LMFDB - Embedded newform 225.4.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.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:	$13.2754297513$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.70156 q^{2} +14.1047 q^{4} -16.2094 q^{7} -28.7016 q^{8} +O(q^{10})$	$q-4.70156 q^{2} +14.1047 q^{4} -16.2094 q^{7} -28.7016 q^{8} +40.2094 q^{11} +19.7906 q^{13} +76.2094 q^{14} +22.1047 q^{16} -83.0156 q^{17} -48.8375 q^{19} -189.047 q^{22} -1.61250 q^{23} -93.0469 q^{26} -228.628 q^{28} +24.5344 q^{29} -12.4187 q^{31} +125.686 q^{32} +390.303 q^{34} +325.884 q^{37} +229.612 q^{38} +242.419 q^{41} +367.350 q^{43} +567.141 q^{44} +7.58125 q^{46} +204.544 q^{47} -80.2562 q^{49} +279.141 q^{52} +61.5281 q^{53} +465.234 q^{56} -115.350 q^{58} +112.209 q^{59} +477.350 q^{61} +58.3875 q^{62} -767.758 q^{64} +558.094 q^{67} -1170.91 q^{68} -558.281 q^{71} +1011.77 q^{73} -1532.17 q^{74} -688.837 q^{76} -651.769 q^{77} +1150.47 q^{79} -1139.75 q^{82} +1157.92 q^{83} -1727.12 q^{86} -1154.07 q^{88} -96.9751 q^{89} -320.794 q^{91} -22.7438 q^{92} -961.675 q^{94} -1152.37 q^{97} +377.330 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 + 9 * q^4 + 6 * q^7 - 51 * q^8	$ 2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8} + 42 q^{11} + 78 q^{13} + 114 q^{14} + 25 q^{16} - 102 q^{17} + 56 q^{19} - 186 q^{22} + 48 q^{23} + 6 q^{26} - 342 q^{28} + 318 q^{29} + 52 q^{31} + 309 q^{32} + 358 q^{34} + 306 q^{37} + 408 q^{38} + 408 q^{41} + 120 q^{43} + 558 q^{44} + 92 q^{46} - 180 q^{47} + 70 q^{49} - 18 q^{52} - 402 q^{53} - 30 q^{56} + 384 q^{58} + 186 q^{59} + 340 q^{61} + 168 q^{62} - 479 q^{64} + 732 q^{67} - 1074 q^{68} + 36 q^{71} + 1332 q^{73} - 1566 q^{74} - 1224 q^{76} - 612 q^{77} + 380 q^{79} - 858 q^{82} + 984 q^{83} - 2148 q^{86} - 1194 q^{88} - 1116 q^{89} + 972 q^{91} - 276 q^{92} - 1616 q^{94} - 768 q^{97} + 633 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 + 9 * q^4 + 6 * q^7 - 51 * q^8 + 42 * q^11 + 78 * q^13 + 114 * q^14 + 25 * q^16 - 102 * q^17 + 56 * q^19 - 186 * q^22 + 48 * q^23 + 6 * q^26 - 342 * q^28 + 318 * q^29 + 52 * q^31 + 309 * q^32 + 358 * q^34 + 306 * q^37 + 408 * q^38 + 408 * q^41 + 120 * q^43 + 558 * q^44 + 92 * q^46 - 180 * q^47 + 70 * q^49 - 18 * q^52 - 402 * q^53 - 30 * q^56 + 384 * q^58 + 186 * q^59 + 340 * q^61 + 168 * q^62 - 479 * q^64 + 732 * q^67 - 1074 * q^68 + 36 * q^71 + 1332 * q^73 - 1566 * q^74 - 1224 * q^76 - 612 * q^77 + 380 * q^79 - 858 * q^82 + 984 * q^83 - 2148 * q^86 - 1194 * q^88 - 1116 * q^89 + 972 * q^91 - 276 * q^92 - 1616 * q^94 - 768 * q^97 + 633 * 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 $

$2$	−4.70156	−1.66225	−0.831127	−	0.556083i	$-0.812304\pi$
$2$	−4.70156	−1.66225	−0.831127	+	0.556083i	$0.812304\pi$
$3$	0	0
$3$	0	0
$4$	14.1047	1.76309
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−16.2094	−0.875224	−0.437612	−	0.899164i	$-0.644176\pi$
$7$	−16.2094	−0.875224	−0.437612	+	0.899164i	$0.644176\pi$
$8$	−28.7016	−1.26844
$9$	0	0
$10$	0	0
$11$	40.2094	1.10214	0.551072	−	0.834458i	$-0.314219\pi$
$11$	40.2094	1.10214	0.551072	+	0.834458i	$0.314219\pi$
$12$	0	0
$13$	19.7906	0.422226	0.211113	−	0.977462i	$-0.432291\pi$
$13$	19.7906	0.422226	0.211113	+	0.977462i	$0.432291\pi$
$14$	76.2094	1.45484
$15$	0	0
$16$	22.1047	0.345386
$17$	−83.0156	−1.18437	−0.592184	−	0.805803i	$-0.701734\pi$
$17$	−83.0156	−1.18437	−0.592184	+	0.805803i	$0.701734\pi$
$18$	0	0
$19$	−48.8375	−0.589689	−0.294844	−	0.955545i	$-0.595268\pi$
$19$	−48.8375	−0.589689	−0.294844	+	0.955545i	$0.595268\pi$
$20$	0	0
$21$	0	0
$22$	−189.047	−1.83204
$23$	−1.61250	−0.0146186	−0.00730932	−	0.999973i	$-0.502327\pi$
$23$	−1.61250	−0.0146186	−0.00730932	+	0.999973i	$0.502327\pi$
$24$	0	0
$25$	0	0
$26$	−93.0469	−0.701846
$27$	0	0
$28$	−228.628	−1.54309
$29$	24.5344	0.157101	0.0785504	−	0.996910i	$-0.474971\pi$
$29$	24.5344	0.157101	0.0785504	+	0.996910i	$0.474971\pi$
$30$	0	0
$31$	−12.4187	−0.0719507	−0.0359754	−	0.999353i	$-0.511454\pi$
$31$	−12.4187	−0.0719507	−0.0359754	+	0.999353i	$0.511454\pi$
$32$	125.686	0.694323
$33$	0	0
$34$	390.303	1.96872
$35$	0	0
$36$	0	0
$37$	325.884	1.44797	0.723987	−	0.689813i	$-0.242307\pi$
$37$	325.884	1.44797	0.723987	+	0.689813i	$0.242307\pi$
$38$	229.612	0.980212
$39$	0	0
$40$	0	0
$41$	242.419	0.923401	0.461701	−	0.887036i	$-0.347239\pi$
$41$	242.419	0.923401	0.461701	+	0.887036i	$0.347239\pi$
$42$	0	0
$43$	367.350	1.30280	0.651399	−	0.758735i	$-0.274182\pi$
$43$	367.350	1.30280	0.651399	+	0.758735i	$0.274182\pi$
$44$	567.141	1.94317
$45$	0	0
$46$	7.58125	0.0242999
$47$	204.544	0.634804	0.317402	−	0.948291i	$-0.397190\pi$
$47$	204.544	0.634804	0.317402	+	0.948291i	$0.397190\pi$
$48$	0	0
$49$	−80.2562	−0.233983
$50$	0	0
$51$	0	0
$52$	279.141	0.744420
$53$	61.5281	0.159463	0.0797314	−	0.996816i	$-0.474594\pi$
$53$	61.5281	0.159463	0.0797314	+	0.996816i	$0.474594\pi$
$54$	0	0
$55$	0	0
$56$	465.234	1.11017
$57$	0	0
$58$	−115.350	−0.261141
$59$	112.209	0.247600	0.123800	−	0.992307i	$-0.460492\pi$
$59$	112.209	0.247600	0.123800	+	0.992307i	$0.460492\pi$
$60$	0	0
$61$	477.350	1.00194	0.500970	−	0.865464i	$-0.332977\pi$
$61$	477.350	1.00194	0.500970	+	0.865464i	$0.332977\pi$
$62$	58.3875	0.119600
$63$	0	0
$64$	−767.758	−1.49953
$65$	0	0
$66$	0	0
$67$	558.094	1.01764	0.508821	−	0.860872i	$-0.330082\pi$
$67$	558.094	1.01764	0.508821	+	0.860872i	$0.330082\pi$
$68$	−1170.91	−2.08814
$69$	0	0
$70$	0	0
$71$	−558.281	−0.933180	−0.466590	−	0.884474i	$-0.654518\pi$
$71$	−558.281	−0.933180	−0.466590	+	0.884474i	$0.654518\pi$
$72$	0	0
$73$	1011.77	1.62217	0.811086	−	0.584927i	$-0.198877\pi$
$73$	1011.77	1.62217	0.811086	+	0.584927i	$0.198877\pi$
$74$	−1532.17	−2.40690
$75$	0	0
$76$	−688.837	−1.03967
$77$	−651.769	−0.964623
$78$	0	0
$79$	1150.47	1.63845	0.819227	−	0.573470i	$-0.194403\pi$
$79$	1150.47	1.63845	0.819227	+	0.573470i	$0.194403\pi$
$80$	0	0
$81$	0	0
$82$	−1139.75	−1.53493
$83$	1157.92	1.53131	0.765655	−	0.643251i	$-0.222415\pi$
$83$	1157.92	1.53131	0.765655	+	0.643251i	$0.222415\pi$
$84$	0	0
$85$	0	0
$86$	−1727.12	−2.16558
$87$	0	0
$88$	−1154.07	−1.39801
$89$	−96.9751	−0.115498	−0.0577491	−	0.998331i	$-0.518392\pi$
$89$	−96.9751	−0.115498	−0.0577491	+	0.998331i	$0.518392\pi$
$90$	0	0
$91$	−320.794	−0.369542
$92$	−22.7438	−0.0257739
$93$	0	0
$94$	−961.675	−1.05520
$95$	0	0
$96$	0	0
$97$	−1152.37	−1.20625	−0.603123	−	0.797648i	$-0.706077\pi$
$97$	−1152.37	−1.20625	−0.603123	+	0.797648i	$0.706077\pi$
$98$	377.330	0.388939
$99$	0	0
$100$	0	0
$101$	1156.49	1.13936	0.569679	−	0.821867i	$-0.307068\pi$
$101$	1156.49	1.13936	0.569679	+	0.821867i	$0.307068\pi$
$102$	0	0
$103$	−1333.70	−1.27585	−0.637927	−	0.770096i	$-0.720208\pi$
$103$	−1333.70	−1.27585	−0.637927	+	0.770096i	$0.720208\pi$
$104$	−568.022	−0.535569
$105$	0	0
$106$	−289.278	−0.265068
$107$	798.263	0.721224	0.360612	−	0.932716i	$-0.382568\pi$
$107$	798.263	0.721224	0.360612	+	0.932716i	$0.382568\pi$
$108$	0	0
$109$	−985.119	−0.865663	−0.432831	−	0.901475i	$-0.642485\pi$
$109$	−985.119	−0.865663	−0.432831	+	0.901475i	$0.642485\pi$
$110$	0	0
$111$	0	0
$112$	−358.303	−0.302290
$113$	−1888.25	−1.57196	−0.785979	−	0.618253i	$-0.787841\pi$
$113$	−1888.25	−1.57196	−0.785979	+	0.618253i	$0.787841\pi$
$114$	0	0
$115$	0	0
$116$	346.050	0.276982
$117$	0	0
$118$	−527.559	−0.411574
$119$	1345.63	1.03659
$120$	0	0
$121$	285.794	0.214721
$122$	−2244.29	−1.66548
$123$	0	0
$124$	−175.163	−0.126855
$125$	0	0
$126$	0	0
$127$	−620.859	−0.433798	−0.216899	−	0.976194i	$-0.569594\pi$
$127$	−620.859	−0.433798	−0.216899	+	0.976194i	$0.569594\pi$
$128$	2604.17	1.79827
$129$	0	0
$130$	0	0
$131$	2588.35	1.72630	0.863151	−	0.504947i	$-0.168488\pi$
$131$	2588.35	1.72630	0.863151	+	0.504947i	$0.168488\pi$
$132$	0	0
$133$	791.625	0.516110
$134$	−2623.91	−1.69158
$135$	0	0
$136$	2382.68	1.50230
$137$	1656.29	1.03289	0.516447	−	0.856319i	$-0.327254\pi$
$137$	1656.29	1.03289	0.516447	+	0.856319i	$0.327254\pi$
$138$	0	0
$139$	−153.256	−0.0935182	−0.0467591	−	0.998906i	$-0.514889\pi$
$139$	−153.256	−0.0935182	−0.0467591	+	0.998906i	$0.514889\pi$
$140$	0	0
$141$	0	0
$142$	2624.79	1.55118
$143$	795.769	0.465353
$144$	0	0
$145$	0	0
$146$	−4756.89	−2.69646
$147$	0	0
$148$	4596.50	2.55290
$149$	−1483.38	−0.815591	−0.407795	−	0.913073i	$-0.633702\pi$
$149$	−1483.38	−0.815591	−0.407795	+	0.913073i	$0.633702\pi$
$150$	0	0
$151$	−394.281	−0.212491	−0.106246	−	0.994340i	$-0.533883\pi$
$151$	−394.281	−0.212491	−0.106246	+	0.994340i	$0.533883\pi$
$152$	1401.71	0.747986
$153$	0	0
$154$	3064.33	1.60345
$155$	0	0
$156$	0	0
$157$	1727.05	0.877922	0.438961	−	0.898506i	$-0.355347\pi$
$157$	1727.05	0.877922	0.438961	+	0.898506i	$0.355347\pi$
$158$	−5409.00	−2.72352
$159$	0	0
$160$	0	0
$161$	26.1376	0.0127946
$162$	0	0
$163$	−2034.28	−0.977529	−0.488764	−	0.872416i	$-0.662552\pi$
$163$	−2034.28	−0.977529	−0.488764	+	0.872416i	$0.662552\pi$
$164$	3419.24	1.62804
$165$	0	0
$166$	−5444.06	−2.54543
$167$	−192.900	−0.0893835	−0.0446918	−	0.999001i	$-0.514231\pi$
$167$	−192.900	−0.0893835	−0.0446918	+	0.999001i	$0.514231\pi$
$168$	0	0
$169$	−1805.33	−0.821726
$170$	0	0
$171$	0	0
$172$	5181.36	2.29695
$173$	1239.91	0.544905	0.272452	−	0.962169i	$-0.412165\pi$
$173$	1239.91	0.544905	0.272452	+	0.962169i	$0.412165\pi$
$174$	0	0
$175$	0	0
$176$	888.816	0.380665
$177$	0	0
$178$	455.934	0.191987
$179$	2636.86	1.10105	0.550525	−	0.834818i	$-0.314427\pi$
$179$	2636.86	1.10105	0.550525	+	0.834818i	$0.314427\pi$
$180$	0	0
$181$	3317.58	1.36240	0.681199	−	0.732099i	$-0.261459\pi$
$181$	3317.58	1.36240	0.681199	+	0.732099i	$0.261459\pi$
$182$	1508.23	0.614272
$183$	0	0
$184$	46.2812	0.0185429
$185$	0	0
$186$	0	0
$187$	−3338.01	−1.30534
$188$	2885.02	1.11921
$189$	0	0
$190$	0	0
$191$	624.506	0.236585	0.118292	−	0.992979i	$-0.462258\pi$
$191$	624.506	0.236585	0.118292	+	0.992979i	$0.462258\pi$
$192$	0	0
$193$	−436.144	−0.162665	−0.0813324	−	0.996687i	$-0.525918\pi$
$193$	−436.144	−0.162665	−0.0813324	+	0.996687i	$0.525918\pi$
$194$	5417.96	2.00509
$195$	0	0
$196$	−1131.99	−0.412532
$197$	−3355.81	−1.21366	−0.606831	−	0.794831i	$-0.707560\pi$
$197$	−3355.81	−1.21366	−0.606831	+	0.794831i	$0.707560\pi$
$198$	0	0
$199$	3799.77	1.35356	0.676780	−	0.736185i	$-0.263375\pi$
$199$	3799.77	1.35356	0.676780	+	0.736185i	$0.263375\pi$
$200$	0	0
$201$	0	0
$202$	−5437.31	−1.89390
$203$	−397.687	−0.137498
$204$	0	0
$205$	0	0
$206$	6270.46	2.12079
$207$	0	0
$208$	437.466	0.145831
$209$	−1963.72	−0.649922
$210$	0	0
$211$	2365.27	0.771715	0.385857	−	0.922558i	$-0.373906\pi$
$211$	2365.27	0.771715	0.385857	+	0.922558i	$0.373906\pi$
$212$	867.834	0.281147
$213$	0	0
$214$	−3753.08	−1.19886
$215$	0	0
$216$	0	0
$217$	201.300	0.0629730
$218$	4631.60	1.43895
$219$	0	0
$220$	0	0
$221$	−1642.93	−0.500070
$222$	0	0
$223$	−3328.58	−0.999545	−0.499772	−	0.866157i	$-0.666583\pi$
$223$	−3328.58	−0.999545	−0.499772	+	0.866157i	$0.666583\pi$
$224$	−2037.29	−0.607688
$225$	0	0
$226$	8877.71	2.61299
$227$	527.100	0.154118	0.0770592	−	0.997027i	$-0.475447\pi$
$227$	527.100	0.154118	0.0770592	+	0.997027i	$0.475447\pi$
$228$	0	0
$229$	2566.06	0.740479	0.370240	−	0.928936i	$-0.379276\pi$
$229$	2566.06	0.740479	0.370240	+	0.928936i	$0.379276\pi$
$230$	0	0
$231$	0	0
$232$	−704.175	−0.199273
$233$	−5534.99	−1.55626	−0.778132	−	0.628101i	$-0.783832\pi$
$233$	−5534.99	−1.55626	−0.778132	+	0.628101i	$0.783832\pi$
$234$	0	0
$235$	0	0
$236$	1582.68	0.436541
$237$	0	0
$238$	−6326.57	−1.72307
$239$	−1010.01	−0.273355	−0.136678	−	0.990616i	$-0.543642\pi$
$239$	−1010.01	−0.273355	−0.136678	+	0.990616i	$0.543642\pi$
$240$	0	0
$241$	−4074.29	−1.08900	−0.544498	−	0.838762i	$-0.683280\pi$
$241$	−4074.29	−1.08900	−0.544498	+	0.838762i	$0.683280\pi$
$242$	−1343.68	−0.356921
$243$	0	0
$244$	6732.87	1.76651
$245$	0	0
$246$	0	0
$247$	−966.525	−0.248982
$248$	356.437	0.0912653
$249$	0	0
$250$	0	0
$251$	−1773.98	−0.446107	−0.223054	−	0.974806i	$-0.571602\pi$
$251$	−1773.98	−0.446107	−0.223054	+	0.974806i	$0.571602\pi$
$252$	0	0
$253$	−64.8375	−0.0161119
$254$	2919.01	0.721082
$255$	0	0
$256$	−6101.62	−1.48965
$257$	−662.784	−0.160869	−0.0804345	−	0.996760i	$-0.525631\pi$
$257$	−662.784	−0.160869	−0.0804345	+	0.996760i	$0.525631\pi$
$258$	0	0
$259$	−5282.38	−1.26730
$260$	0	0
$261$	0	0
$262$	−12169.3	−2.86955
$263$	−712.312	−0.167008	−0.0835039	−	0.996507i	$-0.526611\pi$
$263$	−712.312	−0.167008	−0.0835039	+	0.996507i	$0.526611\pi$
$264$	0	0
$265$	0	0
$266$	−3721.87	−0.857905
$267$	0	0
$268$	7871.74	1.79419
$269$	3136.41	0.710894	0.355447	−	0.934696i	$-0.384329\pi$
$269$	3136.41	0.710894	0.355447	+	0.934696i	$0.384329\pi$
$270$	0	0
$271$	−2275.69	−0.510105	−0.255053	−	0.966927i	$-0.582093\pi$
$271$	−2275.69	−0.510105	−0.255053	+	0.966927i	$0.582093\pi$
$272$	−1835.03	−0.409064
$273$	0	0
$274$	−7787.15	−1.71693
$275$	0	0
$276$	0	0
$277$	5171.00	1.12164	0.560821	−	0.827937i	$-0.310485\pi$
$277$	5171.00	1.12164	0.560821	+	0.827937i	$0.310485\pi$
$278$	720.544	0.155451
$279$	0	0
$280$	0	0
$281$	−2240.14	−0.475571	−0.237785	−	0.971318i	$-0.576422\pi$
$281$	−2240.14	−0.475571	−0.237785	+	0.971318i	$0.576422\pi$
$282$	0	0
$283$	225.244	0.0473123	0.0236561	−	0.999720i	$-0.492469\pi$
$283$	225.244	0.0473123	0.0236561	+	0.999720i	$0.492469\pi$
$284$	−7874.38	−1.64528
$285$	0	0
$286$	−3741.36	−0.773535
$287$	−3929.46	−0.808183
$288$	0	0
$289$	1978.59	0.402726
$290$	0	0
$291$	0	0
$292$	14270.7	2.86003
$293$	−1139.86	−0.227274	−0.113637	−	0.993522i	$-0.536250\pi$
$293$	−1139.86	−0.227274	−0.113637	+	0.993522i	$0.536250\pi$
$294$	0	0
$295$	0	0
$296$	−9353.39	−1.83667
$297$	0	0
$298$	6974.19	1.35572
$299$	−31.9123	−0.00617237
$300$	0	0
$301$	−5954.51	−1.14024
$302$	1853.74	0.353214
$303$	0	0
$304$	−1079.54	−0.203670
$305$	0	0
$306$	0	0
$307$	5244.86	0.975049	0.487525	−	0.873109i	$-0.337900\pi$
$307$	5244.86	0.975049	0.487525	+	0.873109i	$0.337900\pi$
$308$	−9192.99	−1.70071
$309$	0	0
$310$	0	0
$311$	5188.26	0.945977	0.472989	−	0.881068i	$-0.343175\pi$
$311$	5188.26	0.945977	0.472989	+	0.881068i	$0.343175\pi$
$312$	0	0
$313$	486.656	0.0878832	0.0439416	−	0.999034i	$-0.486008\pi$
$313$	486.656	0.0878832	0.0439416	+	0.999034i	$0.486008\pi$
$314$	−8119.85	−1.45933
$315$	0	0
$316$	16227.0	2.88873
$317$	4218.87	0.747493	0.373747	−	0.927531i	$-0.378073\pi$
$317$	4218.87	0.747493	0.373747	+	0.927531i	$0.378073\pi$
$318$	0	0
$319$	986.512	0.173148
$320$	0	0
$321$	0	0
$322$	−122.887	−0.0212678
$323$	4054.27	0.698408
$324$	0	0
$325$	0	0
$326$	9564.30	1.62490
$327$	0	0
$328$	−6957.80	−1.17128
$329$	−3315.53	−0.555595
$330$	0	0
$331$	7439.94	1.23546	0.617728	−	0.786392i	$-0.288053\pi$
$331$	7439.94	1.23546	0.617728	+	0.786392i	$0.288053\pi$
$332$	16332.2	2.69983
$333$	0	0
$334$	906.931	0.148578
$335$	0	0
$336$	0	0
$337$	−6555.39	−1.05963	−0.529815	−	0.848113i	$-0.677739\pi$
$337$	−6555.39	−1.05963	−0.529815	+	0.848113i	$0.677739\pi$
$338$	8487.88	1.36592
$339$	0	0
$340$	0	0
$341$	−499.350	−0.0793000
$342$	0	0
$343$	6860.72	1.08001
$344$	−10543.5	−1.65252
$345$	0	0
$346$	−5829.51	−0.905770
$347$	−1950.56	−0.301763	−0.150881	−	0.988552i	$-0.548211\pi$
$347$	−1950.56	−0.301763	−0.150881	+	0.988552i	$0.548211\pi$
$348$	0	0
$349$	−1426.74	−0.218830	−0.109415	−	0.993996i	$-0.534898\pi$
$349$	−1426.74	−0.218830	−0.109415	+	0.993996i	$0.534898\pi$
$350$	0	0
$351$	0	0
$352$	5053.75	0.765244
$353$	7078.96	1.06735	0.533676	−	0.845689i	$-0.320810\pi$
$353$	7078.96	1.06735	0.533676	+	0.845689i	$0.320810\pi$
$354$	0	0
$355$	0	0
$356$	−1367.80	−0.203633
$357$	0	0
$358$	−12397.4	−1.83023
$359$	5409.79	0.795314	0.397657	−	0.917534i	$-0.369823\pi$
$359$	5409.79	0.795314	0.397657	+	0.917534i	$0.369823\pi$
$360$	0	0
$361$	−4473.90	−0.652267
$362$	−15597.8	−2.26465
$363$	0	0
$364$	−4524.69	−0.651534
$365$	0	0
$366$	0	0
$367$	4940.09	0.702645	0.351322	−	0.936255i	$-0.385732\pi$
$367$	4940.09	0.702645	0.351322	+	0.936255i	$0.385732\pi$
$368$	−35.6437	−0.00504907
$369$	0	0
$370$	0	0
$371$	−997.332	−0.139566
$372$	0	0
$373$	12891.9	1.78959	0.894797	−	0.446473i	$-0.147320\pi$
$373$	12891.9	1.78959	0.894797	+	0.446473i	$0.147320\pi$
$374$	15693.8	2.16981
$375$	0	0
$376$	−5870.72	−0.805211
$377$	485.551	0.0663320
$378$	0	0
$379$	−9475.15	−1.28418	−0.642092	−	0.766627i	$-0.721933\pi$
$379$	−9475.15	−1.28418	−0.642092	+	0.766627i	$0.721933\pi$
$380$	0	0
$381$	0	0
$382$	−2936.16	−0.393264
$383$	5800.97	0.773931	0.386966	−	0.922094i	$-0.373523\pi$
$383$	5800.97	0.773931	0.386966	+	0.922094i	$0.373523\pi$
$384$	0	0
$385$	0	0
$386$	2050.56	0.270390
$387$	0	0
$388$	−16253.9	−2.12672
$389$	−13779.7	−1.79603	−0.898016	−	0.439962i	$-0.854992\pi$
$389$	−13779.7	−1.79603	−0.898016	+	0.439962i	$0.854992\pi$
$390$	0	0
$391$	133.862	0.0173138
$392$	2303.48	0.296794
$393$	0	0
$394$	15777.5	2.01741
$395$	0	0
$396$	0	0
$397$	−2816.46	−0.356056	−0.178028	−	0.984025i	$-0.556972\pi$
$397$	−2816.46	−0.356056	−0.178028	+	0.984025i	$0.556972\pi$
$398$	−17864.8	−2.24996
$399$	0	0
$400$	0	0
$401$	−11986.4	−1.49270	−0.746352	−	0.665551i	$-0.768196\pi$
$401$	−11986.4	−1.49270	−0.746352	+	0.665551i	$0.768196\pi$
$402$	0	0
$403$	−245.775	−0.0303794
$404$	16311.9	2.00879
$405$	0	0
$406$	1869.75	0.228557
$407$	13103.6	1.59588
$408$	0	0
$409$	−3339.07	−0.403683	−0.201841	−	0.979418i	$-0.564693\pi$
$409$	−3339.07	−0.403683	−0.201841	+	0.979418i	$0.564693\pi$
$410$	0	0
$411$	0	0
$412$	−18811.4	−2.24944
$413$	−1818.84	−0.216706
$414$	0	0
$415$	0	0
$416$	2487.40	0.293161
$417$	0	0
$418$	9232.57	1.08033
$419$	−1688.52	−0.196873	−0.0984363	−	0.995143i	$-0.531384\pi$
$419$	−1688.52	−0.196873	−0.0984363	+	0.995143i	$0.531384\pi$
$420$	0	0
$421$	−2664.27	−0.308429	−0.154214	−	0.988037i	$-0.549285\pi$
$421$	−2664.27	−0.308429	−0.154214	+	0.988037i	$0.549285\pi$
$422$	−11120.5	−1.28279
$423$	0	0
$424$	−1765.95	−0.202269
$425$	0	0
$426$	0	0
$427$	−7737.54	−0.876923
$428$	11259.2	1.27158
$429$	0	0
$430$	0	0
$431$	−12266.0	−1.37084	−0.685420	−	0.728148i	$-0.740381\pi$
$431$	−12266.0	−1.37084	−0.685420	+	0.728148i	$0.740381\pi$
$432$	0	0
$433$	−15647.3	−1.73664	−0.868318	−	0.496008i	$-0.834799\pi$
$433$	−15647.3	−1.73664	−0.868318	+	0.496008i	$0.834799\pi$
$434$	−946.425	−0.104677
$435$	0	0
$436$	−13894.8	−1.52624
$437$	78.7503	0.00862045
$438$	0	0
$439$	16131.0	1.75373	0.876867	−	0.480733i	$-0.159629\pi$
$439$	16131.0	1.75373	0.876867	+	0.480733i	$0.159629\pi$
$440$	0	0
$441$	0	0
$442$	7724.34	0.831243
$443$	−10053.7	−1.07825	−0.539127	−	0.842225i	$-0.681246\pi$
$443$	−10053.7	−1.07825	−0.539127	+	0.842225i	$0.681246\pi$
$444$	0	0
$445$	0	0
$446$	15649.5	1.66150
$447$	0	0
$448$	12444.9	1.31242
$449$	−7477.71	−0.785957	−0.392979	−	0.919548i	$-0.628555\pi$
$449$	−7477.71	−0.785957	−0.392979	+	0.919548i	$0.628555\pi$
$450$	0	0
$451$	9747.51	1.01772
$452$	−26633.1	−2.77150
$453$	0	0
$454$	−2478.19	−0.256184
$455$	0	0
$456$	0	0
$457$	1363.46	0.139562	0.0697812	−	0.997562i	$-0.477770\pi$
$457$	1363.46	0.139562	0.0697812	+	0.997562i	$0.477770\pi$
$458$	−12064.5	−1.23086
$459$	0	0
$460$	0	0
$461$	−5276.77	−0.533109	−0.266555	−	0.963820i	$-0.585885\pi$
$461$	−5276.77	−0.533109	−0.266555	+	0.963820i	$0.585885\pi$
$462$	0	0
$463$	5740.02	0.576159	0.288079	−	0.957607i	$-0.406983\pi$
$463$	5740.02	0.576159	0.288079	+	0.957607i	$0.406983\pi$
$464$	542.325	0.0542604
$465$	0	0
$466$	26023.1	2.58690
$467$	−6233.36	−0.617657	−0.308828	−	0.951118i	$-0.599937\pi$
$467$	−6233.36	−0.617657	−0.308828	+	0.951118i	$0.599937\pi$
$468$	0	0
$469$	−9046.35	−0.890664
$470$	0	0
$471$	0	0
$472$	−3220.58	−0.314067
$473$	14770.9	1.43587
$474$	0	0
$475$	0	0
$476$	18979.7	1.82759
$477$	0	0
$478$	4748.61	0.454385
$479$	19688.2	1.87803	0.939013	−	0.343881i	$-0.111742\pi$
$479$	19688.2	1.87803	0.939013	+	0.343881i	$0.111742\pi$
$480$	0	0
$481$	6449.46	0.611372
$482$	19155.5	1.81019
$483$	0	0
$484$	4031.03	0.378572
$485$	0	0
$486$	0	0
$487$	−3955.08	−0.368012	−0.184006	−	0.982925i	$-0.558907\pi$
$487$	−3955.08	−0.368012	−0.184006	+	0.982925i	$0.558907\pi$
$488$	−13700.7	−1.27090
$489$	0	0
$490$	0	0
$491$	13893.5	1.27699	0.638497	−	0.769624i	$-0.279557\pi$
$491$	13893.5	1.27699	0.638497	+	0.769624i	$0.279557\pi$
$492$	0	0
$493$	−2036.74	−0.186065
$494$	4544.18	0.413871
$495$	0	0
$496$	−274.512	−0.0248508
$497$	9049.39	0.816741
$498$	0	0
$499$	13523.7	1.21324	0.606618	−	0.794993i	$-0.292526\pi$
$499$	13523.7	1.21324	0.606618	+	0.794993i	$0.292526\pi$
$500$	0	0
$501$	0	0
$502$	8340.50	0.741543
$503$	−13135.4	−1.16437	−0.582184	−	0.813057i	$-0.697802\pi$
$503$	−13135.4	−1.16437	−0.582184	+	0.813057i	$0.697802\pi$
$504$	0	0
$505$	0	0
$506$	304.837	0.0267820
$507$	0	0
$508$	−8757.03	−0.764823
$509$	2222.71	0.193556	0.0967778	−	0.995306i	$-0.469146\pi$
$509$	2222.71	0.193556	0.0967778	+	0.995306i	$0.469146\pi$
$510$	0	0
$511$	−16400.1	−1.41976
$512$	7853.76	0.677911
$513$	0	0
$514$	3116.12	0.267405
$515$	0	0
$516$	0	0
$517$	8224.57	0.699645
$518$	24835.4	2.10658
$519$	0	0
$520$	0	0
$521$	4916.42	0.413421	0.206710	−	0.978402i	$-0.433724\pi$
$521$	4916.42	0.413421	0.206710	+	0.978402i	$0.433724\pi$
$522$	0	0
$523$	17743.4	1.48349	0.741746	−	0.670681i	$-0.233998\pi$
$523$	17743.4	1.48349	0.741746	+	0.670681i	$0.233998\pi$
$524$	36507.9	3.04362
$525$	0	0
$526$	3348.98	0.277609
$527$	1030.95	0.0852161
$528$	0	0
$529$	−12164.4	−0.999786
$530$	0	0
$531$	0	0
$532$	11165.6	0.909946
$533$	4797.62	0.389884
$534$	0	0
$535$	0	0
$536$	−16018.2	−1.29082
$537$	0	0
$538$	−14746.0	−1.18169
$539$	−3227.05	−0.257883
$540$	0	0
$541$	−12671.3	−1.00699	−0.503495	−	0.863998i	$-0.667953\pi$
$541$	−12671.3	−1.00699	−0.503495	+	0.863998i	$0.667953\pi$
$542$	10699.3	0.847924
$543$	0	0
$544$	−10433.9	−0.822334
$545$	0	0
$546$	0	0
$547$	5250.90	0.410443	0.205221	−	0.978716i	$-0.434209\pi$
$547$	5250.90	0.410443	0.205221	+	0.978716i	$0.434209\pi$
$548$	23361.5	1.82108
$549$	0	0
$550$	0	0
$551$	−1198.20	−0.0926406
$552$	0	0
$553$	−18648.4	−1.43401
$554$	−24311.8	−1.86445
$555$	0	0
$556$	−2161.63	−0.164881
$557$	−25830.2	−1.96492	−0.982462	−	0.186465i	$-0.940297\pi$
$557$	−25830.2	−1.96492	−0.982462	+	0.186465i	$0.940297\pi$
$558$	0	0
$559$	7270.09	0.550075
$560$	0	0
$561$	0	0
$562$	10532.1	0.790519
$563$	−2021.14	−0.151298	−0.0756490	−	0.997135i	$-0.524103\pi$
$563$	−2021.14	−0.151298	−0.0756490	+	0.997135i	$0.524103\pi$
$564$	0	0
$565$	0	0
$566$	−1059.00	−0.0786450
$567$	0	0
$568$	16023.5	1.18368
$569$	−8706.51	−0.641469	−0.320734	−	0.947169i	$-0.603930\pi$
$569$	−8706.51	−0.641469	−0.320734	+	0.947169i	$0.603930\pi$
$570$	0	0
$571$	−12194.5	−0.893740	−0.446870	−	0.894599i	$-0.647461\pi$
$571$	−12194.5	−0.893740	−0.446870	+	0.894599i	$0.647461\pi$
$572$	11224.1	0.820458
$573$	0	0
$574$	18474.6	1.34340
$575$	0	0
$576$	0	0
$577$	−15264.0	−1.10130	−0.550649	−	0.834737i	$-0.685620\pi$
$577$	−15264.0	−1.10130	−0.550649	+	0.834737i	$0.685620\pi$
$578$	−9302.48	−0.669433
$579$	0	0
$580$	0	0
$581$	−18769.2	−1.34024
$582$	0	0
$583$	2474.01	0.175751
$584$	−29039.3	−2.05763
$585$	0	0
$586$	5359.12	0.377787
$587$	8456.89	0.594639	0.297319	−	0.954778i	$-0.403907\pi$
$587$	8456.89	0.594639	0.297319	+	0.954778i	$0.403907\pi$
$588$	0	0
$589$	606.500	0.0424285
$590$	0	0
$591$	0	0
$592$	7203.57	0.500110
$593$	1225.23	0.0848467	0.0424234	−	0.999100i	$-0.486492\pi$
$593$	1225.23	0.0848467	0.0424234	+	0.999100i	$0.486492\pi$
$594$	0	0
$595$	0	0
$596$	−20922.6	−1.43796
$597$	0	0
$598$	150.038	0.0102600
$599$	16060.0	1.09548	0.547741	−	0.836648i	$-0.315488\pi$
$599$	16060.0	1.09548	0.547741	+	0.836648i	$0.315488\pi$
$600$	0	0
$601$	9699.93	0.658350	0.329175	−	0.944269i	$-0.393229\pi$
$601$	9699.93	0.658350	0.329175	+	0.944269i	$0.393229\pi$
$602$	27995.5	1.89537
$603$	0	0
$604$	−5561.21	−0.374640
$605$	0	0
$606$	0	0
$607$	23661.2	1.58217	0.791087	−	0.611703i	$-0.209515\pi$
$607$	23661.2	1.58217	0.791087	+	0.611703i	$0.209515\pi$
$608$	−6138.19	−0.409435
$609$	0	0
$610$	0	0
$611$	4048.05	0.268030
$612$	0	0
$613$	−8085.63	−0.532749	−0.266375	−	0.963870i	$-0.585826\pi$
$613$	−8085.63	−0.532749	−0.266375	+	0.963870i	$0.585826\pi$
$614$	−24659.0	−1.62078
$615$	0	0
$616$	18706.8	1.22357
$617$	11035.1	0.720029	0.360014	−	0.932947i	$-0.382772\pi$
$617$	11035.1	0.720029	0.360014	+	0.932947i	$0.382772\pi$
$618$	0	0
$619$	−16826.3	−1.09258	−0.546290	−	0.837596i	$-0.683960\pi$
$619$	−16826.3	−1.09258	−0.546290	+	0.837596i	$0.683960\pi$
$620$	0	0
$621$	0	0
$622$	−24392.9	−1.57245
$623$	1571.90	0.101087
$624$	0	0
$625$	0	0
$626$	−2288.04	−0.146084
$627$	0	0
$628$	24359.5	1.54785
$629$	−27053.5	−1.71493
$630$	0	0
$631$	−3705.91	−0.233803	−0.116902	−	0.993144i	$-0.537296\pi$
$631$	−3705.91	−0.233803	−0.116902	+	0.993144i	$0.537296\pi$
$632$	−33020.2	−2.07828
$633$	0	0
$634$	−19835.3	−1.24252
$635$	0	0
$636$	0	0
$637$	−1588.32	−0.0987937
$638$	−4638.15	−0.287815
$639$	0	0
$640$	0	0
$641$	24597.4	1.51566	0.757829	−	0.652453i	$-0.226260\pi$
$641$	24597.4	1.51566	0.757829	+	0.652453i	$0.226260\pi$
$642$	0	0
$643$	−21479.5	−1.31737	−0.658685	−	0.752419i	$-0.728887\pi$
$643$	−21479.5	−1.31737	−0.658685	+	0.752419i	$0.728887\pi$
$644$	368.662	0.0225580
$645$	0	0
$646$	−19061.4	−1.16093
$647$	−27119.7	−1.64789	−0.823946	−	0.566668i	$-0.808232\pi$
$647$	−27119.7	−1.64789	−0.823946	+	0.566668i	$0.808232\pi$
$648$	0	0
$649$	4511.87	0.272891
$650$	0	0
$651$	0	0
$652$	−28692.9	−1.72347
$653$	−18476.4	−1.10725	−0.553627	−	0.832765i	$-0.686757\pi$
$653$	−18476.4	−1.10725	−0.553627	+	0.832765i	$0.686757\pi$
$654$	0	0
$655$	0	0
$656$	5358.59	0.318930
$657$	0	0
$658$	15588.1	0.923540
$659$	19273.5	1.13928	0.569641	−	0.821894i	$-0.307082\pi$
$659$	19273.5	1.13928	0.569641	+	0.821894i	$0.307082\pi$
$660$	0	0
$661$	25605.3	1.50670	0.753352	−	0.657618i	$-0.228436\pi$
$661$	25605.3	1.50670	0.753352	+	0.657618i	$0.228436\pi$
$662$	−34979.3	−2.05364
$663$	0	0
$664$	−33234.3	−1.94238
$665$	0	0
$666$	0	0
$667$	−39.5616	−0.00229660
$668$	−2720.79	−0.157591
$669$	0	0
$670$	0	0
$671$	19193.9	1.10428
$672$	0	0
$673$	7855.52	0.449938	0.224969	−	0.974366i	$-0.427772\pi$
$673$	7855.52	0.449938	0.224969	+	0.974366i	$0.427772\pi$
$674$	30820.6	1.76137
$675$	0	0
$676$	−25463.6	−1.44877
$677$	6763.09	0.383939	0.191970	−	0.981401i	$-0.438512\pi$
$677$	6763.09	0.383939	0.191970	+	0.981401i	$0.438512\pi$
$678$	0	0
$679$	18679.3	1.05574
$680$	0	0
$681$	0	0
$682$	2347.72	0.131817
$683$	−15608.6	−0.874447	−0.437224	−	0.899353i	$-0.644038\pi$
$683$	−15608.6	−0.874447	−0.437224	+	0.899353i	$0.644038\pi$
$684$	0	0
$685$	0	0
$686$	−32256.1	−1.79525
$687$	0	0
$688$	8120.16	0.449968
$689$	1217.68	0.0673293
$690$	0	0
$691$	6203.15	0.341504	0.170752	−	0.985314i	$-0.445380\pi$
$691$	6203.15	0.341504	0.170752	+	0.985314i	$0.445380\pi$
$692$	17488.5	0.960714
$693$	0	0
$694$	9170.69	0.501606
$695$	0	0
$696$	0	0
$697$	−20124.5	−1.09365
$698$	6707.90	0.363750
$699$	0	0
$700$	0	0
$701$	16507.9	0.889435	0.444718	−	0.895671i	$-0.353304\pi$
$701$	16507.9	0.889435	0.444718	+	0.895671i	$0.353304\pi$
$702$	0	0
$703$	−15915.4	−0.853854
$704$	−30871.1	−1.65269
$705$	0	0
$706$	−33282.2	−1.77421
$707$	−18746.0	−0.997193
$708$	0	0
$709$	−25539.6	−1.35283	−0.676416	−	0.736520i	$-0.736468\pi$
$709$	−25539.6	−1.35283	−0.676416	+	0.736520i	$0.736468\pi$
$710$	0	0
$711$	0	0
$712$	2783.34	0.146503
$713$	20.0252	0.00105182
$714$	0	0
$715$	0	0
$716$	37192.1	1.94125
$717$	0	0
$718$	−25434.4	−1.32201
$719$	7353.45	0.381415	0.190708	−	0.981647i	$-0.438922\pi$
$719$	7353.45	0.381415	0.190708	+	0.981647i	$0.438922\pi$
$720$	0	0
$721$	21618.4	1.11666
$722$	21034.3	1.08423
$723$	0	0
$724$	46793.4	2.40202
$725$	0	0
$726$	0	0
$727$	−21696.5	−1.10685	−0.553424	−	0.832900i	$-0.686679\pi$
$727$	−21696.5	−1.10685	−0.553424	+	0.832900i	$0.686679\pi$
$728$	9207.28	0.468742
$729$	0	0
$730$	0	0
$731$	−30495.8	−1.54299
$732$	0	0
$733$	90.2714	0.00454877	0.00227439	−	0.999997i	$-0.499276\pi$
$733$	90.2714	0.00454877	0.00227439	+	0.999997i	$0.499276\pi$
$734$	−23226.1	−1.16797
$735$	0	0
$736$	−202.668	−0.0101501
$737$	22440.6	1.12159
$738$	0	0
$739$	14273.1	0.710479	0.355239	−	0.934775i	$-0.384399\pi$
$739$	14273.1	0.710479	0.355239	+	0.934775i	$0.384399\pi$
$740$	0	0
$741$	0	0
$742$	4689.02	0.231994
$743$	−15866.6	−0.783429	−0.391715	−	0.920087i	$-0.628118\pi$
$743$	−15866.6	−0.783429	−0.391715	+	0.920087i	$0.628118\pi$
$744$	0	0
$745$	0	0
$746$	−60612.2	−2.97476
$747$	0	0
$748$	−47081.5	−2.30143
$749$	−12939.3	−0.631232
$750$	0	0
$751$	26776.9	1.30107	0.650534	−	0.759477i	$-0.274545\pi$
$751$	26776.9	1.30107	0.650534	+	0.759477i	$0.274545\pi$
$752$	4521.37	0.219252
$753$	0	0
$754$	−2282.85	−0.110261
$755$	0	0
$756$	0	0
$757$	30478.0	1.46333	0.731666	−	0.681663i	$-0.238743\pi$
$757$	30478.0	1.46333	0.731666	+	0.681663i	$0.238743\pi$
$758$	44548.0	2.13464
$759$	0	0
$760$	0	0
$761$	−29104.7	−1.38639	−0.693195	−	0.720750i	$-0.743798\pi$
$761$	−29104.7	−1.38639	−0.693195	+	0.720750i	$0.743798\pi$
$762$	0	0
$763$	15968.2	0.757649
$764$	8808.47	0.417119
$765$	0	0
$766$	−27273.6	−1.28647
$767$	2220.69	0.104543
$768$	0	0
$769$	4170.65	0.195575	0.0977876	−	0.995207i	$-0.468823\pi$
$769$	4170.65	0.195575	0.0977876	+	0.995207i	$0.468823\pi$
$770$	0	0
$771$	0	0
$772$	−6151.67	−0.286792
$773$	17738.5	0.825367	0.412684	−	0.910874i	$-0.364591\pi$
$773$	17738.5	0.825367	0.412684	+	0.910874i	$0.364591\pi$
$774$	0	0
$775$	0	0
$776$	33075.0	1.53005
$777$	0	0
$778$	64786.0	2.98546
$779$	−11839.1	−0.544519
$780$	0	0
$781$	−22448.1	−1.02850
$782$	−629.363	−0.0287800
$783$	0	0
$784$	−1774.04	−0.0808145
$785$	0	0
$786$	0	0
$787$	−3807.92	−0.172475	−0.0862374	−	0.996275i	$-0.527484\pi$
$787$	−3807.92	−0.172475	−0.0862374	+	0.996275i	$0.527484\pi$
$788$	−47332.6	−2.13979
$789$	0	0
$790$	0	0
$791$	30607.3	1.37582
$792$	0	0
$793$	9447.06	0.423045
$794$	13241.8	0.591856
$795$	0	0
$796$	53594.5	2.38644
$797$	−23840.3	−1.05956	−0.529779	−	0.848136i	$-0.677725\pi$
$797$	−23840.3	−1.05956	−0.529779	+	0.848136i	$0.677725\pi$
$798$	0	0
$799$	−16980.3	−0.751841
$800$	0	0
$801$	0	0
$802$	56355.0	2.48125
$803$	40682.6	1.78787
$804$	0	0
$805$	0	0
$806$	1155.53	0.0504983
$807$	0	0
$808$	−33193.1	−1.44521
$809$	1984.22	0.0862316	0.0431158	−	0.999070i	$-0.486272\pi$
$809$	1984.22	0.0862316	0.0431158	+	0.999070i	$0.486272\pi$
$810$	0	0
$811$	−9713.78	−0.420588	−0.210294	−	0.977638i	$-0.567442\pi$
$811$	−9713.78	−0.420588	−0.210294	+	0.977638i	$0.567442\pi$
$812$	−5609.25	−0.242421
$813$	0	0
$814$	−61607.4	−2.65275
$815$	0	0
$816$	0	0
$817$	−17940.5	−0.768246
$818$	15698.8	0.671023
$819$	0	0
$820$	0	0
$821$	19235.4	0.817686	0.408843	−	0.912605i	$-0.365932\pi$
$821$	19235.4	0.817686	0.408843	+	0.912605i	$0.365932\pi$
$822$	0	0
$823$	−12717.6	−0.538650	−0.269325	−	0.963049i	$-0.586801\pi$
$823$	−12717.6	−0.538650	−0.269325	+	0.963049i	$0.586801\pi$
$824$	38279.2	1.61835
$825$	0	0
$826$	8551.41	0.360220
$827$	6744.75	0.283601	0.141800	−	0.989895i	$-0.454711\pi$
$827$	6744.75	0.283601	0.141800	+	0.989895i	$0.454711\pi$
$828$	0	0
$829$	3404.22	0.142622	0.0713108	−	0.997454i	$-0.477282\pi$
$829$	3404.22	0.142622	0.0713108	+	0.997454i	$0.477282\pi$
$830$	0	0
$831$	0	0
$832$	−15194.4	−0.633139
$833$	6662.52	0.277122
$834$	0	0
$835$	0	0
$836$	−27697.7	−1.14587
$837$	0	0
$838$	7938.69	0.327252
$839$	21361.9	0.879015	0.439508	−	0.898239i	$-0.355153\pi$
$839$	21361.9	0.879015	0.439508	+	0.898239i	$0.355153\pi$
$840$	0	0
$841$	−23787.1	−0.975319
$842$	12526.2	0.512687
$843$	0	0
$844$	33361.4	1.36060
$845$	0	0
$846$	0	0
$847$	−4632.54	−0.187929
$848$	1360.06	0.0550762
$849$	0	0
$850$	0	0
$851$	−525.488	−0.0211674
$852$	0	0
$853$	−10728.9	−0.430657	−0.215328	−	0.976542i	$-0.569082\pi$
$853$	−10728.9	−0.430657	−0.215328	+	0.976542i	$0.569082\pi$
$854$	36378.5	1.45767
$855$	0	0
$856$	−22911.4	−0.914830
$857$	42895.2	1.70977	0.854885	−	0.518817i	$-0.173627\pi$
$857$	42895.2	1.70977	0.854885	+	0.518817i	$0.173627\pi$
$858$	0	0
$859$	35530.5	1.41127	0.705637	−	0.708574i	$-0.250661\pi$
$859$	35530.5	1.41127	0.705637	+	0.708574i	$0.250661\pi$
$860$	0	0
$861$	0	0
$862$	57669.3	2.27868
$863$	−5704.35	−0.225004	−0.112502	−	0.993652i	$-0.535886\pi$
$863$	−5704.35	−0.225004	−0.112502	+	0.993652i	$0.535886\pi$
$864$	0	0
$865$	0	0
$866$	73567.0	2.88673
$867$	0	0
$868$	2839.27	0.111027
$869$	46259.6	1.80581
$870$	0	0
$871$	11045.0	0.429674
$872$	28274.4	1.09804
$873$	0	0
$874$	−370.249	−0.0143294
$875$	0	0
$876$	0	0
$877$	50249.0	1.93476	0.967382	−	0.253324i	$-0.0815237\pi$
$877$	50249.0	1.93476	0.967382	+	0.253324i	$0.0815237\pi$
$878$	−75840.7	−2.91515
$879$	0	0
$880$	0	0
$881$	26864.5	1.02734	0.513672	−	0.857987i	$-0.328285\pi$
$881$	26864.5	1.02734	0.513672	+	0.857987i	$0.328285\pi$
$882$	0	0
$883$	18942.1	0.721918	0.360959	−	0.932582i	$-0.382449\pi$
$883$	18942.1	0.721918	0.360959	+	0.932582i	$0.382449\pi$
$884$	−23173.0	−0.881667
$885$	0	0
$886$	47268.2	1.79233
$887$	25344.8	0.959409	0.479705	−	0.877430i	$-0.340744\pi$
$887$	25344.8	0.959409	0.479705	+	0.877430i	$0.340744\pi$
$888$	0	0
$889$	10063.7	0.379670
$890$	0	0
$891$	0	0
$892$	−46948.6	−1.76228
$893$	−9989.40	−0.374337
$894$	0	0
$895$	0	0
$896$	−42212.0	−1.57389
$897$	0	0
$898$	35156.9	1.30646
$899$	−304.686	−0.0113035
$900$	0	0
$901$	−5107.79	−0.188863
$902$	−45828.5	−1.69171
$903$	0	0
$904$	54195.6	1.99394
$905$	0	0
$906$	0	0
$907$	4800.11	0.175728	0.0878639	−	0.996132i	$-0.471996\pi$
$907$	4800.11	0.175728	0.0878639	+	0.996132i	$0.471996\pi$
$908$	7434.58	0.271724
$909$	0	0
$910$	0	0
$911$	25731.7	0.935819	0.467909	−	0.883776i	$-0.345007\pi$
$911$	25731.7	0.935819	0.467909	+	0.883776i	$0.345007\pi$
$912$	0	0
$913$	46559.4	1.68772
$914$	−6410.40	−0.231988
$915$	0	0
$916$	36193.4	1.30553
$917$	−41955.6	−1.51090
$918$	0	0
$919$	−12751.9	−0.457722	−0.228861	−	0.973459i	$-0.573500\pi$
$919$	−12751.9	−0.457722	−0.228861	+	0.973459i	$0.573500\pi$
$920$	0	0
$921$	0	0
$922$	24809.0	0.886163
$923$	−11048.7	−0.394012
$924$	0	0
$925$	0	0
$926$	−26987.1	−0.957722
$927$	0	0
$928$	3083.63	0.109079
$929$	−15557.8	−0.549444	−0.274722	−	0.961524i	$-0.588586\pi$
$929$	−15557.8	−0.549444	−0.274722	+	0.961524i	$0.588586\pi$
$930$	0	0
$931$	3919.51	0.137977
$932$	−78069.3	−2.74383
$933$	0	0
$934$	29306.5	1.02670
$935$	0	0
$936$	0	0
$937$	−23858.0	−0.831811	−0.415905	−	0.909408i	$-0.636535\pi$
$937$	−23858.0	−0.831811	−0.415905	+	0.909408i	$0.636535\pi$
$938$	42532.0	1.48051
$939$	0	0
$940$	0	0
$941$	−9748.00	−0.337700	−0.168850	−	0.985642i	$-0.554005\pi$
$941$	−9748.00	−0.337700	−0.168850	+	0.985642i	$0.554005\pi$
$942$	0	0
$943$	−390.899	−0.0134989
$944$	2480.35	0.0855176
$945$	0	0
$946$	−69446.4	−2.38678
$947$	51537.0	1.76845	0.884227	−	0.467057i	$-0.154686\pi$
$947$	51537.0	1.76845	0.884227	+	0.467057i	$0.154686\pi$
$948$	0	0
$949$	20023.5	0.684923
$950$	0	0
$951$	0	0
$952$	−38621.7	−1.31485
$953$	5631.36	0.191414	0.0957071	−	0.995410i	$-0.469489\pi$
$953$	5631.36	0.191414	0.0957071	+	0.995410i	$0.469489\pi$
$954$	0	0
$955$	0	0
$956$	−14245.8	−0.481948
$957$	0	0
$958$	−92565.1	−3.12176
$959$	−26847.4	−0.904013
$960$	0	0
$961$	−29636.8	−0.994823
$962$	−30322.5	−1.01625
$963$	0	0
$964$	−57466.5	−1.91999
$965$	0	0
$966$	0	0
$967$	−43360.9	−1.44198	−0.720989	−	0.692946i	$-0.756312\pi$
$967$	−43360.9	−1.44198	−0.720989	+	0.692946i	$0.756312\pi$
$968$	−8202.72	−0.272361
$969$	0	0
$970$	0	0
$971$	−12920.0	−0.427007	−0.213503	−	0.976942i	$-0.568487\pi$
$971$	−12920.0	−0.427007	−0.213503	+	0.976942i	$0.568487\pi$
$972$	0	0
$973$	2484.19	0.0818493
$974$	18595.0	0.611728
$975$	0	0
$976$	10551.7	0.346056
$977$	10650.4	0.348759	0.174379	−	0.984679i	$-0.444208\pi$
$977$	10650.4	0.348759	0.174379	+	0.984679i	$0.444208\pi$
$978$	0	0
$979$	−3899.31	−0.127296
$980$	0	0
$981$	0	0
$982$	−65321.0	−2.12269
$983$	49450.3	1.60450	0.802248	−	0.596991i	$-0.203637\pi$
$983$	49450.3	1.60450	0.802248	+	0.596991i	$0.203637\pi$
$984$	0	0
$985$	0	0
$986$	9575.85	0.309287
$987$	0	0
$988$	−13632.5	−0.438976
$989$	−592.351	−0.0190452
$990$	0	0
$991$	9410.47	0.301648	0.150824	−	0.988561i	$-0.451807\pi$
$991$	9410.47	0.301648	0.150824	+	0.988561i	$0.451807\pi$
$992$	−1560.86	−0.0499571
$993$	0	0
$994$	−42546.3	−1.35763
$995$	0	0
$996$	0	0
$997$	532.117	0.0169030	0.00845151	−	0.999964i	$-0.497310\pi$
$997$	532.117	0.0169030	0.00845151	+	0.999964i	$0.497310\pi$
$998$	−63582.6	−2.01671
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.a.i.1.1		2
3.2	odd	2		75.4.a.f.1.2		2
5.2	odd	4		45.4.b.b.19.1		4
5.3	odd	4		45.4.b.b.19.4		4
5.4	even	2		225.4.a.o.1.2		2
12.11	even	2		1200.4.a.bn.1.2		2
15.2	even	4		15.4.b.a.4.4	yes	4
15.8	even	4		15.4.b.a.4.1	✓	4
15.14	odd	2		75.4.a.c.1.1		2
20.3	even	4		720.4.f.j.289.2		4
20.7	even	4		720.4.f.j.289.1		4
60.23	odd	4		240.4.f.f.49.2		4
60.47	odd	4		240.4.f.f.49.4		4
60.59	even	2		1200.4.a.bt.1.1		2
120.53	even	4		960.4.f.q.769.1		4
120.77	even	4		960.4.f.q.769.3		4
120.83	odd	4		960.4.f.p.769.3		4
120.107	odd	4		960.4.f.p.769.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.b.a.4.1	✓	4	15.8	even	4
15.4.b.a.4.4	yes	4	15.2	even	4
45.4.b.b.19.1		4	5.2	odd	4
45.4.b.b.19.4		4	5.3	odd	4
75.4.a.c.1.1		2	15.14	odd	2
75.4.a.f.1.2		2	3.2	odd	2
225.4.a.i.1.1		2	1.1	even	1	trivial
225.4.a.o.1.2		2	5.4	even	2
240.4.f.f.49.2		4	60.23	odd	4
240.4.f.f.49.4		4	60.47	odd	4
720.4.f.j.289.1		4	20.7	even	4
720.4.f.j.289.2		4	20.3	even	4
960.4.f.p.769.1		4	120.107	odd	4
960.4.f.p.769.3		4	120.83	odd	4
960.4.f.q.769.1		4	120.53	even	4
960.4.f.q.769.3		4	120.77	even	4
1200.4.a.bn.1.2		2	12.11	even	2
1200.4.a.bt.1.1		2	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-4.70156 q^{2} +14.1047 q^{4} -16.2094 q^{7} -28.7016 q^{8} +O(q^{10})\)	\(q-4.70156 q^{2} +14.1047 q^{4} -16.2094 q^{7} -28.7016 q^{8} +40.2094 q^{11} +19.7906 q^{13} +76.2094 q^{14} +22.1047 q^{16} -83.0156 q^{17} -48.8375 q^{19} -189.047 q^{22} -1.61250 q^{23} -93.0469 q^{26} -228.628 q^{28} +24.5344 q^{29} -12.4187 q^{31} +125.686 q^{32} +390.303 q^{34} +325.884 q^{37} +229.612 q^{38} +242.419 q^{41} +367.350 q^{43} +567.141 q^{44} +7.58125 q^{46} +204.544 q^{47} -80.2562 q^{49} +279.141 q^{52} +61.5281 q^{53} +465.234 q^{56} -115.350 q^{58} +112.209 q^{59} +477.350 q^{61} +58.3875 q^{62} -767.758 q^{64} +558.094 q^{67} -1170.91 q^{68} -558.281 q^{71} +1011.77 q^{73} -1532.17 q^{74} -688.837 q^{76} -651.769 q^{77} +1150.47 q^{79} -1139.75 q^{82} +1157.92 q^{83} -1727.12 q^{86} -1154.07 q^{88} -96.9751 q^{89} -320.794 q^{91} -22.7438 q^{92} -961.675 q^{94} -1152.37 q^{97} +377.330 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 + 9 * q^4 + 6 * q^7 - 51 * q^8	\( 2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8} + 42 q^{11} + 78 q^{13} + 114 q^{14} + 25 q^{16} - 102 q^{17} + 56 q^{19} - 186 q^{22} + 48 q^{23} + 6 q^{26} - 342 q^{28} + 318 q^{29} + 52 q^{31} + 309 q^{32} + 358 q^{34} + 306 q^{37} + 408 q^{38} + 408 q^{41} + 120 q^{43} + 558 q^{44} + 92 q^{46} - 180 q^{47} + 70 q^{49} - 18 q^{52} - 402 q^{53} - 30 q^{56} + 384 q^{58} + 186 q^{59} + 340 q^{61} + 168 q^{62} - 479 q^{64} + 732 q^{67} - 1074 q^{68} + 36 q^{71} + 1332 q^{73} - 1566 q^{74} - 1224 q^{76} - 612 q^{77} + 380 q^{79} - 858 q^{82} + 984 q^{83} - 2148 q^{86} - 1194 q^{88} - 1116 q^{89} + 972 q^{91} - 276 q^{92} - 1616 q^{94} - 768 q^{97} + 633 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 + 9 * q^4 + 6 * q^7 - 51 * q^8 + 42 * q^11 + 78 * q^13 + 114 * q^14 + 25 * q^16 - 102 * q^17 + 56 * q^19 - 186 * q^22 + 48 * q^23 + 6 * q^26 - 342 * q^28 + 318 * q^29 + 52 * q^31 + 309 * q^32 + 358 * q^34 + 306 * q^37 + 408 * q^38 + 408 * q^41 + 120 * q^43 + 558 * q^44 + 92 * q^46 - 180 * q^47 + 70 * q^49 - 18 * q^52 - 402 * q^53 - 30 * q^56 + 384 * q^58 + 186 * q^59 + 340 * q^61 + 168 * q^62 - 479 * q^64 + 732 * q^67 - 1074 * q^68 + 36 * q^71 + 1332 * q^73 - 1566 * q^74 - 1224 * q^76 - 612 * q^77 + 380 * q^79 - 858 * q^82 + 984 * q^83 - 2148 * q^86 - 1194 * q^88 - 1116 * q^89 + 972 * q^91 - 276 * q^92 - 1616 * q^94 - 768 * q^97 + 633 * q^98

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