LMFDB - Embedded newform 2175.2.c.h.349.2

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:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.2
Root			$-1.56155i$ of defining polynomial
Character	$\chi$	$=$	2175.349
Dual form			2175.2.c.h.349.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.56155i q^{2} -1.00000i q^{3} -0.438447 q^{4} -1.56155 q^{6} +5.12311i q^{7} -2.43845i q^{8} -1.00000 q^{9} -1.43845 q^{11} +0.438447i q^{12} +2.00000i q^{13} +8.00000 q^{14} -4.68466 q^{16} -7.12311i q^{17} +1.56155i q^{18} -5.12311 q^{19} +5.12311 q^{21} +2.24621i q^{22} -6.56155i q^{23} -2.43845 q^{24} +3.12311 q^{26} +1.00000i q^{27} -2.24621i q^{28} -1.00000 q^{29} +4.00000 q^{31} +2.43845i q^{32} +1.43845i q^{33} -11.1231 q^{34} +0.438447 q^{36} -1.68466i q^{37} +8.00000i q^{38} +2.00000 q^{39} -1.68466 q^{41} -8.00000i q^{42} -7.68466i q^{43} +0.630683 q^{44} -10.2462 q^{46} -13.1231i q^{47} +4.68466i q^{48} -19.2462 q^{49} -7.12311 q^{51} -0.876894i q^{52} +3.43845i q^{53} +1.56155 q^{54} +12.4924 q^{56} +5.12311i q^{57} +1.56155i q^{58} -12.0000 q^{59} +0.876894 q^{61} -6.24621i q^{62} -5.12311i q^{63} -5.56155 q^{64} +2.24621 q^{66} -11.3693i q^{67} +3.12311i q^{68} -6.56155 q^{69} -2.87689 q^{71} +2.43845i q^{72} -1.68466i q^{73} -2.63068 q^{74} +2.24621 q^{76} -7.36932i q^{77} -3.12311i q^{78} +12.0000 q^{79} +1.00000 q^{81} +2.63068i q^{82} +2.56155i q^{83} -2.24621 q^{84} -12.0000 q^{86} +1.00000i q^{87} +3.50758i q^{88} -12.2462 q^{89} -10.2462 q^{91} +2.87689i q^{92} -4.00000i q^{93} -20.4924 q^{94} +2.43845 q^{96} -5.68466i q^{97} +30.0540i q^{98} +1.43845 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 10 q^{4} + 2 q^{6} - 4 q^{9} - 14 q^{11} + 32 q^{14} + 6 q^{16} - 4 q^{19} + 4 q^{21} - 18 q^{24} - 4 q^{26} - 4 q^{29} + 16 q^{31} - 28 q^{34} + 10 q^{36} + 8 q^{39} + 18 q^{41} + 52 q^{44} - 8 q^{46}+ \cdots + 14 q^{99}+O(q^{100}) $ 4 * q - 10 * q^4 + 2 * q^6 - 4 * q^9 - 14 * q^11 + 32 * q^14 + 6 * q^16 - 4 * q^19 + 4 * q^21 - 18 * q^24 - 4 * q^26 - 4 * q^29 + 16 * q^31 - 28 * q^34 + 10 * q^36 + 8 * q^39 + 18 * q^41 + 52 * q^44 - 8 * q^46 - 44 * q^49 - 12 * q^51 - 2 * q^54 - 16 * q^56 - 48 * q^59 + 20 * q^61 - 14 * q^64 - 24 * q^66 - 18 * q^69 - 28 * q^71 - 60 * q^74 - 24 * q^76 + 48 * q^79 + 4 * q^81 + 24 * q^84 - 48 * q^86 - 16 * q^89 - 8 * q^91 - 16 * q^94 + 18 * q^96 + 14 * 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 $

$2$	− 1.56155i	− 1.10418i	−0.833783	−	0.552092i	$-0.813830\pi$
$2$	− 1.56155i	− 1.10418i	0.833783	−	0.552092i	$-0.186170\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−0.438447	−0.219224
$5$	0	0
$5$	0	0
$6$	−1.56155	−0.637501
$7$	5.12311i	1.93635i	0.250270	+	0.968176i	$0.419480\pi$
$7$	5.12311i	1.93635i	−0.250270	+	0.968176i	$0.580520\pi$
$8$	− 2.43845i	− 0.862121i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−1.43845	−0.433708	−0.216854	−	0.976204i	$-0.569580\pi$
$11$	−1.43845	−0.433708	−0.216854	+	0.976204i	$0.569580\pi$
$12$	0.438447i	0.126569i
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	8.00000	2.13809
$15$	0	0
$16$	−4.68466	−1.17116
$17$	− 7.12311i	− 1.72761i	−0.503829	−	0.863803i	$-0.668076\pi$
$17$	− 7.12311i	− 1.72761i	0.503829	−	0.863803i	$-0.331924\pi$
$18$	1.56155i	0.368062i
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	0	0
$21$	5.12311	1.11795
$22$	2.24621i	0.478894i
$23$	− 6.56155i	− 1.36818i	−0.729398	−	0.684089i	$-0.760200\pi$
$23$	− 6.56155i	− 1.36818i	0.729398	−	0.684089i	$-0.239800\pi$
$24$	−2.43845	−0.497746
$25$	0	0
$26$	3.12311	0.612491
$27$	1.00000i	0.192450i
$28$	− 2.24621i	− 0.424494i
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	2.43845i	0.431061i
$33$	1.43845i	0.250402i
$34$	−11.1231	−1.90760
$35$	0	0
$36$	0.438447	0.0730745
$37$	− 1.68466i	− 0.276956i	−0.990366	−	0.138478i	$-0.955779\pi$
$37$	− 1.68466i	− 0.276956i	0.990366	−	0.138478i	$-0.0442210\pi$
$38$	8.00000i	1.29777i
$39$	2.00000	0.320256
$40$	0	0
$41$	−1.68466	−0.263099	−0.131550	−	0.991310i	$-0.541995\pi$
$41$	−1.68466	−0.263099	−0.131550	+	0.991310i	$0.541995\pi$
$42$	− 8.00000i	− 1.23443i
$43$	− 7.68466i	− 1.17190i	−0.810347	−	0.585950i	$-0.800722\pi$
$43$	− 7.68466i	− 1.17190i	0.810347	−	0.585950i	$-0.199278\pi$
$44$	0.630683	0.0950791
$45$	0	0
$46$	−10.2462	−1.51072
$47$	− 13.1231i	− 1.91420i	−0.289755	−	0.957101i	$-0.593574\pi$
$47$	− 13.1231i	− 1.91420i	0.289755	−	0.957101i	$-0.406426\pi$
$48$	4.68466i	0.676172i
$49$	−19.2462	−2.74946
$50$	0	0
$51$	−7.12311	−0.997434
$52$	− 0.876894i	− 0.121603i
$53$	3.43845i	0.472307i	0.971716	+	0.236154i	$0.0758868\pi$
$53$	3.43845i	0.472307i	−0.971716	+	0.236154i	$0.924113\pi$
$54$	1.56155	0.212500
$55$	0	0
$56$	12.4924	1.66937
$57$	5.12311i	0.678572i
$58$	1.56155i	0.205042i
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	− 6.24621i	− 0.793270i
$63$	− 5.12311i	− 0.645451i
$64$	−5.56155	−0.695194
$65$	0	0
$66$	2.24621	0.276489
$67$	− 11.3693i	− 1.38898i	−0.719501	−	0.694492i	$-0.755629\pi$
$67$	− 11.3693i	− 1.38898i	0.719501	−	0.694492i	$-0.244371\pi$
$68$	3.12311i	0.378732i
$69$	−6.56155	−0.789918
$70$	0	0
$71$	−2.87689	−0.341425	−0.170712	−	0.985321i	$-0.554607\pi$
$71$	−2.87689	−0.341425	−0.170712	+	0.985321i	$0.554607\pi$
$72$	2.43845i	0.287374i
$73$	− 1.68466i	− 0.197174i	−0.995128	−	0.0985872i	$-0.968568\pi$
$73$	− 1.68466i	− 0.197174i	0.995128	−	0.0985872i	$-0.0314323\pi$
$74$	−2.63068	−0.305811
$75$	0	0
$76$	2.24621	0.257658
$77$	− 7.36932i	− 0.839812i
$78$	− 3.12311i	− 0.353622i
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.63068i	0.290510i
$83$	2.56155i	0.281167i	0.990069	+	0.140583i	$0.0448978\pi$
$83$	2.56155i	0.281167i	−0.990069	+	0.140583i	$0.955102\pi$
$84$	−2.24621	−0.245082
$85$	0	0
$86$	−12.0000	−1.29399
$87$	1.00000i	0.107211i
$88$	3.50758i	0.373909i
$89$	−12.2462	−1.29810	−0.649048	−	0.760748i	$-0.724833\pi$
$89$	−12.2462	−1.29810	−0.649048	+	0.760748i	$0.724833\pi$
$90$	0	0
$91$	−10.2462	−1.07409
$92$	2.87689i	0.299937i
$93$	− 4.00000i	− 0.414781i
$94$	−20.4924	−2.11363
$95$	0	0
$96$	2.43845	0.248873
$97$	− 5.68466i	− 0.577190i	−0.957451	−	0.288595i	$-0.906812\pi$
$97$	− 5.68466i	− 0.577190i	0.957451	−	0.288595i	$-0.0931880\pi$
$98$	30.0540i	3.03591i
$99$	1.43845	0.144569
$100$	0	0
$101$	−8.56155	−0.851906	−0.425953	−	0.904745i	$-0.640061\pi$
$101$	−8.56155	−0.851906	−0.425953	+	0.904745i	$0.640061\pi$
$102$	11.1231i	1.10135i
$103$	2.87689i	0.283469i	0.989905	+	0.141734i	$0.0452679\pi$
$103$	2.87689i	0.283469i	−0.989905	+	0.141734i	$0.954732\pi$
$104$	4.87689	0.478219
$105$	0	0
$106$	5.36932	0.521514
$107$	16.4924i	1.59438i	0.603727	+	0.797191i	$0.293682\pi$
$107$	16.4924i	1.59438i	−0.603727	+	0.797191i	$0.706318\pi$
$108$	− 0.438447i	− 0.0421896i
$109$	5.68466	0.544492	0.272246	−	0.962228i	$-0.412234\pi$
$109$	5.68466	0.544492	0.272246	+	0.962228i	$0.412234\pi$
$110$	0	0
$111$	−1.68466	−0.159901
$112$	− 24.0000i	− 2.26779i
$113$	4.87689i	0.458780i	0.973335	+	0.229390i	$0.0736731\pi$
$113$	4.87689i	0.458780i	−0.973335	+	0.229390i	$0.926327\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	0.438447	0.0407088
$117$	− 2.00000i	− 0.184900i
$118$	18.7386i	1.72503i
$119$	36.4924	3.34525
$120$	0	0
$121$	−8.93087	−0.811897
$122$	− 1.36932i	− 0.123972i
$123$	1.68466i	0.151901i
$124$	−1.75379	−0.157495
$125$	0	0
$126$	−8.00000	−0.712697
$127$	4.31534i	0.382925i	0.981500	+	0.191462i	$0.0613230\pi$
$127$	4.31534i	0.382925i	−0.981500	+	0.191462i	$0.938677\pi$
$128$	13.5616i	1.19868i
$129$	−7.68466	−0.676596
$130$	0	0
$131$	10.2462	0.895216	0.447608	−	0.894230i	$-0.352276\pi$
$131$	10.2462	0.895216	0.447608	+	0.894230i	$0.352276\pi$
$132$	− 0.630683i	− 0.0548939i
$133$	− 26.2462i	− 2.27584i
$134$	−17.7538	−1.53369
$135$	0	0
$136$	−17.3693	−1.48941
$137$	− 15.1231i	− 1.29205i	−0.763315	−	0.646027i	$-0.776429\pi$
$137$	− 15.1231i	− 1.29205i	0.763315	−	0.646027i	$-0.223571\pi$
$138$	10.2462i	0.872215i
$139$	17.9309	1.52088	0.760438	−	0.649410i	$-0.224984\pi$
$139$	17.9309	1.52088	0.760438	+	0.649410i	$0.224984\pi$
$140$	0	0
$141$	−13.1231	−1.10516
$142$	4.49242i	0.376996i
$143$	− 2.87689i	− 0.240578i
$144$	4.68466	0.390388
$145$	0	0
$146$	−2.63068	−0.217717
$147$	19.2462i	1.58740i
$148$	0.738634i	0.0607153i
$149$	−0.246211	−0.0201704	−0.0100852	−	0.999949i	$-0.503210\pi$
$149$	−0.246211	−0.0201704	−0.0100852	+	0.999949i	$0.503210\pi$
$150$	0	0
$151$	4.31534	0.351178	0.175589	−	0.984464i	$-0.443817\pi$
$151$	4.31534	0.351178	0.175589	+	0.984464i	$0.443817\pi$
$152$	12.4924i	1.01327i
$153$	7.12311i	0.575869i
$154$	−11.5076	−0.927307
$155$	0	0
$156$	−0.876894	−0.0702077
$157$	− 14.0000i	− 1.11732i	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	− 14.0000i	− 1.11732i	0.829396	−	0.558661i	$-0.188685\pi$
$158$	− 18.7386i	− 1.49077i
$159$	3.43845	0.272687
$160$	0	0
$161$	33.6155	2.64927
$162$	− 1.56155i	− 0.122687i
$163$	17.9309i	1.40445i	0.711953	+	0.702227i	$0.247811\pi$
$163$	17.9309i	1.40445i	−0.711953	+	0.702227i	$0.752189\pi$
$164$	0.738634	0.0576776
$165$	0	0
$166$	4.00000	0.310460
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	− 12.4924i	− 0.963811i
$169$	9.00000	0.692308
$170$	0	0
$171$	5.12311	0.391774
$172$	3.36932i	0.256908i
$173$	− 12.5616i	− 0.955037i	−0.878622	−	0.477519i	$-0.841536\pi$
$173$	− 12.5616i	− 0.955037i	0.878622	−	0.477519i	$-0.158464\pi$
$174$	1.56155	0.118381
$175$	0	0
$176$	6.73863	0.507944
$177$	12.0000i	0.901975i
$178$	19.1231i	1.43334i
$179$	−1.12311	−0.0839449	−0.0419724	−	0.999119i	$-0.513364\pi$
$179$	−1.12311	−0.0839449	−0.0419724	+	0.999119i	$0.513364\pi$
$180$	0	0
$181$	25.6847	1.90913	0.954563	−	0.298010i	$-0.0963228\pi$
$181$	25.6847	1.90913	0.954563	+	0.298010i	$0.0963228\pi$
$182$	16.0000i	1.18600i
$183$	− 0.876894i	− 0.0648219i
$184$	−16.0000	−1.17954
$185$	0	0
$186$	−6.24621	−0.457994
$187$	10.2462i	0.749277i
$188$	5.75379i	0.419638i
$189$	−5.12311	−0.372651
$190$	0	0
$191$	−9.93087	−0.718573	−0.359286	−	0.933227i	$-0.616980\pi$
$191$	−9.93087	−0.718573	−0.359286	+	0.933227i	$0.616980\pi$
$192$	5.56155i	0.401371i
$193$	10.0000i	0.719816i	0.932988	+	0.359908i	$0.117192\pi$
$193$	10.0000i	0.719816i	−0.932988	+	0.359908i	$0.882808\pi$
$194$	−8.87689	−0.637324
$195$	0	0
$196$	8.43845	0.602746
$197$	4.56155i	0.324997i	0.986709	+	0.162499i	$0.0519553\pi$
$197$	4.56155i	0.324997i	−0.986709	+	0.162499i	$0.948045\pi$
$198$	− 2.24621i	− 0.159631i
$199$	11.0540	0.783596	0.391798	−	0.920051i	$-0.371853\pi$
$199$	11.0540	0.783596	0.391798	+	0.920051i	$0.371853\pi$
$200$	0	0
$201$	−11.3693	−0.801930
$202$	13.3693i	0.940662i
$203$	− 5.12311i	− 0.359572i
$204$	3.12311	0.218661
$205$	0	0
$206$	4.49242	0.313002
$207$	6.56155i	0.456059i
$208$	− 9.36932i	− 0.649645i
$209$	7.36932	0.509746
$210$	0	0
$211$	−2.87689	−0.198054	−0.0990268	−	0.995085i	$-0.531573\pi$
$211$	−2.87689	−0.198054	−0.0990268	+	0.995085i	$0.531573\pi$
$212$	− 1.50758i	− 0.103541i
$213$	2.87689i	0.197122i
$214$	25.7538	1.76049
$215$	0	0
$216$	2.43845	0.165915
$217$	20.4924i	1.39112i
$218$	− 8.87689i	− 0.601219i
$219$	−1.68466	−0.113839
$220$	0	0
$221$	14.2462	0.958304
$222$	2.63068i	0.176560i
$223$	18.2462i	1.22186i	0.791686	+	0.610928i	$0.209204\pi$
$223$	18.2462i	1.22186i	−0.791686	+	0.610928i	$0.790796\pi$
$224$	−12.4924	−0.834685
$225$	0	0
$226$	7.61553	0.506577
$227$	− 3.19224i	− 0.211876i	−0.994373	−	0.105938i	$-0.966215\pi$
$227$	− 3.19224i	− 0.211876i	0.994373	−	0.105938i	$-0.0337845\pi$
$228$	− 2.24621i	− 0.148759i
$229$	−22.0000	−1.45380	−0.726900	−	0.686743i	$-0.759040\pi$
$229$	−22.0000	−1.45380	−0.726900	+	0.686743i	$0.759040\pi$
$230$	0	0
$231$	−7.36932	−0.484865
$232$	2.43845i	0.160092i
$233$	10.3153i	0.675780i	0.941186	+	0.337890i	$0.109713\pi$
$233$	10.3153i	0.675780i	−0.941186	+	0.337890i	$0.890287\pi$
$234$	−3.12311	−0.204164
$235$	0	0
$236$	5.26137	0.342486
$237$	− 12.0000i	− 0.779484i
$238$	− 56.9848i	− 3.69378i
$239$	13.1231	0.848863	0.424432	−	0.905460i	$-0.360474\pi$
$239$	13.1231	0.848863	0.424432	+	0.905460i	$0.360474\pi$
$240$	0	0
$241$	−25.0540	−1.61387	−0.806934	−	0.590641i	$-0.798875\pi$
$241$	−25.0540	−1.61387	−0.806934	+	0.590641i	$0.798875\pi$
$242$	13.9460i	0.896484i
$243$	− 1.00000i	− 0.0641500i
$244$	−0.384472	−0.0246133
$245$	0	0
$246$	2.63068	0.167726
$247$	− 10.2462i	− 0.651951i
$248$	− 9.75379i	− 0.619366i
$249$	2.56155	0.162332
$250$	0	0
$251$	2.24621	0.141780	0.0708898	−	0.997484i	$-0.477416\pi$
$251$	2.24621	0.141780	0.0708898	+	0.997484i	$0.477416\pi$
$252$	2.24621i	0.141498i
$253$	9.43845i	0.593390i
$254$	6.73863	0.422819
$255$	0	0
$256$	10.0540	0.628373
$257$	11.4384i	0.713511i	0.934198	+	0.356755i	$0.116117\pi$
$257$	11.4384i	0.713511i	−0.934198	+	0.356755i	$0.883883\pi$
$258$	12.0000i	0.747087i
$259$	8.63068	0.536285
$260$	0	0
$261$	1.00000	0.0618984
$262$	− 16.0000i	− 0.988483i
$263$	5.75379i	0.354794i	0.984139	+	0.177397i	$0.0567676\pi$
$263$	5.75379i	0.354794i	−0.984139	+	0.177397i	$0.943232\pi$
$264$	3.50758	0.215876
$265$	0	0
$266$	−40.9848	−2.51294
$267$	12.2462i	0.749456i
$268$	4.98485i	0.304498i
$269$	−11.7538	−0.716641	−0.358321	−	0.933599i	$-0.616651\pi$
$269$	−11.7538	−0.716641	−0.358321	+	0.933599i	$0.616651\pi$
$270$	0	0
$271$	17.1231	1.04015	0.520077	−	0.854119i	$-0.325903\pi$
$271$	17.1231	1.04015	0.520077	+	0.854119i	$0.325903\pi$
$272$	33.3693i	2.02331i
$273$	10.2462i	0.620129i
$274$	−23.6155	−1.42667
$275$	0	0
$276$	2.87689	0.173169
$277$	0.876894i	0.0526875i	0.999653	+	0.0263437i	$0.00838644\pi$
$277$	0.876894i	0.0526875i	−0.999653	+	0.0263437i	$0.991614\pi$
$278$	− 28.0000i	− 1.67933i
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−23.6155	−1.40878	−0.704392	−	0.709811i	$-0.748780\pi$
$281$	−23.6155	−1.40878	−0.704392	+	0.709811i	$0.748780\pi$
$282$	20.4924i	1.22031i
$283$	− 6.87689i	− 0.408789i	−0.978889	−	0.204394i	$-0.934477\pi$
$283$	− 6.87689i	− 0.408789i	0.978889	−	0.204394i	$-0.0655225\pi$
$284$	1.26137	0.0748483
$285$	0	0
$286$	−4.49242	−0.265643
$287$	− 8.63068i	− 0.509453i
$288$	− 2.43845i	− 0.143687i
$289$	−33.7386	−1.98463
$290$	0	0
$291$	−5.68466	−0.333241
$292$	0.738634i	0.0432253i
$293$	21.3693i	1.24841i	0.781261	+	0.624204i	$0.214577\pi$
$293$	21.3693i	1.24841i	−0.781261	+	0.624204i	$0.785423\pi$
$294$	30.0540	1.75278
$295$	0	0
$296$	−4.10795	−0.238770
$297$	− 1.43845i	− 0.0834672i
$298$	0.384472i	0.0222719i
$299$	13.1231	0.758929
$300$	0	0
$301$	39.3693	2.26921
$302$	− 6.73863i	− 0.387765i
$303$	8.56155i	0.491848i
$304$	24.0000	1.37649
$305$	0	0
$306$	11.1231	0.635866
$307$	− 31.6847i	− 1.80834i	−0.427174	−	0.904169i	$-0.640491\pi$
$307$	− 31.6847i	− 1.80834i	0.427174	−	0.904169i	$-0.359509\pi$
$308$	3.23106i	0.184107i
$309$	2.87689	0.163661
$310$	0	0
$311$	16.3153	0.925158	0.462579	−	0.886578i	$-0.346924\pi$
$311$	16.3153	0.925158	0.462579	+	0.886578i	$0.346924\pi$
$312$	− 4.87689i	− 0.276100i
$313$	21.3693i	1.20787i	0.797035	+	0.603933i	$0.206400\pi$
$313$	21.3693i	1.20787i	−0.797035	+	0.603933i	$0.793600\pi$
$314$	−21.8617	−1.23373
$315$	0	0
$316$	−5.26137	−0.295975
$317$	4.87689i	0.273914i	0.990577	+	0.136957i	$0.0437322\pi$
$317$	4.87689i	0.273914i	−0.990577	+	0.136957i	$0.956268\pi$
$318$	− 5.36932i	− 0.301096i
$319$	1.43845	0.0805376
$320$	0	0
$321$	16.4924	0.920517
$322$	− 52.4924i	− 2.92529i
$323$	36.4924i	2.03049i
$324$	−0.438447	−0.0243582
$325$	0	0
$326$	28.0000	1.55078
$327$	− 5.68466i	− 0.314362i
$328$	4.10795i	0.226824i
$329$	67.2311	3.70657
$330$	0	0
$331$	10.2462	0.563183	0.281591	−	0.959534i	$-0.409138\pi$
$331$	10.2462	0.563183	0.281591	+	0.959534i	$0.409138\pi$
$332$	− 1.12311i	− 0.0616384i
$333$	1.68466i	0.0923187i
$334$	0	0
$335$	0	0
$336$	−24.0000	−1.30931
$337$	− 7.75379i	− 0.422376i	−0.977445	−	0.211188i	$-0.932267\pi$
$337$	− 7.75379i	− 0.422376i	0.977445	−	0.211188i	$-0.0677332\pi$
$338$	− 14.0540i	− 0.764435i
$339$	4.87689	0.264877
$340$	0	0
$341$	−5.75379	−0.311585
$342$	− 8.00000i	− 0.432590i
$343$	− 62.7386i	− 3.38757i
$344$	−18.7386	−1.01032
$345$	0	0
$346$	−19.6155	−1.05454
$347$	− 12.1771i	− 0.653700i	−0.945076	−	0.326850i	$-0.894013\pi$
$347$	− 12.1771i	− 0.653700i	0.945076	−	0.326850i	$-0.105987\pi$
$348$	− 0.438447i	− 0.0235032i
$349$	−0.0691303	−0.00370046	−0.00185023	−	0.999998i	$-0.500589\pi$
$349$	−0.0691303	−0.00370046	−0.00185023	+	0.999998i	$0.500589\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	− 3.50758i	− 0.186955i
$353$	− 30.4924i	− 1.62295i	−0.584389	−	0.811474i	$-0.698666\pi$
$353$	− 30.4924i	− 1.62295i	0.584389	−	0.811474i	$-0.301334\pi$
$354$	18.7386	0.995947
$355$	0	0
$356$	5.36932	0.284573
$357$	− 36.4924i	− 1.93138i
$358$	1.75379i	0.0926906i
$359$	3.19224	0.168480	0.0842399	−	0.996446i	$-0.473154\pi$
$359$	3.19224	0.168480	0.0842399	+	0.996446i	$0.473154\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	− 40.1080i	− 2.10803i
$363$	8.93087i	0.468749i
$364$	4.49242	0.235467
$365$	0	0
$366$	−1.36932	−0.0715753
$367$	− 19.6847i	− 1.02753i	−0.857931	−	0.513765i	$-0.828250\pi$
$367$	− 19.6847i	− 1.02753i	0.857931	−	0.513765i	$-0.171750\pi$
$368$	30.7386i	1.60236i
$369$	1.68466	0.0876998
$370$	0	0
$371$	−17.6155	−0.914553
$372$	1.75379i	0.0909297i
$373$	− 13.3693i	− 0.692237i	−0.938191	−	0.346118i	$-0.887500\pi$
$373$	− 13.3693i	− 0.692237i	0.938191	−	0.346118i	$-0.112500\pi$
$374$	16.0000	0.827340
$375$	0	0
$376$	−32.0000	−1.65027
$377$	− 2.00000i	− 0.103005i
$378$	8.00000i	0.411476i
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	0	0
$381$	4.31534	0.221082
$382$	15.5076i	0.793437i
$383$	− 11.0540i	− 0.564832i	−0.959292	−	0.282416i	$-0.908864\pi$
$383$	− 11.0540i	− 0.564832i	0.959292	−	0.282416i	$-0.0911358\pi$
$384$	13.5616	0.692060
$385$	0	0
$386$	15.6155	0.794809
$387$	7.68466i	0.390633i
$388$	2.49242i	0.126534i
$389$	−14.8078	−0.750783	−0.375392	−	0.926866i	$-0.622492\pi$
$389$	−14.8078	−0.750783	−0.375392	+	0.926866i	$0.622492\pi$
$390$	0	0
$391$	−46.7386	−2.36367
$392$	46.9309i	2.37037i
$393$	− 10.2462i	− 0.516853i
$394$	7.12311	0.358857
$395$	0	0
$396$	−0.630683	−0.0316930
$397$	24.2462i	1.21688i	0.793599	+	0.608441i	$0.208205\pi$
$397$	24.2462i	1.21688i	−0.793599	+	0.608441i	$0.791795\pi$
$398$	− 17.2614i	− 0.865234i
$399$	−26.2462	−1.31395
$400$	0	0
$401$	35.6155	1.77855	0.889277	−	0.457368i	$-0.151208\pi$
$401$	35.6155	1.77855	0.889277	+	0.457368i	$0.151208\pi$
$402$	17.7538i	0.885479i
$403$	8.00000i	0.398508i
$404$	3.75379	0.186758
$405$	0	0
$406$	−8.00000	−0.397033
$407$	2.42329i	0.120118i
$408$	17.3693i	0.859909i
$409$	−17.3693	−0.858857	−0.429429	−	0.903101i	$-0.641285\pi$
$409$	−17.3693	−0.858857	−0.429429	+	0.903101i	$0.641285\pi$
$410$	0	0
$411$	−15.1231	−0.745968
$412$	− 1.26137i	− 0.0621431i
$413$	− 61.4773i	− 3.02510i
$414$	10.2462	0.503574
$415$	0	0
$416$	−4.87689	−0.239109
$417$	− 17.9309i	− 0.878078i
$418$	− 11.5076i	− 0.562854i
$419$	27.3693	1.33708	0.668539	−	0.743677i	$-0.266920\pi$
$419$	27.3693	1.33708	0.668539	+	0.743677i	$0.266920\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	4.49242i	0.218688i
$423$	13.1231i	0.638067i
$424$	8.38447	0.407186
$425$	0	0
$426$	4.49242	0.217659
$427$	4.49242i	0.217404i
$428$	− 7.23106i	− 0.349526i
$429$	−2.87689	−0.138898
$430$	0	0
$431$	−21.1231	−1.01746	−0.508732	−	0.860925i	$-0.669886\pi$
$431$	−21.1231	−1.01746	−0.508732	+	0.860925i	$0.669886\pi$
$432$	− 4.68466i	− 0.225391i
$433$	4.06913i	0.195550i	0.995209	+	0.0977750i	$0.0311725\pi$
$433$	4.06913i	0.195550i	−0.995209	+	0.0977750i	$0.968827\pi$
$434$	32.0000	1.53605
$435$	0	0
$436$	−2.49242	−0.119365
$437$	33.6155i	1.60805i
$438$	2.63068i	0.125699i
$439$	−12.4924	−0.596231	−0.298115	−	0.954530i	$-0.596358\pi$
$439$	−12.4924	−0.596231	−0.298115	+	0.954530i	$0.596358\pi$
$440$	0	0
$441$	19.2462	0.916486
$442$	− 22.2462i	− 1.05814i
$443$	− 6.24621i	− 0.296766i	−0.988930	−	0.148383i	$-0.952593\pi$
$443$	− 6.24621i	− 0.296766i	0.988930	−	0.148383i	$-0.0474069\pi$
$444$	0.738634	0.0350540
$445$	0	0
$446$	28.4924	1.34916
$447$	0.246211i	0.0116454i
$448$	− 28.4924i	− 1.34614i
$449$	−11.4384	−0.539814	−0.269907	−	0.962886i	$-0.586993\pi$
$449$	−11.4384	−0.539814	−0.269907	+	0.962886i	$0.586993\pi$
$450$	0	0
$451$	2.42329	0.114108
$452$	− 2.13826i	− 0.100575i
$453$	− 4.31534i	− 0.202752i
$454$	−4.98485	−0.233950
$455$	0	0
$456$	12.4924	0.585011
$457$	− 8.87689i	− 0.415244i	−0.978209	−	0.207622i	$-0.933428\pi$
$457$	− 8.87689i	− 0.415244i	0.978209	−	0.207622i	$-0.0665723\pi$
$458$	34.3542i	1.60526i
$459$	7.12311	0.332478
$460$	0	0
$461$	−18.1771	−0.846591	−0.423296	−	0.905992i	$-0.639127\pi$
$461$	−18.1771	−0.846591	−0.423296	+	0.905992i	$0.639127\pi$
$462$	11.5076i	0.535381i
$463$	− 25.6155i	− 1.19045i	−0.803557	−	0.595227i	$-0.797062\pi$
$463$	− 25.6155i	− 1.19045i	0.803557	−	0.595227i	$-0.202938\pi$
$464$	4.68466	0.217480
$465$	0	0
$466$	16.1080	0.746186
$467$	− 3.36932i	− 0.155913i	−0.996957	−	0.0779567i	$-0.975160\pi$
$467$	− 3.36932i	− 0.155913i	0.996957	−	0.0779567i	$-0.0248396\pi$
$468$	0.876894i	0.0405345i
$469$	58.2462	2.68956
$470$	0	0
$471$	−14.0000	−0.645086
$472$	29.2614i	1.34686i
$473$	11.0540i	0.508262i
$474$	−18.7386	−0.860694
$475$	0	0
$476$	−16.0000	−0.733359
$477$	− 3.43845i	− 0.157436i
$478$	− 20.4924i	− 0.937302i
$479$	22.2462	1.01646	0.508228	−	0.861223i	$-0.330301\pi$
$479$	22.2462	1.01646	0.508228	+	0.861223i	$0.330301\pi$
$480$	0	0
$481$	3.36932	0.153628
$482$	39.1231i	1.78201i
$483$	− 33.6155i	− 1.52956i
$484$	3.91571	0.177987
$485$	0	0
$486$	−1.56155	−0.0708335
$487$	34.2462i	1.55184i	0.630829	+	0.775922i	$0.282715\pi$
$487$	34.2462i	1.55184i	−0.630829	+	0.775922i	$0.717285\pi$
$488$	− 2.13826i	− 0.0967945i
$489$	17.9309	0.810862
$490$	0	0
$491$	6.73863	0.304110	0.152055	−	0.988372i	$-0.451411\pi$
$491$	6.73863	0.304110	0.152055	+	0.988372i	$0.451411\pi$
$492$	− 0.738634i	− 0.0333002i
$493$	7.12311i	0.320809i
$494$	−16.0000	−0.719874
$495$	0	0
$496$	−18.7386	−0.841389
$497$	− 14.7386i	− 0.661118i
$498$	− 4.00000i	− 0.179244i
$499$	−42.7386	−1.91324	−0.956622	−	0.291332i	$-0.905902\pi$
$499$	−42.7386	−1.91324	−0.956622	+	0.291332i	$0.905902\pi$
$500$	0	0
$501$	0	0
$502$	− 3.50758i	− 0.156551i
$503$	39.3693i	1.75539i	0.479219	+	0.877696i	$0.340920\pi$
$503$	39.3693i	1.75539i	−0.479219	+	0.877696i	$0.659080\pi$
$504$	−12.4924	−0.556457
$505$	0	0
$506$	14.7386	0.655212
$507$	− 9.00000i	− 0.399704i
$508$	− 1.89205i	− 0.0839461i
$509$	−18.4924	−0.819662	−0.409831	−	0.912161i	$-0.634412\pi$
$509$	−18.4924	−0.819662	−0.409831	+	0.912161i	$0.634412\pi$
$510$	0	0
$511$	8.63068	0.381799
$512$	11.4233i	0.504843i
$513$	− 5.12311i	− 0.226191i
$514$	17.8617	0.787848
$515$	0	0
$516$	3.36932	0.148326
$517$	18.8769i	0.830205i
$518$	− 13.4773i	− 0.592157i
$519$	−12.5616	−0.551391
$520$	0	0
$521$	−0.246211	−0.0107867	−0.00539336	−	0.999985i	$-0.501717\pi$
$521$	−0.246211	−0.0107867	−0.00539336	+	0.999985i	$0.501717\pi$
$522$	− 1.56155i	− 0.0683473i
$523$	− 18.7386i	− 0.819383i	−0.912224	−	0.409692i	$-0.865636\pi$
$523$	− 18.7386i	− 0.819383i	0.912224	−	0.409692i	$-0.134364\pi$
$524$	−4.49242	−0.196252
$525$	0	0
$526$	8.98485	0.391758
$527$	− 28.4924i	− 1.24115i
$528$	− 6.73863i	− 0.293261i
$529$	−20.0540	−0.871912
$530$	0	0
$531$	12.0000	0.520756
$532$	11.5076i	0.498917i
$533$	− 3.36932i	− 0.145941i
$534$	19.1231	0.827538
$535$	0	0
$536$	−27.7235	−1.19747
$537$	1.12311i	0.0484656i
$538$	18.3542i	0.791304i
$539$	27.6847	1.19246
$540$	0	0
$541$	−40.7386	−1.75149	−0.875745	−	0.482773i	$-0.839629\pi$
$541$	−40.7386	−1.75149	−0.875745	+	0.482773i	$0.839629\pi$
$542$	− 26.7386i	− 1.14852i
$543$	− 25.6847i	− 1.10223i
$544$	17.3693	0.744703
$545$	0	0
$546$	16.0000	0.684737
$547$	− 42.7386i	− 1.82737i	−0.406421	−	0.913686i	$-0.633223\pi$
$547$	− 42.7386i	− 1.82737i	0.406421	−	0.913686i	$-0.366777\pi$
$548$	6.63068i	0.283249i
$549$	−0.876894	−0.0374249
$550$	0	0
$551$	5.12311	0.218252
$552$	16.0000i	0.681005i
$553$	61.4773i	2.61428i
$554$	1.36932	0.0581767
$555$	0	0
$556$	−7.86174	−0.333412
$557$	− 20.4233i	− 0.865363i	−0.901547	−	0.432681i	$-0.857568\pi$
$557$	− 20.4233i	− 0.865363i	0.901547	−	0.432681i	$-0.142432\pi$
$558$	6.24621i	0.264423i
$559$	15.3693	0.650053
$560$	0	0
$561$	10.2462	0.432595
$562$	36.8769i	1.55556i
$563$	− 9.12311i	− 0.384493i	−0.981347	−	0.192247i	$-0.938423\pi$
$563$	− 9.12311i	− 0.384493i	0.981347	−	0.192247i	$-0.0615773\pi$
$564$	5.75379	0.242278
$565$	0	0
$566$	−10.7386	−0.451378
$567$	5.12311i	0.215150i
$568$	7.01515i	0.294349i
$569$	16.2462	0.681077	0.340538	−	0.940231i	$-0.389391\pi$
$569$	16.2462	0.681077	0.340538	+	0.940231i	$0.389391\pi$
$570$	0	0
$571$	39.0540	1.63436	0.817179	−	0.576384i	$-0.195537\pi$
$571$	39.0540	1.63436	0.817179	+	0.576384i	$0.195537\pi$
$572$	1.26137i	0.0527404i
$573$	9.93087i	0.414868i
$574$	−13.4773	−0.562530
$575$	0	0
$576$	5.56155	0.231731
$577$	− 30.4924i	− 1.26942i	−0.772752	−	0.634708i	$-0.781120\pi$
$577$	− 30.4924i	− 1.26942i	0.772752	−	0.634708i	$-0.218880\pi$
$578$	52.6847i	2.19139i
$579$	10.0000	0.415586
$580$	0	0
$581$	−13.1231	−0.544438
$582$	8.87689i	0.367959i
$583$	− 4.94602i	− 0.204843i
$584$	−4.10795	−0.169988
$585$	0	0
$586$	33.3693	1.37847
$587$	24.4924i	1.01091i	0.862853	+	0.505455i	$0.168675\pi$
$587$	24.4924i	1.01091i	−0.862853	+	0.505455i	$0.831325\pi$
$588$	− 8.43845i	− 0.347996i
$589$	−20.4924	−0.844376
$590$	0	0
$591$	4.56155	0.187637
$592$	7.89205i	0.324361i
$593$	− 36.2462i	− 1.48845i	−0.667927	−	0.744227i	$-0.732818\pi$
$593$	− 36.2462i	− 1.48845i	0.667927	−	0.744227i	$-0.267182\pi$
$594$	−2.24621	−0.0921632
$595$	0	0
$596$	0.107951	0.00442183
$597$	− 11.0540i	− 0.452409i
$598$	− 20.4924i	− 0.837997i
$599$	20.9848	0.857418	0.428709	−	0.903443i	$-0.358969\pi$
$599$	20.9848	0.857418	0.428709	+	0.903443i	$0.358969\pi$
$600$	0	0
$601$	41.3693	1.68749	0.843745	−	0.536745i	$-0.180346\pi$
$601$	41.3693	1.68749	0.843745	+	0.536745i	$0.180346\pi$
$602$	− 61.4773i	− 2.50563i
$603$	11.3693i	0.462994i
$604$	−1.89205	−0.0769864
$605$	0	0
$606$	13.3693	0.543091
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	− 12.4924i	− 0.506635i
$609$	−5.12311	−0.207599
$610$	0	0
$611$	26.2462	1.06181
$612$	− 3.12311i	− 0.126244i
$613$	2.00000i	0.0807792i	0.999184	+	0.0403896i	$0.0128599\pi$
$613$	2.00000i	0.0807792i	−0.999184	+	0.0403896i	$0.987140\pi$
$614$	−49.4773	−1.99674
$615$	0	0
$616$	−17.9697	−0.724019
$617$	8.24621i	0.331980i	0.986127	+	0.165990i	$0.0530819\pi$
$617$	8.24621i	0.331980i	−0.986127	+	0.165990i	$0.946918\pi$
$618$	− 4.49242i	− 0.180712i
$619$	−29.1231	−1.17056	−0.585278	−	0.810833i	$-0.699015\pi$
$619$	−29.1231	−1.17056	−0.585278	+	0.810833i	$0.699015\pi$
$620$	0	0
$621$	6.56155	0.263306
$622$	− 25.4773i	− 1.02155i
$623$	− 62.7386i	− 2.51357i
$624$	−9.36932	−0.375073
$625$	0	0
$626$	33.3693	1.33371
$627$	− 7.36932i	− 0.294302i
$628$	6.13826i	0.244943i
$629$	−12.0000	−0.478471
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	− 29.2614i	− 1.16395i
$633$	2.87689i	0.114346i
$634$	7.61553	0.302451
$635$	0	0
$636$	−1.50758	−0.0597793
$637$	− 38.4924i	− 1.52513i
$638$	− 2.24621i	− 0.0889284i
$639$	2.87689	0.113808
$640$	0	0
$641$	19.4384	0.767773	0.383886	−	0.923380i	$-0.374585\pi$
$641$	19.4384	0.767773	0.383886	+	0.923380i	$0.374585\pi$
$642$	− 25.7538i	− 1.01642i
$643$	18.7386i	0.738980i	0.929235	+	0.369490i	$0.120468\pi$
$643$	18.7386i	0.738980i	−0.929235	+	0.369490i	$0.879532\pi$
$644$	−14.7386	−0.580784
$645$	0	0
$646$	56.9848	2.24204
$647$	5.93087i	0.233167i	0.993181	+	0.116583i	$0.0371942\pi$
$647$	5.93087i	0.233167i	−0.993181	+	0.116583i	$0.962806\pi$
$648$	− 2.43845i	− 0.0957913i
$649$	17.2614	0.677568
$650$	0	0
$651$	20.4924	0.803161
$652$	− 7.86174i	− 0.307889i
$653$	− 12.2462i	− 0.479231i	−0.970868	−	0.239616i	$-0.922979\pi$
$653$	− 12.2462i	− 0.479231i	0.970868	−	0.239616i	$-0.0770214\pi$
$654$	−8.87689	−0.347114
$655$	0	0
$656$	7.89205	0.308133
$657$	1.68466i	0.0657248i
$658$	− 104.985i	− 4.09274i
$659$	−6.56155	−0.255602	−0.127801	−	0.991800i	$-0.540792\pi$
$659$	−6.56155	−0.255602	−0.127801	+	0.991800i	$0.540792\pi$
$660$	0	0
$661$	1.05398	0.0409949	0.0204974	−	0.999790i	$-0.493475\pi$
$661$	1.05398	0.0409949	0.0204974	+	0.999790i	$0.493475\pi$
$662$	− 16.0000i	− 0.621858i
$663$	− 14.2462i	− 0.553277i
$664$	6.24621	0.242400
$665$	0	0
$666$	2.63068	0.101937
$667$	6.56155i	0.254064i
$668$	0	0
$669$	18.2462	0.705439
$670$	0	0
$671$	−1.26137	−0.0486945
$672$	12.4924i	0.481906i
$673$	6.00000i	0.231283i	0.993291	+	0.115642i	$0.0368924\pi$
$673$	6.00000i	0.231283i	−0.993291	+	0.115642i	$0.963108\pi$
$674$	−12.1080	−0.466381
$675$	0	0
$676$	−3.94602	−0.151770
$677$	13.8617i	0.532750i	0.963869	+	0.266375i	$0.0858259\pi$
$677$	13.8617i	0.532750i	−0.963869	+	0.266375i	$0.914174\pi$
$678$	− 7.61553i	− 0.292473i
$679$	29.1231	1.11764
$680$	0	0
$681$	−3.19224	−0.122327
$682$	8.98485i	0.344047i
$683$	21.4384i	0.820319i	0.912014	+	0.410160i	$0.134527\pi$
$683$	21.4384i	0.820319i	−0.912014	+	0.410160i	$0.865473\pi$
$684$	−2.24621	−0.0858860
$685$	0	0
$686$	−97.9697	−3.74050
$687$	22.0000i	0.839352i
$688$	36.0000i	1.37249i
$689$	−6.87689	−0.261989
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	5.50758i	0.209367i
$693$	7.36932i	0.279937i
$694$	−19.0152	−0.721805
$695$	0	0
$696$	2.43845	0.0924291
$697$	12.0000i	0.454532i
$698$	0.107951i	0.00408599i
$699$	10.3153	0.390162
$700$	0	0
$701$	3.75379	0.141779	0.0708893	−	0.997484i	$-0.477416\pi$
$701$	3.75379	0.141779	0.0708893	+	0.997484i	$0.477416\pi$
$702$	3.12311i	0.117874i
$703$	8.63068i	0.325512i
$704$	8.00000	0.301511
$705$	0	0
$706$	−47.6155	−1.79203
$707$	− 43.8617i	− 1.64959i
$708$	− 5.26137i	− 0.197734i
$709$	12.4233	0.466567	0.233283	−	0.972409i	$-0.425053\pi$
$709$	12.4233	0.466567	0.233283	+	0.972409i	$0.425053\pi$
$710$	0	0
$711$	−12.0000	−0.450035
$712$	29.8617i	1.11912i
$713$	− 26.2462i	− 0.982928i
$714$	−56.9848	−2.13260
$715$	0	0
$716$	0.492423	0.0184027
$717$	− 13.1231i	− 0.490091i
$718$	− 4.98485i	− 0.186033i
$719$	4.49242	0.167539	0.0837695	−	0.996485i	$-0.473304\pi$
$719$	4.49242	0.167539	0.0837695	+	0.996485i	$0.473304\pi$
$720$	0	0
$721$	−14.7386	−0.548895
$722$	− 11.3153i	− 0.421113i
$723$	25.0540i	0.931767i
$724$	−11.2614	−0.418525
$725$	0	0
$726$	13.9460	0.517586
$727$	− 38.7386i	− 1.43674i	−0.695663	−	0.718368i	$-0.744889\pi$
$727$	− 38.7386i	− 1.43674i	0.695663	−	0.718368i	$-0.255111\pi$
$728$	24.9848i	0.926000i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−54.7386	−2.02458
$732$	0.384472i	0.0142105i
$733$	22.9848i	0.848965i	0.905436	+	0.424482i	$0.139544\pi$
$733$	22.9848i	0.848965i	−0.905436	+	0.424482i	$0.860456\pi$
$734$	−30.7386	−1.13458
$735$	0	0
$736$	16.0000	0.589768
$737$	16.3542i	0.602413i
$738$	− 2.63068i	− 0.0968368i
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	−10.2462	−0.376404
$742$	27.5076i	1.00983i
$743$	− 34.8769i	− 1.27951i	−0.768579	−	0.639755i	$-0.779036\pi$
$743$	− 34.8769i	− 1.27951i	0.768579	−	0.639755i	$-0.220964\pi$
$744$	−9.75379	−0.357591
$745$	0	0
$746$	−20.8769	−0.764357
$747$	− 2.56155i	− 0.0937223i
$748$	− 4.49242i	− 0.164259i
$749$	−84.4924	−3.08729
$750$	0	0
$751$	48.4924	1.76951	0.884757	−	0.466053i	$-0.154324\pi$
$751$	48.4924	1.76951	0.884757	+	0.466053i	$0.154324\pi$
$752$	61.4773i	2.24185i
$753$	− 2.24621i	− 0.0818565i
$754$	−3.12311	−0.113737
$755$	0	0
$756$	2.24621	0.0816939
$757$	5.68466i	0.206612i	0.994650	+	0.103306i	$0.0329422\pi$
$757$	5.68466i	0.206612i	−0.994650	+	0.103306i	$0.967058\pi$
$758$	12.4924i	0.453745i
$759$	9.43845	0.342594
$760$	0	0
$761$	−24.8769	−0.901787	−0.450893	−	0.892578i	$-0.648895\pi$
$761$	−24.8769	−0.901787	−0.450893	+	0.892578i	$0.648895\pi$
$762$	− 6.73863i	− 0.244115i
$763$	29.1231i	1.05433i
$764$	4.35416	0.157528
$765$	0	0
$766$	−17.2614	−0.623679
$767$	− 24.0000i	− 0.866590i
$768$	− 10.0540i	− 0.362792i
$769$	−43.6155	−1.57282	−0.786408	−	0.617707i	$-0.788062\pi$
$769$	−43.6155	−1.57282	−0.786408	+	0.617707i	$0.788062\pi$
$770$	0	0
$771$	11.4384	0.411946
$772$	− 4.38447i	− 0.157801i
$773$	− 24.7386i	− 0.889787i	−0.895584	−	0.444893i	$-0.853242\pi$
$773$	− 24.7386i	− 0.889787i	0.895584	−	0.444893i	$-0.146758\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	−13.8617	−0.497607
$777$	− 8.63068i	− 0.309624i
$778$	23.1231i	0.829004i
$779$	8.63068	0.309226
$780$	0	0
$781$	4.13826	0.148079
$782$	72.9848i	2.60993i
$783$	− 1.00000i	− 0.0357371i
$784$	90.1619	3.22007
$785$	0	0
$786$	−16.0000	−0.570701
$787$	− 31.2311i	− 1.11327i	−0.830758	−	0.556633i	$-0.812093\pi$
$787$	− 31.2311i	− 1.11327i	0.830758	−	0.556633i	$-0.187907\pi$
$788$	− 2.00000i	− 0.0712470i
$789$	5.75379	0.204840
$790$	0	0
$791$	−24.9848	−0.888359
$792$	− 3.50758i	− 0.124636i
$793$	1.75379i	0.0622789i
$794$	37.8617	1.34366
$795$	0	0
$796$	−4.84658	−0.171783
$797$	38.4924i	1.36347i	0.731598	+	0.681736i	$0.238775\pi$
$797$	38.4924i	1.36347i	−0.731598	+	0.681736i	$0.761225\pi$
$798$	40.9848i	1.45085i
$799$	−93.4773	−3.30699
$800$	0	0
$801$	12.2462	0.432699
$802$	− 55.6155i	− 1.96385i
$803$	2.42329i	0.0855161i
$804$	4.98485	0.175802
$805$	0	0
$806$	12.4924	0.440027
$807$	11.7538i	0.413753i
$808$	20.8769i	0.734447i
$809$	22.1771	0.779705	0.389852	−	0.920877i	$-0.372526\pi$
$809$	22.1771	0.779705	0.389852	+	0.920877i	$0.372526\pi$
$810$	0	0
$811$	44.1771	1.55127	0.775634	−	0.631183i	$-0.217431\pi$
$811$	44.1771	1.55127	0.775634	+	0.631183i	$0.217431\pi$
$812$	2.24621i	0.0788266i
$813$	− 17.1231i	− 0.600534i
$814$	3.78410	0.132633
$815$	0	0
$816$	33.3693	1.16816
$817$	39.3693i	1.37736i
$818$	27.1231i	0.948337i
$819$	10.2462	0.358032
$820$	0	0
$821$	55.6155	1.94100	0.970498	−	0.241111i	$-0.0775116\pi$
$821$	55.6155	1.94100	0.970498	+	0.241111i	$0.0775116\pi$
$822$	23.6155i	0.823686i
$823$	− 16.0000i	− 0.557725i	−0.960331	−	0.278862i	$-0.910043\pi$
$823$	− 16.0000i	− 0.557725i	0.960331	−	0.278862i	$-0.0899574\pi$
$824$	7.01515	0.244385
$825$	0	0
$826$	−96.0000	−3.34027
$827$	53.6155i	1.86439i	0.361951	+	0.932197i	$0.382111\pi$
$827$	53.6155i	1.86439i	−0.361951	+	0.932197i	$0.617889\pi$
$828$	− 2.87689i	− 0.0999790i
$829$	4.24621	0.147477	0.0737385	−	0.997278i	$-0.476507\pi$
$829$	4.24621	0.147477	0.0737385	+	0.997278i	$0.476507\pi$
$830$	0	0
$831$	0.876894	0.0304191
$832$	− 11.1231i	− 0.385624i
$833$	137.093i	4.74998i
$834$	−28.0000	−0.969561
$835$	0	0
$836$	−3.23106	−0.111748
$837$	4.00000i	0.138260i
$838$	− 42.7386i	− 1.47638i
$839$	−42.7386	−1.47550	−0.737751	−	0.675073i	$-0.764112\pi$
$839$	−42.7386	−1.47550	−0.737751	+	0.675073i	$0.764112\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	15.6155i	0.538147i
$843$	23.6155i	0.813362i
$844$	1.26137	0.0434180
$845$	0	0
$846$	20.4924	0.704544
$847$	− 45.7538i	− 1.57212i
$848$	− 16.1080i	− 0.553149i
$849$	−6.87689	−0.236014
$850$	0	0
$851$	−11.0540	−0.378925
$852$	− 1.26137i	− 0.0432137i
$853$	− 19.4384i	− 0.665560i	−0.943005	−	0.332780i	$-0.892013\pi$
$853$	− 19.4384i	− 0.665560i	0.943005	−	0.332780i	$-0.107987\pi$
$854$	7.01515	0.240054
$855$	0	0
$856$	40.2159	1.37455
$857$	− 15.4384i	− 0.527367i	−0.964609	−	0.263684i	$-0.915062\pi$
$857$	− 15.4384i	− 0.527367i	0.964609	−	0.263684i	$-0.0849375\pi$
$858$	4.49242i	0.153369i
$859$	23.3693	0.797351	0.398675	−	0.917092i	$-0.369470\pi$
$859$	23.3693	0.797351	0.398675	+	0.917092i	$0.369470\pi$
$860$	0	0
$861$	−8.63068	−0.294133
$862$	32.9848i	1.12347i
$863$	− 40.9848i	− 1.39514i	−0.716516	−	0.697570i	$-0.754265\pi$
$863$	− 40.9848i	− 1.39514i	0.716516	−	0.697570i	$-0.245735\pi$
$864$	−2.43845	−0.0829577
$865$	0	0
$866$	6.35416	0.215923
$867$	33.7386i	1.14582i
$868$	− 8.98485i	− 0.304966i
$869$	−17.2614	−0.585552
$870$	0	0
$871$	22.7386	0.770469
$872$	− 13.8617i	− 0.469418i
$873$	5.68466i	0.192397i
$874$	52.4924	1.77558
$875$	0	0
$876$	0.738634	0.0249561
$877$	8.87689i	0.299751i	0.988705	+	0.149876i	$0.0478874\pi$
$877$	8.87689i	0.299751i	−0.988705	+	0.149876i	$0.952113\pi$
$878$	19.5076i	0.658349i
$879$	21.3693	0.720769
$880$	0	0
$881$	−8.06913	−0.271856	−0.135928	−	0.990719i	$-0.543402\pi$
$881$	−8.06913	−0.271856	−0.135928	+	0.990719i	$0.543402\pi$
$882$	− 30.0540i	− 1.01197i
$883$	42.7386i	1.43827i	0.694871	+	0.719135i	$0.255462\pi$
$883$	42.7386i	1.43827i	−0.694871	+	0.719135i	$0.744538\pi$
$884$	−6.24621	−0.210083
$885$	0	0
$886$	−9.75379	−0.327685
$887$	− 43.8617i	− 1.47273i	−0.676583	−	0.736367i	$-0.736540\pi$
$887$	− 43.8617i	− 1.47273i	0.676583	−	0.736367i	$-0.263460\pi$
$888$	4.10795i	0.137854i
$889$	−22.1080	−0.741477
$890$	0	0
$891$	−1.43845	−0.0481898
$892$	− 8.00000i	− 0.267860i
$893$	67.2311i	2.24980i
$894$	0.384472	0.0128587
$895$	0	0
$896$	−69.4773	−2.32107
$897$	− 13.1231i	− 0.438168i
$898$	17.8617i	0.596054i
$899$	−4.00000	−0.133407
$900$	0	0
$901$	24.4924	0.815961
$902$	− 3.78410i	− 0.125997i
$903$	− 39.3693i	− 1.31013i
$904$	11.8920	0.395524
$905$	0	0
$906$	−6.73863	−0.223876
$907$	31.0540i	1.03113i	0.856850	+	0.515565i	$0.172418\pi$
$907$	31.0540i	1.03113i	−0.856850	+	0.515565i	$0.827582\pi$
$908$	1.39963i	0.0464482i
$909$	8.56155	0.283969
$910$	0	0
$911$	−18.5616	−0.614972	−0.307486	−	0.951553i	$-0.599488\pi$
$911$	−18.5616	−0.614972	−0.307486	+	0.951553i	$0.599488\pi$
$912$	− 24.0000i	− 0.794719i
$913$	− 3.68466i	− 0.121944i
$914$	−13.8617	−0.458506
$915$	0	0
$916$	9.64584	0.318707
$917$	52.4924i	1.73345i
$918$	− 11.1231i	− 0.367117i
$919$	−30.7386	−1.01397	−0.506987	−	0.861954i	$-0.669241\pi$
$919$	−30.7386	−1.01397	−0.506987	+	0.861954i	$0.669241\pi$
$920$	0	0
$921$	−31.6847	−1.04404
$922$	28.3845i	0.934793i
$923$	− 5.75379i	− 0.189388i
$924$	3.23106	0.106294
$925$	0	0
$926$	−40.0000	−1.31448
$927$	− 2.87689i	− 0.0944896i
$928$	− 2.43845i	− 0.0800460i
$929$	−4.87689	−0.160006	−0.0800029	−	0.996795i	$-0.525493\pi$
$929$	−4.87689	−0.160006	−0.0800029	+	0.996795i	$0.525493\pi$
$930$	0	0
$931$	98.6004	3.23150
$932$	− 4.52273i	− 0.148147i
$933$	− 16.3153i	− 0.534140i
$934$	−5.26137	−0.172157
$935$	0	0
$936$	−4.87689	−0.159406
$937$	39.1231i	1.27810i	0.769167	+	0.639048i	$0.220672\pi$
$937$	39.1231i	1.27810i	−0.769167	+	0.639048i	$0.779328\pi$
$938$	− 90.9545i	− 2.96977i
$939$	21.3693	0.697361
$940$	0	0
$941$	−0.738634	−0.0240788	−0.0120394	−	0.999928i	$-0.503832\pi$
$941$	−0.738634	−0.0240788	−0.0120394	+	0.999928i	$0.503832\pi$
$942$	21.8617i	0.712294i
$943$	11.0540i	0.359967i
$944$	56.2159	1.82967
$945$	0	0
$946$	17.2614	0.561215
$947$	− 3.36932i	− 0.109488i	−0.998500	−	0.0547440i	$-0.982566\pi$
$947$	− 3.36932i	− 0.109488i	0.998500	−	0.0547440i	$-0.0174343\pi$
$948$	5.26137i	0.170881i
$949$	3.36932	0.109373
$950$	0	0
$951$	4.87689	0.158144
$952$	− 88.9848i	− 2.88402i
$953$	34.4924i	1.11732i	0.829397	+	0.558660i	$0.188684\pi$
$953$	34.4924i	1.11732i	−0.829397	+	0.558660i	$0.811316\pi$
$954$	−5.36932	−0.173838
$955$	0	0
$956$	−5.75379	−0.186091
$957$	− 1.43845i	− 0.0464984i
$958$	− 34.7386i	− 1.12235i
$959$	77.4773	2.50187
$960$	0	0
$961$	−15.0000	−0.483871
$962$	− 5.26137i	− 0.169633i
$963$	− 16.4924i	− 0.531461i
$964$	10.9848	0.353798
$965$	0	0
$966$	−52.4924	−1.68892
$967$	27.0540i	0.869997i	0.900431	+	0.434999i	$0.143251\pi$
$967$	27.0540i	0.869997i	−0.900431	+	0.434999i	$0.856749\pi$
$968$	21.7775i	0.699954i
$969$	36.4924	1.17231
$970$	0	0
$971$	−44.6695	−1.43351	−0.716756	−	0.697324i	$-0.754374\pi$
$971$	−44.6695	−1.43351	−0.716756	+	0.697324i	$0.754374\pi$
$972$	0.438447i	0.0140632i
$973$	91.8617i	2.94495i
$974$	53.4773	1.71352
$975$	0	0
$976$	−4.10795	−0.131492
$977$	− 7.43845i	− 0.237977i	−0.992896	−	0.118989i	$-0.962035\pi$
$977$	− 7.43845i	− 0.237977i	0.992896	−	0.118989i	$-0.0379652\pi$
$978$	− 28.0000i	− 0.895341i
$979$	17.6155	0.562995
$980$	0	0
$981$	−5.68466	−0.181497
$982$	− 10.5227i	− 0.335794i
$983$	− 27.8617i	− 0.888651i	−0.895865	−	0.444326i	$-0.853443\pi$
$983$	− 27.8617i	− 0.888651i	0.895865	−	0.444326i	$-0.146557\pi$
$984$	4.10795	0.130957
$985$	0	0
$986$	11.1231	0.354232
$987$	− 67.2311i	− 2.13999i
$988$	4.49242i	0.142923i
$989$	−50.4233	−1.60337
$990$	0	0
$991$	4.94602	0.157116	0.0785578	−	0.996910i	$-0.474968\pi$
$991$	4.94602	0.157116	0.0785578	+	0.996910i	$0.474968\pi$
$992$	9.75379i	0.309683i
$993$	− 10.2462i	− 0.325154i
$994$	−23.0152	−0.729996
$995$	0	0
$996$	−1.12311	−0.0355870
$997$	54.0388i	1.71143i	0.517450	+	0.855713i	$0.326881\pi$
$997$	54.0388i	1.71143i	−0.517450	+	0.855713i	$0.673119\pi$
$998$	66.7386i	2.11257i
$999$	1.68466	0.0533002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.c.h.349.2		4
5.2	odd	4		2175.2.a.m.1.2		2
5.3	odd	4		435.2.a.h.1.1	✓	2
5.4	even	2	inner	2175.2.c.h.349.3		4
15.2	even	4		6525.2.a.bc.1.1		2
15.8	even	4		1305.2.a.i.1.2		2
20.3	even	4		6960.2.a.bx.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.h.1.1	✓	2	5.3	odd	4
1305.2.a.i.1.2		2	15.8	even	4
2175.2.a.m.1.2		2	5.2	odd	4
2175.2.c.h.349.2		4	1.1	even	1	trivial
2175.2.c.h.349.3		4	5.4	even	2	inner
6525.2.a.bc.1.1		2	15.2	even	4
6960.2.a.bx.1.1		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.56155i q^{2} -1.00000i q^{3} -0.438447 q^{4} -1.56155 q^{6} +5.12311i q^{7} -2.43845i q^{8} -1.00000 q^{9} -1.43845 q^{11} +0.438447i q^{12} +2.00000i q^{13} +8.00000 q^{14} -4.68466 q^{16} -7.12311i q^{17} +1.56155i q^{18} -5.12311 q^{19} +5.12311 q^{21} +2.24621i q^{22} -6.56155i q^{23} -2.43845 q^{24} +3.12311 q^{26} +1.00000i q^{27} -2.24621i q^{28} -1.00000 q^{29} +4.00000 q^{31} +2.43845i q^{32} +1.43845i q^{33} -11.1231 q^{34} +0.438447 q^{36} -1.68466i q^{37} +8.00000i q^{38} +2.00000 q^{39} -1.68466 q^{41} -8.00000i q^{42} -7.68466i q^{43} +0.630683 q^{44} -10.2462 q^{46} -13.1231i q^{47} +4.68466i q^{48} -19.2462 q^{49} -7.12311 q^{51} -0.876894i q^{52} +3.43845i q^{53} +1.56155 q^{54} +12.4924 q^{56} +5.12311i q^{57} +1.56155i q^{58} -12.0000 q^{59} +0.876894 q^{61} -6.24621i q^{62} -5.12311i q^{63} -5.56155 q^{64} +2.24621 q^{66} -11.3693i q^{67} +3.12311i q^{68} -6.56155 q^{69} -2.87689 q^{71} +2.43845i q^{72} -1.68466i q^{73} -2.63068 q^{74} +2.24621 q^{76} -7.36932i q^{77} -3.12311i q^{78} +12.0000 q^{79} +1.00000 q^{81} +2.63068i q^{82} +2.56155i q^{83} -2.24621 q^{84} -12.0000 q^{86} +1.00000i q^{87} +3.50758i q^{88} -12.2462 q^{89} -10.2462 q^{91} +2.87689i q^{92} -4.00000i q^{93} -20.4924 q^{94} +2.43845 q^{96} -5.68466i q^{97} +30.0540i q^{98} +1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 10 q^{4} + 2 q^{6} - 4 q^{9} - 14 q^{11} + 32 q^{14} + 6 q^{16} - 4 q^{19} + 4 q^{21} - 18 q^{24} - 4 q^{26} - 4 q^{29} + 16 q^{31} - 28 q^{34} + 10 q^{36} + 8 q^{39} + 18 q^{41} + 52 q^{44} - 8 q^{46}+ \cdots + 14 q^{99}+O(q^{100}) \) 4 * q - 10 * q^4 + 2 * q^6 - 4 * q^9 - 14 * q^11 + 32 * q^14 + 6 * q^16 - 4 * q^19 + 4 * q^21 - 18 * q^24 - 4 * q^26 - 4 * q^29 + 16 * q^31 - 28 * q^34 + 10 * q^36 + 8 * q^39 + 18 * q^41 + 52 * q^44 - 8 * q^46 - 44 * q^49 - 12 * q^51 - 2 * q^54 - 16 * q^56 - 48 * q^59 + 20 * q^61 - 14 * q^64 - 24 * q^66 - 18 * q^69 - 28 * q^71 - 60 * q^74 - 24 * q^76 + 48 * q^79 + 4 * q^81 + 24 * q^84 - 48 * q^86 - 16 * q^89 - 8 * q^91 - 16 * q^94 + 18 * q^96 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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