LMFDB - Embedded newform 2175.2.c.o.349.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2175,2,Mod(349,2175)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2175, 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("2175.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.c (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:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 24x^{12} + 224x^{10} + 1023x^{8} + 2364x^{6} + 2612x^{4} + 1241x^{2} + 196 $ x^14 + 24x^12 + 224x^10 + 1023x^8 + 2364x^6 + 2612x^4 + 1241x^2 + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.7
Root			$-0.560139i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.o.349.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.560139i q^{2} -1.00000i q^{3} +1.68624 q^{4} -0.560139 q^{6} +4.36313i q^{7} -2.06481i q^{8} -1.00000 q^{9} -3.72689 q^{11} -1.68624i q^{12} +2.53237i q^{13} +2.44396 q^{14} +2.21591 q^{16} +5.11270i q^{17} +0.560139i q^{18} -2.09916 q^{19} +4.36313 q^{21} +2.08758i q^{22} -6.47404i q^{23} -2.06481 q^{24} +1.41848 q^{26} +1.00000i q^{27} +7.35730i q^{28} +1.00000 q^{29} -9.94570 q^{31} -5.37083i q^{32} +3.72689i q^{33} +2.86382 q^{34} -1.68624 q^{36} +5.37249i q^{37} +1.17582i q^{38} +2.53237 q^{39} +7.96557 q^{41} -2.44396i q^{42} +10.7771i q^{43} -6.28445 q^{44} -3.62636 q^{46} +9.38424i q^{47} -2.21591i q^{48} -12.0369 q^{49} +5.11270 q^{51} +4.27019i q^{52} +5.64265i q^{53} +0.560139 q^{54} +9.00902 q^{56} +2.09916i q^{57} -0.560139i q^{58} -4.24056 q^{59} +9.81848 q^{61} +5.57097i q^{62} -4.36313i q^{63} +1.42341 q^{64} +2.08758 q^{66} +0.520614i q^{67} +8.62126i q^{68} -6.47404 q^{69} +3.22415 q^{71} +2.06481i q^{72} +10.1718i q^{73} +3.00934 q^{74} -3.53970 q^{76} -16.2609i q^{77} -1.41848i q^{78} -1.81484 q^{79} +1.00000 q^{81} -4.46182i q^{82} +1.35377i q^{83} +7.35730 q^{84} +6.03667 q^{86} -1.00000i q^{87} +7.69532i q^{88} -14.2314 q^{89} -11.0490 q^{91} -10.9168i q^{92} +9.94570i q^{93} +5.25648 q^{94} -5.37083 q^{96} +14.0094i q^{97} +6.74233i q^{98} +3.72689 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 20 q^{4} + 4 q^{6} - 14 q^{9} + 8 q^{11} - 30 q^{14} + 24 q^{16} - 30 q^{19} - 2 q^{21} - 6 q^{24} + 12 q^{26} + 14 q^{29} + 10 q^{31} - 14 q^{34} + 20 q^{36} + 2 q^{39} + 44 q^{41} - 30 q^{44} - 8 q^{46}+ \cdots - 8 q^{99}+O(q^{100}) $ 14 * q - 20 * q^4 + 4 * q^6 - 14 * q^9 + 8 * q^11 - 30 * q^14 + 24 * q^16 - 30 * q^19 - 2 * q^21 - 6 * q^24 + 12 * q^26 + 14 * q^29 + 10 * q^31 - 14 * q^34 + 20 * q^36 + 2 * q^39 + 44 * q^41 - 30 * q^44 - 8 * q^46 - 24 * q^49 + 16 * q^51 - 4 * q^54 + 28 * q^56 - 12 * q^59 + 46 * q^61 - 10 * q^64 + 6 * q^66 - 28 * q^69 + 52 * q^71 + 20 * q^74 + 92 * q^76 - 28 * q^79 + 14 * q^81 + 48 * q^84 + 88 * q^86 - 28 * q^89 + 26 * q^91 + 6 * q^94 + 36 * q^96 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times$.

$n$	$901$	$1451$	$2002$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 0.560139i	− 0.396078i	−0.980194	−	0.198039i	$-0.936543\pi$
$2$	− 0.560139i	− 0.396078i	0.980194	−	0.198039i	$-0.0634573\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.68624	0.843122
$5$	0	0
$5$	0	0
$6$	−0.560139	−0.228676
$7$	4.36313i	1.64911i	0.565784	+	0.824554i	$0.308574\pi$
$7$	4.36313i	1.64911i	−0.565784	+	0.824554i	$0.691426\pi$
$8$	− 2.06481i	− 0.730020i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.72689	−1.12370	−0.561850	−	0.827239i	$-0.689910\pi$
$11$	−3.72689	−1.12370	−0.561850	+	0.827239i	$0.689910\pi$
$12$	− 1.68624i	− 0.486777i
$13$	2.53237i	0.702352i	0.936309	+	0.351176i	$0.114218\pi$
$13$	2.53237i	0.702352i	−0.936309	+	0.351176i	$0.885782\pi$
$14$	2.44396	0.653175
$15$	0	0
$16$	2.21591	0.553977
$17$	5.11270i	1.24001i	0.784597	+	0.620006i	$0.212870\pi$
$17$	5.11270i	1.24001i	−0.784597	+	0.620006i	$0.787130\pi$
$18$	0.560139i	0.132026i
$19$	−2.09916	−0.481581	−0.240791	−	0.970577i	$-0.577407\pi$
$19$	−2.09916	−0.481581	−0.240791	+	0.970577i	$0.577407\pi$
$20$	0	0
$21$	4.36313	0.952113
$22$	2.08758i	0.445073i
$23$	− 6.47404i	− 1.34993i	−0.737849	−	0.674966i	$-0.764158\pi$
$23$	− 6.47404i	− 1.34993i	0.737849	−	0.674966i	$-0.235842\pi$
$24$	−2.06481	−0.421477
$25$	0	0
$26$	1.41848	0.278186
$27$	1.00000i	0.192450i
$28$	7.35730i	1.39040i
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−9.94570	−1.78630	−0.893150	−	0.449759i	$-0.851510\pi$
$31$	−9.94570	−1.78630	−0.893150	+	0.449759i	$0.851510\pi$
$32$	− 5.37083i	− 0.949438i
$33$	3.72689i	0.648768i
$34$	2.86382	0.491141
$35$	0	0
$36$	−1.68624	−0.281041
$37$	5.37249i	0.883232i	0.897204	+	0.441616i	$0.145595\pi$
$37$	5.37249i	0.883232i	−0.897204	+	0.441616i	$0.854405\pi$
$38$	1.17582i	0.190744i
$39$	2.53237	0.405503
$40$	0	0
$41$	7.96557	1.24401	0.622006	−	0.783012i	$-0.286318\pi$
$41$	7.96557	1.24401	0.622006	+	0.783012i	$0.286318\pi$
$42$	− 2.44396i	− 0.377111i
$43$	10.7771i	1.64349i	0.569854	+	0.821746i	$0.307000\pi$
$43$	10.7771i	1.64349i	−0.569854	+	0.821746i	$0.693000\pi$
$44$	−6.28445	−0.947416
$45$	0	0
$46$	−3.62636	−0.534678
$47$	9.38424i	1.36883i	0.729092	+	0.684416i	$0.239943\pi$
$47$	9.38424i	1.36883i	−0.729092	+	0.684416i	$0.760057\pi$
$48$	− 2.21591i	− 0.319839i
$49$	−12.0369	−1.71955
$50$	0	0
$51$	5.11270	0.715921
$52$	4.27019i	0.592169i
$53$	5.64265i	0.775077i	0.921853	+	0.387539i	$0.126675\pi$
$53$	5.64265i	0.775077i	−0.921853	+	0.387539i	$0.873325\pi$
$54$	0.560139	0.0762252
$55$	0	0
$56$	9.00902	1.20388
$57$	2.09916i	0.278041i
$58$	− 0.560139i	− 0.0735498i
$59$	−4.24056	−0.552073	−0.276037	−	0.961147i	$-0.589021\pi$
$59$	−4.24056	−0.552073	−0.276037	+	0.961147i	$0.589021\pi$
$60$	0	0
$61$	9.81848	1.25713	0.628564	−	0.777758i	$-0.283643\pi$
$61$	9.81848	1.25713	0.628564	+	0.777758i	$0.283643\pi$
$62$	5.57097i	0.707514i
$63$	− 4.36313i	− 0.549702i
$64$	1.42341	0.177926
$65$	0	0
$66$	2.08758	0.256963
$67$	0.520614i	0.0636031i	0.999494	+	0.0318015i	$0.0101245\pi$
$67$	0.520614i	0.0636031i	−0.999494	+	0.0318015i	$0.989876\pi$
$68$	8.62126i	1.04548i
$69$	−6.47404	−0.779383
$70$	0	0
$71$	3.22415	0.382636	0.191318	−	0.981528i	$-0.438724\pi$
$71$	3.22415	0.382636	0.191318	+	0.981528i	$0.438724\pi$
$72$	2.06481i	0.243340i
$73$	10.1718i	1.19052i	0.803532	+	0.595262i	$0.202952\pi$
$73$	10.1718i	1.19052i	−0.803532	+	0.595262i	$0.797048\pi$
$74$	3.00934	0.349829
$75$	0	0
$76$	−3.53970	−0.406032
$77$	− 16.2609i	− 1.85310i
$78$	− 1.41848i	− 0.160611i
$79$	−1.81484	−0.204185	−0.102093	−	0.994775i	$-0.532554\pi$
$79$	−1.81484	−0.204185	−0.102093	+	0.994775i	$0.532554\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 4.46182i	− 0.492726i
$83$	1.35377i	0.148595i	0.997236	+	0.0742976i	$0.0236715\pi$
$83$	1.35377i	0.148595i	−0.997236	+	0.0742976i	$0.976329\pi$
$84$	7.35730	0.802747
$85$	0	0
$86$	6.03667	0.650951
$87$	− 1.00000i	− 0.107211i
$88$	7.69532i	0.820323i
$89$	−14.2314	−1.50853	−0.754263	−	0.656572i	$-0.772006\pi$
$89$	−14.2314	−1.50853	−0.754263	+	0.656572i	$0.772006\pi$
$90$	0	0
$91$	−11.0490	−1.15825
$92$	− 10.9168i	− 1.13816i
$93$	9.94570i	1.03132i
$94$	5.25648	0.542164
$95$	0	0
$96$	−5.37083	−0.548159
$97$	14.0094i	1.42244i	0.702968	+	0.711221i	$0.251858\pi$
$97$	14.0094i	1.42244i	−0.702968	+	0.711221i	$0.748142\pi$
$98$	6.74233i	0.681078i
$99$	3.72689	0.374567
$100$	0	0
$101$	−1.82364	−0.181459	−0.0907294	−	0.995876i	$-0.528920\pi$
$101$	−1.82364	−0.181459	−0.0907294	+	0.995876i	$0.528920\pi$
$102$	− 2.86382i	− 0.283561i
$103$	− 10.6248i	− 1.04689i	−0.852059	−	0.523445i	$-0.824647\pi$
$103$	− 10.6248i	− 1.04689i	0.852059	−	0.523445i	$-0.175353\pi$
$104$	5.22885	0.512731
$105$	0	0
$106$	3.16067	0.306991
$107$	− 12.2121i	− 1.18059i	−0.807189	−	0.590293i	$-0.799012\pi$
$107$	− 12.2121i	− 1.18059i	0.807189	−	0.590293i	$-0.200988\pi$
$108$	1.68624i	0.162259i
$109$	7.10102	0.680155	0.340077	−	0.940397i	$-0.389547\pi$
$109$	7.10102	0.680155	0.340077	+	0.940397i	$0.389547\pi$
$110$	0	0
$111$	5.37249	0.509934
$112$	9.66829i	0.913568i
$113$	− 2.21008i	− 0.207907i	−0.994582	−	0.103953i	$-0.966851\pi$
$113$	− 2.21008i	− 0.207907i	0.994582	−	0.103953i	$-0.0331493\pi$
$114$	1.17582	0.110126
$115$	0	0
$116$	1.68624	0.156564
$117$	− 2.53237i	− 0.234117i
$118$	2.37530i	0.218664i
$119$	−22.3074	−2.04491
$120$	0	0
$121$	2.88971	0.262701
$122$	− 5.49971i	− 0.497920i
$123$	− 7.96557i	− 0.718231i
$124$	−16.7709	−1.50607
$125$	0	0
$126$	−2.44396	−0.217725
$127$	− 12.1699i	− 1.07990i	−0.841697	−	0.539950i	$-0.818443\pi$
$127$	− 12.1699i	− 1.07990i	0.841697	−	0.539950i	$-0.181557\pi$
$128$	− 11.5390i	− 1.01991i
$129$	10.7771	0.948870
$130$	0	0
$131$	0.754636	0.0659329	0.0329664	−	0.999456i	$-0.489505\pi$
$131$	0.754636	0.0659329	0.0329664	+	0.999456i	$0.489505\pi$
$132$	6.28445i	0.546991i
$133$	− 9.15892i	− 0.794179i
$134$	0.291616	0.0251918
$135$	0	0
$136$	10.5567	0.905234
$137$	19.2792i	1.64713i	0.567219	+	0.823567i	$0.308019\pi$
$137$	19.2792i	1.64713i	−0.567219	+	0.823567i	$0.691981\pi$
$138$	3.62636i	0.308697i
$139$	−21.1062	−1.79020	−0.895102	−	0.445862i	$-0.852897\pi$
$139$	−21.1062	−1.79020	−0.895102	+	0.445862i	$0.852897\pi$
$140$	0	0
$141$	9.38424	0.790296
$142$	− 1.80597i	− 0.151554i
$143$	− 9.43785i	− 0.789233i
$144$	−2.21591	−0.184659
$145$	0	0
$146$	5.69765	0.471540
$147$	12.0369i	0.992786i
$148$	9.05933i	0.744672i
$149$	7.12028	0.583316	0.291658	−	0.956523i	$-0.405793\pi$
$149$	7.12028	0.583316	0.291658	+	0.956523i	$0.405793\pi$
$150$	0	0
$151$	5.46682	0.444884	0.222442	−	0.974946i	$-0.428597\pi$
$151$	5.46682	0.444884	0.222442	+	0.974946i	$0.428597\pi$
$152$	4.33437i	0.351564i
$153$	− 5.11270i	− 0.413337i
$154$	−9.10836	−0.733973
$155$	0	0
$156$	4.27019	0.341889
$157$	− 14.6519i	− 1.16935i	−0.811268	−	0.584675i	$-0.801222\pi$
$157$	− 14.6519i	− 1.16935i	0.811268	−	0.584675i	$-0.198778\pi$
$158$	1.01656i	0.0808732i
$159$	5.64265	0.447491
$160$	0	0
$161$	28.2471	2.22618
$162$	− 0.560139i	− 0.0440087i
$163$	− 13.9352i	− 1.09149i	−0.837952	−	0.545743i	$-0.816247\pi$
$163$	− 13.9352i	− 1.09149i	0.837952	−	0.545743i	$-0.183753\pi$
$164$	13.4319	1.04885
$165$	0	0
$166$	0.758297	0.0588553
$167$	25.1850i	1.94887i	0.224663	+	0.974436i	$0.427872\pi$
$167$	25.1850i	1.94887i	−0.224663	+	0.974436i	$0.572128\pi$
$168$	− 9.00902i	− 0.695061i
$169$	6.58712	0.506702
$170$	0	0
$171$	2.09916	0.160527
$172$	18.1728i	1.38566i
$173$	− 12.7333i	− 0.968097i	−0.875041	−	0.484048i	$-0.839166\pi$
$173$	− 12.7333i	− 0.968097i	0.875041	−	0.484048i	$-0.160834\pi$
$174$	−0.560139	−0.0424640
$175$	0	0
$176$	−8.25845	−0.622504
$177$	4.24056i	0.318740i
$178$	7.97157i	0.597494i
$179$	7.19567	0.537830	0.268915	−	0.963164i	$-0.413335\pi$
$179$	7.19567	0.537830	0.268915	+	0.963164i	$0.413335\pi$
$180$	0	0
$181$	6.47508	0.481289	0.240644	−	0.970613i	$-0.422641\pi$
$181$	6.47508	0.481289	0.240644	+	0.970613i	$0.422641\pi$
$182$	6.18899i	0.458759i
$183$	− 9.81848i	− 0.725803i
$184$	−13.3677	−0.985477
$185$	0	0
$186$	5.57097	0.408483
$187$	− 19.0545i	− 1.39340i
$188$	15.8241i	1.15409i
$189$	−4.36313	−0.317371
$190$	0	0
$191$	18.0212	1.30397	0.651983	−	0.758234i	$-0.273937\pi$
$191$	18.0212	1.30397	0.651983	+	0.758234i	$0.273937\pi$
$192$	− 1.42341i	− 0.102725i
$193$	10.6003i	0.763025i	0.924364	+	0.381512i	$0.124597\pi$
$193$	10.6003i	0.763025i	−0.924364	+	0.381512i	$0.875403\pi$
$194$	7.84723	0.563398
$195$	0	0
$196$	−20.2971	−1.44979
$197$	− 2.28624i	− 0.162888i	−0.996678	−	0.0814441i	$-0.974047\pi$
$197$	− 2.28624i	− 0.162888i	0.996678	−	0.0814441i	$-0.0259532\pi$
$198$	− 2.08758i	− 0.148358i
$199$	22.7385	1.61189	0.805946	−	0.591989i	$-0.201657\pi$
$199$	22.7385	1.61189	0.805946	+	0.591989i	$0.201657\pi$
$200$	0	0
$201$	0.520614	0.0367213
$202$	1.02149i	0.0718718i
$203$	4.36313i	0.306232i
$204$	8.62126	0.603609
$205$	0	0
$206$	−5.95135	−0.414650
$207$	6.47404i	0.449977i
$208$	5.61149i	0.389087i
$209$	7.82336	0.541153
$210$	0	0
$211$	16.9446	1.16651	0.583256	−	0.812289i	$-0.301779\pi$
$211$	16.9446	1.16651	0.583256	+	0.812289i	$0.301779\pi$
$212$	9.51488i	0.653485i
$213$	− 3.22415i	− 0.220915i
$214$	−6.84046	−0.467604
$215$	0	0
$216$	2.06481	0.140492
$217$	− 43.3944i	− 2.94580i
$218$	− 3.97756i	− 0.269394i
$219$	10.1718	0.687350
$220$	0	0
$221$	−12.9472	−0.870925
$222$	− 3.00934i	− 0.201974i
$223$	− 7.13150i	− 0.477560i	−0.971074	−	0.238780i	$-0.923252\pi$
$223$	− 7.13150i	− 0.477560i	0.971074	−	0.238780i	$-0.0767475\pi$
$224$	23.4336	1.56573
$225$	0	0
$226$	−1.23795	−0.0823474
$227$	2.42109i	0.160693i	0.996767	+	0.0803467i	$0.0256028\pi$
$227$	2.42109i	0.160693i	−0.996767	+	0.0803467i	$0.974397\pi$
$228$	3.53970i	0.234423i
$229$	−14.4538	−0.955132	−0.477566	−	0.878596i	$-0.658481\pi$
$229$	−14.4538	−0.955132	−0.477566	+	0.878596i	$0.658481\pi$
$230$	0	0
$231$	−16.2609	−1.06989
$232$	− 2.06481i	− 0.135561i
$233$	− 10.0591i	− 0.658994i	−0.944157	−	0.329497i	$-0.893121\pi$
$233$	− 10.0591i	− 0.658994i	0.944157	−	0.329497i	$-0.106879\pi$
$234$	−1.41848	−0.0927287
$235$	0	0
$236$	−7.15061	−0.465465
$237$	1.81484i	0.117886i
$238$	12.4952i	0.809945i
$239$	14.3621	0.929006	0.464503	−	0.885572i	$-0.346233\pi$
$239$	14.3621	0.929006	0.464503	+	0.885572i	$0.346233\pi$
$240$	0	0
$241$	8.58457	0.552981	0.276490	−	0.961017i	$-0.410829\pi$
$241$	8.58457	0.552981	0.276490	+	0.961017i	$0.410829\pi$
$242$	− 1.61864i	− 0.104050i
$243$	− 1.00000i	− 0.0641500i
$244$	16.5564	1.05991
$245$	0	0
$246$	−4.46182	−0.284476
$247$	− 5.31585i	− 0.338240i
$248$	20.5360i	1.30404i
$249$	1.35377	0.0857915
$250$	0	0
$251$	20.6767	1.30510	0.652552	−	0.757744i	$-0.273699\pi$
$251$	20.6767	1.30510	0.652552	+	0.757744i	$0.273699\pi$
$252$	− 7.35730i	− 0.463466i
$253$	24.1281i	1.51692i
$254$	−6.81681	−0.427725
$255$	0	0
$256$	−3.61662	−0.226039
$257$	− 22.6024i	− 1.40990i	−0.709258	−	0.704949i	$-0.750970\pi$
$257$	− 22.6024i	− 1.40990i	0.709258	−	0.704949i	$-0.249030\pi$
$258$	− 6.03667i	− 0.375827i
$259$	−23.4409	−1.45654
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	− 0.422701i	− 0.0261146i
$263$	− 18.7013i	− 1.15317i	−0.817037	−	0.576585i	$-0.804385\pi$
$263$	− 18.7013i	− 1.15317i	0.817037	−	0.576585i	$-0.195615\pi$
$264$	7.69532	0.473614
$265$	0	0
$266$	−5.13027	−0.314557
$267$	14.2314i	0.870948i
$268$	0.877882i	0.0536252i
$269$	−28.1751	−1.71787	−0.858933	−	0.512088i	$-0.828872\pi$
$269$	−28.1751	−1.71787	−0.858933	+	0.512088i	$0.828872\pi$
$270$	0	0
$271$	−19.1366	−1.16246	−0.581232	−	0.813738i	$-0.697429\pi$
$271$	−19.1366	−1.16246	−0.581232	+	0.813738i	$0.697429\pi$
$272$	11.3293i	0.686938i
$273$	11.0490i	0.668718i
$274$	10.7990	0.652393
$275$	0	0
$276$	−10.9168	−0.657115
$277$	18.7312i	1.12545i	0.826644	+	0.562725i	$0.190247\pi$
$277$	18.7312i	1.12545i	−0.826644	+	0.562725i	$0.809753\pi$
$278$	11.8224i	0.709060i
$279$	9.94570	0.595433
$280$	0	0
$281$	−9.02236	−0.538229	−0.269114	−	0.963108i	$-0.586731\pi$
$281$	−9.02236	−0.538229	−0.269114	+	0.963108i	$0.586731\pi$
$282$	− 5.25648i	− 0.313019i
$283$	23.2193i	1.38025i	0.723692	+	0.690123i	$0.242444\pi$
$283$	23.2193i	1.38025i	−0.723692	+	0.690123i	$0.757556\pi$
$284$	5.43670	0.322609
$285$	0	0
$286$	−5.28651	−0.312598
$287$	34.7548i	2.05151i
$288$	5.37083i	0.316479i
$289$	−9.13969	−0.537629
$290$	0	0
$291$	14.0094	0.821248
$292$	17.1522i	1.00376i
$293$	− 4.66226i	− 0.272372i	−0.990683	−	0.136186i	$-0.956516\pi$
$293$	− 4.66226i	− 0.272372i	0.990683	−	0.136186i	$-0.0434845\pi$
$294$	6.74233	0.393221
$295$	0	0
$296$	11.0932	0.644777
$297$	− 3.72689i	− 0.216256i
$298$	− 3.98834i	− 0.231039i
$299$	16.3946	0.948127
$300$	0	0
$301$	−47.0218	−2.71029
$302$	− 3.06218i	− 0.176209i
$303$	1.82364i	0.104765i
$304$	−4.65156	−0.266785
$305$	0	0
$306$	−2.86382	−0.163714
$307$	− 27.9383i	− 1.59452i	−0.603633	−	0.797262i	$-0.706281\pi$
$307$	− 27.9383i	− 1.59452i	0.603633	−	0.797262i	$-0.293719\pi$
$308$	− 27.4199i	− 1.56239i
$309$	−10.6248	−0.604422
$310$	0	0
$311$	−34.7023	−1.96779	−0.983895	−	0.178748i	$-0.942795\pi$
$311$	−34.7023	−1.96779	−0.983895	+	0.178748i	$0.942795\pi$
$312$	− 5.22885i	− 0.296025i
$313$	19.1085i	1.08008i	0.841641	+	0.540038i	$0.181590\pi$
$313$	19.1085i	1.08008i	−0.841641	+	0.540038i	$0.818410\pi$
$314$	−8.20711	−0.463154
$315$	0	0
$316$	−3.06026	−0.172153
$317$	− 16.7578i	− 0.941214i	−0.882343	−	0.470607i	$-0.844035\pi$
$317$	− 16.7578i	− 0.941214i	0.882343	−	0.470607i	$-0.155965\pi$
$318$	− 3.16067i	− 0.177241i
$319$	−3.72689	−0.208666
$320$	0	0
$321$	−12.2121	−0.681612
$322$	− 15.8223i	− 0.881742i
$323$	− 10.7324i	− 0.597167i
$324$	1.68624	0.0936802
$325$	0	0
$326$	−7.80563	−0.432314
$327$	− 7.10102i	− 0.392687i
$328$	− 16.4474i	− 0.908154i
$329$	−40.9446	−2.25735
$330$	0	0
$331$	2.42129	0.133086	0.0665430	−	0.997784i	$-0.478803\pi$
$331$	2.42129	0.133086	0.0665430	+	0.997784i	$0.478803\pi$
$332$	2.28278i	0.125284i
$333$	− 5.37249i	− 0.294411i
$334$	14.1071	0.771906
$335$	0	0
$336$	9.66829	0.527449
$337$	− 5.94477i	− 0.323832i	−0.986805	−	0.161916i	$-0.948233\pi$
$337$	− 5.94477i	− 0.323832i	0.986805	−	0.161916i	$-0.0517674\pi$
$338$	− 3.68970i	− 0.200693i
$339$	−2.21008	−0.120035
$340$	0	0
$341$	37.0665	2.00727
$342$	− 1.17582i	− 0.0635813i
$343$	− 21.9766i	− 1.18662i
$344$	22.2526	1.19978
$345$	0	0
$346$	−7.13243	−0.383442
$347$	− 24.4339i	− 1.31168i	−0.754899	−	0.655841i	$-0.772314\pi$
$347$	− 24.4339i	− 1.31168i	0.754899	−	0.655841i	$-0.227686\pi$
$348$	− 1.68624i	− 0.0903922i
$349$	15.3090	0.819472	0.409736	−	0.912204i	$-0.365621\pi$
$349$	15.3090	0.819472	0.409736	+	0.912204i	$0.365621\pi$
$350$	0	0
$351$	−2.53237	−0.135168
$352$	20.0165i	1.06688i
$353$	− 17.8418i	− 0.949621i	−0.880088	−	0.474810i	$-0.842517\pi$
$353$	− 17.8418i	− 0.949621i	0.880088	−	0.474810i	$-0.157483\pi$
$354$	2.37530	0.126246
$355$	0	0
$356$	−23.9976	−1.27187
$357$	22.3074i	1.18063i
$358$	− 4.03058i	− 0.213023i
$359$	13.7962	0.728136	0.364068	−	0.931372i	$-0.381388\pi$
$359$	13.7962	0.728136	0.364068	+	0.931372i	$0.381388\pi$
$360$	0	0
$361$	−14.5935	−0.768079
$362$	− 3.62694i	− 0.190628i
$363$	− 2.88971i	− 0.151671i
$364$	−18.6314	−0.976549
$365$	0	0
$366$	−5.49971	−0.287474
$367$	− 0.0965282i	− 0.00503873i	−0.999997	−	0.00251936i	$-0.999198\pi$
$367$	− 0.0965282i	− 0.00503873i	0.999997	−	0.00251936i	$-0.000801940\pi$
$368$	− 14.3459i	− 0.747831i
$369$	−7.96557	−0.414671
$370$	0	0
$371$	−24.6196	−1.27819
$372$	16.7709i	0.869530i
$373$	36.5838i	1.89424i	0.320882	+	0.947119i	$0.396021\pi$
$373$	36.5838i	1.89424i	−0.320882	+	0.947119i	$0.603979\pi$
$374$	−10.6731	−0.551895
$375$	0	0
$376$	19.3767	0.999275
$377$	2.53237i	0.130423i
$378$	2.44396i	0.125704i
$379$	0.562286	0.0288827	0.0144413	−	0.999896i	$-0.495403\pi$
$379$	0.562286	0.0288827	0.0144413	+	0.999896i	$0.495403\pi$
$380$	0	0
$381$	−12.1699	−0.623480
$382$	− 10.0944i	− 0.516472i
$383$	3.31052i	0.169159i	0.996417	+	0.0845797i	$0.0269548\pi$
$383$	3.31052i	0.169159i	−0.996417	+	0.0845797i	$0.973045\pi$
$384$	−11.5390	−0.588846
$385$	0	0
$386$	5.93763	0.302217
$387$	− 10.7771i	− 0.547831i
$388$	23.6233i	1.19929i
$389$	8.36922	0.424336	0.212168	−	0.977233i	$-0.431948\pi$
$389$	8.36922	0.424336	0.212168	+	0.977233i	$0.431948\pi$
$390$	0	0
$391$	33.0998	1.67393
$392$	24.8539i	1.25531i
$393$	− 0.754636i	− 0.0380664i
$394$	−1.28061	−0.0645164
$395$	0	0
$396$	6.28445	0.315805
$397$	− 1.72859i	− 0.0867555i	−0.999059	−	0.0433778i	$-0.986188\pi$
$397$	− 1.72859i	− 0.0867555i	0.999059	−	0.0433778i	$-0.0138119\pi$
$398$	− 12.7367i	− 0.638435i
$399$	−9.15892	−0.458520
$400$	0	0
$401$	−20.6867	−1.03304	−0.516522	−	0.856274i	$-0.672774\pi$
$401$	−20.6867	−1.03304	−0.516522	+	0.856274i	$0.672774\pi$
$402$	− 0.291616i	− 0.0145445i
$403$	− 25.1861i	− 1.25461i
$404$	−3.07510	−0.152992
$405$	0	0
$406$	2.44396	0.121292
$407$	− 20.0227i	− 0.992487i
$408$	− 10.5567i	− 0.522637i
$409$	36.4935	1.80449	0.902243	−	0.431228i	$-0.141919\pi$
$409$	36.4935	1.80449	0.902243	+	0.431228i	$0.141919\pi$
$410$	0	0
$411$	19.2792	0.950973
$412$	− 17.9160i	− 0.882656i
$413$	− 18.5021i	− 0.910428i
$414$	3.62636	0.178226
$415$	0	0
$416$	13.6009	0.666840
$417$	21.1062i	1.03357i
$418$	− 4.38217i	− 0.214339i
$419$	28.1697	1.37618	0.688090	−	0.725625i	$-0.258449\pi$
$419$	28.1697	1.37618	0.688090	+	0.725625i	$0.258449\pi$
$420$	0	0
$421$	−11.1213	−0.542017	−0.271008	−	0.962577i	$-0.587357\pi$
$421$	−11.1213	−0.542017	−0.271008	+	0.962577i	$0.587357\pi$
$422$	− 9.49130i	− 0.462029i
$423$	− 9.38424i	− 0.456277i
$424$	11.6510	0.565822
$425$	0	0
$426$	−1.80597	−0.0874996
$427$	42.8393i	2.07314i
$428$	− 20.5925i	− 0.995378i
$429$	−9.43785	−0.455664
$430$	0	0
$431$	6.56816	0.316377	0.158189	−	0.987409i	$-0.449435\pi$
$431$	6.56816	0.316377	0.158189	+	0.987409i	$0.449435\pi$
$432$	2.21591i	0.106613i
$433$	− 1.10855i	− 0.0532734i	−0.999645	−	0.0266367i	$-0.991520\pi$
$433$	− 1.10855i	− 0.0532734i	0.999645	−	0.0266367i	$-0.00847972\pi$
$434$	−24.3069	−1.16677
$435$	0	0
$436$	11.9741	0.573453
$437$	13.5901i	0.650102i
$438$	− 5.69765i	− 0.272244i
$439$	−26.4625	−1.26299	−0.631494	−	0.775381i	$-0.717558\pi$
$439$	−26.4625	−1.26299	−0.631494	+	0.775381i	$0.717558\pi$
$440$	0	0
$441$	12.0369	0.573185
$442$	7.25224i	0.344954i
$443$	10.7820i	0.512267i	0.966641	+	0.256133i	$0.0824486\pi$
$443$	10.7820i	0.512267i	−0.966641	+	0.256133i	$0.917551\pi$
$444$	9.05933	0.429937
$445$	0	0
$446$	−3.99463	−0.189151
$447$	− 7.12028i	− 0.336778i
$448$	6.21050i	0.293418i
$449$	3.56594	0.168287	0.0841436	−	0.996454i	$-0.473185\pi$
$449$	3.56594	0.168287	0.0841436	+	0.996454i	$0.473185\pi$
$450$	0	0
$451$	−29.6868	−1.39790
$452$	− 3.72674i	− 0.175291i
$453$	− 5.46682i	− 0.256854i
$454$	1.35615	0.0636471
$455$	0	0
$456$	4.33437	0.202976
$457$	0.384118i	0.0179683i	0.999960	+	0.00898415i	$0.00285978\pi$
$457$	0.384118i	0.0179683i	−0.999960	+	0.00898415i	$0.997140\pi$
$458$	8.09612i	0.378307i
$459$	−5.11270	−0.238640
$460$	0	0
$461$	0.168528	0.00784912	0.00392456	−	0.999992i	$-0.498751\pi$
$461$	0.168528	0.00784912	0.00392456	+	0.999992i	$0.498751\pi$
$462$	9.10836i	0.423759i
$463$	11.6694i	0.542325i	0.962534	+	0.271163i	$0.0874081\pi$
$463$	11.6694i	0.542325i	−0.962534	+	0.271163i	$0.912592\pi$
$464$	2.21591	0.102871
$465$	0	0
$466$	−5.63450	−0.261013
$467$	14.6874i	0.679651i	0.940488	+	0.339826i	$0.110368\pi$
$467$	14.6874i	0.679651i	−0.940488	+	0.339826i	$0.889632\pi$
$468$	− 4.27019i	− 0.197390i
$469$	−2.27150	−0.104888
$470$	0	0
$471$	−14.6519	−0.675124
$472$	8.75594i	0.403025i
$473$	− 40.1651i	− 1.84679i
$474$	1.01656	0.0466922
$475$	0	0
$476$	−37.6157	−1.72411
$477$	− 5.64265i	− 0.258359i
$478$	− 8.04476i	− 0.367959i
$479$	−12.8124	−0.585412	−0.292706	−	0.956202i	$-0.594556\pi$
$479$	−12.8124	−0.585412	−0.292706	+	0.956202i	$0.594556\pi$
$480$	0	0
$481$	−13.6051	−0.620340
$482$	− 4.80855i	− 0.219023i
$483$	− 28.2471i	− 1.28529i
$484$	4.87276	0.221489
$485$	0	0
$486$	−0.560139	−0.0254084
$487$	− 12.9556i	− 0.587076i	−0.955947	−	0.293538i	$-0.905167\pi$
$487$	− 12.9556i	− 0.587076i	0.955947	−	0.293538i	$-0.0948327\pi$
$488$	− 20.2733i	− 0.917728i
$489$	−13.9352	−0.630170
$490$	0	0
$491$	−18.6710	−0.842609	−0.421304	−	0.906919i	$-0.638428\pi$
$491$	−18.6710	−0.842609	−0.421304	+	0.906919i	$0.638428\pi$
$492$	− 13.4319i	− 0.605557i
$493$	5.11270i	0.230264i
$494$	−2.97762	−0.133969
$495$	0	0
$496$	−22.0388	−0.989570
$497$	14.0674i	0.631008i
$498$	− 0.758297i	− 0.0339801i
$499$	31.9283	1.42931	0.714653	−	0.699479i	$-0.246585\pi$
$499$	31.9283	1.42931	0.714653	+	0.699479i	$0.246585\pi$
$500$	0	0
$501$	25.1850	1.12518
$502$	− 11.5818i	− 0.516923i
$503$	23.7384i	1.05844i	0.848484	+	0.529221i	$0.177516\pi$
$503$	23.7384i	1.05844i	−0.848484	+	0.529221i	$0.822484\pi$
$504$	−9.00902	−0.401294
$505$	0	0
$506$	13.5151	0.600818
$507$	− 6.58712i	− 0.292544i
$508$	− 20.5213i	− 0.910487i
$509$	9.32935	0.413516	0.206758	−	0.978392i	$-0.433709\pi$
$509$	9.32935	0.413516	0.206758	+	0.978392i	$0.433709\pi$
$510$	0	0
$511$	−44.3811	−1.96330
$512$	− 21.0521i	− 0.930382i
$513$	− 2.09916i	− 0.0926804i
$514$	−12.6605	−0.558430
$515$	0	0
$516$	18.1728	0.800014
$517$	− 34.9740i	− 1.53816i
$518$	13.1301i	0.576905i
$519$	−12.7333	−0.558931
$520$	0	0
$521$	7.78334	0.340994	0.170497	−	0.985358i	$-0.445463\pi$
$521$	7.78334	0.340994	0.170497	+	0.985358i	$0.445463\pi$
$522$	0.560139i	0.0245166i
$523$	21.8373i	0.954879i	0.878665	+	0.477439i	$0.158435\pi$
$523$	21.8373i	0.954879i	−0.878665	+	0.477439i	$0.841565\pi$
$524$	1.27250	0.0555895
$525$	0	0
$526$	−10.4753	−0.456746
$527$	− 50.8494i	− 2.21503i
$528$	8.25845i	0.359403i
$529$	−18.9133	−0.822315
$530$	0	0
$531$	4.24056	0.184024
$532$	− 15.4442i	− 0.669590i
$533$	20.1717i	0.873735i
$534$	7.97157	0.344963
$535$	0	0
$536$	1.07497	0.0464315
$537$	− 7.19567i	− 0.310516i
$538$	15.7820i	0.680409i
$539$	44.8602	1.93226
$540$	0	0
$541$	8.09048	0.347837	0.173918	−	0.984760i	$-0.444357\pi$
$541$	8.09048	0.347837	0.173918	+	0.984760i	$0.444357\pi$
$542$	10.7191i	0.460426i
$543$	− 6.47508i	− 0.277872i
$544$	27.4595	1.17731
$545$	0	0
$546$	6.18899	0.264865
$547$	− 26.3530i	− 1.12677i	−0.826194	−	0.563386i	$-0.809498\pi$
$547$	− 26.3530i	− 1.12677i	0.826194	−	0.563386i	$-0.190502\pi$
$548$	32.5094i	1.38873i
$549$	−9.81848	−0.419042
$550$	0	0
$551$	−2.09916	−0.0894274
$552$	13.3677i	0.568966i
$553$	− 7.91836i	− 0.336723i
$554$	10.4921	0.445766
$555$	0	0
$556$	−35.5902	−1.50936
$557$	− 45.0121i	− 1.90722i	−0.301038	−	0.953612i	$-0.597333\pi$
$557$	− 45.0121i	− 1.90722i	0.301038	−	0.953612i	$-0.402667\pi$
$558$	− 5.57097i	− 0.235838i
$559$	−27.2915	−1.15431
$560$	0	0
$561$	−19.0545	−0.804480
$562$	5.05377i	0.213181i
$563$	− 11.0447i	− 0.465480i	−0.972539	−	0.232740i	$-0.925231\pi$
$563$	− 11.0447i	− 0.465480i	0.972539	−	0.232740i	$-0.0747691\pi$
$564$	15.8241	0.666316
$565$	0	0
$566$	13.0061	0.546685
$567$	4.36313i	0.183234i
$568$	− 6.65725i	− 0.279332i
$569$	−44.8201	−1.87896	−0.939479	−	0.342607i	$-0.888690\pi$
$569$	−44.8201	−1.87896	−0.939479	+	0.342607i	$0.888690\pi$
$570$	0	0
$571$	17.4506	0.730287	0.365143	−	0.930951i	$-0.381020\pi$
$571$	17.4506	0.730287	0.365143	+	0.930951i	$0.381020\pi$
$572$	− 15.9145i	− 0.665420i
$573$	− 18.0212i	− 0.752845i
$574$	19.4675	0.812558
$575$	0	0
$576$	−1.42341	−0.0593085
$577$	− 7.31868i	− 0.304681i	−0.988328	−	0.152340i	$-0.951319\pi$
$577$	− 7.31868i	− 0.304681i	0.988328	−	0.152340i	$-0.0486810\pi$
$578$	5.11950i	0.212943i
$579$	10.6003	0.440533
$580$	0	0
$581$	−5.90666	−0.245049
$582$	− 7.84723i	− 0.325278i
$583$	− 21.0295i	− 0.870954i
$584$	21.0029	0.869107
$585$	0	0
$586$	−2.61151	−0.107881
$587$	− 8.44561i	− 0.348587i	−0.984694	−	0.174294i	$-0.944236\pi$
$587$	− 8.44561i	− 0.348587i	0.984694	−	0.174294i	$-0.0557642\pi$
$588$	20.2971i	0.837040i
$589$	20.8777	0.860249
$590$	0	0
$591$	−2.28624	−0.0940435
$592$	11.9049i	0.489290i
$593$	21.1951i	0.870377i	0.900339	+	0.435189i	$0.143318\pi$
$593$	21.1951i	0.870377i	−0.900339	+	0.435189i	$0.856682\pi$
$594$	−2.08758	−0.0856543
$595$	0	0
$596$	12.0065	0.491807
$597$	− 22.7385i	− 0.930627i
$598$	− 9.18328i	− 0.375532i
$599$	−37.1252	−1.51689	−0.758447	−	0.651735i	$-0.774041\pi$
$599$	−37.1252	−1.51689	−0.758447	+	0.651735i	$0.774041\pi$
$600$	0	0
$601$	40.9314	1.66963	0.834813	−	0.550534i	$-0.185576\pi$
$601$	40.9314	1.66963	0.834813	+	0.550534i	$0.185576\pi$
$602$	26.3388i	1.07349i
$603$	− 0.520614i	− 0.0212010i
$604$	9.21840	0.375091
$605$	0	0
$606$	1.02149	0.0414952
$607$	− 17.6432i	− 0.716114i	−0.933700	−	0.358057i	$-0.883439\pi$
$607$	− 17.6432i	− 0.716114i	0.933700	−	0.358057i	$-0.116561\pi$
$608$	11.2743i	0.457232i
$609$	4.36313	0.176803
$610$	0	0
$611$	−23.7643	−0.961402
$612$	− 8.62126i	− 0.348494i
$613$	− 5.40476i	− 0.218296i	−0.994026	−	0.109148i	$-0.965188\pi$
$613$	− 5.40476i	− 0.218296i	0.994026	−	0.109148i	$-0.0348123\pi$
$614$	−15.6493	−0.631556
$615$	0	0
$616$	−33.5756	−1.35280
$617$	5.47583i	0.220449i	0.993907	+	0.110224i	$0.0351569\pi$
$617$	5.47583i	0.220449i	−0.993907	+	0.110224i	$0.964843\pi$
$618$	5.95135i	0.239398i
$619$	−7.53825	−0.302988	−0.151494	−	0.988458i	$-0.548408\pi$
$619$	−7.53825	−0.302988	−0.151494	+	0.988458i	$0.548408\pi$
$620$	0	0
$621$	6.47404	0.259794
$622$	19.4381i	0.779398i
$623$	− 62.0935i	− 2.48772i
$624$	5.61149	0.224639
$625$	0	0
$626$	10.7034	0.427794
$627$	− 7.82336i	− 0.312435i
$628$	− 24.7067i	− 0.985905i
$629$	−27.4679	−1.09522
$630$	0	0
$631$	7.52265	0.299472	0.149736	−	0.988726i	$-0.452158\pi$
$631$	7.52265	0.299472	0.149736	+	0.988726i	$0.452158\pi$
$632$	3.74729i	0.149059i
$633$	− 16.9446i	− 0.673485i
$634$	−9.38672	−0.372794
$635$	0	0
$636$	9.51488	0.377290
$637$	− 30.4818i	− 1.20773i
$638$	2.08758i	0.0826479i
$639$	−3.22415	−0.127545
$640$	0	0
$641$	−27.1885	−1.07388	−0.536941	−	0.843620i	$-0.680420\pi$
$641$	−27.1885	−1.07388	−0.536941	+	0.843620i	$0.680420\pi$
$642$	6.84046i	0.269971i
$643$	9.59932i	0.378560i	0.981923	+	0.189280i	$0.0606154\pi$
$643$	9.59932i	0.378560i	−0.981923	+	0.189280i	$0.939385\pi$
$644$	47.6315	1.87694
$645$	0	0
$646$	−6.01163	−0.236525
$647$	36.3223i	1.42798i	0.700157	+	0.713989i	$0.253113\pi$
$647$	36.3223i	1.42798i	−0.700157	+	0.713989i	$0.746887\pi$
$648$	− 2.06481i	− 0.0811134i
$649$	15.8041	0.620365
$650$	0	0
$651$	−43.3944	−1.70076
$652$	− 23.4981i	− 0.920257i
$653$	12.3135i	0.481866i	0.970542	+	0.240933i	$0.0774534\pi$
$653$	12.3135i	0.481866i	−0.970542	+	0.240933i	$0.922547\pi$
$654$	−3.97756	−0.155535
$655$	0	0
$656$	17.6510	0.689155
$657$	− 10.1718i	− 0.396841i
$658$	22.9347i	0.894087i
$659$	12.8085	0.498947	0.249473	−	0.968382i	$-0.419742\pi$
$659$	12.8085	0.498947	0.249473	+	0.968382i	$0.419742\pi$
$660$	0	0
$661$	−14.1700	−0.551148	−0.275574	−	0.961280i	$-0.588868\pi$
$661$	−14.1700	−0.551148	−0.275574	+	0.961280i	$0.588868\pi$
$662$	− 1.35626i	− 0.0527125i
$663$	12.9472i	0.502829i
$664$	2.79527	0.108478
$665$	0	0
$666$	−3.00934	−0.116610
$667$	− 6.47404i	− 0.250676i
$668$	42.4680i	1.64314i
$669$	−7.13150	−0.275720
$670$	0	0
$671$	−36.5924	−1.41263
$672$	− 23.4336i	− 0.903972i
$673$	− 4.26452i	− 0.164385i	−0.996616	−	0.0821927i	$-0.973808\pi$
$673$	− 4.26452i	− 0.164385i	0.996616	−	0.0821927i	$-0.0261923\pi$
$674$	−3.32990	−0.128263
$675$	0	0
$676$	11.1075	0.427212
$677$	30.5652i	1.17472i	0.809327	+	0.587358i	$0.199832\pi$
$677$	30.5652i	1.17472i	−0.809327	+	0.587358i	$0.800168\pi$
$678$	1.23795i	0.0475433i
$679$	−61.1250	−2.34576
$680$	0	0
$681$	2.42109	0.0927764
$682$	− 20.7624i	− 0.795034i
$683$	− 34.6493i	− 1.32582i	−0.748700	−	0.662909i	$-0.769321\pi$
$683$	− 34.6493i	− 1.32582i	0.748700	−	0.662909i	$-0.230679\pi$
$684$	3.53970	0.135344
$685$	0	0
$686$	−12.3099	−0.469995
$687$	14.4538i	0.551446i
$688$	23.8811i	0.910457i
$689$	−14.2892	−0.544377
$690$	0	0
$691$	36.5754	1.39140	0.695698	−	0.718335i	$-0.255095\pi$
$691$	36.5754	1.39140	0.695698	+	0.718335i	$0.255095\pi$
$692$	− 21.4715i	− 0.816224i
$693$	16.2609i	0.617701i
$694$	−13.6864	−0.519528
$695$	0	0
$696$	−2.06481	−0.0782664
$697$	40.7255i	1.54259i
$698$	− 8.57517i	− 0.324575i
$699$	−10.0591	−0.380470
$700$	0	0
$701$	30.4230	1.14906	0.574530	−	0.818483i	$-0.305185\pi$
$701$	30.4230	1.14906	0.574530	+	0.818483i	$0.305185\pi$
$702$	1.41848i	0.0535369i
$703$	− 11.2777i	− 0.425348i
$704$	−5.30487	−0.199935
$705$	0	0
$706$	−9.99386	−0.376124
$707$	− 7.95677i	− 0.299245i
$708$	7.15061i	0.268736i
$709$	3.88470	0.145893	0.0729464	−	0.997336i	$-0.476760\pi$
$709$	3.88470	0.145893	0.0729464	+	0.997336i	$0.476760\pi$
$710$	0	0
$711$	1.81484	0.0680617
$712$	29.3851i	1.10125i
$713$	64.3889i	2.41138i
$714$	12.4952	0.467622
$715$	0	0
$716$	12.1337	0.453456
$717$	− 14.3621i	− 0.536362i
$718$	− 7.72779i	− 0.288399i
$719$	8.31279	0.310015	0.155007	−	0.987913i	$-0.450460\pi$
$719$	8.31279	0.310015	0.155007	+	0.987913i	$0.450460\pi$
$720$	0	0
$721$	46.3573	1.72643
$722$	8.17439i	0.304219i
$723$	− 8.58457i	− 0.319263i
$724$	10.9186	0.405785
$725$	0	0
$726$	−1.61864	−0.0600734
$727$	− 10.5865i	− 0.392633i	−0.980541	−	0.196316i	$-0.937102\pi$
$727$	− 10.5865i	− 0.392633i	0.980541	−	0.196316i	$-0.0628979\pi$
$728$	22.8141i	0.845549i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−55.1000	−2.03795
$732$	− 16.5564i	− 0.611940i
$733$	22.7841i	0.841549i	0.907165	+	0.420775i	$0.138242\pi$
$733$	22.7841i	0.841549i	−0.907165	+	0.420775i	$0.861758\pi$
$734$	−0.0540692	−0.00199573
$735$	0	0
$736$	−34.7710	−1.28168
$737$	− 1.94027i	− 0.0714708i
$738$	4.46182i	0.164242i
$739$	−28.1480	−1.03544	−0.517720	−	0.855550i	$-0.673219\pi$
$739$	−28.1480	−1.03544	−0.517720	+	0.855550i	$0.673219\pi$
$740$	0	0
$741$	−5.31585	−0.195283
$742$	13.7904i	0.506261i
$743$	47.6786i	1.74916i	0.484883	+	0.874579i	$0.338862\pi$
$743$	47.6786i	1.74916i	−0.484883	+	0.874579i	$0.661138\pi$
$744$	20.5360	0.752885
$745$	0	0
$746$	20.4920	0.750266
$747$	− 1.35377i	− 0.0495317i
$748$	− 32.1305i	− 1.17481i
$749$	53.2829	1.94691
$750$	0	0
$751$	10.8278	0.395111	0.197556	−	0.980292i	$-0.436700\pi$
$751$	10.8278	0.395111	0.197556	+	0.980292i	$0.436700\pi$
$752$	20.7946i	0.758302i
$753$	− 20.6767i	− 0.753502i
$754$	1.41848	0.0516579
$755$	0	0
$756$	−7.35730	−0.267582
$757$	45.1777i	1.64201i	0.570919	+	0.821007i	$0.306587\pi$
$757$	45.1777i	1.64201i	−0.570919	+	0.821007i	$0.693413\pi$
$758$	− 0.314958i	− 0.0114398i
$759$	24.1281	0.875793
$760$	0	0
$761$	−8.52024	−0.308858	−0.154429	−	0.988004i	$-0.549354\pi$
$761$	−8.52024	−0.308858	−0.154429	+	0.988004i	$0.549354\pi$
$762$	6.81681i	0.246947i
$763$	30.9827i	1.12165i
$764$	30.3881	1.09940
$765$	0	0
$766$	1.85435	0.0670003
$767$	− 10.7386i	− 0.387750i
$768$	3.61662i	0.130503i
$769$	−17.7232	−0.639114	−0.319557	−	0.947567i	$-0.603534\pi$
$769$	−17.7232	−0.639114	−0.319557	+	0.947567i	$0.603534\pi$
$770$	0	0
$771$	−22.6024	−0.814005
$772$	17.8747i	0.643323i
$773$	29.8111i	1.07223i	0.844144	+	0.536116i	$0.180109\pi$
$773$	29.8111i	1.07223i	−0.844144	+	0.536116i	$0.819891\pi$
$774$	−6.03667	−0.216984
$775$	0	0
$776$	28.9268	1.03841
$777$	23.4409i	0.840936i
$778$	− 4.68793i	− 0.168070i
$779$	−16.7210	−0.599093
$780$	0	0
$781$	−12.0161	−0.429968
$782$	− 18.5405i	− 0.663007i
$783$	1.00000i	0.0357371i
$784$	−26.6726	−0.952594
$785$	0	0
$786$	−0.422701	−0.0150772
$787$	4.46894i	0.159301i	0.996823	+	0.0796503i	$0.0253804\pi$
$787$	4.46894i	0.159301i	−0.996823	+	0.0796503i	$0.974620\pi$
$788$	− 3.85517i	− 0.137335i
$789$	−18.7013	−0.665783
$790$	0	0
$791$	9.64287	0.342861
$792$	− 7.69532i	− 0.273441i
$793$	24.8640i	0.882946i
$794$	−0.968252	−0.0343620
$795$	0	0
$796$	38.3427	1.35902
$797$	− 13.4589i	− 0.476737i	−0.971175	−	0.238368i	$-0.923387\pi$
$797$	− 13.4589i	− 0.476737i	0.971175	−	0.238368i	$-0.0766126\pi$
$798$	5.13027i	0.181610i
$799$	−47.9788	−1.69737
$800$	0	0
$801$	14.2314	0.502842
$802$	11.5874i	0.409166i
$803$	− 37.9093i	− 1.33779i
$804$	0.877882	0.0309605
$805$	0	0
$806$	−14.1077	−0.496924
$807$	28.1751i	0.991810i
$808$	3.76546i	0.132469i
$809$	47.1877	1.65903	0.829516	−	0.558484i	$-0.188617\pi$
$809$	47.1877	1.65903	0.829516	+	0.558484i	$0.188617\pi$
$810$	0	0
$811$	32.6796	1.14754	0.573769	−	0.819017i	$-0.305481\pi$
$811$	32.6796	1.14754	0.573769	+	0.819017i	$0.305481\pi$
$812$	7.35730i	0.258191i
$813$	19.1366i	0.671149i
$814$	−11.2155	−0.393102
$815$	0	0
$816$	11.3293	0.396604
$817$	− 22.6229i	− 0.791475i
$818$	− 20.4414i	− 0.714717i
$819$	11.0490	0.386085
$820$	0	0
$821$	20.0420	0.699470	0.349735	−	0.936849i	$-0.386272\pi$
$821$	20.0420	0.699470	0.349735	+	0.936849i	$0.386272\pi$
$822$	− 10.7990i	− 0.376659i
$823$	− 1.64700i	− 0.0574108i	−0.999588	−	0.0287054i	$-0.990862\pi$
$823$	− 1.64700i	− 0.0574108i	0.999588	−	0.0287054i	$-0.00913846\pi$
$824$	−21.9381	−0.764251
$825$	0	0
$826$	−10.3637	−0.360601
$827$	− 1.65948i	− 0.0577057i	−0.999584	−	0.0288528i	$-0.990815\pi$
$827$	− 1.65948i	− 0.0577057i	0.999584	−	0.0288528i	$-0.00918542\pi$
$828$	10.9168i	0.379386i
$829$	30.2298	1.04992	0.524962	−	0.851126i	$-0.324080\pi$
$829$	30.2298	1.04992	0.524962	+	0.851126i	$0.324080\pi$
$830$	0	0
$831$	18.7312	0.649779
$832$	3.60458i	0.124966i
$833$	− 61.5410i	− 2.13227i
$834$	11.8224	0.409376
$835$	0	0
$836$	13.1921	0.456258
$837$	− 9.94570i	− 0.343774i
$838$	− 15.7789i	− 0.545075i
$839$	1.43638	0.0495895	0.0247947	−	0.999693i	$-0.492107\pi$
$839$	1.43638	0.0495895	0.0247947	+	0.999693i	$0.492107\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	6.22944i	0.214681i
$843$	9.02236i	0.310746i
$844$	28.5727	0.983511
$845$	0	0
$846$	−5.25648	−0.180721
$847$	12.6082i	0.433222i
$848$	12.5036i	0.429375i
$849$	23.2193	0.796886
$850$	0	0
$851$	34.7817	1.19230
$852$	− 5.43670i	− 0.186258i
$853$	− 0.913705i	− 0.0312847i	−0.999878	−	0.0156423i	$-0.995021\pi$
$853$	− 0.913705i	− 0.0312847i	0.999878	−	0.0156423i	$-0.00497931\pi$
$854$	23.9959	0.821124
$855$	0	0
$856$	−25.2156	−0.861852
$857$	53.1479i	1.81550i	0.419516	+	0.907748i	$0.362200\pi$
$857$	53.1479i	1.81550i	−0.419516	+	0.907748i	$0.637800\pi$
$858$	5.28651i	0.180478i
$859$	10.2236	0.348826	0.174413	−	0.984673i	$-0.444197\pi$
$859$	10.2236	0.348826	0.174413	+	0.984673i	$0.444197\pi$
$860$	0	0
$861$	34.7548	1.18444
$862$	− 3.67908i	− 0.125310i
$863$	45.2465i	1.54021i	0.637918	+	0.770104i	$0.279796\pi$
$863$	45.2465i	1.54021i	−0.637918	+	0.770104i	$0.720204\pi$
$864$	5.37083	0.182720
$865$	0	0
$866$	−0.620940	−0.0211004
$867$	9.13969i	0.310400i
$868$	− 73.1735i	− 2.48367i
$869$	6.76370	0.229443
$870$	0	0
$871$	−1.31838	−0.0446718
$872$	− 14.6623i	− 0.496527i
$873$	− 14.0094i	− 0.474148i
$874$	7.61234	0.257491
$875$	0	0
$876$	17.1522	0.579520
$877$	19.3183i	0.652332i	0.945312	+	0.326166i	$0.105757\pi$
$877$	19.3183i	0.652332i	−0.945312	+	0.326166i	$0.894243\pi$
$878$	14.8227i	0.500242i
$879$	−4.66226	−0.157254
$880$	0	0
$881$	5.54756	0.186902	0.0934510	−	0.995624i	$-0.470210\pi$
$881$	5.54756	0.186902	0.0934510	+	0.995624i	$0.470210\pi$
$882$	− 6.74233i	− 0.227026i
$883$	32.1271i	1.08116i	0.841292	+	0.540581i	$0.181795\pi$
$883$	32.1271i	1.08116i	−0.841292	+	0.540581i	$0.818205\pi$
$884$	−21.8322	−0.734296
$885$	0	0
$886$	6.03940	0.202898
$887$	− 18.8095i	− 0.631560i	−0.948832	−	0.315780i	$-0.897734\pi$
$887$	− 18.8095i	− 0.631560i	0.948832	−	0.315780i	$-0.102266\pi$
$888$	− 11.0932i	− 0.372262i
$889$	53.0986	1.78087
$890$	0	0
$891$	−3.72689	−0.124856
$892$	− 12.0254i	− 0.402642i
$893$	− 19.6991i	− 0.659204i
$894$	−3.98834	−0.133390
$895$	0	0
$896$	50.3460	1.68194
$897$	− 16.3946i	− 0.547401i
$898$	− 1.99742i	− 0.0666549i
$899$	−9.94570	−0.331708
$900$	0	0
$901$	−28.8492	−0.961105
$902$	16.6287i	0.553676i
$903$	47.0218i	1.56479i
$904$	−4.56340	−0.151776
$905$	0	0
$906$	−3.06218	−0.101734
$907$	3.04313i	0.101045i	0.998723	+	0.0505227i	$0.0160887\pi$
$907$	3.04313i	0.101045i	−0.998723	+	0.0505227i	$0.983911\pi$
$908$	4.08255i	0.135484i
$909$	1.82364	0.0604863
$910$	0	0
$911$	−52.8072	−1.74958	−0.874790	−	0.484501i	$-0.839001\pi$
$911$	−52.8072	−1.74958	−0.874790	+	0.484501i	$0.839001\pi$
$912$	4.65156i	0.154028i
$913$	− 5.04534i	− 0.166976i
$914$	0.215160	0.00711685
$915$	0	0
$916$	−24.3726	−0.805293
$917$	3.29257i	0.108730i
$918$	2.86382i	0.0945202i
$919$	−33.4001	−1.10177	−0.550884	−	0.834582i	$-0.685709\pi$
$919$	−33.4001	−1.10177	−0.550884	+	0.834582i	$0.685709\pi$
$920$	0	0
$921$	−27.9383	−0.920599
$922$	− 0.0943990i	− 0.00310886i
$923$	8.16473i	0.268745i
$924$	−27.4199	−0.902047
$925$	0	0
$926$	6.53651	0.214803
$927$	10.6248i	0.348963i
$928$	− 5.37083i	− 0.176306i
$929$	4.79788	0.157413	0.0787067	−	0.996898i	$-0.474921\pi$
$929$	4.79788	0.157413	0.0787067	+	0.996898i	$0.474921\pi$
$930$	0	0
$931$	25.2674	0.828106
$932$	− 16.9621i	− 0.555613i
$933$	34.7023i	1.13610i
$934$	8.22698	0.269195
$935$	0	0
$936$	−5.22885	−0.170910
$937$	− 27.5261i	− 0.899238i	−0.893220	−	0.449619i	$-0.851560\pi$
$937$	− 27.5261i	− 0.899238i	0.893220	−	0.449619i	$-0.148440\pi$
$938$	1.27236i	0.0415440i
$939$	19.1085	0.623582
$940$	0	0
$941$	16.6083	0.541415	0.270707	−	0.962662i	$-0.412742\pi$
$941$	16.6083	0.541415	0.270707	+	0.962662i	$0.412742\pi$
$942$	8.20711i	0.267402i
$943$	− 51.5694i	− 1.67933i
$944$	−9.39669	−0.305836
$945$	0	0
$946$	−22.4980	−0.731473
$947$	37.8994i	1.23156i	0.787916	+	0.615782i	$0.211160\pi$
$947$	37.8994i	1.23156i	−0.787916	+	0.615782i	$0.788840\pi$
$948$	3.06026i	0.0993926i
$949$	−25.7588	−0.836167
$950$	0	0
$951$	−16.7578	−0.543410
$952$	46.0604i	1.49283i
$953$	− 39.3478i	− 1.27460i	−0.770615	−	0.637301i	$-0.780051\pi$
$953$	− 39.3478i	− 1.27460i	0.770615	−	0.637301i	$-0.219949\pi$
$954$	−3.16067	−0.102330
$955$	0	0
$956$	24.2180	0.783265
$957$	3.72689i	0.120473i
$958$	7.17671i	0.231869i
$959$	−84.1176	−2.71630
$960$	0	0
$961$	67.9169	2.19087
$962$	7.62075i	0.245703i
$963$	12.2121i	0.393529i
$964$	14.4757	0.466230
$965$	0	0
$966$	−15.8223	−0.509074
$967$	− 34.8122i	− 1.11949i	−0.828666	−	0.559743i	$-0.810900\pi$
$967$	− 34.8122i	− 1.11949i	0.828666	−	0.559743i	$-0.189100\pi$
$968$	− 5.96671i	− 0.191777i
$969$	−10.7324	−0.344774
$970$	0	0
$971$	−16.7398	−0.537207	−0.268603	−	0.963251i	$-0.586562\pi$
$971$	−16.7398	−0.537207	−0.268603	+	0.963251i	$0.586562\pi$
$972$	− 1.68624i	− 0.0540863i
$973$	− 92.0890i	− 2.95224i
$974$	−7.25696	−0.232528
$975$	0	0
$976$	21.7569	0.696420
$977$	− 12.8886i	− 0.412344i	−0.978516	−	0.206172i	$-0.933899\pi$
$977$	− 12.8886i	− 0.412344i	0.978516	−	0.206172i	$-0.0661006\pi$
$978$	7.80563i	0.249597i
$979$	53.0389	1.69513
$980$	0	0
$981$	−7.10102	−0.226718
$982$	10.4583i	0.333739i
$983$	− 20.3218i	− 0.648166i	−0.946029	−	0.324083i	$-0.894944\pi$
$983$	− 20.3218i	− 0.648166i	0.946029	−	0.324083i	$-0.105056\pi$
$984$	−16.4474	−0.524323
$985$	0	0
$986$	2.86382	0.0912027
$987$	40.9446i	1.30328i
$988$	− 8.96383i	− 0.285177i
$989$	69.7714	2.21860
$990$	0	0
$991$	−13.1802	−0.418683	−0.209341	−	0.977843i	$-0.567132\pi$
$991$	−13.1802	−0.418683	−0.209341	+	0.977843i	$0.567132\pi$
$992$	53.4167i	1.69598i
$993$	− 2.42129i	− 0.0768373i
$994$	7.87968	0.249928
$995$	0	0
$996$	2.28278	0.0723327
$997$	57.5645i	1.82309i	0.411206	+	0.911543i	$0.365108\pi$
$997$	57.5645i	1.82309i	−0.411206	+	0.911543i	$0.634892\pi$
$998$	− 17.8843i	− 0.566116i
$999$	−5.37249	−0.169978

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.o.349.7		14
5.2	odd	4		2175.2.a.ba.1.5	✓	7
5.3	odd	4		2175.2.a.bb.1.3	yes	7
5.4	even	2	inner	2175.2.c.o.349.8		14
15.2	even	4		6525.2.a.bx.1.3		7
15.8	even	4		6525.2.a.bu.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2175.2.a.ba.1.5	✓	7	5.2	odd	4
2175.2.a.bb.1.3	yes	7	5.3	odd	4
2175.2.c.o.349.7		14	1.1	even	1	trivial
2175.2.c.o.349.8		14	5.4	even	2	inner
6525.2.a.bu.1.5		7	15.8	even	4
6525.2.a.bx.1.3		7	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.560139i q^{2} -1.00000i q^{3} +1.68624 q^{4} -0.560139 q^{6} +4.36313i q^{7} -2.06481i q^{8} -1.00000 q^{9} -3.72689 q^{11} -1.68624i q^{12} +2.53237i q^{13} +2.44396 q^{14} +2.21591 q^{16} +5.11270i q^{17} +0.560139i q^{18} -2.09916 q^{19} +4.36313 q^{21} +2.08758i q^{22} -6.47404i q^{23} -2.06481 q^{24} +1.41848 q^{26} +1.00000i q^{27} +7.35730i q^{28} +1.00000 q^{29} -9.94570 q^{31} -5.37083i q^{32} +3.72689i q^{33} +2.86382 q^{34} -1.68624 q^{36} +5.37249i q^{37} +1.17582i q^{38} +2.53237 q^{39} +7.96557 q^{41} -2.44396i q^{42} +10.7771i q^{43} -6.28445 q^{44} -3.62636 q^{46} +9.38424i q^{47} -2.21591i q^{48} -12.0369 q^{49} +5.11270 q^{51} +4.27019i q^{52} +5.64265i q^{53} +0.560139 q^{54} +9.00902 q^{56} +2.09916i q^{57} -0.560139i q^{58} -4.24056 q^{59} +9.81848 q^{61} +5.57097i q^{62} -4.36313i q^{63} +1.42341 q^{64} +2.08758 q^{66} +0.520614i q^{67} +8.62126i q^{68} -6.47404 q^{69} +3.22415 q^{71} +2.06481i q^{72} +10.1718i q^{73} +3.00934 q^{74} -3.53970 q^{76} -16.2609i q^{77} -1.41848i q^{78} -1.81484 q^{79} +1.00000 q^{81} -4.46182i q^{82} +1.35377i q^{83} +7.35730 q^{84} +6.03667 q^{86} -1.00000i q^{87} +7.69532i q^{88} -14.2314 q^{89} -11.0490 q^{91} -10.9168i q^{92} +9.94570i q^{93} +5.25648 q^{94} -5.37083 q^{96} +14.0094i q^{97} +6.74233i q^{98} +3.72689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 20 q^{4} + 4 q^{6} - 14 q^{9} + 8 q^{11} - 30 q^{14} + 24 q^{16} - 30 q^{19} - 2 q^{21} - 6 q^{24} + 12 q^{26} + 14 q^{29} + 10 q^{31} - 14 q^{34} + 20 q^{36} + 2 q^{39} + 44 q^{41} - 30 q^{44} - 8 q^{46}+ \cdots - 8 q^{99}+O(q^{100}) \) 14 * q - 20 * q^4 + 4 * q^6 - 14 * q^9 + 8 * q^11 - 30 * q^14 + 24 * q^16 - 30 * q^19 - 2 * q^21 - 6 * q^24 + 12 * q^26 + 14 * q^29 + 10 * q^31 - 14 * q^34 + 20 * q^36 + 2 * q^39 + 44 * q^41 - 30 * q^44 - 8 * q^46 - 24 * q^49 + 16 * q^51 - 4 * q^54 + 28 * q^56 - 12 * q^59 + 46 * q^61 - 10 * q^64 + 6 * q^66 - 28 * q^69 + 52 * q^71 + 20 * q^74 + 92 * q^76 - 28 * q^79 + 14 * q^81 + 48 * q^84 + 88 * q^86 - 28 * q^89 + 26 * q^91 + 6 * q^94 + 36 * q^96 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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