LMFDB - Embedded newform 225.4.a.o.1.2

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			1.2
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	$-0.187696\pi$
$2$	4.70156	1.66225	0.831127	+	0.556083i	$0.187696\pi$
$3$	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	$-0.355824\pi$
$7$	16.2094	0.875224	0.437612	+	0.899164i	$0.355824\pi$
$8$	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	$-0.567709\pi$
$13$	−19.7906	−0.422226	−0.211113	+	0.977462i	$0.567709\pi$
$14$	76.2094	1.45484
$15$	0	0
$16$	22.1047	0.345386
$17$	83.0156	1.18437	0.592184	−	0.805803i	$-0.298266\pi$
$17$	83.0156	1.18437	0.592184	+	0.805803i	$0.298266\pi$
$18$	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	$-0.497673\pi$
$23$	1.61250	0.0146186	0.00730932	+	0.999973i	$0.497673\pi$
$24$	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	$-0.757693\pi$
$37$	−325.884	−1.44797	−0.723987	+	0.689813i	$0.757693\pi$
$38$	−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	$-0.725818\pi$
$43$	−367.350	−1.30280	−0.651399	+	0.758735i	$0.725818\pi$
$44$	567.141	1.94317
$45$	0	0
$46$	7.58125	0.0242999
$47$	−204.544	−0.634804	−0.317402	−	0.948291i	$-0.602810\pi$
$47$	−204.544	−0.634804	−0.317402	+	0.948291i	$0.602810\pi$
$48$	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	$-0.525406\pi$
$53$	−61.5281	−0.159463	−0.0797314	+	0.996816i	$0.525406\pi$
$54$	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	$-0.669918\pi$
$67$	−558.094	−1.01764	−0.508821	+	0.860872i	$0.669918\pi$
$68$	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	$-0.801123\pi$
$73$	−1011.77	−1.62217	−0.811086	+	0.584927i	$0.801123\pi$
$74$	−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	$-0.777585\pi$
$83$	−1157.92	−1.53131	−0.765655	+	0.643251i	$0.777585\pi$
$84$	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	$-0.293923\pi$
$97$	1152.37	1.20625	0.603123	+	0.797648i	$0.293923\pi$
$98$	−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	$-0.279792\pi$
$103$	1333.70	1.27585	0.637927	+	0.770096i	$0.279792\pi$
$104$	−568.022	−0.535569
$105$	0	0
$106$	−289.278	−0.265068
$107$	−798.263	−0.721224	−0.360612	−	0.932716i	$-0.617432\pi$
$107$	−798.263	−0.721224	−0.360612	+	0.932716i	$0.617432\pi$
$108$	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	$-0.212159\pi$
$113$	1888.25	1.57196	0.785979	+	0.618253i	$0.212159\pi$
$114$	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	$-0.430406\pi$
$127$	620.859	0.433798	0.216899	+	0.976194i	$0.430406\pi$
$128$	−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	$-0.672746\pi$
$137$	−1656.29	−1.03289	−0.516447	+	0.856319i	$0.672746\pi$
$138$	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	$-0.644653\pi$
$157$	−1727.05	−0.877922	−0.438961	+	0.898506i	$0.644653\pi$
$158$	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	$-0.337448\pi$
$163$	2034.28	0.977529	0.488764	+	0.872416i	$0.337448\pi$
$164$	3419.24	1.62804
$165$	0	0
$166$	−5444.06	−2.54543
$167$	192.900	0.0893835	0.0446918	−	0.999001i	$-0.485769\pi$
$167$	192.900	0.0893835	0.0446918	+	0.999001i	$0.485769\pi$
$168$	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	$-0.587835\pi$
$173$	−1239.91	−0.544905	−0.272452	+	0.962169i	$0.587835\pi$
$174$	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	$-0.474082\pi$
$193$	436.144	0.162665	0.0813324	+	0.996687i	$0.474082\pi$
$194$	5417.96	2.00509
$195$	0	0
$196$	−1131.99	−0.412532
$197$	3355.81	1.21366	0.606831	−	0.794831i	$-0.292440\pi$
$197$	3355.81	1.21366	0.606831	+	0.794831i	$0.292440\pi$
$198$	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	$-0.333417\pi$
$223$	3328.58	0.999545	0.499772	+	0.866157i	$0.333417\pi$
$224$	−2037.29	−0.607688
$225$	0	0
$226$	8877.71	2.61299
$227$	−527.100	−0.154118	−0.0770592	−	0.997027i	$-0.524553\pi$
$227$	−527.100	−0.154118	−0.0770592	+	0.997027i	$0.524553\pi$
$228$	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	$-0.216168\pi$
$233$	5534.99	1.55626	0.778132	+	0.628101i	$0.216168\pi$
$234$	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	$-0.474369\pi$
$257$	662.784	0.160869	0.0804345	+	0.996760i	$0.474369\pi$
$258$	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	$-0.473389\pi$
$263$	712.312	0.167008	0.0835039	+	0.996507i	$0.473389\pi$
$264$	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	$-0.689515\pi$
$277$	−5171.00	−1.12164	−0.560821	+	0.827937i	$0.689515\pi$
$278$	−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	$-0.507531\pi$
$283$	−225.244	−0.0473123	−0.0236561	+	0.999720i	$0.507531\pi$
$284$	−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	$-0.463750\pi$
$293$	1139.86	0.227274	0.113637	+	0.993522i	$0.463750\pi$
$294$	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	$-0.662100\pi$
$307$	−5244.86	−0.975049	−0.487525	+	0.873109i	$0.662100\pi$
$308$	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	$-0.513992\pi$
$313$	−486.656	−0.0878832	−0.0439416	+	0.999034i	$0.513992\pi$
$314$	−8119.85	−1.45933
$315$	0	0
$316$	16227.0	2.88873
$317$	−4218.87	−0.747493	−0.373747	−	0.927531i	$-0.621927\pi$
$317$	−4218.87	−0.747493	−0.373747	+	0.927531i	$0.621927\pi$
$318$	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	$-0.322261\pi$
$337$	6555.39	1.05963	0.529815	+	0.848113i	$0.322261\pi$
$338$	−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	$-0.451789\pi$
$347$	1950.56	0.301763	0.150881	+	0.988552i	$0.451789\pi$
$348$	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	$-0.679190\pi$
$353$	−7078.96	−1.06735	−0.533676	+	0.845689i	$0.679190\pi$
$354$	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	$-0.614268\pi$
$367$	−4940.09	−0.702645	−0.351322	+	0.936255i	$0.614268\pi$
$368$	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	$-0.852680\pi$
$373$	−12891.9	−1.78959	−0.894797	+	0.446473i	$0.852680\pi$
$374$	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	$-0.626477\pi$
$383$	−5800.97	−0.773931	−0.386966	+	0.922094i	$0.626477\pi$
$384$	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	$-0.443028\pi$
$397$	2816.46	0.356056	0.178028	+	0.984025i	$0.443028\pi$
$398$	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	$-0.165201\pi$
$433$	15647.3	1.73664	0.868318	+	0.496008i	$0.165201\pi$
$434$	−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	$-0.318754\pi$
$443$	10053.7	1.07825	0.539127	+	0.842225i	$0.318754\pi$
$444$	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	$-0.522230\pi$
$457$	−1363.46	−0.139562	−0.0697812	+	0.997562i	$0.522230\pi$
$458$	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	$-0.593017\pi$
$463$	−5740.02	−0.576159	−0.288079	+	0.957607i	$0.593017\pi$
$464$	542.325	0.0542604
$465$	0	0
$466$	26023.1	2.58690
$467$	6233.36	0.617657	0.308828	−	0.951118i	$-0.400063\pi$
$467$	6233.36	0.617657	0.308828	+	0.951118i	$0.400063\pi$
$468$	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	$-0.441093\pi$
$487$	3955.08	0.368012	0.184006	+	0.982925i	$0.441093\pi$
$488$	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	$-0.302198\pi$
$503$	13135.4	1.16437	0.582184	+	0.813057i	$0.302198\pi$
$504$	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	$-0.766002\pi$
$523$	−17743.4	−1.48349	−0.741746	+	0.670681i	$0.766002\pi$
$524$	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	$-0.565791\pi$
$547$	−5250.90	−0.410443	−0.205221	+	0.978716i	$0.565791\pi$
$548$	−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	$-0.0597032\pi$
$557$	25830.2	1.96492	0.982462	+	0.186465i	$0.0597032\pi$
$558$	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	$-0.475897\pi$
$563$	2021.14	0.151298	0.0756490	+	0.997135i	$0.475897\pi$
$564$	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	$-0.314380\pi$
$577$	15264.0	1.10130	0.550649	+	0.834737i	$0.314380\pi$
$578$	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	$-0.596093\pi$
$587$	−8456.89	−0.594639	−0.297319	+	0.954778i	$0.596093\pi$
$588$	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	$-0.513508\pi$
$593$	−1225.23	−0.0848467	−0.0424234	+	0.999100i	$0.513508\pi$
$594$	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	$-0.790485\pi$
$607$	−23661.2	−1.58217	−0.791087	+	0.611703i	$0.790485\pi$
$608$	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	$-0.414174\pi$
$613$	8085.63	0.532749	0.266375	+	0.963870i	$0.414174\pi$
$614$	−24659.0	−1.62078
$615$	0	0
$616$	18706.8	1.22357
$617$	−11035.1	−0.720029	−0.360014	−	0.932947i	$-0.617228\pi$
$617$	−11035.1	−0.720029	−0.360014	+	0.932947i	$0.617228\pi$
$618$	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	$-0.271113\pi$
$643$	21479.5	1.31737	0.658685	+	0.752419i	$0.271113\pi$
$644$	368.662	0.0225580
$645$	0	0
$646$	−19061.4	−1.16093
$647$	27119.7	1.64789	0.823946	−	0.566668i	$-0.191768\pi$
$647$	27119.7	1.64789	0.823946	+	0.566668i	$0.191768\pi$
$648$	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	$-0.313243\pi$
$653$	18476.4	1.10725	0.553627	+	0.832765i	$0.313243\pi$
$654$	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	$-0.572228\pi$
$673$	−7855.52	−0.449938	−0.224969	+	0.974366i	$0.572228\pi$
$674$	30820.6	1.76137
$675$	0	0
$676$	−25463.6	−1.44877
$677$	−6763.09	−0.383939	−0.191970	−	0.981401i	$-0.561488\pi$
$677$	−6763.09	−0.383939	−0.191970	+	0.981401i	$0.561488\pi$
$678$	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	$-0.355962\pi$
$683$	15608.6	0.874447	0.437224	+	0.899353i	$0.355962\pi$
$684$	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	$-0.313321\pi$
$727$	21696.5	1.10685	0.553424	+	0.832900i	$0.313321\pi$
$728$	−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	$-0.500724\pi$
$733$	−90.2714	−0.00454877	−0.00227439	+	0.999997i	$0.500724\pi$
$734$	−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	$-0.371882\pi$
$743$	15866.6	0.783429	0.391715	+	0.920087i	$0.371882\pi$
$744$	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	$-0.761257\pi$
$757$	−30478.0	−1.46333	−0.731666	+	0.681663i	$0.761257\pi$
$758$	−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	$-0.635409\pi$
$773$	−17738.5	−0.825367	−0.412684	+	0.910874i	$0.635409\pi$
$774$	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	$-0.472516\pi$
$787$	3807.92	0.172475	0.0862374	+	0.996275i	$0.472516\pi$
$788$	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	$-0.322275\pi$
$797$	23840.3	1.05956	0.529779	+	0.848136i	$0.322275\pi$
$798$	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	$-0.413199\pi$
$823$	12717.6	0.538650	0.269325	+	0.963049i	$0.413199\pi$
$824$	38279.2	1.61835
$825$	0	0
$826$	8551.41	0.360220
$827$	−6744.75	−0.283601	−0.141800	−	0.989895i	$-0.545289\pi$
$827$	−6744.75	−0.283601	−0.141800	+	0.989895i	$0.545289\pi$
$828$	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	$-0.430918\pi$
$853$	10728.9	0.430657	0.215328	+	0.976542i	$0.430918\pi$
$854$	36378.5	1.45767
$855$	0	0
$856$	−22911.4	−0.914830
$857$	−42895.2	−1.70977	−0.854885	−	0.518817i	$-0.826373\pi$
$857$	−42895.2	−1.70977	−0.854885	+	0.518817i	$0.826373\pi$
$858$	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	$-0.464114\pi$
$863$	5704.35	0.225004	0.112502	+	0.993652i	$0.464114\pi$
$864$	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	$-0.918476\pi$
$877$	−50249.0	−1.93476	−0.967382	+	0.253324i	$0.918476\pi$
$878$	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	$-0.617551\pi$
$883$	−18942.1	−0.721918	−0.360959	+	0.932582i	$0.617551\pi$
$884$	−23173.0	−0.881667
$885$	0	0
$886$	47268.2	1.79233
$887$	−25344.8	−0.959409	−0.479705	−	0.877430i	$-0.659256\pi$
$887$	−25344.8	−0.959409	−0.479705	+	0.877430i	$0.659256\pi$
$888$	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	$-0.528004\pi$
$907$	−4800.11	−0.175728	−0.0878639	+	0.996132i	$0.528004\pi$
$908$	−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	$-0.363465\pi$
$937$	23858.0	0.831811	0.415905	+	0.909408i	$0.363465\pi$
$938$	−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	$-0.845314\pi$
$947$	−51537.0	−1.76845	−0.884227	+	0.467057i	$0.845314\pi$
$948$	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	$-0.530511\pi$
$953$	−5631.36	−0.191414	−0.0957071	+	0.995410i	$0.530511\pi$
$954$	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	$-0.243688\pi$
$967$	43360.9	1.44198	0.720989	+	0.692946i	$0.243688\pi$
$968$	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	$-0.555792\pi$
$977$	−10650.4	−0.348759	−0.174379	+	0.984679i	$0.555792\pi$
$978$	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	$-0.796363\pi$
$983$	−49450.3	−1.60450	−0.802248	+	0.596991i	$0.796363\pi$
$984$	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	$-0.502690\pi$
$997$	−532.117	−0.0169030	−0.00845151	+	0.999964i	$0.502690\pi$
$998$	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.o.1.2		2
3.2	odd	2		75.4.a.c.1.1		2
5.2	odd	4		45.4.b.b.19.4		4
5.3	odd	4		45.4.b.b.19.1		4
5.4	even	2		225.4.a.i.1.1		2
12.11	even	2		1200.4.a.bt.1.1		2
15.2	even	4		15.4.b.a.4.1	✓	4
15.8	even	4		15.4.b.a.4.4	yes	4
15.14	odd	2		75.4.a.f.1.2		2
20.3	even	4		720.4.f.j.289.1		4
20.7	even	4		720.4.f.j.289.2		4
60.23	odd	4		240.4.f.f.49.4		4
60.47	odd	4		240.4.f.f.49.2		4
60.59	even	2		1200.4.a.bn.1.2		2
120.53	even	4		960.4.f.q.769.3		4
120.77	even	4		960.4.f.q.769.1		4
120.83	odd	4		960.4.f.p.769.1		4
120.107	odd	4		960.4.f.p.769.3		4

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