LMFDB - Embedded newform 2475.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} +6.56155 q^{8} +O(q^{10})$	$q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} +6.56155 q^{8} -1.00000 q^{11} -0.438447 q^{13} +2.56155 q^{14} +7.68466 q^{16} +1.43845 q^{17} +3.00000 q^{19} -2.56155 q^{22} +6.56155 q^{23} -1.12311 q^{26} +4.56155 q^{28} +5.12311 q^{29} +4.68466 q^{31} +6.56155 q^{32} +3.68466 q^{34} -10.8078 q^{37} +7.68466 q^{38} -7.68466 q^{41} +4.68466 q^{43} -4.56155 q^{44} +16.8078 q^{46} +5.43845 q^{47} -6.00000 q^{49} -2.00000 q^{52} -1.12311 q^{53} +6.56155 q^{56} +13.1231 q^{58} +10.5616 q^{59} -7.56155 q^{61} +12.0000 q^{62} +1.43845 q^{64} -6.68466 q^{67} +6.56155 q^{68} +8.80776 q^{71} +16.2462 q^{73} -27.6847 q^{74} +13.6847 q^{76} -1.00000 q^{77} -15.6847 q^{79} -19.6847 q^{82} +12.0000 q^{86} -6.56155 q^{88} -17.1231 q^{89} -0.438447 q^{91} +29.9309 q^{92} +13.9309 q^{94} +2.12311 q^{97} -15.3693 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8}+O(q^{10}) $ 2 * q + q^2 + 5 * q^4 + 2 * q^7 + 9 * q^8	$ 2 q + q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8} - 2 q^{11} - 5 q^{13} + q^{14} + 3 q^{16} + 7 q^{17} + 6 q^{19} - q^{22} + 9 q^{23} + 6 q^{26} + 5 q^{28} + 2 q^{29} - 3 q^{31} + 9 q^{32} - 5 q^{34} - q^{37} + 3 q^{38} - 3 q^{41} - 3 q^{43} - 5 q^{44} + 13 q^{46} + 15 q^{47} - 12 q^{49} - 4 q^{52} + 6 q^{53} + 9 q^{56} + 18 q^{58} + 17 q^{59} - 11 q^{61} + 24 q^{62} + 7 q^{64} - q^{67} + 9 q^{68} - 3 q^{71} + 16 q^{73} - 43 q^{74} + 15 q^{76} - 2 q^{77} - 19 q^{79} - 27 q^{82} + 24 q^{86} - 9 q^{88} - 26 q^{89} - 5 q^{91} + 31 q^{92} - q^{94} - 4 q^{97} - 6 q^{98}+O(q^{100}) $ 2 * q + q^2 + 5 * q^4 + 2 * q^7 + 9 * q^8 - 2 * q^11 - 5 * q^13 + q^14 + 3 * q^16 + 7 * q^17 + 6 * q^19 - q^22 + 9 * q^23 + 6 * q^26 + 5 * q^28 + 2 * q^29 - 3 * q^31 + 9 * q^32 - 5 * q^34 - q^37 + 3 * q^38 - 3 * q^41 - 3 * q^43 - 5 * q^44 + 13 * q^46 + 15 * q^47 - 12 * q^49 - 4 * q^52 + 6 * q^53 + 9 * q^56 + 18 * q^58 + 17 * q^59 - 11 * q^61 + 24 * q^62 + 7 * q^64 - q^67 + 9 * q^68 - 3 * q^71 + 16 * q^73 - 43 * q^74 + 15 * q^76 - 2 * q^77 - 19 * q^79 - 27 * q^82 + 24 * q^86 - 9 * q^88 - 26 * q^89 - 5 * q^91 + 31 * q^92 - q^94 - 4 * q^97 - 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.56155	1.81129	0.905646	−	0.424035i	$-0.139387\pi$
$2$	2.56155	1.81129	0.905646	+	0.424035i	$0.139387\pi$
$3$	0	0
$3$	0	0
$4$	4.56155	2.28078
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	6.56155	2.31986
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.438447	−0.121603	−0.0608017	−	0.998150i	$-0.519366\pi$
$13$	−0.438447	−0.121603	−0.0608017	+	0.998150i	$0.519366\pi$
$14$	2.56155	0.684604
$15$	0	0
$16$	7.68466	1.92116
$17$	1.43845	0.348875	0.174437	−	0.984668i	$-0.444189\pi$
$17$	1.43845	0.348875	0.174437	+	0.984668i	$0.444189\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0	0
$22$	−2.56155	−0.546125
$23$	6.56155	1.36818	0.684089	−	0.729398i	$-0.260200\pi$
$23$	6.56155	1.36818	0.684089	+	0.729398i	$0.260200\pi$
$24$	0	0
$25$	0	0
$26$	−1.12311	−0.220259
$27$	0	0
$28$	4.56155	0.862052
$29$	5.12311	0.951337	0.475668	−	0.879625i	$-0.342206\pi$
$29$	5.12311	0.951337	0.475668	+	0.879625i	$0.342206\pi$
$30$	0	0
$31$	4.68466	0.841389	0.420695	−	0.907202i	$-0.361786\pi$
$31$	4.68466	0.841389	0.420695	+	0.907202i	$0.361786\pi$
$32$	6.56155	1.15993
$33$	0	0
$34$	3.68466	0.631914
$35$	0	0
$36$	0	0
$37$	−10.8078	−1.77679	−0.888393	−	0.459084i	$-0.848178\pi$
$37$	−10.8078	−1.77679	−0.888393	+	0.459084i	$0.848178\pi$
$38$	7.68466	1.24662
$39$	0	0
$40$	0	0
$41$	−7.68466	−1.20014	−0.600071	−	0.799947i	$-0.704861\pi$
$41$	−7.68466	−1.20014	−0.600071	+	0.799947i	$0.704861\pi$
$42$	0	0
$43$	4.68466	0.714404	0.357202	−	0.934027i	$-0.383731\pi$
$43$	4.68466	0.714404	0.357202	+	0.934027i	$0.383731\pi$
$44$	−4.56155	−0.687680
$45$	0	0
$46$	16.8078	2.47817
$47$	5.43845	0.793279	0.396640	−	0.917974i	$-0.370176\pi$
$47$	5.43845	0.793279	0.396640	+	0.917974i	$0.370176\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−1.12311	−0.154270	−0.0771352	−	0.997021i	$-0.524577\pi$
$53$	−1.12311	−0.154270	−0.0771352	+	0.997021i	$0.524577\pi$
$54$	0	0
$55$	0	0
$56$	6.56155	0.876824
$57$	0	0
$58$	13.1231	1.72315
$59$	10.5616	1.37500	0.687499	−	0.726186i	$-0.258709\pi$
$59$	10.5616	1.37500	0.687499	+	0.726186i	$0.258709\pi$
$60$	0	0
$61$	−7.56155	−0.968158	−0.484079	−	0.875024i	$-0.660845\pi$
$61$	−7.56155	−0.968158	−0.484079	+	0.875024i	$0.660845\pi$
$62$	12.0000	1.52400
$63$	0	0
$64$	1.43845	0.179806
$65$	0	0
$66$	0	0
$67$	−6.68466	−0.816661	−0.408331	−	0.912834i	$-0.633889\pi$
$67$	−6.68466	−0.816661	−0.408331	+	0.912834i	$0.633889\pi$
$68$	6.56155	0.795705
$69$	0	0
$70$	0	0
$71$	8.80776	1.04529	0.522645	−	0.852551i	$-0.324945\pi$
$71$	8.80776	1.04529	0.522645	+	0.852551i	$0.324945\pi$
$72$	0	0
$73$	16.2462	1.90148	0.950738	−	0.309997i	$-0.100328\pi$
$73$	16.2462	1.90148	0.950738	+	0.309997i	$0.100328\pi$
$74$	−27.6847	−3.21828
$75$	0	0
$76$	13.6847	1.56974
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−15.6847	−1.76466	−0.882331	−	0.470629i	$-0.844027\pi$
$79$	−15.6847	−1.76466	−0.882331	+	0.470629i	$0.844027\pi$
$80$	0	0
$81$	0	0
$82$	−19.6847	−2.17381
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	12.0000	1.29399
$87$	0	0
$88$	−6.56155	−0.699464
$89$	−17.1231	−1.81505	−0.907523	−	0.420003i	$-0.862029\pi$
$89$	−17.1231	−1.81505	−0.907523	+	0.420003i	$0.862029\pi$
$90$	0	0
$91$	−0.438447	−0.0459618
$92$	29.9309	3.12051
$93$	0	0
$94$	13.9309	1.43686
$95$	0	0
$96$	0	0
$97$	2.12311	0.215569	0.107784	−	0.994174i	$-0.465624\pi$
$97$	2.12311	0.215569	0.107784	+	0.994174i	$0.465624\pi$
$98$	−15.3693	−1.55254
$99$	0	0
$100$	0	0
$101$	1.43845	0.143131	0.0715654	−	0.997436i	$-0.477201\pi$
$101$	1.43845	0.143131	0.0715654	+	0.997436i	$0.477201\pi$
$102$	0	0
$103$	−9.12311	−0.898926	−0.449463	−	0.893299i	$-0.648385\pi$
$103$	−9.12311	−0.898926	−0.449463	+	0.893299i	$0.648385\pi$
$104$	−2.87689	−0.282103
$105$	0	0
$106$	−2.87689	−0.279429
$107$	−9.12311	−0.881964	−0.440982	−	0.897516i	$-0.645370\pi$
$107$	−9.12311	−0.881964	−0.440982	+	0.897516i	$0.645370\pi$
$108$	0	0
$109$	−7.80776	−0.747848	−0.373924	−	0.927459i	$-0.621988\pi$
$109$	−7.80776	−0.747848	−0.373924	+	0.927459i	$0.621988\pi$
$110$	0	0
$111$	0	0
$112$	7.68466	0.726132
$113$	−9.12311	−0.858230	−0.429115	−	0.903250i	$-0.641174\pi$
$113$	−9.12311	−0.858230	−0.429115	+	0.903250i	$0.641174\pi$
$114$	0	0
$115$	0	0
$116$	23.3693	2.16979
$117$	0	0
$118$	27.0540	2.49052
$119$	1.43845	0.131862
$120$	0	0
$121$	1.00000	0.0909091
$122$	−19.3693	−1.75362
$123$	0	0
$124$	21.3693	1.91902
$125$	0	0
$126$	0	0
$127$	−4.80776	−0.426620	−0.213310	−	0.976985i	$-0.568424\pi$
$127$	−4.80776	−0.426620	−0.213310	+	0.976985i	$0.568424\pi$
$128$	−9.43845	−0.834249
$129$	0	0
$130$	0	0
$131$	−16.4924	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$131$	−16.4924	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$132$	0	0
$133$	3.00000	0.260133
$134$	−17.1231	−1.47921
$135$	0	0
$136$	9.43845	0.809340
$137$	−3.36932	−0.287860	−0.143930	−	0.989588i	$-0.545974\pi$
$137$	−3.36932	−0.287860	−0.143930	+	0.989588i	$0.545974\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	0	0
$142$	22.5616	1.89332
$143$	0.438447	0.0366648
$144$	0	0
$145$	0	0
$146$	41.6155	3.44413
$147$	0	0
$148$	−49.3002	−4.05245
$149$	8.80776	0.721560	0.360780	−	0.932651i	$-0.382510\pi$
$149$	8.80776	0.721560	0.360780	+	0.932651i	$0.382510\pi$
$150$	0	0
$151$	−16.6847	−1.35778	−0.678889	−	0.734241i	$-0.737538\pi$
$151$	−16.6847	−1.35778	−0.678889	+	0.734241i	$0.737538\pi$
$152$	19.6847	1.59664
$153$	0	0
$154$	−2.56155	−0.206416
$155$	0	0
$156$	0	0
$157$	2.68466	0.214259	0.107130	−	0.994245i	$-0.465834\pi$
$157$	2.68466	0.214259	0.107130	+	0.994245i	$0.465834\pi$
$158$	−40.1771	−3.19632
$159$	0	0
$160$	0	0
$161$	6.56155	0.517123
$162$	0	0
$163$	−5.31534	−0.416330	−0.208165	−	0.978094i	$-0.566749\pi$
$163$	−5.31534	−0.416330	−0.208165	+	0.978094i	$0.566749\pi$
$164$	−35.0540	−2.73726
$165$	0	0
$166$	0	0
$167$	−15.3693	−1.18931	−0.594657	−	0.803980i	$-0.702712\pi$
$167$	−15.3693	−1.18931	−0.594657	+	0.803980i	$0.702712\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	0	0
$171$	0	0
$172$	21.3693	1.62940
$173$	13.4384	1.02171	0.510853	−	0.859668i	$-0.329330\pi$
$173$	13.4384	1.02171	0.510853	+	0.859668i	$0.329330\pi$
$174$	0	0
$175$	0	0
$176$	−7.68466	−0.579253
$177$	0	0
$178$	−43.8617	−3.28758
$179$	14.5616	1.08838	0.544191	−	0.838961i	$-0.316837\pi$
$179$	14.5616	1.08838	0.544191	+	0.838961i	$0.316837\pi$
$180$	0	0
$181$	11.8769	0.882803	0.441401	−	0.897310i	$-0.354482\pi$
$181$	11.8769	0.882803	0.441401	+	0.897310i	$0.354482\pi$
$182$	−1.12311	−0.0832501
$183$	0	0
$184$	43.0540	3.17398
$185$	0	0
$186$	0	0
$187$	−1.43845	−0.105190
$188$	24.8078	1.80929
$189$	0	0
$190$	0	0
$191$	−1.93087	−0.139713	−0.0698564	−	0.997557i	$-0.522254\pi$
$191$	−1.93087	−0.139713	−0.0698564	+	0.997557i	$0.522254\pi$
$192$	0	0
$193$	17.5616	1.26411	0.632054	−	0.774924i	$-0.282212\pi$
$193$	17.5616	1.26411	0.632054	+	0.774924i	$0.282212\pi$
$194$	5.43845	0.390458
$195$	0	0
$196$	−27.3693	−1.95495
$197$	17.9309	1.27752	0.638761	−	0.769405i	$-0.279447\pi$
$197$	17.9309	1.27752	0.638761	+	0.769405i	$0.279447\pi$
$198$	0	0
$199$	−13.8078	−0.978806	−0.489403	−	0.872058i	$-0.662785\pi$
$199$	−13.8078	−0.978806	−0.489403	+	0.872058i	$0.662785\pi$
$200$	0	0
$201$	0	0
$202$	3.68466	0.259252
$203$	5.12311	0.359572
$204$	0	0
$205$	0	0
$206$	−23.3693	−1.62822
$207$	0	0
$208$	−3.36932	−0.233620
$209$	−3.00000	−0.207514
$210$	0	0
$211$	3.31534	0.228238	0.114119	−	0.993467i	$-0.463596\pi$
$211$	3.31534	0.228238	0.114119	+	0.993467i	$0.463596\pi$
$212$	−5.12311	−0.351856
$213$	0	0
$214$	−23.3693	−1.59749
$215$	0	0
$216$	0	0
$217$	4.68466	0.318015
$218$	−20.0000	−1.35457
$219$	0	0
$220$	0	0
$221$	−0.630683	−0.0424243
$222$	0	0
$223$	21.8078	1.46036	0.730178	−	0.683257i	$-0.239437\pi$
$223$	21.8078	1.46036	0.730178	+	0.683257i	$0.239437\pi$
$224$	6.56155	0.438412
$225$	0	0
$226$	−23.3693	−1.55450
$227$	20.4924	1.36013	0.680065	−	0.733152i	$-0.261952\pi$
$227$	20.4924	1.36013	0.680065	+	0.733152i	$0.261952\pi$
$228$	0	0
$229$	−8.12311	−0.536790	−0.268395	−	0.963309i	$-0.586493\pi$
$229$	−8.12311	−0.536790	−0.268395	+	0.963309i	$0.586493\pi$
$230$	0	0
$231$	0	0
$232$	33.6155	2.20697
$233$	24.8078	1.62521	0.812605	−	0.582814i	$-0.198049\pi$
$233$	24.8078	1.62521	0.812605	+	0.582814i	$0.198049\pi$
$234$	0	0
$235$	0	0
$236$	48.1771	3.13606
$237$	0	0
$238$	3.68466	0.238841
$239$	−16.4924	−1.06681	−0.533403	−	0.845861i	$-0.679087\pi$
$239$	−16.4924	−1.06681	−0.533403	+	0.845861i	$0.679087\pi$
$240$	0	0
$241$	−14.0540	−0.905296	−0.452648	−	0.891689i	$-0.649521\pi$
$241$	−14.0540	−0.905296	−0.452648	+	0.891689i	$0.649521\pi$
$242$	2.56155	0.164663
$243$	0	0
$244$	−34.4924	−2.20815
$245$	0	0
$246$	0	0
$247$	−1.31534	−0.0836932
$248$	30.7386	1.95191
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	−6.56155	−0.412521
$254$	−12.3153	−0.772733
$255$	0	0
$256$	−27.0540	−1.69087
$257$	18.2462	1.13817	0.569084	−	0.822280i	$-0.307298\pi$
$257$	18.2462	1.13817	0.569084	+	0.822280i	$0.307298\pi$
$258$	0	0
$259$	−10.8078	−0.671562
$260$	0	0
$261$	0	0
$262$	−42.2462	−2.60998
$263$	−3.36932	−0.207761	−0.103880	−	0.994590i	$-0.533126\pi$
$263$	−3.36932	−0.207761	−0.103880	+	0.994590i	$0.533126\pi$
$264$	0	0
$265$	0	0
$266$	7.68466	0.471177
$267$	0	0
$268$	−30.4924	−1.86262
$269$	28.4924	1.73721	0.868607	−	0.495502i	$-0.165016\pi$
$269$	28.4924	1.73721	0.868607	+	0.495502i	$0.165016\pi$
$270$	0	0
$271$	−28.1771	−1.71164	−0.855818	−	0.517277i	$-0.826946\pi$
$271$	−28.1771	−1.71164	−0.855818	+	0.517277i	$0.826946\pi$
$272$	11.0540	0.670246
$273$	0	0
$274$	−8.63068	−0.521399
$275$	0	0
$276$	0	0
$277$	−23.1771	−1.39258	−0.696288	−	0.717763i	$-0.745166\pi$
$277$	−23.1771	−1.39258	−0.696288	+	0.717763i	$0.745166\pi$
$278$	20.4924	1.22905
$279$	0	0
$280$	0	0
$281$	−3.19224	−0.190433	−0.0952164	−	0.995457i	$-0.530354\pi$
$281$	−3.19224	−0.190433	−0.0952164	+	0.995457i	$0.530354\pi$
$282$	0	0
$283$	32.1231	1.90952	0.954760	−	0.297377i	$-0.0961117\pi$
$283$	32.1231	1.90952	0.954760	+	0.297377i	$0.0961117\pi$
$284$	40.1771	2.38407
$285$	0	0
$286$	1.12311	0.0664106
$287$	−7.68466	−0.453611
$288$	0	0
$289$	−14.9309	−0.878286
$290$	0	0
$291$	0	0
$292$	74.1080	4.33684
$293$	2.06913	0.120880	0.0604399	−	0.998172i	$-0.480750\pi$
$293$	2.06913	0.120880	0.0604399	+	0.998172i	$0.480750\pi$
$294$	0	0
$295$	0	0
$296$	−70.9157	−4.12189
$297$	0	0
$298$	22.5616	1.30696
$299$	−2.87689	−0.166375
$300$	0	0
$301$	4.68466	0.270019
$302$	−42.7386	−2.45933
$303$	0	0
$304$	23.0540	1.32224
$305$	0	0
$306$	0	0
$307$	−22.9309	−1.30873	−0.654367	−	0.756177i	$-0.727065\pi$
$307$	−22.9309	−1.30873	−0.654367	+	0.756177i	$0.727065\pi$
$308$	−4.56155	−0.259919
$309$	0	0
$310$	0	0
$311$	−13.7538	−0.779906	−0.389953	−	0.920835i	$-0.627509\pi$
$311$	−13.7538	−0.779906	−0.389953	+	0.920835i	$0.627509\pi$
$312$	0	0
$313$	−31.2462	−1.76614	−0.883070	−	0.469241i	$-0.844528\pi$
$313$	−31.2462	−1.76614	−0.883070	+	0.469241i	$0.844528\pi$
$314$	6.87689	0.388086
$315$	0	0
$316$	−71.5464	−4.02480
$317$	6.87689	0.386245	0.193122	−	0.981175i	$-0.438139\pi$
$317$	6.87689	0.386245	0.193122	+	0.981175i	$0.438139\pi$
$318$	0	0
$319$	−5.12311	−0.286839
$320$	0	0
$321$	0	0
$322$	16.8078	0.936660
$323$	4.31534	0.240112
$324$	0	0
$325$	0	0
$326$	−13.6155	−0.754094
$327$	0	0
$328$	−50.4233	−2.78416
$329$	5.43845	0.299831
$330$	0	0
$331$	−10.2462	−0.563183	−0.281591	−	0.959534i	$-0.590862\pi$
$331$	−10.2462	−0.563183	−0.281591	+	0.959534i	$0.590862\pi$
$332$	0	0
$333$	0	0
$334$	−39.3693	−2.15419
$335$	0	0
$336$	0	0
$337$	−10.6847	−0.582030	−0.291015	−	0.956718i	$-0.593993\pi$
$337$	−10.6847	−0.582030	−0.291015	+	0.956718i	$0.593993\pi$
$338$	−32.8078	−1.78451
$339$	0	0
$340$	0	0
$341$	−4.68466	−0.253688
$342$	0	0
$343$	−13.0000	−0.701934
$344$	30.7386	1.65732
$345$	0	0
$346$	34.4233	1.85061
$347$	−27.3693	−1.46926	−0.734631	−	0.678467i	$-0.762645\pi$
$347$	−27.3693	−1.46926	−0.734631	+	0.678467i	$0.762645\pi$
$348$	0	0
$349$	6.63068	0.354932	0.177466	−	0.984127i	$-0.443210\pi$
$349$	6.63068	0.354932	0.177466	+	0.984127i	$0.443210\pi$
$350$	0	0
$351$	0	0
$352$	−6.56155	−0.349732
$353$	27.8617	1.48293	0.741465	−	0.670991i	$-0.234131\pi$
$353$	27.8617	1.48293	0.741465	+	0.670991i	$0.234131\pi$
$354$	0	0
$355$	0	0
$356$	−78.1080	−4.13971
$357$	0	0
$358$	37.3002	1.97138
$359$	26.2462	1.38522	0.692611	−	0.721311i	$-0.256460\pi$
$359$	26.2462	1.38522	0.692611	+	0.721311i	$0.256460\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	30.4233	1.59901
$363$	0	0
$364$	−2.00000	−0.104828
$365$	0	0
$366$	0	0
$367$	14.0540	0.733612	0.366806	−	0.930298i	$-0.380451\pi$
$367$	14.0540	0.733612	0.366806	+	0.930298i	$0.380451\pi$
$368$	50.4233	2.62850
$369$	0	0
$370$	0	0
$371$	−1.12311	−0.0583087
$372$	0	0
$373$	−13.8078	−0.714939	−0.357469	−	0.933925i	$-0.616360\pi$
$373$	−13.8078	−0.714939	−0.357469	+	0.933925i	$0.616360\pi$
$374$	−3.68466	−0.190529
$375$	0	0
$376$	35.6847	1.84030
$377$	−2.24621	−0.115686
$378$	0	0
$379$	−24.3002	−1.24822	−0.624108	−	0.781338i	$-0.714538\pi$
$379$	−24.3002	−1.24822	−0.624108	+	0.781338i	$0.714538\pi$
$380$	0	0
$381$	0	0
$382$	−4.94602	−0.253061
$383$	22.2462	1.13673	0.568364	−	0.822777i	$-0.307576\pi$
$383$	22.2462	1.13673	0.568364	+	0.822777i	$0.307576\pi$
$384$	0	0
$385$	0	0
$386$	44.9848	2.28967
$387$	0	0
$388$	9.68466	0.491664
$389$	−23.8617	−1.20984	−0.604919	−	0.796287i	$-0.706795\pi$
$389$	−23.8617	−1.20984	−0.604919	+	0.796287i	$0.706795\pi$
$390$	0	0
$391$	9.43845	0.477323
$392$	−39.3693	−1.98845
$393$	0	0
$394$	45.9309	2.31396
$395$	0	0
$396$	0	0
$397$	28.4384	1.42728	0.713642	−	0.700510i	$-0.247044\pi$
$397$	28.4384	1.42728	0.713642	+	0.700510i	$0.247044\pi$
$398$	−35.3693	−1.77290
$399$	0	0
$400$	0	0
$401$	5.61553	0.280426	0.140213	−	0.990121i	$-0.455221\pi$
$401$	5.61553	0.280426	0.140213	+	0.990121i	$0.455221\pi$
$402$	0	0
$403$	−2.05398	−0.102316
$404$	6.56155	0.326449
$405$	0	0
$406$	13.1231	0.651289
$407$	10.8078	0.535721
$408$	0	0
$409$	8.05398	0.398243	0.199122	−	0.979975i	$-0.436191\pi$
$409$	8.05398	0.398243	0.199122	+	0.979975i	$0.436191\pi$
$410$	0	0
$411$	0	0
$412$	−41.6155	−2.05025
$413$	10.5616	0.519700
$414$	0	0
$415$	0	0
$416$	−2.87689	−0.141051
$417$	0	0
$418$	−7.68466	−0.375869
$419$	25.4384	1.24275	0.621375	−	0.783514i	$-0.286574\pi$
$419$	25.4384	1.24275	0.621375	+	0.783514i	$0.286574\pi$
$420$	0	0
$421$	−17.0540	−0.831160	−0.415580	−	0.909557i	$-0.636421\pi$
$421$	−17.0540	−0.831160	−0.415580	+	0.909557i	$0.636421\pi$
$422$	8.49242	0.413405
$423$	0	0
$424$	−7.36932	−0.357886
$425$	0	0
$426$	0	0
$427$	−7.56155	−0.365929
$428$	−41.6155	−2.01156
$429$	0	0
$430$	0	0
$431$	−21.7538	−1.04784	−0.523922	−	0.851767i	$-0.675532\pi$
$431$	−21.7538	−1.04784	−0.523922	+	0.851767i	$0.675532\pi$
$432$	0	0
$433$	16.9309	0.813646	0.406823	−	0.913507i	$-0.366637\pi$
$433$	16.9309	0.813646	0.406823	+	0.913507i	$0.366637\pi$
$434$	12.0000	0.576018
$435$	0	0
$436$	−35.6155	−1.70567
$437$	19.6847	0.941645
$438$	0	0
$439$	11.7386	0.560254	0.280127	−	0.959963i	$-0.409623\pi$
$439$	11.7386	0.560254	0.280127	+	0.959963i	$0.409623\pi$
$440$	0	0
$441$	0	0
$442$	−1.61553	−0.0768428
$443$	25.3002	1.20205	0.601024	−	0.799231i	$-0.294760\pi$
$443$	25.3002	1.20205	0.601024	+	0.799231i	$0.294760\pi$
$444$	0	0
$445$	0	0
$446$	55.8617	2.64513
$447$	0	0
$448$	1.43845	0.0679602
$449$	−28.4924	−1.34464	−0.672320	−	0.740260i	$-0.734702\pi$
$449$	−28.4924	−1.34464	−0.672320	+	0.740260i	$0.734702\pi$
$450$	0	0
$451$	7.68466	0.361856
$452$	−41.6155	−1.95743
$453$	0	0
$454$	52.4924	2.46359
$455$	0	0
$456$	0	0
$457$	13.3693	0.625390	0.312695	−	0.949854i	$-0.398768\pi$
$457$	13.3693	0.625390	0.312695	+	0.949854i	$0.398768\pi$
$458$	−20.8078	−0.972283
$459$	0	0
$460$	0	0
$461$	−15.8617	−0.738755	−0.369377	−	0.929279i	$-0.620429\pi$
$461$	−15.8617	−0.738755	−0.369377	+	0.929279i	$0.620429\pi$
$462$	0	0
$463$	−1.75379	−0.0815055	−0.0407527	−	0.999169i	$-0.512976\pi$
$463$	−1.75379	−0.0815055	−0.0407527	+	0.999169i	$0.512976\pi$
$464$	39.3693	1.82767
$465$	0	0
$466$	63.5464	2.94373
$467$	2.24621	0.103942	0.0519711	−	0.998649i	$-0.483450\pi$
$467$	2.24621	0.103942	0.0519711	+	0.998649i	$0.483450\pi$
$468$	0	0
$469$	−6.68466	−0.308669
$470$	0	0
$471$	0	0
$472$	69.3002	3.18980
$473$	−4.68466	−0.215401
$474$	0	0
$475$	0	0
$476$	6.56155	0.300748
$477$	0	0
$478$	−42.2462	−1.93230
$479$	3.36932	0.153948	0.0769740	−	0.997033i	$-0.475474\pi$
$479$	3.36932	0.153948	0.0769740	+	0.997033i	$0.475474\pi$
$480$	0	0
$481$	4.73863	0.216063
$482$	−36.0000	−1.63976
$483$	0	0
$484$	4.56155	0.207343
$485$	0	0
$486$	0	0
$487$	−12.0540	−0.546218	−0.273109	−	0.961983i	$-0.588052\pi$
$487$	−12.0540	−0.546218	−0.273109	+	0.961983i	$0.588052\pi$
$488$	−49.6155	−2.24599
$489$	0	0
$490$	0	0
$491$	16.4924	0.744293	0.372146	−	0.928174i	$-0.378622\pi$
$491$	16.4924	0.744293	0.372146	+	0.928174i	$0.378622\pi$
$492$	0	0
$493$	7.36932	0.331897
$494$	−3.36932	−0.151593
$495$	0	0
$496$	36.0000	1.61645
$497$	8.80776	0.395082
$498$	0	0
$499$	3.80776	0.170459	0.0852295	−	0.996361i	$-0.472838\pi$
$499$	3.80776	0.170459	0.0852295	+	0.996361i	$0.472838\pi$
$500$	0	0
$501$	0	0
$502$	−30.7386	−1.37193
$503$	−16.4924	−0.735361	−0.367680	−	0.929952i	$-0.619848\pi$
$503$	−16.4924	−0.735361	−0.367680	+	0.929952i	$0.619848\pi$
$504$	0	0
$505$	0	0
$506$	−16.8078	−0.747196
$507$	0	0
$508$	−21.9309	−0.973025
$509$	−17.1231	−0.758968	−0.379484	−	0.925198i	$-0.623899\pi$
$509$	−17.1231	−0.758968	−0.379484	+	0.925198i	$0.623899\pi$
$510$	0	0
$511$	16.2462	0.718690
$512$	−50.4233	−2.22842
$513$	0	0
$514$	46.7386	2.06155
$515$	0	0
$516$	0	0
$517$	−5.43845	−0.239183
$518$	−27.6847	−1.21639
$519$	0	0
$520$	0	0
$521$	−25.6155	−1.12224	−0.561118	−	0.827736i	$-0.689629\pi$
$521$	−25.6155	−1.12224	−0.561118	+	0.827736i	$0.689629\pi$
$522$	0	0
$523$	9.24621	0.404309	0.202154	−	0.979354i	$-0.435206\pi$
$523$	9.24621	0.404309	0.202154	+	0.979354i	$0.435206\pi$
$524$	−75.2311	−3.28648
$525$	0	0
$526$	−8.63068	−0.376316
$527$	6.73863	0.293539
$528$	0	0
$529$	20.0540	0.871912
$530$	0	0
$531$	0	0
$532$	13.6847	0.593305
$533$	3.36932	0.145941
$534$	0	0
$535$	0	0
$536$	−43.8617	−1.89454
$537$	0	0
$538$	72.9848	3.14660
$539$	6.00000	0.258438
$540$	0	0
$541$	−7.31534	−0.314511	−0.157256	−	0.987558i	$-0.550265\pi$
$541$	−7.31534	−0.314511	−0.157256	+	0.987558i	$0.550265\pi$
$542$	−72.1771	−3.10027
$543$	0	0
$544$	9.43845	0.404670
$545$	0	0
$546$	0	0
$547$	3.68466	0.157545	0.0787723	−	0.996893i	$-0.474900\pi$
$547$	3.68466	0.157545	0.0787723	+	0.996893i	$0.474900\pi$
$548$	−15.3693	−0.656545
$549$	0	0
$550$	0	0
$551$	15.3693	0.654755
$552$	0	0
$553$	−15.6847	−0.666980
$554$	−59.3693	−2.52236
$555$	0	0
$556$	36.4924	1.54762
$557$	1.12311	0.0475875	0.0237938	−	0.999717i	$-0.492425\pi$
$557$	1.12311	0.0475875	0.0237938	+	0.999717i	$0.492425\pi$
$558$	0	0
$559$	−2.05398	−0.0868739
$560$	0	0
$561$	0	0
$562$	−8.17708	−0.344929
$563$	−10.8769	−0.458406	−0.229203	−	0.973379i	$-0.573612\pi$
$563$	−10.8769	−0.458406	−0.229203	+	0.973379i	$0.573612\pi$
$564$	0	0
$565$	0	0
$566$	82.2850	3.45870
$567$	0	0
$568$	57.7926	2.42492
$569$	37.9309	1.59014	0.795072	−	0.606515i	$-0.207433\pi$
$569$	37.9309	1.59014	0.795072	+	0.606515i	$0.207433\pi$
$570$	0	0
$571$	38.0540	1.59251	0.796255	−	0.604962i	$-0.206812\pi$
$571$	38.0540	1.59251	0.796255	+	0.604962i	$0.206812\pi$
$572$	2.00000	0.0836242
$573$	0	0
$574$	−19.6847	−0.821622
$575$	0	0
$576$	0	0
$577$	−28.3693	−1.18103	−0.590515	−	0.807027i	$-0.701075\pi$
$577$	−28.3693	−1.18103	−0.590515	+	0.807027i	$0.701075\pi$
$578$	−38.2462	−1.59083
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	1.12311	0.0465143
$584$	106.600	4.41115
$585$	0	0
$586$	5.30019	0.218949
$587$	−27.1922	−1.12234	−0.561172	−	0.827699i	$-0.689649\pi$
$587$	−27.1922	−1.12234	−0.561172	+	0.827699i	$0.689649\pi$
$588$	0	0
$589$	14.0540	0.579084
$590$	0	0
$591$	0	0
$592$	−83.0540	−3.41350
$593$	−34.1080	−1.40065	−0.700323	−	0.713826i	$-0.746961\pi$
$593$	−34.1080	−1.40065	−0.700323	+	0.713826i	$0.746961\pi$
$594$	0	0
$595$	0	0
$596$	40.1771	1.64572
$597$	0	0
$598$	−7.36932	−0.301354
$599$	24.1771	0.987849	0.493924	−	0.869505i	$-0.335562\pi$
$599$	24.1771	0.987849	0.493924	+	0.869505i	$0.335562\pi$
$600$	0	0
$601$	37.4233	1.52653	0.763264	−	0.646087i	$-0.223596\pi$
$601$	37.4233	1.52653	0.763264	+	0.646087i	$0.223596\pi$
$602$	12.0000	0.489083
$603$	0	0
$604$	−76.1080	−3.09679
$605$	0	0
$606$	0	0
$607$	6.24621	0.253526	0.126763	−	0.991933i	$-0.459541\pi$
$607$	6.24621	0.253526	0.126763	+	0.991933i	$0.459541\pi$
$608$	19.6847	0.798318
$609$	0	0
$610$	0	0
$611$	−2.38447	−0.0964654
$612$	0	0
$613$	28.1080	1.13527	0.567635	−	0.823281i	$-0.307859\pi$
$613$	28.1080	1.13527	0.567635	+	0.823281i	$0.307859\pi$
$614$	−58.7386	−2.37050
$615$	0	0
$616$	−6.56155	−0.264372
$617$	21.7538	0.875775	0.437887	−	0.899030i	$-0.355727\pi$
$617$	21.7538	0.875775	0.437887	+	0.899030i	$0.355727\pi$
$618$	0	0
$619$	7.17708	0.288471	0.144236	−	0.989543i	$-0.453928\pi$
$619$	7.17708	0.288471	0.144236	+	0.989543i	$0.453928\pi$
$620$	0	0
$621$	0	0
$622$	−35.2311	−1.41264
$623$	−17.1231	−0.686023
$624$	0	0
$625$	0	0
$626$	−80.0388	−3.19899
$627$	0	0
$628$	12.2462	0.488677
$629$	−15.5464	−0.619875
$630$	0	0
$631$	−22.6847	−0.903062	−0.451531	−	0.892255i	$-0.649122\pi$
$631$	−22.6847	−0.903062	−0.451531	+	0.892255i	$0.649122\pi$
$632$	−102.916	−4.09377
$633$	0	0
$634$	17.6155	0.699602
$635$	0	0
$636$	0	0
$637$	2.63068	0.104231
$638$	−13.1231	−0.519549
$639$	0	0
$640$	0	0
$641$	42.7386	1.68807	0.844037	−	0.536285i	$-0.180173\pi$
$641$	42.7386	1.68807	0.844037	+	0.536285i	$0.180173\pi$
$642$	0	0
$643$	15.3693	0.606107	0.303053	−	0.952974i	$-0.401994\pi$
$643$	15.3693	0.606107	0.303053	+	0.952974i	$0.401994\pi$
$644$	29.9309	1.17944
$645$	0	0
$646$	11.0540	0.434913
$647$	−40.8078	−1.60432	−0.802159	−	0.597110i	$-0.796316\pi$
$647$	−40.8078	−1.60432	−0.802159	+	0.597110i	$0.796316\pi$
$648$	0	0
$649$	−10.5616	−0.414577
$650$	0	0
$651$	0	0
$652$	−24.2462	−0.949555
$653$	7.50758	0.293794	0.146897	−	0.989152i	$-0.453071\pi$
$653$	7.50758	0.293794	0.146897	+	0.989152i	$0.453071\pi$
$654$	0	0
$655$	0	0
$656$	−59.0540	−2.30567
$657$	0	0
$658$	13.9309	0.543082
$659$	8.63068	0.336204	0.168102	−	0.985770i	$-0.446236\pi$
$659$	8.63068	0.336204	0.168102	+	0.985770i	$0.446236\pi$
$660$	0	0
$661$	6.80776	0.264791	0.132396	−	0.991197i	$-0.457733\pi$
$661$	6.80776	0.264791	0.132396	+	0.991197i	$0.457733\pi$
$662$	−26.2462	−1.02009
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	33.6155	1.30160
$668$	−70.1080	−2.71256
$669$	0	0
$670$	0	0
$671$	7.56155	0.291911
$672$	0	0
$673$	10.6307	0.409783	0.204891	−	0.978785i	$-0.434316\pi$
$673$	10.6307	0.409783	0.204891	+	0.978785i	$0.434316\pi$
$674$	−27.3693	−1.05423
$675$	0	0
$676$	−58.4233	−2.24705
$677$	29.1231	1.11929	0.559646	−	0.828732i	$-0.310937\pi$
$677$	29.1231	1.11929	0.559646	+	0.828732i	$0.310937\pi$
$678$	0	0
$679$	2.12311	0.0814773
$680$	0	0
$681$	0	0
$682$	−12.0000	−0.459504
$683$	−20.8078	−0.796187	−0.398093	−	0.917345i	$-0.630328\pi$
$683$	−20.8078	−0.796187	−0.398093	+	0.917345i	$0.630328\pi$
$684$	0	0
$685$	0	0
$686$	−33.3002	−1.27141
$687$	0	0
$688$	36.0000	1.37249
$689$	0.492423	0.0187598
$690$	0	0
$691$	12.4924	0.475234	0.237617	−	0.971359i	$-0.423634\pi$
$691$	12.4924	0.475234	0.237617	+	0.971359i	$0.423634\pi$
$692$	61.3002	2.33028
$693$	0	0
$694$	−70.1080	−2.66126
$695$	0	0
$696$	0	0
$697$	−11.0540	−0.418699
$698$	16.9848	0.642886
$699$	0	0
$700$	0	0
$701$	−1.93087	−0.0729279	−0.0364640	−	0.999335i	$-0.511609\pi$
$701$	−1.93087	−0.0729279	−0.0364640	+	0.999335i	$0.511609\pi$
$702$	0	0
$703$	−32.4233	−1.22287
$704$	−1.43845	−0.0542135
$705$	0	0
$706$	71.3693	2.68602
$707$	1.43845	0.0540984
$708$	0	0
$709$	6.12311	0.229958	0.114979	−	0.993368i	$-0.463320\pi$
$709$	6.12311	0.229958	0.114979	+	0.993368i	$0.463320\pi$
$710$	0	0
$711$	0	0
$712$	−112.354	−4.21065
$713$	30.7386	1.15117
$714$	0	0
$715$	0	0
$716$	66.4233	2.48235
$717$	0	0
$718$	67.2311	2.50904
$719$	−32.9848	−1.23013	−0.615064	−	0.788478i	$-0.710870\pi$
$719$	−32.9848	−1.23013	−0.615064	+	0.788478i	$0.710870\pi$
$720$	0	0
$721$	−9.12311	−0.339762
$722$	−25.6155	−0.953311
$723$	0	0
$724$	54.1771	2.01348
$725$	0	0
$726$	0	0
$727$	−30.0540	−1.11464	−0.557320	−	0.830298i	$-0.688170\pi$
$727$	−30.0540	−1.11464	−0.557320	+	0.830298i	$0.688170\pi$
$728$	−2.87689	−0.106625
$729$	0	0
$730$	0	0
$731$	6.73863	0.249237
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	36.0000	1.32878
$735$	0	0
$736$	43.0540	1.58699
$737$	6.68466	0.246233
$738$	0	0
$739$	−11.6847	−0.429827	−0.214914	−	0.976633i	$-0.568947\pi$
$739$	−11.6847	−0.429827	−0.214914	+	0.976633i	$0.568947\pi$
$740$	0	0
$741$	0	0
$742$	−2.87689	−0.105614
$743$	47.8617	1.75588	0.877938	−	0.478773i	$-0.158918\pi$
$743$	47.8617	1.75588	0.877938	+	0.478773i	$0.158918\pi$
$744$	0	0
$745$	0	0
$746$	−35.3693	−1.29496
$747$	0	0
$748$	−6.56155	−0.239914
$749$	−9.12311	−0.333351
$750$	0	0
$751$	26.7386	0.975707	0.487853	−	0.872926i	$-0.337780\pi$
$751$	26.7386	0.975707	0.487853	+	0.872926i	$0.337780\pi$
$752$	41.7926	1.52402
$753$	0	0
$754$	−5.75379	−0.209541
$755$	0	0
$756$	0	0
$757$	6.30019	0.228984	0.114492	−	0.993424i	$-0.463476\pi$
$757$	6.30019	0.228984	0.114492	+	0.993424i	$0.463476\pi$
$758$	−62.2462	−2.26088
$759$	0	0
$760$	0	0
$761$	20.6307	0.747862	0.373931	−	0.927457i	$-0.378010\pi$
$761$	20.6307	0.747862	0.373931	+	0.927457i	$0.378010\pi$
$762$	0	0
$763$	−7.80776	−0.282660
$764$	−8.80776	−0.318654
$765$	0	0
$766$	56.9848	2.05895
$767$	−4.63068	−0.167204
$768$	0	0
$769$	6.05398	0.218312	0.109156	−	0.994025i	$-0.465185\pi$
$769$	6.05398	0.218312	0.109156	+	0.994025i	$0.465185\pi$
$770$	0	0
$771$	0	0
$772$	80.1080	2.88315
$773$	48.4924	1.74415	0.872076	−	0.489371i	$-0.162774\pi$
$773$	48.4924	1.74415	0.872076	+	0.489371i	$0.162774\pi$
$774$	0	0
$775$	0	0
$776$	13.9309	0.500089
$777$	0	0
$778$	−61.1231	−2.19137
$779$	−23.0540	−0.825994
$780$	0	0
$781$	−8.80776	−0.315167
$782$	24.1771	0.864571
$783$	0	0
$784$	−46.1080	−1.64671
$785$	0	0
$786$	0	0
$787$	4.26137	0.151901	0.0759507	−	0.997112i	$-0.475801\pi$
$787$	4.26137	0.151901	0.0759507	+	0.997112i	$0.475801\pi$
$788$	81.7926	2.91374
$789$	0	0
$790$	0	0
$791$	−9.12311	−0.324380
$792$	0	0
$793$	3.31534	0.117731
$794$	72.8466	2.58523
$795$	0	0
$796$	−62.9848	−2.23244
$797$	42.8769	1.51878	0.759389	−	0.650637i	$-0.225498\pi$
$797$	42.8769	1.51878	0.759389	+	0.650637i	$0.225498\pi$
$798$	0	0
$799$	7.82292	0.276755
$800$	0	0
$801$	0	0
$802$	14.3845	0.507933
$803$	−16.2462	−0.573316
$804$	0	0
$805$	0	0
$806$	−5.26137	−0.185324
$807$	0	0
$808$	9.43845	0.332043
$809$	19.6847	0.692076	0.346038	−	0.938221i	$-0.387527\pi$
$809$	19.6847	0.692076	0.346038	+	0.938221i	$0.387527\pi$
$810$	0	0
$811$	−1.00000	−0.0351147	−0.0175574	−	0.999846i	$-0.505589\pi$
$811$	−1.00000	−0.0351147	−0.0175574	+	0.999846i	$0.505589\pi$
$812$	23.3693	0.820102
$813$	0	0
$814$	27.6847	0.970347
$815$	0	0
$816$	0	0
$817$	14.0540	0.491686
$818$	20.6307	0.721335
$819$	0	0
$820$	0	0
$821$	−25.1231	−0.876802	−0.438401	−	0.898779i	$-0.644455\pi$
$821$	−25.1231	−0.876802	−0.438401	+	0.898779i	$0.644455\pi$
$822$	0	0
$823$	0.0539753	0.00188146	0.000940731	−	1.00000i	$-0.499701\pi$
$823$	0.0539753	0.00188146	0.000940731		1.00000i	$0.499701\pi$
$824$	−59.8617	−2.08538
$825$	0	0
$826$	27.0540	0.941328
$827$	9.75379	0.339172	0.169586	−	0.985515i	$-0.445757\pi$
$827$	9.75379	0.339172	0.169586	+	0.985515i	$0.445757\pi$
$828$	0	0
$829$	48.7386	1.69276	0.846381	−	0.532577i	$-0.178776\pi$
$829$	48.7386	1.69276	0.846381	+	0.532577i	$0.178776\pi$
$830$	0	0
$831$	0	0
$832$	−0.630683	−0.0218650
$833$	−8.63068	−0.299035
$834$	0	0
$835$	0	0
$836$	−13.6847	−0.473294
$837$	0	0
$838$	65.1619	2.25098
$839$	20.4924	0.707477	0.353738	−	0.935344i	$-0.384910\pi$
$839$	20.4924	0.707477	0.353738	+	0.935344i	$0.384910\pi$
$840$	0	0
$841$	−2.75379	−0.0949582
$842$	−43.6847	−1.50547
$843$	0	0
$844$	15.1231	0.520559
$845$	0	0
$846$	0	0
$847$	1.00000	0.0343604
$848$	−8.63068	−0.296379
$849$	0	0
$850$	0	0
$851$	−70.9157	−2.43096
$852$	0	0
$853$	−35.4233	−1.21287	−0.606435	−	0.795133i	$-0.707401\pi$
$853$	−35.4233	−1.21287	−0.606435	+	0.795133i	$0.707401\pi$
$854$	−19.3693	−0.662804
$855$	0	0
$856$	−59.8617	−2.04603
$857$	5.43845	0.185774	0.0928869	−	0.995677i	$-0.470390\pi$
$857$	5.43845	0.185774	0.0928869	+	0.995677i	$0.470390\pi$
$858$	0	0
$859$	56.3542	1.92278	0.961390	−	0.275191i	$-0.0887411\pi$
$859$	56.3542	1.92278	0.961390	+	0.275191i	$0.0887411\pi$
$860$	0	0
$861$	0	0
$862$	−55.7235	−1.89795
$863$	−2.24621	−0.0764619	−0.0382310	−	0.999269i	$-0.512172\pi$
$863$	−2.24621	−0.0764619	−0.0382310	+	0.999269i	$0.512172\pi$
$864$	0	0
$865$	0	0
$866$	43.3693	1.47375
$867$	0	0
$868$	21.3693	0.725322
$869$	15.6847	0.532066
$870$	0	0
$871$	2.93087	0.0993087
$872$	−51.2311	−1.73490
$873$	0	0
$874$	50.4233	1.70559
$875$	0	0
$876$	0	0
$877$	14.0540	0.474569	0.237285	−	0.971440i	$-0.423743\pi$
$877$	14.0540	0.474569	0.237285	+	0.971440i	$0.423743\pi$
$878$	30.0691	1.01478
$879$	0	0
$880$	0	0
$881$	−9.75379	−0.328613	−0.164307	−	0.986409i	$-0.552539\pi$
$881$	−9.75379	−0.328613	−0.164307	+	0.986409i	$0.552539\pi$
$882$	0	0
$883$	−44.9309	−1.51204	−0.756022	−	0.654546i	$-0.772860\pi$
$883$	−44.9309	−1.51204	−0.756022	+	0.654546i	$0.772860\pi$
$884$	−2.87689	−0.0967604
$885$	0	0
$886$	64.8078	2.17726
$887$	27.8617	0.935506	0.467753	−	0.883859i	$-0.345064\pi$
$887$	27.8617	0.935506	0.467753	+	0.883859i	$0.345064\pi$
$888$	0	0
$889$	−4.80776	−0.161247
$890$	0	0
$891$	0	0
$892$	99.4773	3.33075
$893$	16.3153	0.545972
$894$	0	0
$895$	0	0
$896$	−9.43845	−0.315316
$897$	0	0
$898$	−72.9848	−2.43554
$899$	24.0000	0.800445
$900$	0	0
$901$	−1.61553	−0.0538210
$902$	19.6847	0.655427
$903$	0	0
$904$	−59.8617	−1.99097
$905$	0	0
$906$	0	0
$907$	46.1080	1.53099	0.765495	−	0.643442i	$-0.222494\pi$
$907$	46.1080	1.53099	0.765495	+	0.643442i	$0.222494\pi$
$908$	93.4773	3.10215
$909$	0	0
$910$	0	0
$911$	−57.1619	−1.89386	−0.946930	−	0.321441i	$-0.895833\pi$
$911$	−57.1619	−1.89386	−0.946930	+	0.321441i	$0.895833\pi$
$912$	0	0
$913$	0	0
$914$	34.2462	1.13276
$915$	0	0
$916$	−37.0540	−1.22430
$917$	−16.4924	−0.544628
$918$	0	0
$919$	41.1080	1.35603	0.678013	−	0.735050i	$-0.262841\pi$
$919$	41.1080	1.35603	0.678013	+	0.735050i	$0.262841\pi$
$920$	0	0
$921$	0	0
$922$	−40.6307	−1.33810
$923$	−3.86174	−0.127111
$924$	0	0
$925$	0	0
$926$	−4.49242	−0.147630
$927$	0	0
$928$	33.6155	1.10348
$929$	23.2311	0.762186	0.381093	−	0.924537i	$-0.375548\pi$
$929$	23.2311	0.762186	0.381093	+	0.924537i	$0.375548\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	113.162	3.70674
$933$	0	0
$934$	5.75379	0.188270
$935$	0	0
$936$	0	0
$937$	34.6847	1.13310	0.566549	−	0.824028i	$-0.308278\pi$
$937$	34.6847	1.13310	0.566549	+	0.824028i	$0.308278\pi$
$938$	−17.1231	−0.559089
$939$	0	0
$940$	0	0
$941$	60.1771	1.96172	0.980858	−	0.194722i	$-0.0623806\pi$
$941$	60.1771	1.96172	0.980858	+	0.194722i	$0.0623806\pi$
$942$	0	0
$943$	−50.4233	−1.64201
$944$	81.1619	2.64160
$945$	0	0
$946$	−12.0000	−0.390154
$947$	−32.1771	−1.04561	−0.522807	−	0.852451i	$-0.675115\pi$
$947$	−32.1771	−1.04561	−0.522807	+	0.852451i	$0.675115\pi$
$948$	0	0
$949$	−7.12311	−0.231226
$950$	0	0
$951$	0	0
$952$	9.43845	0.305902
$953$	−4.80776	−0.155739	−0.0778694	−	0.996964i	$-0.524812\pi$
$953$	−4.80776	−0.155739	−0.0778694	+	0.996964i	$0.524812\pi$
$954$	0	0
$955$	0	0
$956$	−75.2311	−2.43315
$957$	0	0
$958$	8.63068	0.278845
$959$	−3.36932	−0.108801
$960$	0	0
$961$	−9.05398	−0.292064
$962$	12.1383	0.391353
$963$	0	0
$964$	−64.1080	−2.06478
$965$	0	0
$966$	0	0
$967$	44.0000	1.41494	0.707472	−	0.706741i	$-0.249835\pi$
$967$	44.0000	1.41494	0.707472	+	0.706741i	$0.249835\pi$
$968$	6.56155	0.210896
$969$	0	0
$970$	0	0
$971$	8.17708	0.262415	0.131208	−	0.991355i	$-0.458115\pi$
$971$	8.17708	0.262415	0.131208	+	0.991355i	$0.458115\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	−30.8769	−0.989360
$975$	0	0
$976$	−58.1080	−1.85999
$977$	−25.6155	−0.819513	−0.409757	−	0.912195i	$-0.634386\pi$
$977$	−25.6155	−0.819513	−0.409757	+	0.912195i	$0.634386\pi$
$978$	0	0
$979$	17.1231	0.547257
$980$	0	0
$981$	0	0
$982$	42.2462	1.34813
$983$	21.9309	0.699486	0.349743	−	0.936846i	$-0.386269\pi$
$983$	21.9309	0.699486	0.349743	+	0.936846i	$0.386269\pi$
$984$	0	0
$985$	0	0
$986$	18.8769	0.601163
$987$	0	0
$988$	−6.00000	−0.190885
$989$	30.7386	0.977432
$990$	0	0
$991$	37.3153	1.18536	0.592680	−	0.805438i	$-0.298070\pi$
$991$	37.3153	1.18536	0.592680	+	0.805438i	$0.298070\pi$
$992$	30.7386	0.975953
$993$	0	0
$994$	22.5616	0.715609
$995$	0	0
$996$	0	0
$997$	20.7386	0.656799	0.328400	−	0.944539i	$-0.393491\pi$
$997$	20.7386	0.656799	0.328400	+	0.944539i	$0.393491\pi$
$998$	9.75379	0.308751
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.v.1.2	yes	2
3.2	odd	2		2475.2.a.q.1.1	yes	2
5.2	odd	4		2475.2.c.i.199.4		4
5.3	odd	4		2475.2.c.i.199.1		4
5.4	even	2		2475.2.a.p.1.1	✓	2
15.2	even	4		2475.2.c.j.199.1		4
15.8	even	4		2475.2.c.j.199.4		4
15.14	odd	2		2475.2.a.u.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2475.2.a.p.1.1	✓	2	5.4	even	2
2475.2.a.q.1.1	yes	2	3.2	odd	2
2475.2.a.u.1.2	yes	2	15.14	odd	2
2475.2.a.v.1.2	yes	2	1.1	even	1	trivial
2475.2.c.i.199.1		4	5.3	odd	4
2475.2.c.i.199.4		4	5.2	odd	4
2475.2.c.j.199.1		4	15.2	even	4
2475.2.c.j.199.4		4	15.8	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} +6.56155 q^{8} +O(q^{10})\)	\(q+2.56155 q^{2} +4.56155 q^{4} +1.00000 q^{7} +6.56155 q^{8} -1.00000 q^{11} -0.438447 q^{13} +2.56155 q^{14} +7.68466 q^{16} +1.43845 q^{17} +3.00000 q^{19} -2.56155 q^{22} +6.56155 q^{23} -1.12311 q^{26} +4.56155 q^{28} +5.12311 q^{29} +4.68466 q^{31} +6.56155 q^{32} +3.68466 q^{34} -10.8078 q^{37} +7.68466 q^{38} -7.68466 q^{41} +4.68466 q^{43} -4.56155 q^{44} +16.8078 q^{46} +5.43845 q^{47} -6.00000 q^{49} -2.00000 q^{52} -1.12311 q^{53} +6.56155 q^{56} +13.1231 q^{58} +10.5616 q^{59} -7.56155 q^{61} +12.0000 q^{62} +1.43845 q^{64} -6.68466 q^{67} +6.56155 q^{68} +8.80776 q^{71} +16.2462 q^{73} -27.6847 q^{74} +13.6847 q^{76} -1.00000 q^{77} -15.6847 q^{79} -19.6847 q^{82} +12.0000 q^{86} -6.56155 q^{88} -17.1231 q^{89} -0.438447 q^{91} +29.9309 q^{92} +13.9309 q^{94} +2.12311 q^{97} -15.3693 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8}+O(q^{10}) \) 2 * q + q^2 + 5 * q^4 + 2 * q^7 + 9 * q^8	\( 2 q + q^{2} + 5 q^{4} + 2 q^{7} + 9 q^{8} - 2 q^{11} - 5 q^{13} + q^{14} + 3 q^{16} + 7 q^{17} + 6 q^{19} - q^{22} + 9 q^{23} + 6 q^{26} + 5 q^{28} + 2 q^{29} - 3 q^{31} + 9 q^{32} - 5 q^{34} - q^{37} + 3 q^{38} - 3 q^{41} - 3 q^{43} - 5 q^{44} + 13 q^{46} + 15 q^{47} - 12 q^{49} - 4 q^{52} + 6 q^{53} + 9 q^{56} + 18 q^{58} + 17 q^{59} - 11 q^{61} + 24 q^{62} + 7 q^{64} - q^{67} + 9 q^{68} - 3 q^{71} + 16 q^{73} - 43 q^{74} + 15 q^{76} - 2 q^{77} - 19 q^{79} - 27 q^{82} + 24 q^{86} - 9 q^{88} - 26 q^{89} - 5 q^{91} + 31 q^{92} - q^{94} - 4 q^{97} - 6 q^{98}+O(q^{100}) \) 2 * q + q^2 + 5 * q^4 + 2 * q^7 + 9 * q^8 - 2 * q^11 - 5 * q^13 + q^14 + 3 * q^16 + 7 * q^17 + 6 * q^19 - q^22 + 9 * q^23 + 6 * q^26 + 5 * q^28 + 2 * q^29 - 3 * q^31 + 9 * q^32 - 5 * q^34 - q^37 + 3 * q^38 - 3 * q^41 - 3 * q^43 - 5 * q^44 + 13 * q^46 + 15 * q^47 - 12 * q^49 - 4 * q^52 + 6 * q^53 + 9 * q^56 + 18 * q^58 + 17 * q^59 - 11 * q^61 + 24 * q^62 + 7 * q^64 - q^67 + 9 * q^68 - 3 * q^71 + 16 * q^73 - 43 * q^74 + 15 * q^76 - 2 * q^77 - 19 * q^79 - 27 * q^82 + 24 * q^86 - 9 * q^88 - 26 * q^89 - 5 * q^91 + 31 * q^92 - q^94 - 4 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more