.7082i	1.37847i
$249$	−7.05573	−0.447139
$250$	0	0
$251$	2.47214	0.156040	0.0780199	−	0.996952i	$-0.475140\pi$
$251$	2.47214	0.156040	0.0780199	+	0.996952i	$0.475140\pi$
$252$	0	0
$253$	− 7.47214i	− 0.469769i
$254$	28.5623	1.79216
$255$	0	0
$256$	13.5623	0.847644
$257$	− 18.6525i	− 1.16351i	−0.813364	−	0.581755i	$-0.802366\pi$
$257$	− 18.6525i	− 1.16351i	0.813364	−	0.581755i	$-0.197634\pi$
$258$	12.4721i	0.776481i
$259$	0	0
$260$	0	0
$261$	−7.36068	−0.455615
$262$	− 1.52786i	− 0.0943918i
$263$	− 11.7639i	− 0.725395i	−0.931907	−	0.362698i	$-0.881856\pi$
$263$	− 11.7639i	− 0.725395i	0.931907	−	0.362698i	$-0.118144\pi$
$264$	−11.7082	−0.720590
$265$	0	0
$266$	0	0
$267$	15.7771i	0.965542i
$268$	− 0.145898i	− 0.00891214i
$269$	1.70820	0.104151	0.0520755	−	0.998643i	$-0.483416\pi$
$269$	1.70820	0.104151	0.0520755	+	0.998643i	$0.483416\pi$
$270$	0	0
$271$	−10.2918	−0.625182	−0.312591	−	0.949888i	$-0.601197\pi$
$271$	−10.2918	−0.625182	−0.312591	+	0.949888i	$0.601197\pi$
$272$	− 31.4164i	− 1.90490i
$273$	0	0
$274$	−11.2361	−0.678796
$275$	0	0
$276$	−1.34752	−0.0811114
$277$	15.8885i	0.954650i	0.878727	+	0.477325i	$0.158394\pi$
$277$	15.8885i	0.954650i	−0.878727	+	0.477325i	$0.841606\pi$
$278$	− 33.4164i	− 2.00418i
$279$	14.2918	0.855627
$280$	0	0
$281$	29.3607	1.75151	0.875756	−	0.482755i	$-0.160364\pi$
$281$	29.3607	1.75151	0.875756	+	0.482755i	$0.160364\pi$
$282$	− 4.00000i	− 0.238197i
$283$	− 9.41641i	− 0.559747i	−0.960037	−	0.279874i	$-0.909707\pi$
$283$	− 9.41641i	− 0.559747i	0.960037	−	0.279874i	$-0.0902926\pi$
$284$	2.90983	0.172667
$285$	0	0
$286$	22.1803	1.31155
$287$	0	0
$288$	− 4.97871i	− 0.293374i
$289$	−24.8885	−1.46403
$290$	0	0
$291$	0.944272	0.0553542
$292$	8.18034i	0.478718i
$293$	9.12461i	0.533066i	0.963826	+	0.266533i	$0.0858781\pi$
$293$	9.12461i	0.533066i	−0.963826	+	0.266533i	$0.914122\pi$
$294$	0	0
$295$	0	0
$296$	−6.70820	−0.389906
$297$	23.4164i	1.35876i
$298$	22.5623i	1.30700i
$299$	−5.70820	−0.330114
$300$	0	0
$301$	0	0
$302$	− 25.5066i	− 1.46774i
$303$	− 11.4164i	− 0.655855i
$304$	−21.7082	−1.24505
$305$	0	0
$306$	−15.4164	−0.881297
$307$	− 31.4164i	− 1.79303i	−0.443014	−	0.896515i	$-0.646091\pi$
$307$	− 31.4164i	− 1.79303i	0.443014	−	0.896515i	$-0.353909\pi$
$308$	0	0
$309$	0.583592	0.0331994
$310$	0	0
$311$	20.3607	1.15455	0.577274	−	0.816550i	$-0.304116\pi$
$311$	20.3607	1.15455	0.577274	+	0.816550i	$0.304116\pi$
$312$	8.94427i	0.506370i
$313$	28.4721i	1.60934i	0.593722	+	0.804670i	$0.297658\pi$
$313$	28.4721i	1.60934i	−0.593722	+	0.804670i	$0.702342\pi$
$314$	8.47214	0.478110
$315$	0	0
$316$	6.90983	0.388708
$317$	− 19.3607i	− 1.08740i	−0.839278	−	0.543702i	$-0.817022\pi$
$317$	− 19.3607i	− 1.08740i	0.839278	−	0.543702i	$-0.182978\pi$
$318$	− 0.944272i	− 0.0529521i
$319$	−21.1803	−1.18587
$320$	0	0
$321$	−9.88854	−0.551925
$322$	0	0
$323$	28.9443i	1.61050i
$324$	1.49342	0.0829679
$325$	0	0
$326$	−16.9443	−0.938456
$327$	22.7639i	1.25885i
$328$	− 20.6525i	− 1.14034i
$329$	0	0
$330$	0	0
$331$	−11.2918	−0.620653	−0.310327	−	0.950630i	$-0.600438\pi$
$331$	−11.2918	−0.620653	−0.310327	+	0.950630i	$0.600438\pi$
$332$	− 3.52786i	− 0.193617i
$333$	4.41641i	0.242018i
$334$	1.23607	0.0676346
$335$	0	0
$336$	0	0
$337$	− 7.52786i	− 0.410069i	−0.978755	−	0.205034i	$-0.934269\pi$
$337$	− 7.52786i	− 0.410069i	0.978755	−	0.205034i	$-0.0657306\pi$
$338$	4.09017i	0.222476i
$339$	15.3475	0.833563
$340$	0	0
$341$	41.1246	2.22702
$342$	10.6525i	0.576020i
$343$	0	0
$344$	13.9443	0.751825
$345$	0	0
$346$	33.1246	1.78079
$347$	15.7639i	0.846252i	0.906071	+	0.423126i	$0.139067\pi$
$347$	15.7639i	0.846252i	−0.906071	+	0.423126i	$0.860933\pi$
$348$	3.81966i	0.204755i
$349$	−4.47214	−0.239388	−0.119694	−	0.992811i	$-0.538191\pi$
$349$	−4.47214	−0.239388	−0.119694	+	0.992811i	$0.538191\pi$
$350$	0	0
$351$	17.8885	0.954820
$352$	− 14.3262i	− 0.763591i
$353$	20.1803i	1.07409i	0.843553	+	0.537046i	$0.180460\pi$
$353$	20.1803i	1.07409i	−0.843553	+	0.537046i	$0.819540\pi$
$354$	3.41641	0.181580
$355$	0	0
$356$	−7.88854	−0.418092
$357$	0	0
$358$	− 5.52786i	− 0.292157i
$359$	−10.1246	−0.534357	−0.267178	−	0.963647i	$-0.586091\pi$
$359$	−10.1246	−0.534357	−0.267178	+	0.963647i	$0.586091\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	22.9443i	1.20592i
$363$	8.58359i	0.450522i
$364$	0	0
$365$	0	0
$366$	7.41641	0.387662
$367$	− 3.12461i	− 0.163103i	−0.996669	−	0.0815517i	$-0.974012\pi$
$367$	− 3.12461i	− 0.163103i	0.996669	−	0.0815517i	$-0.0259876\pi$
$368$	8.56231i	0.446341i
$369$	−13.5967	−0.707818
$370$	0	0
$371$	0	0
$372$	− 7.41641i	− 0.384523i
$373$	− 15.8328i	− 0.819792i	−0.912132	−	0.409896i	$-0.865565\pi$
$373$	− 15.8328i	− 0.819792i	0.912132	−	0.409896i	$-0.134435\pi$
$374$	−44.3607	−2.29384
$375$	0	0
$376$	−4.47214	−0.230633
$377$	16.1803i	0.833330i
$378$	0	0
$379$	−11.1803	−0.574295	−0.287148	−	0.957886i	$-0.592707\pi$
$379$	−11.1803	−0.574295	−0.287148	+	0.957886i	$0.592707\pi$
$380$	0	0
$381$	21.8197	1.11786
$382$	− 4.00000i	− 0.204658i
$383$	− 28.7639i	− 1.46977i	−0.678193	−	0.734884i	$-0.737237\pi$
$383$	− 28.7639i	− 1.46977i	0.678193	−	0.734884i	$-0.262763\pi$
$384$	16.8328	0.858996
$385$	0	0
$386$	−23.3262	−1.18727
$387$	− 9.18034i	− 0.466663i
$388$	0.472136i	0.0239691i
$389$	32.8885	1.66752	0.833758	−	0.552131i	$-0.186185\pi$
$389$	32.8885	1.66752	0.833758	+	0.552131i	$0.186185\pi$
$390$	0	0
$391$	11.4164	0.577353
$392$	0	0
$393$	− 1.16718i	− 0.0588767i
$394$	−12.0902	−0.609094
$395$	0	0
$396$	−3.85410	−0.193676
$397$	− 26.9443i	− 1.35229i	−0.736767	−	0.676147i	$-0.763648\pi$
$397$	− 26.9443i	− 1.35229i	0.736767	−	0.676147i	$-0.236352\pi$
$398$	4.47214i	0.224168i
$399$	0	0
$400$	0	0
$401$	11.4721	0.572891	0.286446	−	0.958097i	$-0.407526\pi$
$401$	11.4721	0.572891	0.286446	+	0.958097i	$0.407526\pi$
$402$	− 0.472136i	− 0.0235480i
$403$	− 31.4164i	− 1.56496i
$404$	5.70820	0.283994
$405$	0	0
$406$	0	0
$407$	12.7082i	0.629922i
$408$	− 17.8885i	− 0.885615i
$409$	15.5279	0.767803	0.383902	−	0.923374i	$-0.374580\pi$
$409$	15.5279	0.767803	0.383902	+	0.923374i	$0.374580\pi$
$410$	0	0
$411$	−8.58359	−0.423397
$412$	0.291796i	0.0143758i
$413$	0	0
$414$	4.20163	0.206499
$415$	0	0
$416$	−10.9443	−0.536587
$417$	− 25.5279i	− 1.25010i
$418$	30.6525i	1.49926i
$419$	−3.81966	−0.186603	−0.0933013	−	0.995638i	$-0.529742\pi$
$419$	−3.81966	−0.186603	−0.0933013	+	0.995638i	$0.529742\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	19.4164i	0.945176i
$423$	2.94427i	0.143155i
$424$	−1.05573	−0.0512707
$425$	0	0
$426$	9.41641	0.456226
$427$	0	0
$428$	− 4.94427i	− 0.238990i
$429$	16.9443	0.818077
$430$	0	0
$431$	26.4721	1.27512	0.637559	−	0.770402i	$-0.279944\pi$
$431$	26.4721	1.27512	0.637559	+	0.770402i	$0.279944\pi$
$432$	− 26.8328i	− 1.29099i
$433$	16.3607i	0.786244i	0.919486	+	0.393122i	$0.128605\pi$
$433$	16.3607i	0.786244i	−0.919486	+	0.393122i	$0.871395\pi$
$434$	0	0
$435$	0	0
$436$	−11.3820	−0.545097
$437$	− 7.88854i	− 0.377360i
$438$	26.4721i	1.26489i
$439$	−21.7082	−1.03608	−0.518038	−	0.855358i	$-0.673337\pi$
$439$	−21.7082	−1.03608	−0.518038	+	0.855358i	$0.673337\pi$
$440$	0	0
$441$	0	0
$442$	33.8885i	1.61191i
$443$	− 7.41641i	− 0.352364i	−0.984358	−	0.176182i	$-0.943625\pi$
$443$	− 7.41641i	− 0.352364i	0.984358	−	0.176182i	$-0.0563748\pi$
$444$	2.29180	0.108764
$445$	0	0
$446$	3.52786	0.167049
$447$	17.2361i	0.815238i
$448$	0	0
$449$	−29.4721	−1.39088	−0.695438	−	0.718586i	$-0.744790\pi$
$449$	−29.4721	−1.39088	−0.695438	+	0.718586i	$0.744790\pi$
$450$	0	0
$451$	−39.1246	−1.84231
$452$	7.67376i	0.360943i
$453$	− 19.4853i	− 0.915499i
$454$	−8.76393	−0.411312
$455$	0	0
$456$	−12.3607	−0.578842
$457$	− 21.4721i	− 1.00442i	−0.864744	−	0.502212i	$-0.832520\pi$
$457$	− 21.4721i	− 1.00442i	0.864744	−	0.502212i	$-0.167480\pi$
$458$	7.23607i	0.338119i
$459$	−35.7771	−1.66993
$460$	0	0
$461$	−8.18034	−0.380996	−0.190498	−	0.981688i	$-0.561010\pi$
$461$	−8.18034	−0.380996	−0.190498	+	0.981688i	$0.561010\pi$
$462$	0	0
$463$	− 21.8885i	− 1.01725i	−0.860989	−	0.508623i	$-0.830155\pi$
$463$	− 21.8885i	− 1.01725i	0.860989	−	0.508623i	$-0.169845\pi$
$464$	24.2705	1.12673
$465$	0	0
$466$	−16.0902	−0.745363
$467$	10.9443i	0.506441i	0.967409	+	0.253220i	$0.0814897\pi$
$467$	10.9443i	0.506441i	−0.967409	+	0.253220i	$0.918510\pi$
$468$	2.94427i	0.136099i
$469$	0	0
$470$	0	0
$471$	6.47214	0.298220
$472$	− 3.81966i	− 0.175814i
$473$	− 26.4164i	− 1.21463i
$474$	22.3607	1.02706
$475$	0	0
$476$	0	0
$477$	0.695048i	0.0318241i
$478$	23.4164i	1.07104i
$479$	−3.81966	−0.174525	−0.0872624	−	0.996185i	$-0.527812\pi$
$479$	−3.81966	−0.174525	−0.0872624	+	0.996185i	$0.527812\pi$
$480$	0	0
$481$	9.70820	0.442656
$482$	20.1803i	0.919189i
$483$	0	0
$484$	−4.29180	−0.195082
$485$	0	0
$486$	−22.0000	−0.997940
$487$	10.2361i	0.463841i	0.972735	+	0.231920i	$0.0745008\pi$
$487$	10.2361i	0.463841i	−0.972735	+	0.231920i	$0.925499\pi$
$488$	− 8.29180i	− 0.375352i
$489$	−12.9443	−0.585360
$490$	0	0
$491$	−10.2361	−0.461947	−0.230974	−	0.972960i	$-0.574191\pi$
$491$	−10.2361	−0.461947	−0.230974	+	0.972960i	$0.574191\pi$
$492$	7.05573i	0.318097i
$493$	− 32.3607i	− 1.45745i
$494$	23.4164	1.05355
$495$	0	0
$496$	−47.1246	−2.11596
$497$	0	0
$498$	− 11.4164i	− 0.511581i
$499$	−28.9443	−1.29572	−0.647862	−	0.761758i	$-0.724337\pi$
$499$	−28.9443	−1.29572	−0.647862	+	0.761758i	$0.724337\pi$
$500$	0	0
$501$	0.944272	0.0421870
$502$	4.00000i	0.178529i
$503$	− 43.8885i	− 1.95689i	−0.206499	−	0.978447i	$-0.566207\pi$
$503$	− 43.8885i	− 1.95689i	0.206499	−	0.978447i	$-0.433793\pi$
$504$	0	0
$505$	0	0
$506$	12.0902	0.537474
$507$	3.12461i	0.138769i
$508$	10.9098i	0.484045i
$509$	9.34752	0.414322	0.207161	−	0.978307i	$-0.433578\pi$
$509$	9.34752	0.414322	0.207161	+	0.978307i	$0.433578\pi$
$510$	0	0
$511$	0	0
$512$	− 5.29180i	− 0.233867i
$513$	24.7214i	1.09147i
$514$	30.1803	1.33120
$515$	0	0
$516$	−4.76393	−0.209720
$517$	8.47214i	0.372604i
$518$	0	0
$519$	25.3050	1.11076
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	− 11.9098i	− 0.521279i
$523$	− 28.3607i	− 1.24013i	−0.784552	−	0.620063i	$-0.787107\pi$
$523$	− 28.3607i	− 1.24013i	0.784552	−	0.620063i	$-0.212893\pi$
$524$	0.583592	0.0254943
$525$	0	0
$526$	19.0344	0.829941
$527$	62.8328i	2.73704i
$528$	− 25.4164i	− 1.10611i
$529$	19.8885	0.864719
$530$	0	0
$531$	−2.51471	−0.109129
$532$	0	0
$533$	29.8885i	1.29462i
$534$	−25.5279	−1.10470
$535$	0	0
$536$	−0.527864	−0.0228003
$537$	− 4.22291i	− 0.182232i
$538$	2.76393i	0.119162i
$539$	0	0
$540$	0	0
$541$	−1.94427	−0.0835908	−0.0417954	−	0.999126i	$-0.513308\pi$
$541$	−1.94427	−0.0835908	−0.0417954	+	0.999126i	$0.513308\pi$
$542$	− 16.6525i	− 0.715285i
$543$	17.5279i	0.752193i
$544$	21.8885	0.938464
$545$	0	0
$546$	0	0
$547$	− 14.2361i	− 0.608690i	−0.952562	−	0.304345i	$-0.901562\pi$
$547$	− 14.2361i	− 0.608690i	0.952562	−	0.304345i	$-0.0984376\pi$
$548$	− 4.29180i	− 0.183336i
$549$	−5.45898	−0.232984
$550$	0	0
$551$	−22.3607	−0.952597
$552$	4.87539i	0.207510i
$553$	0	0
$554$	−25.7082	−1.09224
$555$	0	0
$556$	12.7639	0.541311
$557$	− 44.8885i	− 1.90199i	−0.309208	−	0.950994i	$-0.600064\pi$
$557$	− 44.8885i	− 1.90199i	0.309208	−	0.950994i	$-0.399936\pi$
$558$	23.1246i	0.978943i
$559$	−20.1803	−0.853537
$560$	0	0
$561$	−33.8885	−1.43078
$562$	47.5066i	2.00394i
$563$	− 9.41641i	− 0.396854i	−0.980116	−	0.198427i	$-0.936417\pi$
$563$	− 9.41641i	− 0.396854i	0.980116	−	0.198427i	$-0.0635833\pi$
$564$	1.52786	0.0643347
$565$	0	0
$566$	15.2361	0.640420
$567$	0	0
$568$	− 10.5279i	− 0.441739i
$569$	−13.9443	−0.584574	−0.292287	−	0.956331i	$-0.594416\pi$
$569$	−13.9443	−0.584574	−0.292287	+	0.956331i	$0.594416\pi$
$570$	0	0
$571$	−12.5967	−0.527157	−0.263579	−	0.964638i	$-0.584903\pi$
$571$	−12.5967	−0.527157	−0.263579	+	0.964638i	$0.584903\pi$
$572$	8.47214i	0.354238i
$573$	− 3.05573i	− 0.127655i
$574$	0	0
$575$	0	0
$576$	−6.23607	−0.259836
$577$	2.00000i	0.0832611i	0.999133	+	0.0416305i	$0.0132552\pi$
$577$	2.00000i	0.0832611i	−0.999133	+	0.0416305i	$0.986745\pi$
$578$	− 40.2705i	− 1.67503i
$579$	−17.8197	−0.740560
$580$	0	0
$581$	0	0
$582$	1.52786i	0.0633320i
$583$	2.00000i	0.0828315i
$584$	29.5967	1.22472
$585$	0	0
$586$	−14.7639	−0.609892
$587$	29.2361i	1.20670i	0.797476	+	0.603351i	$0.206168\pi$
$587$	29.2361i	1.20670i	−0.797476	+	0.603351i	$0.793832\pi$
$588$	0	0
$589$	43.4164	1.78894
$590$	0	0
$591$	−9.23607	−0.379921
$592$	− 14.5623i	− 0.598507i
$593$	25.3050i	1.03915i	0.854425	+	0.519575i	$0.173910\pi$
$593$	25.3050i	1.03915i	−0.854425	+	0.519575i	$0.826090\pi$
$594$	−37.8885	−1.55459
$595$	0	0
$596$	−8.61803	−0.353008
$597$	3.41641i	0.139824i
$598$	− 9.23607i	− 0.377691i
$599$	−11.1803	−0.456816	−0.228408	−	0.973565i	$-0.573352\pi$
$599$	−11.1803	−0.456816	−0.228408	+	0.973565i	$0.573352\pi$
$600$	0	0
$601$	19.0557	0.777299	0.388650	−	0.921386i	$-0.372942\pi$
$601$	19.0557	0.777299	0.388650	+	0.921386i	$0.372942\pi$
$602$	0	0
$603$	0.347524i	0.0141523i
$604$	9.74265	0.396423
$605$	0	0
$606$	18.4721	0.750379
$607$	− 33.1246i	− 1.34449i	−0.740330	−	0.672243i	$-0.765331\pi$
$607$	− 33.1246i	− 1.34449i	0.740330	−	0.672243i	$-0.234669\pi$
$608$	− 15.1246i	− 0.613384i
$609$	0	0
$610$	0	0
$611$	6.47214	0.261835
$612$	− 5.88854i	− 0.238030i
$613$	17.5836i	0.710195i	0.934829	+	0.355097i	$0.115552\pi$
$613$	17.5836i	0.710195i	−0.934829	+	0.355097i	$0.884448\pi$
$614$	50.8328	2.05145
$615$	0	0
$616$	0	0
$617$	11.9443i	0.480858i	0.970667	+	0.240429i	$0.0772882\pi$
$617$	11.9443i	0.480858i	−0.970667	+	0.240429i	$0.922712\pi$
$618$	0.944272i	0.0379842i
$619$	1.70820	0.0686585	0.0343293	−	0.999411i	$-0.489071\pi$
$619$	1.70820	0.0686585	0.0343293	+	0.999411i	$0.489071\pi$
$620$	0	0
$621$	9.75078	0.391285
$622$	32.9443i	1.32094i
$623$	0	0
$624$	−19.4164	−0.777278
$625$	0	0
$626$	−46.0689	−1.84128
$627$	23.4164i	0.935161i
$628$	3.23607i	0.129133i
$629$	−19.4164	−0.774183
$630$	0	0
$631$	−3.65248	−0.145403	−0.0727014	−	0.997354i	$-0.523162\pi$
$631$	−3.65248	−0.145403	−0.0727014	+	0.997354i	$0.523162\pi$
$632$	− 25.0000i	− 0.994447i
$633$	14.8328i	0.589551i
$634$	31.3262	1.24412
$635$	0	0
$636$	0.360680	0.0143019
$637$	0	0
$638$	− 34.2705i	− 1.35678i
$639$	−6.93112	−0.274191
$640$	0	0
$641$	−9.83282	−0.388373	−0.194186	−	0.980965i	$-0.562207\pi$
$641$	−9.83282	−0.388373	−0.194186	+	0.980965i	$0.562207\pi$
$642$	− 16.0000i	− 0.631470i
$643$	9.52786i	0.375742i	0.982194	+	0.187871i	$0.0601587\pi$
$643$	9.52786i	0.375742i	−0.982194	+	0.187871i	$0.939841\pi$
$644$	0	0
$645$	0	0
$646$	−46.8328	−1.84261
$647$	− 15.8885i	− 0.624643i	−0.949976	−	0.312322i	$-0.898893\pi$
$647$	− 15.8885i	− 0.624643i	0.949976	−	0.312322i	$-0.101107\pi$
$648$	− 5.40325i	− 0.212260i
$649$	−7.23607	−0.284041
$650$	0	0
$651$	0	0
$652$	− 6.47214i	− 0.253468i
$653$	− 42.9443i	− 1.68054i	−0.542169	−	0.840270i	$-0.682397\pi$
$653$	− 42.9443i	− 1.68054i	0.542169	−	0.840270i	$-0.317603\pi$
$654$	−36.8328	−1.44028
$655$	0	0
$656$	44.8328	1.75043
$657$	− 19.4853i	− 0.760194i
$658$	0	0
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	−46.7214	−1.81725	−0.908625	−	0.417613i	$-0.862867\pi$
$661$	−46.7214	−1.81725	−0.908625	+	0.417613i	$0.862867\pi$
$662$	− 18.2705i	− 0.710104i
$663$	25.8885i	1.00543i
$664$	−12.7639	−0.495337
$665$	0	0
$666$	−7.14590	−0.276898
$667$	8.81966i	0.341499i
$668$	0.472136i	0.0182675i
$669$	2.69505	0.104197
$670$	0	0
$671$	−15.7082	−0.606408
$672$	0	0
$673$	− 28.4721i	− 1.09752i	−0.835980	−	0.548760i	$-0.815100\pi$
$673$	− 28.4721i	− 1.09752i	0.835980	−	0.548760i	$-0.184900\pi$
$674$	12.1803	0.469169
$675$	0	0
$676$	−1.56231	−0.0600887
$677$	− 30.3607i	− 1.16686i	−0.812165	−	0.583428i	$-0.801711\pi$
$677$	− 30.3607i	− 1.16686i	0.812165	−	0.583428i	$-0.198289\pi$
$678$	24.8328i	0.953699i
$679$	0	0
$680$	0	0
$681$	−6.69505	−0.256555
$682$	66.5410i	2.54799i
$683$	26.1246i	0.999630i	0.866132	+	0.499815i	$0.166599\pi$
$683$	26.1246i	0.999630i	−0.866132	+	0.499815i	$0.833401\pi$
$684$	−4.06888	−0.155578
$685$	0	0
$686$	0	0
$687$	5.52786i	0.210901i
$688$	30.2705i	1.15405i
$689$	1.52786	0.0582070
$690$	0	0
$691$	−18.1803	−0.691613	−0.345806	−	0.938306i	$-0.612395\pi$
$691$	−18.1803	−0.691613	−0.345806	+	0.938306i	$0.612395\pi$
$692$	12.6525i	0.480975i
$693$	0	0
$694$	−25.5066	−0.968216
$695$	0	0
$696$	13.8197	0.523833
$697$	− 59.7771i	− 2.26422i
$698$	− 7.23607i	− 0.273889i
$699$	−12.2918	−0.464918
$700$	0	0
$701$	−46.9443	−1.77306	−0.886530	−	0.462670i	$-0.846891\pi$
$701$	−46.9443	−1.77306	−0.886530	+	0.462670i	$0.846891\pi$
$702$	28.9443i	1.09243i
$703$	13.4164i	0.506009i
$704$	−17.9443	−0.676300
$705$	0	0
$706$	−32.6525	−1.22889
$707$	0	0
$708$	1.30495i	0.0490431i
$709$	47.8885	1.79849	0.899246	−	0.437443i	$-0.144116\pi$
$709$	47.8885	1.79849	0.899246	+	0.437443i	$0.144116\pi$
$710$	0	0
$711$	−16.4590	−0.617260
$712$	28.5410i	1.06962i
$713$	− 17.1246i	− 0.641322i
$714$	0	0
$715$	0	0
$716$	2.11146	0.0789088
$717$	17.8885i	0.668060i
$718$	− 16.3820i	− 0.611370i
$719$	−6.18034	−0.230488	−0.115244	−	0.993337i	$-0.536765\pi$
$719$	−6.18034	−0.230488	−0.115244	+	0.993337i	$0.536765\pi$
$720$	0	0
$721$	0	0
$722$	1.61803i	0.0602170i
$723$	15.4164i	0.573342i
$724$	−8.76393	−0.325709
$725$	0	0
$726$	−13.8885	−0.515452
$727$	20.9443i	0.776780i	0.921495	+	0.388390i	$0.126969\pi$
$727$	20.9443i	0.776780i	−0.921495	+	0.388390i	$0.873031\pi$
$728$	0	0
$729$	−24.0557	−0.890953
$730$	0	0
$731$	40.3607	1.49279
$732$	2.83282i	0.104704i
$733$	4.00000i	0.147743i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	4.00000i	0.147743i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	5.05573	0.186610
$735$	0	0
$736$	−5.96556	−0.219893
$737$	1.00000i	0.0368355i
$738$	− 22.0000i	− 0.809831i
$739$	5.65248	0.207930	0.103965	−	0.994581i	$-0.466847\pi$
$739$	5.65248	0.207930	0.103965	+	0.994581i	$0.466847\pi$
$740$	0	0
$741$	17.8885	0.657152
$742$	0	0
$743$	1.52786i	0.0560519i	0.999607	+	0.0280259i	$0.00892210\pi$
$743$	1.52786i	0.0560519i	−0.999607	+	0.0280259i	$0.991078\pi$
$744$	−26.8328	−0.983739
$745$	0	0
$746$	25.6180	0.937943
$747$	8.40325i	0.307459i
$748$	− 16.9443i	− 0.619544i
$749$	0	0
$750$	0	0
$751$	20.9443	0.764267	0.382134	−	0.924107i	$-0.375189\pi$
$751$	20.9443	0.764267	0.382134	+	0.924107i	$0.375189\pi$
$752$	− 9.70820i	− 0.354022i
$753$	3.05573i	0.111357i
$754$	−26.1803	−0.953432
$755$	0	0
$756$	0	0
$757$	46.4164i	1.68703i	0.537103	+	0.843517i	$0.319519\pi$
$757$	46.4164i	1.68703i	−0.537103	+	0.843517i	$0.680481\pi$
$758$	− 18.0902i	− 0.657065i
$759$	9.23607	0.335248
$760$	0	0
$761$	43.7771	1.58692	0.793459	−	0.608624i	$-0.208278\pi$
$761$	43.7771	1.58692	0.793459	+	0.608624i	$0.208278\pi$
$762$	35.3050i	1.27896i
$763$	0	0
$764$	1.52786	0.0552762
$765$	0	0
$766$	46.5410	1.68160
$767$	5.52786i	0.199600i
$768$	16.7639i	0.604916i
$769$	33.0132	1.19048	0.595242	−	0.803546i	$-0.297056\pi$
$769$	33.0132	1.19048	0.595242	+	0.803546i	$0.297056\pi$
$770$	0	0
$771$	23.0557	0.830332
$772$	− 8.90983i	− 0.320672i
$773$	27.8197i	1.00060i	0.865851	+	0.500302i	$0.166778\pi$
$773$	27.8197i	1.00060i	−0.865851	+	0.500302i	$0.833222\pi$
$774$	14.8541	0.533920
$775$	0	0
$776$	1.70820	0.0613209
$777$	0	0
$778$	53.2148i	1.90784i
$779$	−41.3050	−1.47990
$780$	0	0
$781$	−19.9443	−0.713662
$782$	18.4721i	0.660562i
$783$	− 27.6393i	− 0.987749i
$784$	0	0
$785$	0	0
$786$	1.88854	0.0673621
$787$	− 45.2361i	− 1.61249i	−0.591581	−	0.806246i	$-0.701496\pi$
$787$	− 45.2361i	− 1.61249i	0.591581	−	0.806246i	$-0.298504\pi$
$788$	− 4.61803i	− 0.164511i
$789$	14.5410	0.517674
$790$	0	0
$791$	0	0
$792$	13.9443i	0.495488i
$793$	12.0000i	0.426132i
$794$	43.5967	1.54719
$795$	0	0
$796$	−1.70820	−0.0605457
$797$	8.58359i	0.304046i	0.988377	+	0.152023i	$0.0485789\pi$
$797$	8.58359i	0.304046i	−0.988377	+	0.152023i	$0.951421\pi$
$798$	0	0
$799$	−12.9443	−0.457935
$800$	0	0
$801$	18.7902	0.663921
$802$	18.5623i	0.655458i
$803$	− 56.0689i	− 1.97863i
$804$	0.180340	0.00636010
$805$	0	0
$806$	50.8328	1.79051
$807$	2.11146i	0.0743268i
$808$	− 20.6525i	− 0.726552i
$809$	−20.5279	−0.721721	−0.360861	−	0.932620i	$-0.617517\pi$
$809$	−20.5279	−0.721721	−0.360861	+	0.932620i	$0.617517\pi$
$810$	0	0
$811$	−46.7214	−1.64061	−0.820304	−	0.571927i	$-0.806196\pi$
$811$	−46.7214	−1.64061	−0.820304	+	0.571927i	$0.806196\pi$
$812$	0	0
$813$	− 12.7214i	− 0.446158i
$814$	−20.5623	−0.720708
$815$	0	0
$816$	38.8328	1.35942
$817$	− 27.8885i	− 0.975697i
$818$	25.1246i	0.878461i
$819$	0	0
$820$	0	0
$821$	−24.8328	−0.866671	−0.433336	−	0.901233i	$-0.642664\pi$
$821$	−24.8328	−0.866671	−0.433336	+	0.901233i	$0.642664\pi$
$822$	− 13.8885i	− 0.484419i
$823$	0.347524i	0.0121139i	0.999982	+	0.00605697i	$0.00192800\pi$
$823$	0.347524i	0.0121139i	−0.999982	+	0.00605697i	$0.998072\pi$
$824$	1.05573	0.0367780
$825$	0	0
$826$	0	0
$827$	− 25.5410i	− 0.888148i	−0.895990	−	0.444074i	$-0.853533\pi$
$827$	− 25.5410i	− 0.888148i	0.895990	−	0.444074i	$-0.146467\pi$
$828$	1.60488i	0.0557734i
$829$	−52.3607	−1.81856	−0.909281	−	0.416183i	$-0.863368\pi$
$829$	−52.3607	−1.81856	−0.909281	+	0.416183i	$0.863368\pi$
$830$	0	0
$831$	−19.6393	−0.681280
$832$	13.7082i	0.475246i
$833$	0	0
$834$	41.3050	1.43027
$835$	0	0
$836$	−11.7082	−0.404937
$837$	53.6656i	1.85496i
$838$	− 6.18034i	− 0.213496i
$839$	0.652476	0.0225260	0.0112630	−	0.999937i	$-0.496415\pi$
$839$	0.652476	0.0225260	0.0112630	+	0.999937i	$0.496415\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	− 21.0344i	− 0.724895i
$843$	36.2918i	1.24996i
$844$	−7.41641	−0.255283
$845$	0	0
$846$	−4.76393	−0.163787
$847$	0	0
$848$	− 2.29180i	− 0.0787006i
$849$	11.6393	0.399460
$850$	0	0
$851$	5.29180	0.181400
$852$	3.59675i	0.123223i
$853$	0.583592i	0.0199818i	0.999950	+	0.00999091i	$0.00318026\pi$
$853$	0.583592i	0.0199818i	−0.999950	+	0.00999091i	$0.996820\pi$
$854$	0	0
$855$	0	0
$856$	−17.8885	−0.611418
$857$	38.1803i	1.30422i	0.758126	+	0.652108i	$0.226115\pi$
$857$	38.1803i	1.30422i	−0.758126	+	0.652108i	$0.773885\pi$
$858$	27.4164i	0.935981i
$859$	−22.3607	−0.762937	−0.381468	−	0.924382i	$-0.624581\pi$
$859$	−22.3607	−0.762937	−0.381468	+	0.924382i	$0.624581\pi$
$860$	0	0
$861$	0	0
$862$	42.8328i	1.45889i
$863$	− 49.6525i	− 1.69019i	−0.534616	−	0.845095i	$-0.679544\pi$
$863$	− 49.6525i	− 1.69019i	0.534616	−	0.845095i	$-0.320456\pi$
$864$	18.6950	0.636018
$865$	0	0
$866$	−26.4721	−0.899560
$867$	− 30.7639i	− 1.04480i
$868$	0	0
$869$	−47.3607	−1.60660
$870$	0	0
$871$	0.763932	0.0258848
$872$	41.1803i	1.39454i
$873$	− 1.12461i	− 0.0380623i
$874$	12.7639	0.431746
$875$	0	0
$876$	−10.1115	−0.341634
$877$	− 14.3607i	− 0.484926i	−0.970161	−	0.242463i	$-0.922045\pi$
$877$	− 14.3607i	− 0.484926i	0.970161	−	0.242463i	$-0.0779553\pi$
$878$	− 35.1246i	− 1.18540i
$879$	−11.2786	−0.380419
$880$	0	0
$881$	−28.1803	−0.949420	−0.474710	−	0.880142i	$-0.657447\pi$
$881$	−28.1803	−0.949420	−0.474710	+	0.880142i	$0.657447\pi$
$882$	0	0
$883$	50.5967i	1.70272i	0.524585	+	0.851358i	$0.324220\pi$
$883$	50.5967i	1.70272i	−0.524585	+	0.851358i	$0.675780\pi$
$884$	−12.9443	−0.435363
$885$	0	0
$886$	12.0000	0.403148
$887$	52.6525i	1.76790i	0.467584	+	0.883949i	$0.345124\pi$
$887$	52.6525i	1.76790i	−0.467584	+	0.883949i	$0.654876\pi$
$888$	− 8.29180i	− 0.278254i
$889$	0	0
$890$	0	0
$891$	−10.2361	−0.342921
$892$	1.34752i	0.0451184i
$893$	8.94427i	0.299309i
$894$	−27.8885	−0.932732
$895$	0	0
$896$	0	0
$897$	− 7.05573i	− 0.235584i
$898$	− 47.6869i	− 1.59133i
$899$	−48.5410	−1.61893
$900$	0	0
$901$	−3.05573	−0.101801
$902$	− 63.3050i	− 2.10782i
$903$	0	0
$904$	27.7639	0.923415
$905$	0	0
$906$	31.5279	1.04744
$907$	− 18.8328i	− 0.625333i	−0.949863	−	0.312667i	$-0.898778\pi$
$907$	− 18.8328i	− 0.625333i	0.949863	−	0.312667i	$-0.101222\pi$
$908$	− 3.34752i	− 0.111091i
$909$	−13.5967	−0.450976
$910$	0	0
$911$	23.1803	0.767999	0.383999	−	0.923333i	$-0.374546\pi$
$911$	23.1803	0.767999	0.383999	+	0.923333i	$0.374546\pi$
$912$	− 26.8328i	− 0.888523i
$913$	24.1803i	0.800252i
$914$	34.7426	1.14918
$915$	0	0
$916$	−2.76393	−0.0913229
$917$	0	0
$918$	− 57.8885i	− 1.91061i
$919$	32.2361	1.06337	0.531685	−	0.846942i	$-0.321559\pi$
$919$	32.2361	1.06337	0.531685	+	0.846942i	$0.321559\pi$
$920$	0	0
$921$	38.8328	1.27958
$922$	− 13.2361i	− 0.435907i
$923$	15.2361i	0.501501i
$924$	0	0
$925$	0	0
$926$	35.4164	1.16386
$927$	− 0.695048i	− 0.0228284i
$928$	16.9098i	0.555092i
$929$	51.7082	1.69649	0.848246	−	0.529603i	$-0.177659\pi$
$929$	51.7082	1.69649	0.848246	+	0.529603i	$0.177659\pi$
$930$	0	0
$931$	0	0
$932$	− 6.14590i	− 0.201316i
$933$	25.1672i	0.823937i
$934$	−17.7082	−0.579430
$935$	0	0
$936$	10.6525	0.348187
$937$	− 30.7639i	− 1.00501i	−0.864573	−	0.502507i	$-0.832411\pi$
$937$	− 30.7639i	− 1.00501i	0.864573	−	0.502507i	$-0.167589\pi$
$938$	0	0
$939$	−35.1935	−1.14850
$940$	0	0
$941$	0.763932	0.0249035	0.0124517	−	0.999922i	$-0.496036\pi$
$941$	0.763932	0.0249035	0.0124517	+	0.999922i	$0.496036\pi$
$942$	10.4721i	0.341201i
$943$	16.2918i	0.530534i
$944$	8.29180	0.269875
$945$	0	0
$946$	42.7426	1.38968
$947$	− 18.8328i	− 0.611984i	−0.952034	−	0.305992i	$-0.901012\pi$
$947$	− 18.8328i	− 0.611984i	0.952034	−	0.305992i	$-0.0989881\pi$
$948$	8.54102i	0.277399i
$949$	−42.8328	−1.39041
$950$	0	0
$951$	23.9311	0.776020
$952$	0	0
$953$	5.47214i	0.177260i	0.996065	+	0.0886299i	$0.0282489\pi$
$953$	5.47214i	0.177260i	−0.996065	+	0.0886299i	$0.971751\pi$
$954$	−1.12461	−0.0364107
$955$	0	0
$956$	−8.94427	−0.289278
$957$	− 26.1803i	− 0.846290i
$958$	− 6.18034i	− 0.199678i
$959$	0	0
$960$	0	0
$961$	63.2492	2.04030
$962$	15.7082i	0.506453i
$963$	11.7771i	0.379511i
$964$	−7.70820	−0.248265
$965$	0	0
$966$	0	0
$967$	− 49.8885i	− 1.60431i	−0.597118	−	0.802154i	$-0.703687\pi$
$967$	− 49.8885i	− 1.60431i	0.597118	−	0.802154i	$-0.296313\pi$
$968$	15.5279i	0.499084i
$969$	−35.7771	−1.14933
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	− 8.40325i	− 0.269534i
$973$	0	0
$974$	−16.5623	−0.530691
$975$	0	0
$976$	18.0000	0.576166
$977$	− 2.52786i	− 0.0808735i	−0.999182	−	0.0404368i	$-0.987125\pi$
$977$	− 2.52786i	− 0.0808735i	0.999182	−	0.0404368i	$-0.0128749\pi$
$978$	− 20.9443i	− 0.669724i
$979$	54.0689	1.72805
$980$	0	0
$981$	27.1115	0.865602
$982$	− 16.5623i	− 0.528524i
$983$	32.5410i	1.03790i	0.854805	+	0.518949i	$0.173676\pi$
$983$	32.5410i	1.03790i	−0.854805	+	0.518949i	$0.826324\pi$
$984$	25.5279	0.813799
$985$	0	0
$986$	52.3607	1.66750
$987$	0	0
$988$	8.94427i	0.284555i
$989$	−11.0000	−0.349780
$990$	0	0
$991$	−9.18034	−0.291623	−0.145812	−	0.989312i	$-0.546579\pi$
$991$	−9.18034	−0.291623	−0.145812	+	0.989312i	$0.546579\pi$
$992$	− 32.8328i	− 1.04244i
$993$	− 13.9574i	− 0.442926i
$994$	0	0
$995$	0	0
$996$	4.36068	0.138173
$997$	18.5836i	0.588548i	0.955721	+	0.294274i	$0.0950779\pi$
$997$	18.5836i	0.588548i	−0.955721	+	0.294274i	$0.904922\pi$
$998$	− 46.8328i	− 1.48247i
$999$	−16.5836	−0.524682

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.b.k.99.4		4
5.2	odd	4		1225.2.a.n.1.1		2
5.3	odd	4		1225.2.a.u.1.2		2
5.4	even	2	inner	1225.2.b.k.99.1		4
7.6	odd	2		175.2.b.c.99.4		4
21.20	even	2		1575.2.d.k.1324.1		4
28.27	even	2		2800.2.g.s.449.3		4
35.13	even	4		175.2.a.e.1.2	yes	2
35.27	even	4		175.2.a.d.1.1	✓	2
35.34	odd	2		175.2.b.c.99.1		4
105.62	odd	4		1575.2.a.s.1.2		2
105.83	odd	4		1575.2.a.n.1.1		2
105.104	even	2		1575.2.d.k.1324.4		4
140.27	odd	4		2800.2.a.bh.1.2		2
140.83	odd	4		2800.2.a.bp.1.1		2
140.139	even	2		2800.2.g.s.449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.a.d.1.1	✓	2	35.27	even	4
175.2.a.e.1.2	yes	2	35.13	even	4
175.2.b.c.99.1		4	35.34	odd	2
175.2.b.c.99.4		4	7.6	odd	2
1225.2.a.n.1.1		2	5.2	odd	4
1225.2.a.u.1.2		2	5.3	odd	4
1225.2.b.k.99.1		4	5.4	even	2	inner
1225.2.b.k.99.4		4	1.1	even	1	trivial
1575.2.a.n.1.1		2	105.83	odd	4
1575.2.a.s.1.2		2	105.62	odd	4
1575.2.d.k.1324.1		4	21.20	even	2
1575.2.d.k.1324.4		4	105.104	even	2
2800.2.a.bh.1.2		2	140.27	odd	4
2800.2.a.bp.1.1		2	140.83	odd	4
2800.2.g.s.449.2		4	140.139	even	2
2800.2.g.s.449.3		4	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.61803i q^{2} +1.23607i q^{3} -0.618034 q^{4} -2.00000 q^{6} +2.23607i q^{8} +1.47214 q^{9} +O(q^{10})\)	\(q+1.61803i q^{2} +1.23607i q^{3} -0.618034 q^{4} -2.00000 q^{6} +2.23607i q^{8} +1.47214 q^{9} +4.23607 q^{11} -0.763932i q^{12} -3.23607i q^{13} -4.85410 q^{16} +6.47214i q^{17} +2.38197i q^{18} +4.47214 q^{19} +6.85410i q^{22} -1.76393i q^{23} -2.76393 q^{24} +5.23607 q^{26} +5.52786i q^{27} -5.00000 q^{29} +9.70820 q^{31} -3.38197i q^{32} +5.23607i q^{33} -10.4721 q^{34} -0.909830 q^{36} +3.00000i q^{37} +7.23607i q^{38} +4.00000 q^{39} -9.23607 q^{41} -6.23607i q^{43} -2.61803 q^{44} +2.85410 q^{46} +2.00000i q^{47} -6.00000i q^{48} -8.00000 q^{51} +2.00000i q^{52} +0.472136i q^{53} -8.94427 q^{54} +5.52786i q^{57} -8.09017i q^{58} -1.70820 q^{59} -3.70820 q^{61} +15.7082i q^{62} -4.23607 q^{64} -8.47214 q^{66} +0.236068i q^{67} -4.00000i q^{68} +2.18034 q^{69} -4.70820 q^{71} +3.29180i q^{72} -13.2361i q^{73} -4.85410 q^{74} -2.76393 q^{76} +6.47214i q^{78} -11.1803 q^{79} -2.41641 q^{81} -14.9443i q^{82} +5.70820i q^{83} +10.0902 q^{86} -6.18034i q^{87} +9.47214i q^{88} +12.7639 q^{89} +1.09017i q^{92} +12.0000i q^{93} -3.23607 q^{94} +4.18034 q^{96} -0.763932i q^{97} +6.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 8 q^{6} - 12 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 8 * q^6 - 12 * q^9	\( 4 q + 2 q^{4} - 8 q^{6} - 12 q^{9} + 8 q^{11} - 6 q^{16} - 20 q^{24} + 12 q^{26} - 20 q^{29} + 12 q^{31} - 24 q^{34} - 26 q^{36} + 16 q^{39} - 28 q^{41} - 6 q^{44} - 2 q^{46} - 32 q^{51} + 20 q^{59} + 12 q^{61} - 8 q^{64} - 16 q^{66} - 36 q^{69} + 8 q^{71} - 6 q^{74} - 20 q^{76} + 44 q^{81} + 18 q^{86} + 60 q^{89} - 4 q^{94} - 28 q^{96} + 16 q^{99}+O(q^{100}) \) 4 * q + 2 * q^4 - 8 * q^6 - 12 * q^9 + 8 * q^11 - 6 * q^16 - 20 * q^24 + 12 * q^26 - 20 * q^29 + 12 * q^31 - 24 * q^34 - 26 * q^36 + 16 * q^39 - 28 * q^41 - 6 * q^44 - 2 * q^46 - 32 * q^51 + 20 * q^59 + 12 * q^61 - 8 * q^64 - 16 * q^66 - 36 * q^69 + 8 * q^71 - 6 * q^74 - 20 * q^76 + 44 * q^81 + 18 * q^86 + 60 * q^89 - 4 * q^94 - 28 * q^96 + 16 * q^99

Introduction

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Database

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