LMFDB - Embedded newform 1575.2.d.k.1324.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1324,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1575.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.d (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:	$12.5764383184$
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			1324.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1575.1324
Dual form			1575.2.d.k.1324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034i q^{2} +1.61803 q^{4} +1.00000i q^{7} +2.23607i q^{8} +O(q^{10})$	$q+0.618034i q^{2} +1.61803 q^{4} +1.00000i q^{7} +2.23607i q^{8} +0.236068 q^{11} -1.23607i q^{13} -0.618034 q^{14} +1.85410 q^{16} -2.47214i q^{17} +4.47214 q^{19} +0.145898i q^{22} +6.23607i q^{23} +0.763932 q^{26} +1.61803i q^{28} +5.00000 q^{29} +3.70820 q^{31} +5.61803i q^{32} +1.52786 q^{34} +3.00000i q^{37} +2.76393i q^{38} -4.76393 q^{41} -1.76393i q^{43} +0.381966 q^{44} -3.85410 q^{46} +2.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +8.47214i q^{53} -2.23607 q^{56} +3.09017i q^{58} +11.7082 q^{59} -9.70820 q^{61} +2.29180i q^{62} +0.236068 q^{64} -4.23607i q^{67} -4.00000i q^{68} -8.70820 q^{71} +8.76393i q^{73} -1.85410 q^{74} +7.23607 q^{76} +0.236068i q^{77} +11.1803 q^{79} -2.94427i q^{82} -7.70820i q^{83} +1.09017 q^{86} +0.527864i q^{88} +17.2361 q^{89} +1.23607 q^{91} +10.0902i q^{92} -1.23607 q^{94} +5.23607i q^{97} -0.618034i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4}+O(q^{10}) $ 4 * q + 2 * q^4	$ 4 q + 2 q^{4} - 8 q^{11} + 2 q^{14} - 6 q^{16} + 12 q^{26} + 20 q^{29} - 12 q^{31} + 24 q^{34} - 28 q^{41} + 6 q^{44} - 2 q^{46} - 4 q^{49} + 20 q^{59} - 12 q^{61} - 8 q^{64} - 8 q^{71} + 6 q^{74} + 20 q^{76} - 18 q^{86} + 60 q^{89} - 4 q^{91} + 4 q^{94}+O(q^{100}) $ 4 * q + 2 * q^4 - 8 * q^11 + 2 * q^14 - 6 * q^16 + 12 * q^26 + 20 * q^29 - 12 * q^31 + 24 * q^34 - 28 * q^41 + 6 * q^44 - 2 * q^46 - 4 * q^49 + 20 * q^59 - 12 * q^61 - 8 * q^64 - 8 * q^71 + 6 * q^74 + 20 * q^76 - 18 * q^86 + 60 * q^89 - 4 * q^91 + 4 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0.618034i	0.437016i	0.975835	+	0.218508i	$0.0701190\pi$
$2$	0.618034i	0.437016i	−0.975835	+	0.218508i	$0.929881\pi$
$3$	0	0
$3$	0	0
$4$	1.61803	0.809017
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	2.23607i	0.790569i
$9$	0	0
$10$	0	0
$11$	0.236068	0.0711772	0.0355886	−	0.999367i	$-0.488669\pi$
$11$	0.236068	0.0711772	0.0355886	+	0.999367i	$0.488669\pi$
$12$	0	0
$13$	− 1.23607i	− 0.342824i	−0.985199	−	0.171412i	$-0.945167\pi$
$13$	− 1.23607i	− 0.342824i	0.985199	−	0.171412i	$-0.0548329\pi$
$14$	−0.618034	−0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	− 2.47214i	− 0.599581i	−0.954005	−	0.299791i	$-0.903083\pi$
$17$	− 2.47214i	− 0.599581i	0.954005	−	0.299791i	$-0.0969168\pi$
$18$	0	0
$19$	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
$22$	0.145898i	0.0311056i
$23$	6.23607i	1.30031i	0.759802	+	0.650155i	$0.225296\pi$
$23$	6.23607i	1.30031i	−0.759802	+	0.650155i	$0.774704\pi$
$24$	0	0
$25$	0	0
$26$	0.763932	0.149819
$27$	0	0
$28$	1.61803i	0.305780i
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	3.70820	0.666013	0.333007	−	0.942925i	$-0.391937\pi$
$31$	3.70820	0.666013	0.333007	+	0.942925i	$0.391937\pi$
$32$	5.61803i	0.993137i
$33$	0	0
$34$	1.52786	0.262027
$35$	0	0
$36$	0	0
$37$	3.00000i	0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$	3.00000i	0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$	2.76393i	0.448369i
$39$	0	0
$40$	0	0
$41$	−4.76393	−0.744001	−0.372001	−	0.928232i	$-0.621328\pi$
$41$	−4.76393	−0.744001	−0.372001	+	0.928232i	$0.621328\pi$
$42$	0	0
$43$	− 1.76393i	− 0.268997i	−0.990914	−	0.134499i	$-0.957058\pi$
$43$	− 1.76393i	− 0.268997i	0.990914	−	0.134499i	$-0.0429424\pi$
$44$	0.381966	0.0575835
$45$	0	0
$46$	−3.85410	−0.568256
$47$	2.00000i	0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$	2.00000i	0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 2.00000i	− 0.277350i
$53$	8.47214i	1.16374i	0.813283	+	0.581869i	$0.197678\pi$
$53$	8.47214i	1.16374i	−0.813283	+	0.581869i	$0.802322\pi$
$54$	0	0
$55$	0	0
$56$	−2.23607	−0.298807
$57$	0	0
$58$	3.09017i	0.405759i
$59$	11.7082	1.52428	0.762139	−	0.647413i	$-0.224149\pi$
$59$	11.7082	1.52428	0.762139	+	0.647413i	$0.224149\pi$
$60$	0	0
$61$	−9.70820	−1.24301	−0.621504	−	0.783411i	$-0.713478\pi$
$61$	−9.70820	−1.24301	−0.621504	+	0.783411i	$0.713478\pi$
$62$	2.29180i	0.291058i
$63$	0	0
$64$	0.236068	0.0295085
$65$	0	0
$66$	0	0
$67$	− 4.23607i	− 0.517518i	−0.965942	−	0.258759i	$-0.916686\pi$
$67$	− 4.23607i	− 0.517518i	0.965942	−	0.258759i	$-0.0833136\pi$
$68$	− 4.00000i	− 0.485071i
$69$	0	0
$70$	0	0
$71$	−8.70820	−1.03347	−0.516737	−	0.856144i	$-0.672853\pi$
$71$	−8.70820	−1.03347	−0.516737	+	0.856144i	$0.672853\pi$
$72$	0	0
$73$	8.76393i	1.02574i	0.858466	+	0.512870i	$0.171418\pi$
$73$	8.76393i	1.02574i	−0.858466	+	0.512870i	$0.828582\pi$
$74$	−1.85410	−0.215535
$75$	0	0
$76$	7.23607	0.830034
$77$	0.236068i	0.0269024i
$78$	0	0
$79$	11.1803	1.25789	0.628943	−	0.777451i	$-0.283488\pi$
$79$	11.1803	1.25789	0.628943	+	0.777451i	$0.283488\pi$
$80$	0	0
$81$	0	0
$82$	− 2.94427i	− 0.325140i
$83$	− 7.70820i	− 0.846085i	−0.906110	−	0.423043i	$-0.860962\pi$
$83$	− 7.70820i	− 0.846085i	0.906110	−	0.423043i	$-0.139038\pi$
$84$	0	0
$85$	0	0
$86$	1.09017	0.117556
$87$	0	0
$88$	0.527864i	0.0562705i
$89$	17.2361	1.82702	0.913510	−	0.406817i	$-0.133361\pi$
$89$	17.2361	1.82702	0.913510	+	0.406817i	$0.133361\pi$
$90$	0	0
$91$	1.23607	0.129575
$92$	10.0902i	1.05197i
$93$	0	0
$94$	−1.23607	−0.127491
$95$	0	0
$96$	0	0
$97$	5.23607i	0.531642i	0.964022	+	0.265821i	$0.0856430\pi$
$97$	5.23607i	0.531642i	−0.964022	+	0.265821i	$0.914357\pi$
$98$	− 0.618034i	− 0.0624309i
$99$	0	0
$100$	0	0
$101$	−4.76393	−0.474029	−0.237014	−	0.971506i	$-0.576169\pi$
$101$	−4.76393	−0.474029	−0.237014	+	0.971506i	$0.576169\pi$
$102$	0	0
$103$	− 8.47214i	− 0.834784i	−0.908726	−	0.417392i	$-0.862944\pi$
$103$	− 8.47214i	− 0.834784i	0.908726	−	0.417392i	$-0.137056\pi$
$104$	2.76393	0.271026
$105$	0	0
$106$	−5.23607	−0.508572
$107$	− 8.00000i	− 0.773389i	−0.922208	−	0.386695i	$-0.873617\pi$
$107$	− 8.00000i	− 0.773389i	0.922208	−	0.386695i	$-0.126383\pi$
$108$	0	0
$109$	−8.41641	−0.806146	−0.403073	−	0.915168i	$-0.632058\pi$
$109$	−8.41641	−0.806146	−0.403073	+	0.915168i	$0.632058\pi$
$110$	0	0
$111$	0	0
$112$	1.85410i	0.175196i
$113$	− 14.4164i	− 1.35618i	−0.734978	−	0.678091i	$-0.762808\pi$
$113$	− 14.4164i	− 1.35618i	0.734978	−	0.678091i	$-0.237192\pi$
$114$	0	0
$115$	0	0
$116$	8.09017	0.751153
$117$	0	0
$118$	7.23607i	0.666134i
$119$	2.47214	0.226620
$120$	0	0
$121$	−10.9443	−0.994934
$122$	− 6.00000i	− 0.543214i
$123$	0	0
$124$	6.00000	0.538816
$125$	0	0
$126$	0	0
$127$	13.6525i	1.21146i	0.795670	+	0.605731i	$0.207119\pi$
$127$	13.6525i	1.21146i	−0.795670	+	0.605731i	$0.792881\pi$
$128$	11.3820i	1.00603i
$129$	0	0
$130$	0	0
$131$	16.9443	1.48043	0.740214	−	0.672371i	$-0.234724\pi$
$131$	16.9443	1.48043	0.740214	+	0.672371i	$0.234724\pi$
$132$	0	0
$133$	4.47214i	0.387783i
$134$	2.61803	0.226164
$135$	0	0
$136$	5.52786	0.474010
$137$	10.9443i	0.935032i	0.883985	+	0.467516i	$0.154851\pi$
$137$	10.9443i	0.935032i	−0.883985	+	0.467516i	$0.845149\pi$
$138$	0	0
$139$	−10.6525	−0.903531	−0.451766	−	0.892137i	$-0.649206\pi$
$139$	−10.6525	−0.903531	−0.451766	+	0.892137i	$0.649206\pi$
$140$	0	0
$141$	0	0
$142$	− 5.38197i	− 0.451645i
$143$	− 0.291796i	− 0.0244012i
$144$	0	0
$145$	0	0
$146$	−5.41641	−0.448265
$147$	0	0
$148$	4.85410i	0.399005i
$149$	3.94427	0.323127	0.161564	−	0.986862i	$-0.448346\pi$
$149$	3.94427	0.323127	0.161564	+	0.986862i	$0.448346\pi$
$150$	0	0
$151$	−20.2361	−1.64679	−0.823394	−	0.567470i	$-0.807922\pi$
$151$	−20.2361	−1.64679	−0.823394	+	0.567470i	$0.807922\pi$
$152$	10.0000i	0.811107i
$153$	0	0
$154$	−0.145898	−0.0117568
$155$	0	0
$156$	0	0
$157$	0.763932i	0.0609684i	0.999535	+	0.0304842i	$0.00970493\pi$
$157$	0.763932i	0.0609684i	−0.999535	+	0.0304842i	$0.990295\pi$
$158$	6.90983i	0.549717i
$159$	0	0
$160$	0	0
$161$	−6.23607	−0.491471
$162$	0	0
$163$	1.52786i	0.119672i	0.998208	+	0.0598358i	$0.0190577\pi$
$163$	1.52786i	0.119672i	−0.998208	+	0.0598358i	$0.980942\pi$
$164$	−7.70820	−0.601910
$165$	0	0
$166$	4.76393	0.369753
$167$	− 5.23607i	− 0.405179i	−0.979264	−	0.202590i	$-0.935064\pi$
$167$	− 5.23607i	− 0.405179i	0.979264	−	0.202590i	$-0.0649357\pi$
$168$	0	0
$169$	11.4721	0.882472
$170$	0	0
$171$	0	0
$172$	− 2.85410i	− 0.217623i
$173$	− 11.5279i	− 0.876447i	−0.898866	−	0.438224i	$-0.855608\pi$
$173$	− 11.5279i	− 0.876447i	0.898866	−	0.438224i	$-0.144392\pi$
$174$	0	0
$175$	0	0
$176$	0.437694	0.0329924
$177$	0	0
$178$	10.6525i	0.798437i
$179$	−23.4164	−1.75022	−0.875112	−	0.483920i	$-0.839213\pi$
$179$	−23.4164	−1.75022	−0.875112	+	0.483920i	$0.839213\pi$
$180$	0	0
$181$	8.18034	0.608040	0.304020	−	0.952666i	$-0.401671\pi$
$181$	8.18034	0.608040	0.304020	+	0.952666i	$0.401671\pi$
$182$	0.763932i	0.0566264i
$183$	0	0
$184$	−13.9443	−1.02799
$185$	0	0
$186$	0	0
$187$	− 0.583592i	− 0.0426765i
$188$	3.23607i	0.236015i
$189$	0	0
$190$	0	0
$191$	−6.47214	−0.468307	−0.234154	−	0.972200i	$-0.575232\pi$
$191$	−6.47214	−0.468307	−0.234154	+	0.972200i	$0.575232\pi$
$192$	0	0
$193$	− 12.4164i	− 0.893753i	−0.894596	−	0.446876i	$-0.852536\pi$
$193$	− 12.4164i	− 0.893753i	0.894596	−	0.446876i	$-0.147464\pi$
$194$	−3.23607	−0.232336
$195$	0	0
$196$	−1.61803	−0.115574
$197$	1.47214i	0.104885i	0.998624	+	0.0524427i	$0.0167007\pi$
$197$	1.47214i	0.104885i	−0.998624	+	0.0524427i	$0.983299\pi$
$198$	0	0
$199$	−7.23607	−0.512951	−0.256476	−	0.966551i	$-0.582561\pi$
$199$	−7.23607	−0.512951	−0.256476	+	0.966551i	$0.582561\pi$
$200$	0	0
$201$	0	0
$202$	− 2.94427i	− 0.207158i
$203$	5.00000i	0.350931i
$204$	0	0
$205$	0	0
$206$	5.23607	0.364814
$207$	0	0
$208$	− 2.29180i	− 0.158907i
$209$	1.05573	0.0730262
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	13.7082i	0.941483i
$213$	0	0
$214$	4.94427	0.337983
$215$	0	0
$216$	0	0
$217$	3.70820i	0.251729i
$218$	− 5.20163i	− 0.352299i
$219$	0	0
$220$	0	0
$221$	−3.05573	−0.205551
$222$	0	0
$223$	− 20.1803i	− 1.35138i	−0.737188	−	0.675688i	$-0.763847\pi$
$223$	− 20.1803i	− 1.35138i	0.737188	−	0.675688i	$-0.236153\pi$
$224$	−5.61803	−0.375371
$225$	0	0
$226$	8.90983	0.592673
$227$	− 21.4164i	− 1.42146i	−0.703466	−	0.710728i	$-0.748365\pi$
$227$	− 21.4164i	− 1.42146i	0.703466	−	0.710728i	$-0.251635\pi$
$228$	0	0
$229$	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
$233$	7.94427i	0.520447i	0.965548	+	0.260223i	$0.0837962\pi$
$233$	7.94427i	0.520447i	−0.965548	+	0.260223i	$0.916204\pi$
$234$	0	0
$235$	0	0
$236$	18.9443	1.23317
$237$	0	0
$238$	1.52786i	0.0990367i
$239$	−5.52786	−0.357568	−0.178784	−	0.983888i	$-0.557216\pi$
$239$	−5.52786	−0.357568	−0.178784	+	0.983888i	$0.557216\pi$
$240$	0	0
$241$	−3.52786	−0.227250	−0.113625	−	0.993524i	$-0.536246\pi$
$241$	−3.52786	−0.227250	−0.113625	+	0.993524i	$0.536246\pi$
$242$	− 6.76393i	− 0.434802i
$243$	0	0
$244$	−15.7082	−1.00561
$245$	0	0
$246$	0	0
$247$	− 5.52786i	− 0.351730i
$248$	8.29180i	0.526530i
$249$	0	0
$250$	0	0
$251$	−6.47214	−0.408518	−0.204259	−	0.978917i	$-0.565478\pi$
$251$	−6.47214	−0.408518	−0.204259	+	0.978917i	$0.565478\pi$
$252$	0	0
$253$	1.47214i	0.0925524i
$254$	−8.43769	−0.529428
$255$	0	0
$256$	−6.56231	−0.410144
$257$	12.6525i	0.789240i	0.918844	+	0.394620i	$0.129124\pi$
$257$	12.6525i	0.789240i	−0.918844	+	0.394620i	$0.870876\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	0	0
$262$	10.4721i	0.646971i
$263$	16.2361i	1.00116i	0.865691	+	0.500579i	$0.166880\pi$
$263$	16.2361i	1.00116i	−0.865691	+	0.500579i	$0.833120\pi$
$264$	0	0
$265$	0	0
$266$	−2.76393	−0.169468
$267$	0	0
$268$	− 6.85410i	− 0.418681i
$269$	−11.7082	−0.713862	−0.356931	−	0.934131i	$-0.616177\pi$
$269$	−11.7082	−0.713862	−0.356931	+	0.934131i	$0.616177\pi$
$270$	0	0
$271$	23.7082	1.44017	0.720085	−	0.693885i	$-0.244103\pi$
$271$	23.7082	1.44017	0.720085	+	0.693885i	$0.244103\pi$
$272$	− 4.58359i	− 0.277921i
$273$	0	0
$274$	−6.76393	−0.408624
$275$	0	0
$276$	0	0
$277$	− 19.8885i	− 1.19499i	−0.801874	−	0.597493i	$-0.796163\pi$
$277$	− 19.8885i	− 1.19499i	0.801874	−	0.597493i	$-0.203837\pi$
$278$	− 6.58359i	− 0.394858i
$279$	0	0
$280$	0	0
$281$	15.3607	0.916341	0.458171	−	0.888864i	$-0.348505\pi$
$281$	15.3607	0.916341	0.458171	+	0.888864i	$0.348505\pi$
$282$	0	0
$283$	− 17.4164i	− 1.03530i	−0.855593	−	0.517649i	$-0.826807\pi$
$283$	− 17.4164i	− 1.03530i	0.855593	−	0.517649i	$-0.173193\pi$
$284$	−14.0902	−0.836098
$285$	0	0
$286$	0.180340	0.0106637
$287$	− 4.76393i	− 0.281206i
$288$	0	0
$289$	10.8885	0.640503
$290$	0	0
$291$	0	0
$292$	14.1803i	0.829842i
$293$	− 31.1246i	− 1.81832i	−0.416448	−	0.909160i	$-0.636725\pi$
$293$	− 31.1246i	− 1.81832i	0.416448	−	0.909160i	$-0.363275\pi$
$294$	0	0
$295$	0	0
$296$	−6.70820	−0.389906
$297$	0	0
$298$	2.43769i	0.141212i
$299$	7.70820	0.445777
$300$	0	0
$301$	1.76393	0.101671
$302$	− 12.5066i	− 0.719673i
$303$	0	0
$304$	8.29180	0.475567
$305$	0	0
$306$	0	0
$307$	4.58359i	0.261599i	0.991409	+	0.130800i	$0.0417545\pi$
$307$	4.58359i	0.261599i	−0.991409	+	0.130800i	$0.958246\pi$
$308$	0.381966i	0.0217645i
$309$	0	0
$310$	0	0
$311$	−24.3607	−1.38137	−0.690684	−	0.723157i	$-0.742690\pi$
$311$	−24.3607	−1.38137	−0.690684	+	0.723157i	$0.742690\pi$
$312$	0	0
$313$	− 19.5279i	− 1.10378i	−0.833917	−	0.551890i	$-0.813907\pi$
$313$	− 19.5279i	− 1.10378i	0.833917	−	0.551890i	$-0.186093\pi$
$314$	−0.472136	−0.0266442
$315$	0	0
$316$	18.0902	1.01765
$317$	− 25.3607i	− 1.42440i	−0.701978	−	0.712199i	$-0.747699\pi$
$317$	− 25.3607i	− 1.42440i	0.701978	−	0.712199i	$-0.252301\pi$
$318$	0	0
$319$	1.18034	0.0660863
$320$	0	0
$321$	0	0
$322$	− 3.85410i	− 0.214781i
$323$	− 11.0557i	− 0.615157i
$324$	0	0
$325$	0	0
$326$	−0.944272	−0.0522984
$327$	0	0
$328$	− 10.6525i	− 0.588185i
$329$	−2.00000	−0.110264
$330$	0	0
$331$	−24.7082	−1.35809	−0.679043	−	0.734099i	$-0.737605\pi$
$331$	−24.7082	−1.35809	−0.679043	+	0.734099i	$0.737605\pi$
$332$	− 12.4721i	− 0.684497i
$333$	0	0
$334$	3.23607	0.177070
$335$	0	0
$336$	0	0
$337$	− 16.4721i	− 0.897294i	−0.893709	−	0.448647i	$-0.851906\pi$
$337$	− 16.4721i	− 0.897294i	0.893709	−	0.448647i	$-0.148094\pi$
$338$	7.09017i	0.385654i
$339$	0	0
$340$	0	0
$341$	0.875388	0.0474049
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	3.94427	0.212661
$345$	0	0
$346$	7.12461	0.383022
$347$	− 20.2361i	− 1.08633i	−0.839626	−	0.543165i	$-0.817226\pi$
$347$	− 20.2361i	− 1.08633i	0.839626	−	0.543165i	$-0.182774\pi$
$348$	0	0
$349$	−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
$351$	0	0
$352$	1.32624i	0.0706887i
$353$	− 2.18034i	− 0.116048i	−0.998315	−	0.0580239i	$-0.981520\pi$
$353$	− 2.18034i	− 0.116048i	0.998315	−	0.0580239i	$-0.0184800\pi$
$354$	0	0
$355$	0	0
$356$	27.8885	1.47809
$357$	0	0
$358$	− 14.4721i	− 0.764876i
$359$	−30.1246	−1.58992	−0.794958	−	0.606664i	$-0.792507\pi$
$359$	−30.1246	−1.58992	−0.794958	+	0.606664i	$0.792507\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	5.05573i	0.265723i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	− 37.1246i	− 1.93789i	−0.247278	−	0.968944i	$-0.579536\pi$
$367$	− 37.1246i	− 1.93789i	0.247278	−	0.968944i	$-0.420464\pi$
$368$	11.5623i	0.602727i
$369$	0	0
$370$	0	0
$371$	−8.47214	−0.439851
$372$	0	0
$373$	37.8328i	1.95891i	0.201665	+	0.979454i	$0.435365\pi$
$373$	37.8328i	1.95891i	−0.201665	+	0.979454i	$0.564635\pi$
$374$	0.360680	0.0186503
$375$	0	0
$376$	−4.47214	−0.230633
$377$	− 6.18034i	− 0.318304i
$378$	0	0
$379$	11.1803	0.574295	0.287148	−	0.957886i	$-0.407293\pi$
$379$	11.1803	0.574295	0.287148	+	0.957886i	$0.407293\pi$
$380$	0	0
$381$	0	0
$382$	− 4.00000i	− 0.204658i
$383$	− 33.2361i	− 1.69828i	−0.528165	−	0.849142i	$-0.677120\pi$
$383$	− 33.2361i	− 1.69828i	0.528165	−	0.849142i	$-0.322880\pi$
$384$	0	0
$385$	0	0
$386$	7.67376	0.390584
$387$	0	0
$388$	8.47214i	0.430108i
$389$	2.88854	0.146455	0.0732275	−	0.997315i	$-0.476670\pi$
$389$	2.88854	0.146455	0.0732275	+	0.997315i	$0.476670\pi$
$390$	0	0
$391$	15.4164	0.779641
$392$	− 2.23607i	− 0.112938i
$393$	0	0
$394$	−0.909830	−0.0458366
$395$	0	0
$396$	0	0
$397$	9.05573i	0.454494i	0.973837	+	0.227247i	$0.0729725\pi$
$397$	9.05573i	0.454494i	−0.973837	+	0.227247i	$0.927028\pi$
$398$	− 4.47214i	− 0.224168i
$399$	0	0
$400$	0	0
$401$	−2.52786	−0.126236	−0.0631178	−	0.998006i	$-0.520104\pi$
$401$	−2.52786	−0.126236	−0.0631178	+	0.998006i	$0.520104\pi$
$402$	0	0
$403$	− 4.58359i	− 0.228325i
$404$	−7.70820	−0.383497
$405$	0	0
$406$	−3.09017	−0.153363
$407$	0.708204i	0.0351044i
$408$	0	0
$409$	−24.4721	−1.21007	−0.605035	−	0.796199i	$-0.706841\pi$
$409$	−24.4721	−1.21007	−0.605035	+	0.796199i	$0.706841\pi$
$410$	0	0
$411$	0	0
$412$	− 13.7082i	− 0.675355i
$413$	11.7082i	0.576123i
$414$	0	0
$415$	0	0
$416$	6.94427	0.340471
$417$	0	0
$418$	0.652476i	0.0319136i
$419$	−26.1803	−1.27899	−0.639497	−	0.768794i	$-0.720857\pi$
$419$	−26.1803	−1.27899	−0.639497	+	0.768794i	$0.720857\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	7.41641i	0.361025i
$423$	0	0
$424$	−18.9443	−0.920015
$425$	0	0
$426$	0	0
$427$	− 9.70820i	− 0.469813i
$428$	− 12.9443i	− 0.625685i
$429$	0	0
$430$	0	0
$431$	−17.5279	−0.844288	−0.422144	−	0.906529i	$-0.638722\pi$
$431$	−17.5279	−0.844288	−0.422144	+	0.906529i	$0.638722\pi$
$432$	0	0
$433$	28.3607i	1.36293i	0.731852	+	0.681464i	$0.238656\pi$
$433$	28.3607i	1.36293i	−0.731852	+	0.681464i	$0.761344\pi$
$434$	−2.29180	−0.110010
$435$	0	0
$436$	−13.6180	−0.652186
$437$	27.8885i	1.33409i
$438$	0	0
$439$	8.29180	0.395746	0.197873	−	0.980228i	$-0.436597\pi$
$439$	8.29180	0.395746	0.197873	+	0.980228i	$0.436597\pi$
$440$	0	0
$441$	0	0
$442$	− 1.88854i	− 0.0898289i
$443$	− 19.4164i	− 0.922501i	−0.887270	−	0.461251i	$-0.847401\pi$
$443$	− 19.4164i	− 0.922501i	0.887270	−	0.461251i	$-0.152599\pi$
$444$	0	0
$445$	0	0
$446$	12.4721	0.590573
$447$	0	0
$448$	0.236068i	0.0111532i
$449$	20.5279	0.968770	0.484385	−	0.874855i	$-0.339043\pi$
$449$	20.5279	0.968770	0.484385	+	0.874855i	$0.339043\pi$
$450$	0	0
$451$	−1.12461	−0.0529559
$452$	− 23.3262i	− 1.09717i
$453$	0	0
$454$	13.2361	0.621199
$455$	0	0
$456$	0	0
$457$	− 12.5279i	− 0.586029i	−0.956108	−	0.293014i	$-0.905342\pi$
$457$	− 12.5279i	− 0.586029i	0.956108	−	0.293014i	$-0.0946584\pi$
$458$	2.76393i	0.129150i
$459$	0	0
$460$	0	0
$461$	14.1803	0.660444	0.330222	−	0.943903i	$-0.392876\pi$
$461$	14.1803	0.660444	0.330222	+	0.943903i	$0.392876\pi$
$462$	0	0
$463$	13.8885i	0.645455i	0.946492	+	0.322728i	$0.104600\pi$
$463$	13.8885i	0.645455i	−0.946492	+	0.322728i	$0.895400\pi$
$464$	9.27051	0.430373
$465$	0	0
$466$	−4.90983	−0.227443
$467$	− 6.94427i	− 0.321343i	−0.987008	−	0.160671i	$-0.948634\pi$
$467$	− 6.94427i	− 0.321343i	0.987008	−	0.160671i	$-0.0513659\pi$
$468$	0	0
$469$	4.23607	0.195603
$470$	0	0
$471$	0	0
$472$	26.1803i	1.20505i
$473$	− 0.416408i	− 0.0191465i
$474$	0	0
$475$	0	0
$476$	4.00000	0.183340
$477$	0	0
$478$	− 3.41641i	− 0.156263i
$479$	−26.1803	−1.19621	−0.598105	−	0.801418i	$-0.704079\pi$
$479$	−26.1803	−1.19621	−0.598105	+	0.801418i	$0.704079\pi$
$480$	0	0
$481$	3.70820	0.169080
$482$	− 2.18034i	− 0.0993118i
$483$	0	0
$484$	−17.7082	−0.804918
$485$	0	0
$486$	0	0
$487$	5.76393i	0.261189i	0.991436	+	0.130594i	$0.0416885\pi$
$487$	5.76393i	0.261189i	−0.991436	+	0.130594i	$0.958311\pi$
$488$	− 21.7082i	− 0.982684i
$489$	0	0
$490$	0	0
$491$	5.76393	0.260123	0.130061	−	0.991506i	$-0.458483\pi$
$491$	5.76393	0.260123	0.130061	+	0.991506i	$0.458483\pi$
$492$	0	0
$493$	− 12.3607i	− 0.556697i
$494$	3.41641	0.153711
$495$	0	0
$496$	6.87539	0.308714
$497$	− 8.70820i	− 0.390616i
$498$	0	0
$499$	−11.0557	−0.494922	−0.247461	−	0.968898i	$-0.579596\pi$
$499$	−11.0557	−0.494922	−0.247461	+	0.968898i	$0.579596\pi$
$500$	0	0
$501$	0	0
$502$	− 4.00000i	− 0.178529i
$503$	− 8.11146i	− 0.361672i	−0.983513	−	0.180836i	$-0.942120\pi$
$503$	− 8.11146i	− 0.361672i	0.983513	−	0.180836i	$-0.0578803\pi$
$504$	0	0
$505$	0	0
$506$	−0.909830	−0.0404469
$507$	0	0
$508$	22.0902i	0.980093i
$509$	40.6525	1.80189	0.900945	−	0.433934i	$-0.142875\pi$
$509$	40.6525	1.80189	0.900945	+	0.433934i	$0.142875\pi$
$510$	0	0
$511$	−8.76393	−0.387694
$512$	18.7082i	0.826794i
$513$	0	0
$514$	−7.81966	−0.344910
$515$	0	0
$516$	0	0
$517$	0.472136i	0.0207645i
$518$	− 1.85410i	− 0.0814646i
$519$	0	0
$520$	0	0
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	0	0
$523$	− 16.3607i	− 0.715403i	−0.933836	−	0.357701i	$-0.883561\pi$
$523$	− 16.3607i	− 0.715403i	0.933836	−	0.357701i	$-0.116439\pi$
$524$	27.4164	1.19769
$525$	0	0
$526$	−10.0344	−0.437522
$527$	− 9.16718i	− 0.399329i
$528$	0	0
$529$	−15.8885	−0.690806
$530$	0	0
$531$	0	0
$532$	7.23607i	0.313723i
$533$	5.88854i	0.255061i
$534$	0	0
$535$	0	0
$536$	9.47214	0.409134
$537$	0	0
$538$	− 7.23607i	− 0.311969i
$539$	−0.236068	−0.0101682
$540$	0	0
$541$	15.9443	0.685498	0.342749	−	0.939427i	$-0.388642\pi$
$541$	15.9443	0.685498	0.342749	+	0.939427i	$0.388642\pi$
$542$	14.6525i	0.629378i
$543$	0	0
$544$	13.8885	0.595466
$545$	0	0
$546$	0	0
$547$	− 9.76393i	− 0.417476i	−0.977972	−	0.208738i	$-0.933064\pi$
$547$	− 9.76393i	− 0.417476i	0.977972	−	0.208738i	$-0.0669355\pi$
$548$	17.7082i	0.756457i
$549$	0	0
$550$	0	0
$551$	22.3607	0.952597
$552$	0	0
$553$	11.1803i	0.475436i
$554$	12.2918	0.522228
$555$	0	0
$556$	−17.2361	−0.730972
$557$	9.11146i	0.386065i	0.981192	+	0.193032i	$0.0618322\pi$
$557$	9.11146i	0.386065i	−0.981192	+	0.193032i	$0.938168\pi$
$558$	0	0
$559$	−2.18034	−0.0922186
$560$	0	0
$561$	0	0
$562$	9.49342i	0.400456i
$563$	17.4164i	0.734014i	0.930218	+	0.367007i	$0.119618\pi$
$563$	17.4164i	0.734014i	−0.930218	+	0.367007i	$0.880382\pi$
$564$	0	0
$565$	0	0
$566$	10.7639	0.452442
$567$	0	0
$568$	− 19.4721i	− 0.817033i
$569$	−3.94427	−0.165352	−0.0826762	−	0.996576i	$-0.526347\pi$
$569$	−3.94427	−0.165352	−0.0826762	+	0.996576i	$0.526347\pi$
$570$	0	0
$571$	36.5967	1.53153	0.765763	−	0.643123i	$-0.222362\pi$
$571$	36.5967	1.53153	0.765763	+	0.643123i	$0.222362\pi$
$572$	− 0.472136i	− 0.0197410i
$573$	0	0
$574$	2.94427	0.122892
$575$	0	0
$576$	0	0
$577$	− 2.00000i	− 0.0832611i	−0.999133	−	0.0416305i	$-0.986745\pi$
$577$	− 2.00000i	− 0.0832611i	0.999133	−	0.0416305i	$-0.0132552\pi$
$578$	6.72949i	0.279910i
$579$	0	0
$580$	0	0
$581$	7.70820	0.319790
$582$	0	0
$583$	2.00000i	0.0828315i
$584$	−19.5967	−0.810919
$585$	0	0
$586$	19.2361	0.794635
$587$	24.7639i	1.02212i	0.859546	+	0.511058i	$0.170746\pi$
$587$	24.7639i	1.02212i	−0.859546	+	0.511058i	$0.829254\pi$
$588$	0	0
$589$	16.5836	0.683315
$590$	0	0
$591$	0	0
$592$	5.56231i	0.228609i
$593$	− 37.3050i	− 1.53193i	−0.642882	−	0.765965i	$-0.722261\pi$
$593$	− 37.3050i	− 1.53193i	0.642882	−	0.765965i	$-0.277739\pi$
$594$	0	0
$595$	0	0
$596$	6.38197	0.261416
$597$	0	0
$598$	4.76393i	0.194812i
$599$	−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
$601$	−36.9443	−1.50699	−0.753494	−	0.657455i	$-0.771633\pi$
$601$	−36.9443	−1.50699	−0.753494	+	0.657455i	$0.771633\pi$
$602$	1.09017i	0.0444320i
$603$	0	0
$604$	−32.7426	−1.33228
$605$	0	0
$606$	0	0
$607$	− 7.12461i	− 0.289179i	−0.989492	−	0.144590i	$-0.953814\pi$
$607$	− 7.12461i	− 0.289179i	0.989492	−	0.144590i	$-0.0461862\pi$
$608$	25.1246i	1.01894i
$609$	0	0
$610$	0	0
$611$	2.47214	0.100012
$612$	0	0
$613$	44.4164i	1.79396i	0.442069	+	0.896981i	$0.354245\pi$
$613$	44.4164i	1.79396i	−0.442069	+	0.896981i	$0.645755\pi$
$614$	−2.83282	−0.114323
$615$	0	0
$616$	−0.527864	−0.0212682
$617$	5.94427i	0.239307i	0.992816	+	0.119654i	$0.0381784\pi$
$617$	5.94427i	0.239307i	−0.992816	+	0.119654i	$0.961822\pi$
$618$	0	0
$619$	11.7082	0.470592	0.235296	−	0.971924i	$-0.424394\pi$
$619$	11.7082	0.470592	0.235296	+	0.971924i	$0.424394\pi$
$620$	0	0
$621$	0	0
$622$	− 15.0557i	− 0.603680i
$623$	17.2361i	0.690548i
$624$	0	0
$625$	0	0
$626$	12.0689	0.482370
$627$	0	0
$628$	1.23607i	0.0493245i
$629$	7.41641	0.295712
$630$	0	0
$631$	27.6525	1.10083	0.550414	−	0.834892i	$-0.314470\pi$
$631$	27.6525	1.10083	0.550414	+	0.834892i	$0.314470\pi$
$632$	25.0000i	0.994447i
$633$	0	0
$634$	15.6738	0.622485
$635$	0	0
$636$	0	0
$637$	1.23607i	0.0489748i
$638$	0.729490i	0.0288808i
$639$	0	0
$640$	0	0
$641$	−43.8328	−1.73129	−0.865646	−	0.500656i	$-0.833092\pi$
$641$	−43.8328	−1.73129	−0.865646	+	0.500656i	$0.833092\pi$
$642$	0	0
$643$	− 18.4721i	− 0.728470i	−0.931307	−	0.364235i	$-0.881331\pi$
$643$	− 18.4721i	− 0.728470i	0.931307	−	0.364235i	$-0.118669\pi$
$644$	−10.0902	−0.397608
$645$	0	0
$646$	6.83282	0.268834
$647$	19.8885i	0.781899i	0.920412	+	0.390950i	$0.127853\pi$
$647$	19.8885i	0.781899i	−0.920412	+	0.390950i	$0.872147\pi$
$648$	0	0
$649$	2.76393	0.108494
$650$	0	0
$651$	0	0
$652$	2.47214i	0.0968163i
$653$	25.0557i	0.980506i	0.871580	+	0.490253i	$0.163096\pi$
$653$	25.0557i	0.980506i	−0.871580	+	0.490253i	$0.836904\pi$
$654$	0	0
$655$	0	0
$656$	−8.83282	−0.344864
$657$	0	0
$658$	− 1.23607i	− 0.0481869i
$659$	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
$661$	−42.7214	−1.66167	−0.830834	−	0.556520i	$-0.812136\pi$
$661$	−42.7214	−1.66167	−0.830834	+	0.556520i	$0.812136\pi$
$662$	− 15.2705i	− 0.593505i
$663$	0	0
$664$	17.2361	0.668889
$665$	0	0
$666$	0	0
$667$	31.1803i	1.20731i
$668$	− 8.47214i	− 0.327797i
$669$	0	0
$670$	0	0
$671$	−2.29180	−0.0884738
$672$	0	0
$673$	− 19.5279i	− 0.752744i	−0.926469	−	0.376372i	$-0.877171\pi$
$673$	− 19.5279i	− 0.752744i	0.926469	−	0.376372i	$-0.122829\pi$
$674$	10.1803	0.392132
$675$	0	0
$676$	18.5623	0.713935
$677$	14.3607i	0.551926i	0.961168	+	0.275963i	$0.0889967\pi$
$677$	14.3607i	0.551926i	−0.961168	+	0.275963i	$0.911003\pi$
$678$	0	0
$679$	−5.23607	−0.200942
$680$	0	0
$681$	0	0
$682$	0.541020i	0.0207167i
$683$	14.1246i	0.540463i	0.962795	+	0.270232i	$0.0871003\pi$
$683$	14.1246i	0.540463i	−0.962795	+	0.270232i	$0.912900\pi$
$684$	0	0
$685$	0	0
$686$	0.618034	0.0235966
$687$	0	0
$688$	− 3.27051i	− 0.124687i
$689$	10.4721	0.398957
$690$	0	0
$691$	−4.18034	−0.159028	−0.0795138	−	0.996834i	$-0.525337\pi$
$691$	−4.18034	−0.159028	−0.0795138	+	0.996834i	$0.525337\pi$
$692$	− 18.6525i	− 0.709061i
$693$	0	0
$694$	12.5066	0.474743
$695$	0	0
$696$	0	0
$697$	11.7771i	0.446089i
$698$	− 2.76393i	− 0.104616i
$699$	0	0
$700$	0	0
$701$	29.0557	1.09742	0.548710	−	0.836013i	$-0.315119\pi$
$701$	29.0557	1.09742	0.548710	+	0.836013i	$0.315119\pi$
$702$	0	0
$703$	13.4164i	0.506009i
$704$	0.0557281	0.00210033
$705$	0	0
$706$	1.34752	0.0507147
$707$	− 4.76393i	− 0.179166i
$708$	0	0
$709$	12.1115	0.454855	0.227428	−	0.973795i	$-0.426968\pi$
$709$	12.1115	0.454855	0.227428	+	0.973795i	$0.426968\pi$
$710$	0	0
$711$	0	0
$712$	38.5410i	1.44439i
$713$	23.1246i	0.866024i
$714$	0	0
$715$	0	0
$716$	−37.8885	−1.41596
$717$	0	0
$718$	− 18.6180i	− 0.694819i
$719$	16.1803	0.603425	0.301712	−	0.953399i	$-0.402442\pi$
$719$	16.1803	0.603425	0.301712	+	0.953399i	$0.402442\pi$
$720$	0	0
$721$	8.47214	0.315519
$722$	0.618034i	0.0230008i
$723$	0	0
$724$	13.2361	0.491915
$725$	0	0
$726$	0	0
$727$	− 3.05573i	− 0.113331i	−0.998393	−	0.0566653i	$-0.981953\pi$
$727$	− 3.05573i	− 0.113331i	0.998393	−	0.0566653i	$-0.0180468\pi$
$728$	2.76393i	0.102438i
$729$	0	0
$730$	0	0
$731$	−4.36068	−0.161286
$732$	0	0
$733$	− 4.00000i	− 0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$	− 4.00000i	− 0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	22.9443	0.846889
$735$	0	0
$736$	−35.0344	−1.29139
$737$	− 1.00000i	− 0.0368355i
$738$	0	0
$739$	−25.6525	−0.943642	−0.471821	−	0.881694i	$-0.656403\pi$
$739$	−25.6525	−0.943642	−0.471821	+	0.881694i	$0.656403\pi$
$740$	0	0
$741$	0	0
$742$	− 5.23607i	− 0.192222i
$743$	− 10.4721i	− 0.384185i	−0.981377	−	0.192093i	$-0.938473\pi$
$743$	− 10.4721i	− 0.384185i	0.981377	−	0.192093i	$-0.0615274\pi$
$744$	0	0
$745$	0	0
$746$	−23.3820	−0.856075
$747$	0	0
$748$	− 0.944272i	− 0.0345260i
$749$	8.00000	0.292314
$750$	0	0
$751$	3.05573	0.111505	0.0557526	−	0.998445i	$-0.482244\pi$
$751$	3.05573	0.111505	0.0557526	+	0.998445i	$0.482244\pi$
$752$	3.70820i	0.135224i
$753$	0	0
$754$	3.81966	0.139104
$755$	0	0
$756$	0	0
$757$	19.5836i	0.711778i	0.934528	+	0.355889i	$0.115822\pi$
$757$	19.5836i	0.711778i	−0.934528	+	0.355889i	$0.884178\pi$
$758$	6.90983i	0.250976i
$759$	0	0
$760$	0	0
$761$	−27.7771	−1.00692	−0.503459	−	0.864019i	$-0.667940\pi$
$761$	−27.7771	−1.00692	−0.503459	+	0.864019i	$0.667940\pi$
$762$	0	0
$763$	− 8.41641i	− 0.304694i
$764$	−10.4721	−0.378869
$765$	0	0
$766$	20.5410	0.742177
$767$	− 14.4721i	− 0.522559i
$768$	0	0
$769$	43.0132	1.55109	0.775547	−	0.631290i	$-0.217474\pi$
$769$	43.0132	1.55109	0.775547	+	0.631290i	$0.217474\pi$
$770$	0	0
$771$	0	0
$772$	− 20.0902i	− 0.723061i
$773$	50.1803i	1.80486i	0.430835	+	0.902431i	$0.358219\pi$
$773$	50.1803i	1.80486i	−0.430835	+	0.902431i	$0.641781\pi$
$774$	0	0
$775$	0	0
$776$	−11.7082	−0.420300
$777$	0	0
$778$	1.78522i	0.0640032i
$779$	−21.3050	−0.763329
$780$	0	0
$781$	−2.05573	−0.0735597
$782$	9.52786i	0.340716i
$783$	0	0
$784$	−1.85410	−0.0662179
$785$	0	0
$786$	0	0
$787$	40.7639i	1.45308i	0.687126	+	0.726539i	$0.258872\pi$
$787$	40.7639i	1.45308i	−0.687126	+	0.726539i	$0.741128\pi$
$788$	2.38197i	0.0848540i
$789$	0	0
$790$	0	0
$791$	14.4164	0.512588
$792$	0	0
$793$	12.0000i	0.426132i
$794$	−5.59675	−0.198621
$795$	0	0
$796$	−11.7082	−0.414986
$797$	35.4164i	1.25451i	0.778813	+	0.627257i	$0.215822\pi$
$797$	35.4164i	1.25451i	−0.778813	+	0.627257i	$0.784178\pi$
$798$	0	0
$799$	4.94427	0.174916
$800$	0	0
$801$	0	0
$802$	− 1.56231i	− 0.0551669i
$803$	2.06888i	0.0730093i
$804$	0	0
$805$	0	0
$806$	2.83282	0.0997817
$807$	0	0
$808$	− 10.6525i	− 0.374753i
$809$	29.4721	1.03619	0.518093	−	0.855325i	$-0.326642\pi$
$809$	29.4721	1.03619	0.518093	+	0.855325i	$0.326642\pi$
$810$	0	0
$811$	−42.7214	−1.50015	−0.750075	−	0.661353i	$-0.769983\pi$
$811$	−42.7214	−1.50015	−0.750075	+	0.661353i	$0.769983\pi$
$812$	8.09017i	0.283909i
$813$	0	0
$814$	−0.437694	−0.0153412
$815$	0	0
$816$	0	0
$817$	− 7.88854i	− 0.275985i
$818$	− 15.1246i	− 0.528820i
$819$	0	0
$820$	0	0
$821$	−28.8328	−1.00627	−0.503136	−	0.864207i	$-0.667821\pi$
$821$	−28.8328	−1.00627	−0.503136	+	0.864207i	$0.667821\pi$
$822$	0	0
$823$	31.6525i	1.10334i	0.834064	+	0.551668i	$0.186008\pi$
$823$	31.6525i	1.10334i	−0.834064	+	0.551668i	$0.813992\pi$
$824$	18.9443	0.659955
$825$	0	0
$826$	−7.23607	−0.251775
$827$	− 41.5410i	− 1.44452i	−0.691620	−	0.722261i	$-0.743103\pi$
$827$	− 41.5410i	− 1.44452i	0.691620	−	0.722261i	$-0.256897\pi$
$828$	0	0
$829$	7.63932	0.265325	0.132662	−	0.991161i	$-0.457647\pi$
$829$	7.63932	0.265325	0.132662	+	0.991161i	$0.457647\pi$
$830$	0	0
$831$	0	0
$832$	− 0.291796i	− 0.0101162i
$833$	2.47214i	0.0856544i
$834$	0	0
$835$	0	0
$836$	1.70820	0.0590795
$837$	0	0
$838$	− 16.1803i	− 0.558941i
$839$	−30.6525	−1.05824	−0.529120	−	0.848547i	$-0.677478\pi$
$839$	−30.6525	−1.05824	−0.529120	+	0.848547i	$0.677478\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	− 8.03444i	− 0.276885i
$843$	0	0
$844$	19.4164	0.668340
$845$	0	0
$846$	0	0
$847$	− 10.9443i	− 0.376050i
$848$	15.7082i	0.539422i
$849$	0	0
$850$	0	0
$851$	−18.7082	−0.641309
$852$	0	0
$853$	− 27.4164i	− 0.938720i	−0.883007	−	0.469360i	$-0.844485\pi$
$853$	− 27.4164i	− 0.938720i	0.883007	−	0.469360i	$-0.155515\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	17.8885	0.611418
$857$	15.8197i	0.540389i	0.962806	+	0.270195i	$0.0870881\pi$
$857$	15.8197i	0.540389i	−0.962806	+	0.270195i	$0.912912\pi$
$858$	0	0
$859$	−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
$862$	− 10.8328i	− 0.368967i
$863$	18.3475i	0.624557i	0.949991	+	0.312278i	$0.101092\pi$
$863$	18.3475i	0.624557i	−0.949991	+	0.312278i	$0.898908\pi$
$864$	0	0
$865$	0	0
$866$	−17.5279	−0.595621
$867$	0	0
$868$	6.00000i	0.203653i
$869$	2.63932	0.0895328
$870$	0	0
$871$	−5.23607	−0.177417
$872$	− 18.8197i	− 0.637314i
$873$	0	0
$874$	−17.2361	−0.583019
$875$	0	0
$876$	0	0
$877$	30.3607i	1.02521i	0.858625	+	0.512604i	$0.171319\pi$
$877$	30.3607i	1.02521i	−0.858625	+	0.512604i	$0.828681\pi$
$878$	5.12461i	0.172947i
$879$	0	0
$880$	0	0
$881$	−5.81966	−0.196069	−0.0980347	−	0.995183i	$-0.531256\pi$
$881$	−5.81966	−0.196069	−0.0980347	+	0.995183i	$0.531256\pi$
$882$	0	0
$883$	1.40325i	0.0472232i	0.999721	+	0.0236116i	$0.00751650\pi$
$883$	1.40325i	0.0472232i	−0.999721	+	0.0236116i	$0.992483\pi$
$884$	−4.94427	−0.166294
$885$	0	0
$886$	12.0000	0.403148
$887$	21.3475i	0.716780i	0.933572	+	0.358390i	$0.116674\pi$
$887$	21.3475i	0.716780i	−0.933572	+	0.358390i	$0.883326\pi$
$888$	0	0
$889$	−13.6525	−0.457889
$890$	0	0
$891$	0	0
$892$	− 32.6525i	− 1.09329i
$893$	8.94427i	0.299309i
$894$	0	0
$895$	0	0
$896$	−11.3820	−0.380245
$897$	0	0
$898$	12.6869i	0.423368i
$899$	18.5410	0.618378
$900$	0	0
$901$	20.9443	0.697755
$902$	− 0.695048i	− 0.0231426i
$903$	0	0
$904$	32.2361	1.07216
$905$	0	0
$906$	0	0
$907$	34.8328i	1.15660i	0.815823	+	0.578302i	$0.196285\pi$
$907$	34.8328i	1.15660i	−0.815823	+	0.578302i	$0.803715\pi$
$908$	− 34.6525i	− 1.14998i
$909$	0	0
$910$	0	0
$911$	−0.819660	−0.0271566	−0.0135783	−	0.999908i	$-0.504322\pi$
$911$	−0.819660	−0.0271566	−0.0135783	+	0.999908i	$0.504322\pi$
$912$	0	0
$913$	− 1.81966i	− 0.0602220i
$914$	7.74265	0.256104
$915$	0	0
$916$	7.23607	0.239086
$917$	16.9443i	0.559549i
$918$	0	0
$919$	27.7639	0.915848	0.457924	−	0.888991i	$-0.348593\pi$
$919$	27.7639	0.915848	0.457924	+	0.888991i	$0.348593\pi$
$920$	0	0
$921$	0	0
$922$	8.76393i	0.288625i
$923$	10.7639i	0.354299i
$924$	0	0
$925$	0	0
$926$	−8.58359	−0.282074
$927$	0	0
$928$	28.0902i	0.922105i
$929$	38.2918	1.25631	0.628157	−	0.778087i	$-0.283810\pi$
$929$	38.2918	1.25631	0.628157	+	0.778087i	$0.283810\pi$
$930$	0	0
$931$	−4.47214	−0.146568
$932$	12.8541i	0.421050i
$933$	0	0
$934$	4.29180	0.140432
$935$	0	0
$936$	0	0
$937$	35.2361i	1.15111i	0.817762	+	0.575556i	$0.195214\pi$
$937$	35.2361i	1.15111i	−0.817762	+	0.575556i	$0.804786\pi$
$938$	2.61803i	0.0854818i
$939$	0	0
$940$	0	0
$941$	5.23607	0.170691	0.0853455	−	0.996351i	$-0.472801\pi$
$941$	5.23607	0.170691	0.0853455	+	0.996351i	$0.472801\pi$
$942$	0	0
$943$	− 29.7082i	− 0.967432i
$944$	21.7082	0.706542
$945$	0	0
$946$	0.257354	0.00836731
$947$	− 34.8328i	− 1.13191i	−0.824435	−	0.565957i	$-0.808507\pi$
$947$	− 34.8328i	− 1.13191i	0.824435	−	0.565957i	$-0.191493\pi$
$948$	0	0
$949$	10.8328	0.351648
$950$	0	0
$951$	0	0
$952$	5.52786i	0.179159i
$953$	3.47214i	0.112474i	0.998417	+	0.0562368i	$0.0179102\pi$
$953$	3.47214i	0.112474i	−0.998417	+	0.0562368i	$0.982090\pi$
$954$	0	0
$955$	0	0
$956$	−8.94427	−0.289278
$957$	0	0
$958$	− 16.1803i	− 0.522763i
$959$	−10.9443	−0.353409
$960$	0	0
$961$	−17.2492	−0.556427
$962$	2.29180i	0.0738905i
$963$	0	0
$964$	−5.70820	−0.183849
$965$	0	0
$966$	0	0
$967$	− 14.1115i	− 0.453794i	−0.973919	−	0.226897i	$-0.927142\pi$
$967$	− 14.1115i	− 0.453794i	0.973919	−	0.226897i	$-0.0728580\pi$
$968$	− 24.4721i	− 0.786564i
$969$	0	0
$970$	0	0
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	0	0
$973$	− 10.6525i	− 0.341503i
$974$	−3.56231	−0.114144
$975$	0	0
$976$	−18.0000	−0.576166
$977$	11.4721i	0.367026i	0.983017	+	0.183513i	$0.0587470\pi$
$977$	11.4721i	0.367026i	−0.983017	+	0.183513i	$0.941253\pi$
$978$	0	0
$979$	4.06888	0.130042
$980$	0	0
$981$	0	0
$982$	3.56231i	0.113678i
$983$	− 34.5410i	− 1.10169i	−0.834608	−	0.550844i	$-0.814306\pi$
$983$	− 34.5410i	− 1.10169i	0.834608	−	0.550844i	$-0.185694\pi$
$984$	0	0
$985$	0	0
$986$	7.63932	0.243286
$987$	0	0
$988$	− 8.94427i	− 0.284555i
$989$	11.0000	0.349780
$990$	0	0
$991$	13.1803	0.418687	0.209344	−	0.977842i	$-0.432867\pi$
$991$	13.1803	0.418687	0.209344	+	0.977842i	$0.432867\pi$
$992$	20.8328i	0.661443i
$993$	0	0
$994$	5.38197	0.170706
$995$	0	0
$996$	0	0
$997$	− 45.4164i	− 1.43835i	−0.694828	−	0.719176i	$-0.744519\pi$
$997$	− 45.4164i	− 1.43835i	0.694828	−	0.719176i	$-0.255481\pi$
$998$	− 6.83282i	− 0.216289i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.d.k.1324.3		4
3.2	odd	2		175.2.b.c.99.2		4
5.2	odd	4		1575.2.a.s.1.1		2
5.3	odd	4		1575.2.a.n.1.2		2
5.4	even	2	inner	1575.2.d.k.1324.2		4
12.11	even	2		2800.2.g.s.449.1		4
15.2	even	4		175.2.a.d.1.2	✓	2
15.8	even	4		175.2.a.e.1.1	yes	2
15.14	odd	2		175.2.b.c.99.3		4
21.20	even	2		1225.2.b.k.99.2		4
60.23	odd	4		2800.2.a.bp.1.2		2
60.47	odd	4		2800.2.a.bh.1.1		2
60.59	even	2		2800.2.g.s.449.4		4
105.62	odd	4		1225.2.a.n.1.2		2
105.83	odd	4		1225.2.a.u.1.1		2
105.104	even	2		1225.2.b.k.99.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
175.2.a.d.1.2	✓	2	15.2	even	4
175.2.a.e.1.1	yes	2	15.8	even	4
175.2.b.c.99.2		4	3.2	odd	2
175.2.b.c.99.3		4	15.14	odd	2
1225.2.a.n.1.2		2	105.62	odd	4
1225.2.a.u.1.1		2	105.83	odd	4
1225.2.b.k.99.2		4	21.20	even	2
1225.2.b.k.99.3		4	105.104	even	2
1575.2.a.n.1.2		2	5.3	odd	4
1575.2.a.s.1.1		2	5.2	odd	4
1575.2.d.k.1324.2		4	5.4	even	2	inner
1575.2.d.k.1324.3		4	1.1	even	1	trivial
2800.2.a.bh.1.1		2	60.47	odd	4
2800.2.a.bp.1.2		2	60.23	odd	4
2800.2.g.s.449.1		4	12.11	even	2
2800.2.g.s.449.4		4	60.59	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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.618034i q^{2} +1.61803 q^{4} +1.00000i q^{7} +2.23607i q^{8} +O(q^{10})\)	\(q+0.618034i q^{2} +1.61803 q^{4} +1.00000i q^{7} +2.23607i q^{8} +0.236068 q^{11} -1.23607i q^{13} -0.618034 q^{14} +1.85410 q^{16} -2.47214i q^{17} +4.47214 q^{19} +0.145898i q^{22} +6.23607i q^{23} +0.763932 q^{26} +1.61803i q^{28} +5.00000 q^{29} +3.70820 q^{31} +5.61803i q^{32} +1.52786 q^{34} +3.00000i q^{37} +2.76393i q^{38} -4.76393 q^{41} -1.76393i q^{43} +0.381966 q^{44} -3.85410 q^{46} +2.00000i q^{47} -1.00000 q^{49} -2.00000i q^{52} +8.47214i q^{53} -2.23607 q^{56} +3.09017i q^{58} +11.7082 q^{59} -9.70820 q^{61} +2.29180i q^{62} +0.236068 q^{64} -4.23607i q^{67} -4.00000i q^{68} -8.70820 q^{71} +8.76393i q^{73} -1.85410 q^{74} +7.23607 q^{76} +0.236068i q^{77} +11.1803 q^{79} -2.94427i q^{82} -7.70820i q^{83} +1.09017 q^{86} +0.527864i q^{88} +17.2361 q^{89} +1.23607 q^{91} +10.0902i q^{92} -1.23607 q^{94} +5.23607i q^{97} -0.618034i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4}+O(q^{10}) \) 4 * q + 2 * q^4	\( 4 q + 2 q^{4} - 8 q^{11} + 2 q^{14} - 6 q^{16} + 12 q^{26} + 20 q^{29} - 12 q^{31} + 24 q^{34} - 28 q^{41} + 6 q^{44} - 2 q^{46} - 4 q^{49} + 20 q^{59} - 12 q^{61} - 8 q^{64} - 8 q^{71} + 6 q^{74} + 20 q^{76} - 18 q^{86} + 60 q^{89} - 4 q^{91} + 4 q^{94}+O(q^{100}) \) 4 * q + 2 * q^4 - 8 * q^11 + 2 * q^14 - 6 * q^16 + 12 * q^26 + 20 * q^29 - 12 * q^31 + 24 * q^34 - 28 * q^41 + 6 * q^44 - 2 * q^46 - 4 * q^49 + 20 * q^59 - 12 * q^61 - 8 * q^64 - 8 * q^71 + 6 * q^74 + 20 * q^76 - 18 * q^86 + 60 * q^89 - 4 * q^91 + 4 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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