LMFDB - Embedded newform 225.6.b.k.199.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [225,6,Mod(199,225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("225.199"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	225.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-42,0,0,0,0,0,0,1600] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$36.0863594579$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{145})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 73x^{2} + 1296 $ x^4 + 73*x^2 + 1296
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$-6.52080i$ of defining polynomial
Character	$\chi$	$=$	225.199
Dual form			225.6.b.k.199.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-8.52080i q^{2} -40.6040 q^{4} -160.416i q^{7} +73.3128i q^{8} +279.584 q^{11} -541.664i q^{13} -1366.87 q^{14} -674.644 q^{16} -777.334i q^{17} +2682.66 q^{19} -2382.28i q^{22} -3694.48i q^{23} -4615.41 q^{26} +6513.53i q^{28} -8356.62 q^{29} -262.849 q^{31} +8094.51i q^{32} -6623.51 q^{34} +14949.9i q^{37} -22858.4i q^{38} -7988.28 q^{41} +5133.38i q^{43} -11352.2 q^{44} -31479.9 q^{46} -10567.8i q^{47} -8926.28 q^{49} +21993.7i q^{52} +21069.4i q^{53} +11760.5 q^{56} +71205.1i q^{58} -25699.6 q^{59} +8195.14 q^{61} +2239.68i q^{62} +47383.1 q^{64} -51928.9i q^{67} +31562.9i q^{68} +23689.2 q^{71} +20933.5i q^{73} +127385. q^{74} -108926. q^{76} -44849.7i q^{77} +39279.7 q^{79} +68066.6i q^{82} +35911.1i q^{83} +43740.5 q^{86} +20497.1i q^{88} +94074.7 q^{89} -86891.5 q^{91} +150011. i q^{92} -90046.3 q^{94} +12249.9i q^{97} +76059.0i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 42 q^{4} + 1600 q^{11} - 3300 q^{14} - 1374 q^{16} + 3024 q^{19} - 12200 q^{26} - 2600 q^{29} - 11648 q^{31} - 5060 q^{34} + 800 q^{41} - 2300 q^{44} - 79680 q^{46} + 2828 q^{49} - 5700 q^{56} - 126400 q^{59}+ \cdots - 121280 q^{94}+O(q^{100}) $ 4 * q - 42 * q^4 + 1600 * q^11 - 3300 * q^14 - 1374 * q^16 + 3024 * q^19 - 12200 * q^26 - 2600 * q^29 - 11648 * q^31 - 5060 * q^34 + 800 * q^41 - 2300 * q^44 - 79680 * q^46 + 2828 * q^49 - 5700 * q^56 - 126400 * q^59 - 98232 * q^61 + 53342 * q^64 + 130400 * q^71 + 271600 * q^74 - 263752 * q^76 + 92576 * q^79 - 57200 * q^86 + 174000 * q^89 - 241600 * q^91 - 121280 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	− 8.52080i	− 1.50628i	−0.657861	−	0.753139i	$-0.728539\pi$
$2$	− 8.52080i	− 1.50628i	0.657861	−	0.753139i	$-0.271461\pi$
$3$	0	0
$3$	0	0
$4$	−40.6040	−1.26887
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 160.416i	− 1.23738i	−0.785636	−	0.618689i	$-0.787664\pi$
$7$	− 160.416i	− 1.23738i	0.785636	−	0.618689i	$-0.212336\pi$
$8$	73.3128i	0.405000i
$9$	0	0
$10$	0	0
$11$	279.584	0.696676	0.348338	−	0.937369i	$-0.386746\pi$
$11$	279.584	0.696676	0.348338	+	0.937369i	$0.386746\pi$
$12$	0	0
$13$	− 541.664i	− 0.888938i	−0.895794	−	0.444469i	$-0.853392\pi$
$13$	− 541.664i	− 0.888938i	0.895794	−	0.444469i	$-0.146608\pi$
$14$	−1366.87	−1.86384
$15$	0	0
$16$	−674.644	−0.658832
$17$	− 777.334i	− 0.652357i	−0.945308	−	0.326179i	$-0.894239\pi$
$17$	− 777.334i	− 0.652357i	0.945308	−	0.326179i	$-0.105761\pi$
$18$	0	0
$19$	2682.66	1.70483	0.852415	−	0.522867i	$-0.175137\pi$
$19$	2682.66	1.70483	0.852415	+	0.522867i	$0.175137\pi$
$20$	0	0
$21$	0	0
$22$	− 2382.28i	− 1.04939i
$23$	− 3694.48i	− 1.45624i	−0.685448	−	0.728122i	$-0.740394\pi$
$23$	− 3694.48i	− 1.45624i	0.685448	−	0.728122i	$-0.259606\pi$
$24$	0	0
$25$	0	0
$26$	−4615.41	−1.33899
$27$	0	0
$28$	6513.53i	1.57008i
$29$	−8356.62	−1.84517	−0.922584	−	0.385797i	$-0.873926\pi$
$29$	−8356.62	−1.84517	−0.922584	+	0.385797i	$0.873926\pi$
$30$	0	0
$31$	−262.849	−0.0491250	−0.0245625	−	0.999698i	$-0.507819\pi$
$31$	−262.849	−0.0491250	−0.0245625	+	0.999698i	$0.507819\pi$
$32$	8094.51i	1.39738i
$33$	0	0
$34$	−6623.51	−0.982632
$35$	0	0
$36$	0	0
$37$	14949.9i	1.79529i	0.440716	+	0.897647i	$0.354725\pi$
$37$	14949.9i	1.79529i	−0.440716	+	0.897647i	$0.645275\pi$
$38$	− 22858.4i	− 2.56795i
$39$	0	0
$40$	0	0
$41$	−7988.28	−0.742154	−0.371077	−	0.928602i	$-0.621011\pi$
$41$	−7988.28	−0.742154	−0.371077	+	0.928602i	$0.621011\pi$
$42$	0	0
$43$	5133.38i	0.423382i	0.977337	+	0.211691i	$0.0678970\pi$
$43$	5133.38i	0.423382i	−0.977337	+	0.211691i	$0.932103\pi$
$44$	−11352.2	−0.883994
$45$	0	0
$46$	−31479.9	−2.19351
$47$	− 10567.8i	− 0.697816i	−0.937157	−	0.348908i	$-0.886553\pi$
$47$	− 10567.8i	− 0.697816i	0.937157	−	0.348908i	$-0.113447\pi$
$48$	0	0
$49$	−8926.28	−0.531105
$50$	0	0
$51$	0	0
$52$	21993.7i	1.12795i
$53$	21069.4i	1.03029i	0.857102	+	0.515147i	$0.172263\pi$
$53$	21069.4i	1.03029i	−0.857102	+	0.515147i	$0.827737\pi$
$54$	0	0
$55$	0	0
$56$	11760.5	0.501138
$57$	0	0
$58$	71205.1i	2.77934i
$59$	−25699.6	−0.961162	−0.480581	−	0.876950i	$-0.659574\pi$
$59$	−25699.6	−0.961162	−0.480581	+	0.876950i	$0.659574\pi$
$60$	0	0
$61$	8195.14	0.281989	0.140994	−	0.990010i	$-0.454970\pi$
$61$	8195.14	0.281989	0.140994	+	0.990010i	$0.454970\pi$
$62$	2239.68i	0.0739959i
$63$	0	0
$64$	47383.1	1.44602
$65$	0	0
$66$	0	0
$67$	− 51928.9i	− 1.41326i	−0.707584	−	0.706630i	$-0.750215\pi$
$67$	− 51928.9i	− 1.41326i	0.707584	−	0.706630i	$-0.249785\pi$
$68$	31562.9i	0.827760i
$69$	0	0
$70$	0	0
$71$	23689.2	0.557705	0.278853	−	0.960334i	$-0.410046\pi$
$71$	23689.2	0.557705	0.278853	+	0.960334i	$0.410046\pi$
$72$	0	0
$73$	20933.5i	0.459764i	0.973219	+	0.229882i	$0.0738340\pi$
$73$	20933.5i	0.459764i	−0.973219	+	0.229882i	$0.926166\pi$
$74$	127385.	2.70421
$75$	0	0
$76$	−108926.	−2.16321
$77$	− 44849.7i	− 0.862051i
$78$	0	0
$79$	39279.7	0.708110	0.354055	−	0.935225i	$-0.384803\pi$
$79$	39279.7	0.708110	0.354055	+	0.935225i	$0.384803\pi$
$80$	0	0
$81$	0	0
$82$	68066.6i	1.11789i
$83$	35911.1i	0.572181i	0.958203	+	0.286091i	$0.0923558\pi$
$83$	35911.1i	0.572181i	−0.958203	+	0.286091i	$0.907644\pi$
$84$	0	0
$85$	0	0
$86$	43740.5	0.637731
$87$	0	0
$88$	20497.1i	0.282154i
$89$	94074.7	1.25892	0.629460	−	0.777033i	$-0.283276\pi$
$89$	94074.7	1.25892	0.629460	+	0.777033i	$0.283276\pi$
$90$	0	0
$91$	−86891.5	−1.09995
$92$	150011.i	1.84779i
$93$	0	0
$94$	−90046.3	−1.05111
$95$	0	0
$96$	0	0
$97$	12249.9i	0.132191i	0.997813	+	0.0660957i	$0.0210543\pi$
$97$	12249.9i	0.132191i	−0.997813	+	0.0660957i	$0.978946\pi$
$98$	76059.0i	0.799991i
$99$	0	0
$100$	0	0
$101$	−159360.	−1.55445	−0.777223	−	0.629226i	$-0.783372\pi$
$101$	−159360.	−1.55445	−0.777223	+	0.629226i	$0.783372\pi$
$102$	0	0
$103$	74393.5i	0.690942i	0.938429	+	0.345471i	$0.112281\pi$
$103$	74393.5i	0.690942i	−0.938429	+	0.345471i	$0.887719\pi$
$104$	39710.9	0.360020
$105$	0	0
$106$	179528.	1.55191
$107$	− 170203.i	− 1.43717i	−0.695439	−	0.718585i	$-0.744790\pi$
$107$	− 170203.i	− 1.43717i	0.695439	−	0.718585i	$-0.255210\pi$
$108$	0	0
$109$	−54355.6	−0.438206	−0.219103	−	0.975702i	$-0.570313\pi$
$109$	−54355.6	−0.438206	−0.219103	+	0.975702i	$0.570313\pi$
$110$	0	0
$111$	0	0
$112$	108224.i	0.815224i
$113$	− 9340.66i	− 0.0688148i	−0.999408	−	0.0344074i	$-0.989046\pi$
$113$	− 9340.66i	− 0.0688148i	0.999408	−	0.0344074i	$-0.0109544\pi$
$114$	0	0
$115$	0	0
$116$	339312.	2.34129
$117$	0	0
$118$	218981.i	1.44778i
$119$	−124697.	−0.807213
$120$	0	0
$121$	−82883.8	−0.514643
$122$	− 69829.1i	− 0.424753i
$123$	0	0
$124$	10672.7	0.0623334
$125$	0	0
$126$	0	0
$127$	22872.4i	0.125835i	0.998019	+	0.0629176i	$0.0200405\pi$
$127$	22872.4i	0.125835i	−0.998019	+	0.0629176i	$0.979959\pi$
$128$	− 144717.i	− 0.780721i
$129$	0	0
$130$	0	0
$131$	−345444.	−1.75873	−0.879366	−	0.476146i	$-0.842033\pi$
$131$	−345444.	−1.75873	−0.879366	+	0.476146i	$0.842033\pi$
$132$	0	0
$133$	− 430341.i	− 2.10952i
$134$	−442475.	−2.12876
$135$	0	0
$136$	56988.6	0.264205
$137$	170270.i	0.775064i	0.921856	+	0.387532i	$0.126672\pi$
$137$	170270.i	0.775064i	−0.921856	+	0.387532i	$0.873328\pi$
$138$	0	0
$139$	268230.	1.17753	0.588763	−	0.808305i	$-0.299615\pi$
$139$	268230.	1.17753	0.588763	+	0.808305i	$0.299615\pi$
$140$	0	0
$141$	0	0
$142$	− 201851.i	− 0.840060i
$143$	− 151441.i	− 0.619301i
$144$	0	0
$145$	0	0
$146$	178370.	0.692532
$147$	0	0
$148$	− 607027.i	− 2.27800i
$149$	−293424.	−1.08276	−0.541378	−	0.840779i	$-0.682097\pi$
$149$	−293424.	−1.08276	−0.541378	+	0.840779i	$0.682097\pi$
$150$	0	0
$151$	365971.	1.30618	0.653091	−	0.757279i	$-0.273472\pi$
$151$	365971.	1.30618	0.653091	+	0.757279i	$0.273472\pi$
$152$	196673.i	0.690456i
$153$	0	0
$154$	−382156.	−1.29849
$155$	0	0
$156$	0	0
$157$	− 503049.i	− 1.62878i	−0.580320	−	0.814388i	$-0.697073\pi$
$157$	− 503049.i	− 1.62878i	0.580320	−	0.814388i	$-0.302927\pi$
$158$	− 334695.i	− 1.06661i
$159$	0	0
$160$	0	0
$161$	−592654.	−1.80192
$162$	0	0
$163$	181660.i	0.535538i	0.963483	+	0.267769i	$0.0862864\pi$
$163$	181660.i	0.535538i	−0.963483	+	0.267769i	$0.913714\pi$
$164$	324356.	0.941700
$165$	0	0
$166$	305991.	0.861864
$167$	− 440148.i	− 1.22126i	−0.791917	−	0.610629i	$-0.790917\pi$
$167$	− 440148.i	− 1.22126i	0.791917	−	0.610629i	$-0.209083\pi$
$168$	0	0
$169$	77893.3	0.209789
$170$	0	0
$171$	0	0
$172$	− 208436.i	− 0.537218i
$173$	− 552730.i	− 1.40410i	−0.712129	−	0.702049i	$-0.752269\pi$
$173$	− 552730.i	− 1.40410i	0.712129	−	0.702049i	$-0.247731\pi$
$174$	0	0
$175$	0	0
$176$	−188620.	−0.458992
$177$	0	0
$178$	− 801591.i	− 1.89628i
$179$	−190220.	−0.443735	−0.221867	−	0.975077i	$-0.571215\pi$
$179$	−190220.	−0.443735	−0.221867	+	0.975077i	$0.571215\pi$
$180$	0	0
$181$	113139.	0.256694	0.128347	−	0.991729i	$-0.459033\pi$
$181$	113139.	0.256694	0.128347	+	0.991729i	$0.459033\pi$
$182$	740385.i	1.65683i
$183$	0	0
$184$	270853.	0.589778
$185$	0	0
$186$	0	0
$187$	− 217330.i	− 0.454482i
$188$	429096.i	0.885441i
$189$	0	0
$190$	0	0
$191$	694078.	1.37665	0.688327	−	0.725401i	$-0.258346\pi$
$191$	694078.	1.37665	0.688327	+	0.725401i	$0.258346\pi$
$192$	0	0
$193$	− 51802.2i	− 0.100105i	−0.998747	−	0.0500524i	$-0.984061\pi$
$193$	− 51802.2i	− 0.100105i	0.998747	−	0.0500524i	$-0.0159388\pi$
$194$	104379.	0.199117
$195$	0	0
$196$	362442.	0.673905
$197$	− 217963.i	− 0.400145i	−0.979781	−	0.200073i	$-0.935882\pi$
$197$	− 217963.i	− 0.400145i	0.979781	−	0.200073i	$-0.0641178\pi$
$198$	0	0
$199$	385135.	0.689414	0.344707	−	0.938710i	$-0.387978\pi$
$199$	385135.	0.689414	0.344707	+	0.938710i	$0.387978\pi$
$200$	0	0
$201$	0	0
$202$	1.35787e6i	2.34143i
$203$	1.34054e6i	2.28317i
$204$	0	0
$205$	0	0
$206$	633892.	1.04075
$207$	0	0
$208$	365430.i	0.585661i
$209$	750028.	1.18771
$210$	0	0
$211$	300290.	0.464339	0.232170	−	0.972675i	$-0.425418\pi$
$211$	300290.	0.464339	0.232170	+	0.972675i	$0.425418\pi$
$212$	− 855500.i	− 1.30732i
$213$	0	0
$214$	−1.45027e6	−2.16478
$215$	0	0
$216$	0	0
$217$	42165.2i	0.0607862i
$218$	463153.i	0.660060i
$219$	0	0
$220$	0	0
$221$	−421054.	−0.579905
$222$	0	0
$223$	− 409580.i	− 0.551539i	−0.961224	−	0.275769i	$-0.911067\pi$
$223$	− 409580.i	− 0.551539i	0.961224	−	0.275769i	$-0.0889326\pi$
$224$	1.29849e6	1.72909
$225$	0	0
$226$	−79589.9	−0.103654
$227$	− 386331.i	− 0.497617i	−0.968553	−	0.248808i	$-0.919961\pi$
$227$	− 386331.i	− 0.497617i	0.968553	−	0.248808i	$-0.0800389\pi$
$228$	0	0
$229$	−360206.	−0.453902	−0.226951	−	0.973906i	$-0.572876\pi$
$229$	−360206.	−0.453902	−0.226951	+	0.973906i	$0.572876\pi$
$230$	0	0
$231$	0	0
$232$	− 612647.i	− 0.747293i
$233$	376766.i	0.454655i	0.973818	+	0.227327i	$0.0729988\pi$
$233$	376766.i	0.454655i	−0.973818	+	0.227327i	$0.927001\pi$
$234$	0	0
$235$	0	0
$236$	1.04351e6	1.21959
$237$	0	0
$238$	1.06252e6i	1.21589i
$239$	−416917.	−0.472123	−0.236061	−	0.971738i	$-0.575857\pi$
$239$	−416917.	−0.472123	−0.236061	+	0.971738i	$0.575857\pi$
$240$	0	0
$241$	−1.15082e6	−1.27634	−0.638168	−	0.769897i	$-0.720308\pi$
$241$	−1.15082e6	−1.27634	−0.638168	+	0.769897i	$0.720308\pi$
$242$	706236.i	0.775196i
$243$	0	0
$244$	−332755.	−0.357808
$245$	0	0
$246$	0	0
$247$	− 1.45310e6i	− 1.51549i
$248$	− 19270.2i	− 0.0198956i
$249$	0	0
$250$	0	0
$251$	642245.	0.643452	0.321726	−	0.946833i	$-0.395737\pi$
$251$	642245.	0.643452	0.321726	+	0.946833i	$0.395737\pi$
$252$	0	0
$253$	− 1.03292e6i	− 1.01453i
$254$	194891.	0.189543
$255$	0	0
$256$	283152.	0.270034
$257$	126474.i	0.119446i	0.998215	+	0.0597228i	$0.0190217\pi$
$257$	126474.i	0.119446i	−0.998215	+	0.0597228i	$0.980978\pi$
$258$	0	0
$259$	2.39821e6	2.22146
$260$	0	0
$261$	0	0
$262$	2.94346e6i	2.64914i
$263$	366879.i	0.327064i	0.986538	+	0.163532i	$0.0522887\pi$
$263$	366879.i	0.327064i	−0.986538	+	0.163532i	$0.947711\pi$
$264$	0	0
$265$	0	0
$266$	−3.66685e6	−3.17752
$267$	0	0
$268$	2.10852e6i	1.79325i
$269$	1.21165e6	1.02093	0.510465	−	0.859899i	$-0.329473\pi$
$269$	1.21165e6	1.02093	0.510465	+	0.859899i	$0.329473\pi$
$270$	0	0
$271$	−1.79322e6	−1.48324	−0.741619	−	0.670821i	$-0.765942\pi$
$271$	−1.79322e6	−1.48324	−0.741619	+	0.670821i	$0.765942\pi$
$272$	524424.i	0.429794i
$273$	0	0
$274$	1.45084e6	1.16746
$275$	0	0
$276$	0	0
$277$	− 1.46441e6i	− 1.14674i	−0.819297	−	0.573369i	$-0.805636\pi$
$277$	− 1.46441e6i	− 1.14674i	0.819297	−	0.573369i	$-0.194364\pi$
$278$	− 2.28554e6i	− 1.77368i
$279$	0	0
$280$	0	0
$281$	2.12044e6	1.60199	0.800997	−	0.598669i	$-0.204304\pi$
$281$	2.12044e6	1.60199	0.800997	+	0.598669i	$0.204304\pi$
$282$	0	0
$283$	454281.i	0.337177i	0.985687	+	0.168589i	$0.0539209\pi$
$283$	454281.i	0.337177i	−0.985687	+	0.168589i	$0.946079\pi$
$284$	−961877.	−0.707658
$285$	0	0
$286$	−1.29039e6	−0.932840
$287$	1.28145e6i	0.918325i
$288$	0	0
$289$	815608.	0.574430
$290$	0	0
$291$	0	0
$292$	− 849984.i	− 0.583383i
$293$	− 1.93287e6i	− 1.31533i	−0.753312	−	0.657664i	$-0.771545\pi$
$293$	− 1.93287e6i	− 1.31533i	0.753312	−	0.657664i	$-0.228455\pi$
$294$	0	0
$295$	0	0
$296$	−1.09602e6	−0.727094
$297$	0	0
$298$	2.50021e6i	1.63093i
$299$	−2.00117e6	−1.29451
$300$	0	0
$301$	823476.	0.523883
$302$	− 3.11836e6i	− 1.96747i
$303$	0	0
$304$	−1.80984e6	−1.12320
$305$	0	0
$306$	0	0
$307$	− 224205.i	− 0.135768i	−0.997693	−	0.0678842i	$-0.978375\pi$
$307$	− 224205.i	− 0.135768i	0.997693	−	0.0678842i	$-0.0216248\pi$
$308$	1.82108e6i	1.09384i
$309$	0	0
$310$	0	0
$311$	−592091.	−0.347126	−0.173563	−	0.984823i	$-0.555528\pi$
$311$	−592091.	−0.347126	−0.173563	+	0.984823i	$0.555528\pi$
$312$	0	0
$313$	1.99079e6i	1.14859i	0.818648	+	0.574295i	$0.194724\pi$
$313$	1.99079e6i	1.14859i	−0.818648	+	0.574295i	$0.805276\pi$
$314$	−4.28638e6	−2.45339
$315$	0	0
$316$	−1.59491e6	−0.898503
$317$	2.89884e6i	1.62023i	0.586274	+	0.810113i	$0.300594\pi$
$317$	2.89884e6i	1.62023i	−0.586274	+	0.810113i	$0.699406\pi$
$318$	0	0
$319$	−2.33638e6	−1.28548
$320$	0	0
$321$	0	0
$322$	5.04988e6i	2.71420i
$323$	− 2.08532e6i	− 1.11216i
$324$	0	0
$325$	0	0
$326$	1.54789e6	0.806669
$327$	0	0
$328$	− 585644.i	− 0.300572i
$329$	−1.69525e6	−0.863462
$330$	0	0
$331$	416722.	0.209063	0.104531	−	0.994522i	$-0.466666\pi$
$331$	416722.	0.209063	0.104531	+	0.994522i	$0.466666\pi$
$332$	− 1.45813e6i	− 0.726026i
$333$	0	0
$334$	−3.75041e6	−1.83955
$335$	0	0
$336$	0	0
$337$	− 212393.i	− 0.101874i	−0.998702	−	0.0509371i	$-0.983779\pi$
$337$	− 212393.i	− 0.101874i	0.998702	−	0.0509371i	$-0.0162208\pi$
$338$	− 663713.i	− 0.316001i
$339$	0	0
$340$	0	0
$341$	−73488.4	−0.0342242
$342$	0	0
$343$	− 1.26419e6i	− 0.580201i
$344$	−376343.	−0.171470
$345$	0	0
$346$	−4.70970e6	−2.11496
$347$	− 1.58972e6i	− 0.708755i	−0.935103	−	0.354377i	$-0.884693\pi$
$347$	− 1.58972e6i	− 0.708755i	0.935103	−	0.354377i	$-0.115307\pi$
$348$	0	0
$349$	3.98771e6	1.75251	0.876254	−	0.481850i	$-0.160035\pi$
$349$	3.98771e6	1.75251	0.876254	+	0.481850i	$0.160035\pi$
$350$	0	0
$351$	0	0
$352$	2.26310e6i	0.973524i
$353$	− 585594.i	− 0.250127i	−0.992149	−	0.125063i	$-0.960087\pi$
$353$	− 585594.i	− 0.250127i	0.992149	−	0.125063i	$-0.0399134\pi$
$354$	0	0
$355$	0	0
$356$	−3.81981e6	−1.59741
$357$	0	0
$358$	1.62082e6i	0.668388i
$359$	4.28470e6	1.75463	0.877313	−	0.479918i	$-0.159334\pi$
$359$	4.28470e6	1.75463	0.877313	+	0.479918i	$0.159334\pi$
$360$	0	0
$361$	4.72054e6	1.90644
$362$	− 964033.i	− 0.386652i
$363$	0	0
$364$	3.52814e6	1.39570
$365$	0	0
$366$	0	0
$367$	524307.i	0.203198i	0.994825	+	0.101599i	$0.0323959\pi$
$367$	524307.i	0.203198i	−0.994825	+	0.101599i	$0.967604\pi$
$368$	2.49246e6i	0.959420i
$369$	0	0
$370$	0	0
$371$	3.37986e6	1.27486
$372$	0	0
$373$	− 1.16607e6i	− 0.433964i	−0.976176	−	0.216982i	$-0.930379\pi$
$373$	− 1.16607e6i	− 0.433964i	0.976176	−	0.216982i	$-0.0696213\pi$
$374$	−1.85183e6	−0.684576
$375$	0	0
$376$	774757.	0.282616
$377$	4.52648e6i	1.64024i
$378$	0	0
$379$	−2.48592e6	−0.888973	−0.444487	−	0.895786i	$-0.646614\pi$
$379$	−2.48592e6	−0.888973	−0.444487	+	0.895786i	$0.646614\pi$
$380$	0	0
$381$	0	0
$382$	− 5.91409e6i	− 2.07362i
$383$	1.52361e6i	0.530732i	0.964148	+	0.265366i	$0.0854929\pi$
$383$	1.52361e6i	0.530732i	−0.964148	+	0.265366i	$0.914507\pi$
$384$	0	0
$385$	0	0
$386$	−441396.	−0.150786
$387$	0	0
$388$	− 497395.i	− 0.167734i
$389$	4.59972e6	1.54120	0.770598	−	0.637322i	$-0.219958\pi$
$389$	4.59972e6	1.54120	0.770598	+	0.637322i	$0.219958\pi$
$390$	0	0
$391$	−2.87185e6	−0.949991
$392$	− 654410.i	− 0.215097i
$393$	0	0
$394$	−1.85722e6	−0.602730
$395$	0	0
$396$	0	0
$397$	− 3.54374e6i	− 1.12846i	−0.825618	−	0.564229i	$-0.809173\pi$
$397$	− 3.54374e6i	− 1.12846i	0.825618	−	0.564229i	$-0.190827\pi$
$398$	− 3.28165e6i	− 1.03845i
$399$	0	0
$400$	0	0
$401$	−1.01833e6	−0.316248	−0.158124	−	0.987419i	$-0.550545\pi$
$401$	−1.01833e6	−0.316248	−0.158124	+	0.987419i	$0.550545\pi$
$402$	0	0
$403$	142376.i	0.0436691i
$404$	6.47064e6	1.97240
$405$	0	0
$406$	1.14224e7	3.43909
$407$	4.17977e6i	1.25074i
$408$	0	0
$409$	6.32925e6	1.87087	0.935436	−	0.353497i	$-0.115008\pi$
$409$	6.32925e6	1.87087	0.935436	+	0.353497i	$0.115008\pi$
$410$	0	0
$411$	0	0
$412$	− 3.02067e6i	− 0.876719i
$413$	4.12263e6i	1.18932i
$414$	0	0
$415$	0	0
$416$	4.38451e6	1.24219
$417$	0	0
$418$	− 6.39083e6i	− 1.78903i
$419$	2.41445e6	0.671865	0.335933	−	0.941886i	$-0.390949\pi$
$419$	2.41445e6	0.671865	0.335933	+	0.941886i	$0.390949\pi$
$420$	0	0
$421$	−3.96045e6	−1.08903	−0.544514	−	0.838752i	$-0.683286\pi$
$421$	−3.96045e6	−1.08903	−0.544514	+	0.838752i	$0.683286\pi$
$422$	− 2.55871e6i	− 0.699424i
$423$	0	0
$424$	−1.54465e6	−0.417269
$425$	0	0
$426$	0	0
$427$	− 1.31463e6i	− 0.348927i
$428$	6.91093e6i	1.82359i
$429$	0	0
$430$	0	0
$431$	1.48955e6	0.386245	0.193123	−	0.981175i	$-0.438139\pi$
$431$	1.48955e6	0.386245	0.193123	+	0.981175i	$0.438139\pi$
$432$	0	0
$433$	− 5.66221e6i	− 1.45133i	−0.688049	−	0.725665i	$-0.741532\pi$
$433$	− 5.66221e6i	− 1.45133i	0.688049	−	0.725665i	$-0.258468\pi$
$434$	359281.	0.0915609
$435$	0	0
$436$	2.20706e6	0.556028
$437$	− 9.91102e6i	− 2.48265i
$438$	0	0
$439$	−8764.09	−0.00217043	−0.00108521	−	0.999999i	$-0.500345\pi$
$439$	−8764.09	−0.00217043	−0.00108521	+	0.999999i	$0.500345\pi$
$440$	0	0
$441$	0	0
$442$	3.58772e6i	0.873499i
$443$	− 962960.i	− 0.233130i	−0.993183	−	0.116565i	$-0.962812\pi$
$443$	− 962960.i	− 0.233130i	0.993183	−	0.116565i	$-0.0371884\pi$
$444$	0	0
$445$	0	0
$446$	−3.48994e6	−0.830771
$447$	0	0
$448$	− 7.60101e6i	− 1.78927i
$449$	−3.81934e6	−0.894071	−0.447035	−	0.894516i	$-0.647520\pi$
$449$	−3.81934e6	−0.894071	−0.447035	+	0.894516i	$0.647520\pi$
$450$	0	0
$451$	−2.23340e6	−0.517040
$452$	379268.i	0.0873173i
$453$	0	0
$454$	−3.29185e6	−0.749549
$455$	0	0
$456$	0	0
$457$	2.28095e6i	0.510887i	0.966824	+	0.255443i	$0.0822215\pi$
$457$	2.28095e6i	0.510887i	−0.966824	+	0.255443i	$0.917779\pi$
$458$	3.06924e6i	0.683703i
$459$	0	0
$460$	0	0
$461$	2.56688e6	0.562539	0.281270	−	0.959629i	$-0.409245\pi$
$461$	2.56688e6	0.562539	0.281270	+	0.959629i	$0.409245\pi$
$462$	0	0
$463$	− 2.16948e6i	− 0.470330i	−0.971955	−	0.235165i	$-0.924437\pi$
$463$	− 2.16948e6i	− 0.470330i	0.971955	−	0.235165i	$-0.0755631\pi$
$464$	5.63774e6	1.21565
$465$	0	0
$466$	3.21035e6	0.684837
$467$	1.84499e6i	0.391472i	0.980657	+	0.195736i	$0.0627096\pi$
$467$	1.84499e6i	0.391472i	−0.980657	+	0.195736i	$0.937290\pi$
$468$	0	0
$469$	−8.33022e6	−1.74874
$470$	0	0
$471$	0	0
$472$	− 1.88411e6i	− 0.389271i
$473$	1.43521e6i	0.294960i
$474$	0	0
$475$	0	0
$476$	5.06319e6	1.02425
$477$	0	0
$478$	3.55247e6i	0.711148i
$479$	−2.88467e6	−0.574457	−0.287229	−	0.957862i	$-0.592734\pi$
$479$	−2.88467e6	−0.574457	−0.287229	+	0.957862i	$0.592734\pi$
$480$	0	0
$481$	8.09785e6	1.59590
$482$	9.80591e6i	1.92252i
$483$	0	0
$484$	3.36541e6	0.653017
$485$	0	0
$486$	0	0
$487$	3.37928e6i	0.645657i	0.946457	+	0.322829i	$0.104634\pi$
$487$	3.37928e6i	0.645657i	−0.946457	+	0.322829i	$0.895366\pi$
$488$	600809.i	0.114205i
$489$	0	0
$490$	0	0
$491$	128325.	0.0240218	0.0120109	−	0.999928i	$-0.496177\pi$
$491$	128325.	0.0240218	0.0120109	+	0.999928i	$0.496177\pi$
$492$	0	0
$493$	6.49589e6i	1.20371i
$494$	−1.23815e7	−2.28275
$495$	0	0
$496$	177330.	0.0323651
$497$	− 3.80013e6i	− 0.690093i
$498$	0	0
$499$	2.15334e6	0.387135	0.193567	−	0.981087i	$-0.437994\pi$
$499$	2.15334e6	0.387135	0.193567	+	0.981087i	$0.437994\pi$
$500$	0	0
$501$	0	0
$502$	− 5.47244e6i	− 0.969218i
$503$	− 7.42765e6i	− 1.30898i	−0.756072	−	0.654488i	$-0.772884\pi$
$503$	− 7.42765e6i	− 1.30898i	0.756072	−	0.654488i	$-0.227116\pi$
$504$	0	0
$505$	0	0
$506$	−8.80129e6	−1.52816
$507$	0	0
$508$	− 928710.i	− 0.159669i
$509$	−8.01756e6	−1.37166	−0.685832	−	0.727760i	$-0.740561\pi$
$509$	−8.01756e6	−1.37166	−0.685832	+	0.727760i	$0.740561\pi$
$510$	0	0
$511$	3.35807e6	0.568902
$512$	− 7.04364e6i	− 1.18747i
$513$	0	0
$514$	1.07766e6	0.179918
$515$	0	0
$516$	0	0
$517$	− 2.95460e6i	− 0.486152i
$518$	− 2.04347e7i	− 3.34613i
$519$	0	0
$520$	0	0
$521$	−2.24266e6	−0.361967	−0.180984	−	0.983486i	$-0.557928\pi$
$521$	−2.24266e6	−0.361967	−0.180984	+	0.983486i	$0.557928\pi$
$522$	0	0
$523$	7.70554e6i	1.23182i	0.787815	+	0.615912i	$0.211213\pi$
$523$	7.70554e6i	1.23182i	−0.787815	+	0.615912i	$0.788787\pi$
$524$	1.40264e7	2.23161
$525$	0	0
$526$	3.12610e6	0.492650
$527$	204322.i	0.0320470i
$528$	0	0
$529$	−7.21285e6	−1.12064
$530$	0	0
$531$	0	0
$532$	1.74735e7i	2.67671i
$533$	4.32696e6i	0.659729i
$534$	0	0
$535$	0	0
$536$	3.80705e6	0.572370
$537$	0	0
$538$	− 1.03242e7i	− 1.53780i
$539$	−2.49564e6	−0.370008
$540$	0	0
$541$	−7.64969e6	−1.12370	−0.561850	−	0.827239i	$-0.689910\pi$
$541$	−7.64969e6	−1.12370	−0.561850	+	0.827239i	$0.689910\pi$
$542$	1.52797e7i	2.23417i
$543$	0	0
$544$	6.29214e6	0.911594
$545$	0	0
$546$	0	0
$547$	− 2.76870e6i	− 0.395647i	−0.980238	−	0.197823i	$-0.936613\pi$
$547$	− 2.76870e6i	− 0.395647i	0.980238	−	0.197823i	$-0.0633873\pi$
$548$	− 6.91366e6i	− 0.983459i
$549$	0	0
$550$	0	0
$551$	−2.24179e7	−3.14569
$552$	0	0
$553$	− 6.30110e6i	− 0.876200i
$554$	−1.24780e7	−1.72731
$555$	0	0
$556$	−1.08912e7	−1.49413
$557$	− 1.21255e6i	− 0.165600i	−0.996566	−	0.0828002i	$-0.973614\pi$
$557$	− 1.21255e6i	− 0.165600i	0.996566	−	0.0828002i	$-0.0263863\pi$
$558$	0	0
$559$	2.78057e6	0.376360
$560$	0	0
$561$	0	0
$562$	− 1.80679e7i	− 2.41305i
$563$	6.55586e6i	0.871683i	0.900024	+	0.435841i	$0.143549\pi$
$563$	6.55586e6i	0.871683i	−0.900024	+	0.435841i	$0.856451\pi$
$564$	0	0
$565$	0	0
$566$	3.87083e6	0.507883
$567$	0	0
$568$	1.73672e6i	0.225871i
$569$	−8.97209e6	−1.16175	−0.580875	−	0.813993i	$-0.697290\pi$
$569$	−8.97209e6	−1.16175	−0.580875	+	0.813993i	$0.697290\pi$
$570$	0	0
$571$	−5.46422e6	−0.701354	−0.350677	−	0.936496i	$-0.614049\pi$
$571$	−5.46422e6	−0.701354	−0.350677	+	0.936496i	$0.614049\pi$
$572$	6.14909e6i	0.785816i
$573$	0	0
$574$	1.09190e7	1.38325
$575$	0	0
$576$	0	0
$577$	6.35363e6i	0.794478i	0.917715	+	0.397239i	$0.130032\pi$
$577$	6.35363e6i	0.794478i	−0.917715	+	0.397239i	$0.869968\pi$
$578$	− 6.94963e6i	− 0.865251i
$579$	0	0
$580$	0	0
$581$	5.76072e6	0.708005
$582$	0	0
$583$	5.89066e6i	0.717781i
$584$	−1.53469e6	−0.186204
$585$	0	0
$586$	−1.64696e7	−1.98125
$587$	− 1.33705e7i	− 1.60159i	−0.598938	−	0.800795i	$-0.704410\pi$
$587$	− 1.33705e7i	− 1.60159i	0.598938	−	0.800795i	$-0.295590\pi$
$588$	0	0
$589$	−705134.	−0.0837497
$590$	0	0
$591$	0	0
$592$	− 1.00859e7i	− 1.18280i
$593$	− 1.42274e6i	− 0.166146i	−0.996543	−	0.0830728i	$-0.973527\pi$
$593$	− 1.42274e6i	− 0.166146i	0.996543	−	0.0830728i	$-0.0264734\pi$
$594$	0	0
$595$	0	0
$596$	1.19142e7	1.37388
$597$	0	0
$598$	1.70515e7i	1.94989i
$599$	5.91805e6	0.673925	0.336962	−	0.941518i	$-0.390601\pi$
$599$	5.91805e6	0.673925	0.336962	+	0.941518i	$0.390601\pi$
$600$	0	0
$601$	1.48610e7	1.67827	0.839135	−	0.543924i	$-0.183062\pi$
$601$	1.48610e7	1.67827	0.839135	+	0.543924i	$0.183062\pi$
$602$	− 7.01667e6i	− 0.789114i
$603$	0	0
$604$	−1.48599e7	−1.65738
$605$	0	0
$606$	0	0
$607$	1.16289e7i	1.28105i	0.767937	+	0.640526i	$0.221283\pi$
$607$	1.16289e7i	1.28105i	−0.767937	+	0.640526i	$0.778717\pi$
$608$	2.17148e7i	2.38230i
$609$	0	0
$610$	0	0
$611$	−5.72421e6	−0.620315
$612$	0	0
$613$	− 1.06616e6i	− 0.114596i	−0.998357	−	0.0572980i	$-0.981751\pi$
$613$	− 1.06616e6i	− 0.114596i	0.998357	−	0.0572980i	$-0.0182485\pi$
$614$	−1.91040e6	−0.204505
$615$	0	0
$616$	3.28806e6	0.349131
$617$	− 663411.i	− 0.0701568i	−0.999385	−	0.0350784i	$-0.988832\pi$
$617$	− 663411.i	− 0.0701568i	0.999385	−	0.0350784i	$-0.0111681\pi$
$618$	0	0
$619$	−2.53444e6	−0.265861	−0.132930	−	0.991125i	$-0.542439\pi$
$619$	−2.53444e6	−0.265861	−0.132930	+	0.991125i	$0.542439\pi$
$620$	0	0
$621$	0	0
$622$	5.04508e6i	0.522868i
$623$	− 1.50911e7i	− 1.55776i
$624$	0	0
$625$	0	0
$626$	1.69631e7	1.73010
$627$	0	0
$628$	2.04258e7i	2.06671i
$629$	1.16211e7	1.17117
$630$	0	0
$631$	102057.	0.0102040	0.00510199	−	0.999987i	$-0.498376\pi$
$631$	102057.	0.0102040	0.00510199	+	0.999987i	$0.498376\pi$
$632$	2.87971e6i	0.286785i
$633$	0	0
$634$	2.47004e7	2.44051
$635$	0	0
$636$	0	0
$637$	4.83504e6i	0.472119i
$638$	1.99078e7i	1.93630i
$639$	0	0
$640$	0	0
$641$	−3.67995e6	−0.353750	−0.176875	−	0.984233i	$-0.556599\pi$
$641$	−3.67995e6	−0.353750	−0.176875	+	0.984233i	$0.556599\pi$
$642$	0	0
$643$	8.95581e6i	0.854235i	0.904196	+	0.427118i	$0.140471\pi$
$643$	8.95581e6i	0.854235i	−0.904196	+	0.427118i	$0.859529\pi$
$644$	2.40641e7	2.28642
$645$	0	0
$646$	−1.77686e7	−1.67522
$647$	− 1.36816e6i	− 0.128492i	−0.997934	−	0.0642458i	$-0.979536\pi$
$647$	− 1.36816e6i	− 0.128492i	0.997934	−	0.0642458i	$-0.0204642\pi$
$648$	0	0
$649$	−7.18520e6	−0.669618
$650$	0	0
$651$	0	0
$652$	− 7.37612e6i	− 0.679530i
$653$	3.23673e6i	0.297046i	0.988909	+	0.148523i	$0.0474519\pi$
$653$	3.23673e6i	0.297046i	−0.988909	+	0.148523i	$0.952548\pi$
$654$	0	0
$655$	0	0
$656$	5.38925e6	0.488955
$657$	0	0
$658$	1.44449e7i	1.30061i
$659$	−5.17492e6	−0.464184	−0.232092	−	0.972694i	$-0.574557\pi$
$659$	−5.17492e6	−0.464184	−0.232092	+	0.972694i	$0.574557\pi$
$660$	0	0
$661$	−1.07060e7	−0.953064	−0.476532	−	0.879157i	$-0.658106\pi$
$661$	−1.07060e7	−0.953064	−0.476532	+	0.879157i	$0.658106\pi$
$662$	− 3.55080e6i	− 0.314906i
$663$	0	0
$664$	−2.63275e6	−0.231733
$665$	0	0
$666$	0	0
$667$	3.08734e7i	2.68701i
$668$	1.78717e7i	1.54962i
$669$	0	0
$670$	0	0
$671$	2.29123e6	0.196455
$672$	0	0
$673$	− 2.15653e7i	− 1.83535i	−0.397338	−	0.917673i	$-0.630066\pi$
$673$	− 2.15653e7i	− 1.83535i	0.397338	−	0.917673i	$-0.369934\pi$
$674$	−1.80975e6	−0.153451
$675$	0	0
$676$	−3.16278e6	−0.266196
$677$	− 6.09762e6i	− 0.511315i	−0.966767	−	0.255658i	$-0.917708\pi$
$677$	− 6.09762e6i	− 0.511315i	0.966767	−	0.255658i	$-0.0822920\pi$
$678$	0	0
$679$	1.96508e6	0.163571
$680$	0	0
$681$	0	0
$682$	626180.i	0.0515511i
$683$	2.04240e7i	1.67528i	0.546220	+	0.837642i	$0.316066\pi$
$683$	2.04240e7i	1.67528i	−0.546220	+	0.837642i	$0.683934\pi$
$684$	0	0
$685$	0	0
$686$	−1.07719e7	−0.873944
$687$	0	0
$688$	− 3.46320e6i	− 0.278937i
$689$	1.14125e7	0.915868
$690$	0	0
$691$	−2.91819e6	−0.232497	−0.116249	−	0.993220i	$-0.537087\pi$
$691$	−2.91819e6	−0.232497	−0.116249	+	0.993220i	$0.537087\pi$
$692$	2.24430e7i	1.78162i
$693$	0	0
$694$	−1.35456e7	−1.06758
$695$	0	0
$696$	0	0
$697$	6.20957e6i	0.484150i
$698$	− 3.39785e7i	− 2.63976i
$699$	0	0
$700$	0	0
$701$	−8.78394e6	−0.675141	−0.337570	−	0.941300i	$-0.609605\pi$
$701$	−8.78394e6	−0.675141	−0.337570	+	0.941300i	$0.609605\pi$
$702$	0	0
$703$	4.01056e7i	3.06067i
$704$	1.32476e7	1.00741
$705$	0	0
$706$	−4.98973e6	−0.376760
$707$	2.55639e7i	1.92344i
$708$	0	0
$709$	−1.66475e7	−1.24375	−0.621875	−	0.783116i	$-0.713629\pi$
$709$	−1.66475e7	−1.24375	−0.621875	+	0.783116i	$0.713629\pi$
$710$	0	0
$711$	0	0
$712$	6.89688e6i	0.509862i
$713$	971092.i	0.0715379i
$714$	0	0
$715$	0	0
$716$	7.72368e6	0.563043
$717$	0	0
$718$	− 3.65091e7i	− 2.64296i
$719$	2.55062e7	1.84003	0.920013	−	0.391888i	$-0.128178\pi$
$719$	2.55062e7	1.84003	0.920013	+	0.391888i	$0.128178\pi$
$720$	0	0
$721$	1.19339e7	0.854957
$722$	− 4.02228e7i	− 2.87163i
$723$	0	0
$724$	−4.59389e6	−0.325712
$725$	0	0
$726$	0	0
$727$	− 1.36506e7i	− 0.957890i	−0.877845	−	0.478945i	$-0.841019\pi$
$727$	− 1.36506e7i	− 0.957890i	0.877845	−	0.478945i	$-0.158981\pi$
$728$	− 6.37026e6i	− 0.445481i
$729$	0	0
$730$	0	0
$731$	3.99035e6	0.276196
$732$	0	0
$733$	− 1.38922e7i	− 0.955018i	−0.878627	−	0.477509i	$-0.841540\pi$
$733$	− 1.38922e7i	− 0.955018i	0.878627	−	0.477509i	$-0.158460\pi$
$734$	4.46751e6	0.306073
$735$	0	0
$736$	2.99050e7	2.03493
$737$	− 1.45185e7i	− 0.984584i
$738$	0	0
$739$	−837888.	−0.0564384	−0.0282192	−	0.999602i	$-0.508984\pi$
$739$	−837888.	−0.0564384	−0.0282192	+	0.999602i	$0.508984\pi$
$740$	0	0
$741$	0	0
$742$	− 2.87991e7i	− 1.92030i
$743$	− 8.07258e6i	− 0.536464i	−0.963354	−	0.268232i	$-0.913561\pi$
$743$	− 8.07258e6i	− 0.536464i	0.963354	−	0.268232i	$-0.0864394\pi$
$744$	0	0
$745$	0	0
$746$	−9.93588e6	−0.653671
$747$	0	0
$748$	8.82448e6i	0.576680i
$749$	−2.73033e7	−1.77832
$750$	0	0
$751$	−1.74892e7	−1.13154	−0.565770	−	0.824563i	$-0.691421\pi$
$751$	−1.74892e7	−1.13154	−0.565770	+	0.824563i	$0.691421\pi$
$752$	7.12952e6i	0.459744i
$753$	0	0
$754$	3.85692e7	2.47066
$755$	0	0
$756$	0	0
$757$	− 2.19231e7i	− 1.39047i	−0.718781	−	0.695236i	$-0.755300\pi$
$757$	− 2.19231e7i	− 1.39047i	0.718781	−	0.695236i	$-0.244700\pi$
$758$	2.11820e7i	1.33904i
$759$	0	0
$760$	0	0
$761$	1.18585e7	0.742278	0.371139	−	0.928577i	$-0.378967\pi$
$761$	1.18585e7	0.742278	0.371139	+	0.928577i	$0.378967\pi$
$762$	0	0
$763$	8.71951e6i	0.542227i
$764$	−2.81823e7	−1.74680
$765$	0	0
$766$	1.29823e7	0.799431
$767$	1.39206e7i	0.854413i
$768$	0	0
$769$	2.94181e6	0.179390	0.0896950	−	0.995969i	$-0.471411\pi$
$769$	2.94181e6	0.179390	0.0896950	+	0.995969i	$0.471411\pi$
$770$	0	0
$771$	0	0
$772$	2.10337e6i	0.127020i
$773$	− 2.01745e7i	− 1.21438i	−0.794558	−	0.607188i	$-0.792297\pi$
$773$	− 2.01745e7i	− 1.21438i	0.794558	−	0.607188i	$-0.207703\pi$
$774$	0	0
$775$	0	0
$776$	−898074.	−0.0535375
$777$	0	0
$778$	− 3.91933e7i	− 2.32147i
$779$	−2.14298e7	−1.26525
$780$	0	0
$781$	6.62313e6	0.388540
$782$	2.44704e7i	1.43095i
$783$	0	0
$784$	6.02206e6	0.349909
$785$	0	0
$786$	0	0
$787$	− 8.17269e6i	− 0.470358i	−0.971952	−	0.235179i	$-0.924432\pi$
$787$	− 8.17269e6i	− 0.470358i	0.971952	−	0.235179i	$-0.0755676\pi$
$788$	8.85018e6i	0.507734i
$789$	0	0
$790$	0	0
$791$	−1.49839e6	−0.0851499
$792$	0	0
$793$	− 4.43901e6i	− 0.250670i
$794$	−3.01955e7	−1.69977
$795$	0	0
$796$	−1.56380e7	−0.874779
$797$	1.66457e7i	0.928234i	0.885774	+	0.464117i	$0.153628\pi$
$797$	1.66457e7i	0.928234i	−0.885774	+	0.464117i	$0.846372\pi$
$798$	0	0
$799$	−8.21474e6	−0.455226
$800$	0	0
$801$	0	0
$802$	8.67699e6i	0.476358i
$803$	5.85267e6i	0.320306i
$804$	0	0
$805$	0	0
$806$	1.21316e6	0.0657778
$807$	0	0
$808$	− 1.16831e7i	− 0.629550i
$809$	7.91242e6	0.425048	0.212524	−	0.977156i	$-0.431832\pi$
$809$	7.91242e6	0.425048	0.212524	+	0.977156i	$0.431832\pi$
$810$	0	0
$811$	−4.07133e6	−0.217362	−0.108681	−	0.994077i	$-0.534663\pi$
$811$	−4.07133e6	−0.217362	−0.108681	+	0.994077i	$0.534663\pi$
$812$	− 5.44311e7i	− 2.89706i
$813$	0	0
$814$	3.56149e7	1.88396
$815$	0	0
$816$	0	0
$817$	1.37711e7i	0.721794i
$818$	− 5.39302e7i	− 2.81805i
$819$	0	0
$820$	0	0
$821$	−1.86139e7	−0.963782	−0.481891	−	0.876231i	$-0.660050\pi$
$821$	−1.86139e7	−0.963782	−0.481891	+	0.876231i	$0.660050\pi$
$822$	0	0
$823$	− 3.54074e6i	− 0.182220i	−0.995841	−	0.0911098i	$-0.970959\pi$
$823$	− 3.54074e6i	− 0.182220i	0.995841	−	0.0911098i	$-0.0290414\pi$
$824$	−5.45400e6	−0.279832
$825$	0	0
$826$	3.51281e7	1.79145
$827$	− 3.76580e7i	− 1.91467i	−0.288981	−	0.957335i	$-0.593316\pi$
$827$	− 3.76580e7i	− 1.91467i	0.288981	−	0.957335i	$-0.406684\pi$
$828$	0	0
$829$	−1.16650e7	−0.589518	−0.294759	−	0.955572i	$-0.595239\pi$
$829$	−1.16650e7	−0.589518	−0.294759	+	0.955572i	$0.595239\pi$
$830$	0	0
$831$	0	0
$832$	− 2.56657e7i	− 1.28542i
$833$	6.93870e6i	0.346470i
$834$	0	0
$835$	0	0
$836$	−3.04541e7	−1.50706
$837$	0	0
$838$	− 2.05730e7i	− 1.01202i
$839$	−2.07091e7	−1.01568	−0.507839	−	0.861452i	$-0.669556\pi$
$839$	−2.07091e7	−1.01568	−0.507839	+	0.861452i	$0.669556\pi$
$840$	0	0
$841$	4.93220e7	2.40464
$842$	3.37462e7i	1.64038i
$843$	0	0
$844$	−1.21930e7	−0.589188
$845$	0	0
$846$	0	0
$847$	1.32959e7i	0.636808i
$848$	− 1.42143e7i	− 0.678791i
$849$	0	0
$850$	0	0
$851$	5.52323e7	2.61438
$852$	0	0
$853$	− 1.94473e7i	− 0.915137i	−0.889174	−	0.457569i	$-0.848720\pi$
$853$	− 1.94473e7i	− 0.915137i	0.889174	−	0.457569i	$-0.151280\pi$
$854$	−1.12017e7	−0.525581
$855$	0	0
$856$	1.24781e7	0.582054
$857$	2.10458e7i	0.978843i	0.872047	+	0.489421i	$0.162792\pi$
$857$	2.10458e7i	0.978843i	−0.872047	+	0.489421i	$0.837208\pi$
$858$	0	0
$859$	−2.40179e7	−1.11058	−0.555292	−	0.831656i	$-0.687393\pi$
$859$	−2.40179e7	−1.11058	−0.555292	+	0.831656i	$0.687393\pi$
$860$	0	0
$861$	0	0
$862$	− 1.26922e7i	− 0.581793i
$863$	5.25620e6i	0.240240i	0.992759	+	0.120120i	$0.0383279\pi$
$863$	5.25620e6i	0.240240i	−0.992759	+	0.120120i	$0.961672\pi$
$864$	0	0
$865$	0	0
$866$	−4.82465e7	−2.18611
$867$	0	0
$868$	− 1.71208e6i	− 0.0771300i
$869$	1.09820e7	0.493323
$870$	0	0
$871$	−2.81280e7	−1.25630
$872$	− 3.98497e6i	− 0.177473i
$873$	0	0
$874$	−8.44498e7	−3.73956
$875$	0	0
$876$	0	0
$877$	2.99490e6i	0.131487i	0.997837	+	0.0657436i	$0.0209420\pi$
$877$	2.99490e6i	0.131487i	−0.997837	+	0.0657436i	$0.979058\pi$
$878$	74677.0i	0.00326927i
$879$	0	0
$880$	0	0
$881$	2.19714e7	0.953714	0.476857	−	0.878981i	$-0.341776\pi$
$881$	2.19714e7	0.953714	0.476857	+	0.878981i	$0.341776\pi$
$882$	0	0
$883$	− 1.05962e7i	− 0.457348i	−0.973503	−	0.228674i	$-0.926561\pi$
$883$	− 1.05962e7i	− 0.457348i	0.973503	−	0.228674i	$-0.0734389\pi$
$884$	1.70965e7	0.735827
$885$	0	0
$886$	−8.20519e6	−0.351159
$887$	1.57778e6i	0.0673343i	0.999433	+	0.0336672i	$0.0107186\pi$
$887$	1.57778e6i	0.0673343i	−0.999433	+	0.0336672i	$0.989281\pi$
$888$	0	0
$889$	3.66909e6	0.155706
$890$	0	0
$891$	0	0
$892$	1.66306e7i	0.699834i
$893$	− 2.83498e7i	− 1.18966i
$894$	0	0
$895$	0	0
$896$	−2.32150e7	−0.966047
$897$	0	0
$898$	3.25438e7i	1.34672i
$899$	2.19653e6	0.0906438
$900$	0	0
$901$	1.63779e7	0.672121
$902$	1.90303e7i	0.778807i
$903$	0	0
$904$	684790.	0.0278700
$905$	0	0
$906$	0	0
$907$	3.03048e7i	1.22319i	0.791171	+	0.611594i	$0.209472\pi$
$907$	3.03048e7i	1.22319i	−0.791171	+	0.611594i	$0.790528\pi$
$908$	1.56866e7i	0.631413i
$909$	0	0
$910$	0	0
$911$	1.14226e7	0.456005	0.228003	−	0.973661i	$-0.426781\pi$
$911$	1.14226e7	0.456005	0.228003	+	0.973661i	$0.426781\pi$
$912$	0	0
$913$	1.00402e7i	0.398625i
$914$	1.94355e7	0.769538
$915$	0	0
$916$	1.46258e7	0.575945
$917$	5.54148e7i	2.17622i
$918$	0	0
$919$	4.43959e7	1.73402	0.867010	−	0.498291i	$-0.166039\pi$
$919$	4.43959e7	1.73402	0.867010	+	0.498291i	$0.166039\pi$
$920$	0	0
$921$	0	0
$922$	− 2.18718e7i	− 0.847340i
$923$	− 1.28316e7i	− 0.495766i
$924$	0	0
$925$	0	0
$926$	−1.84857e7	−0.708448
$927$	0	0
$928$	− 6.76428e7i	− 2.57841i
$929$	2.73604e7	1.04012	0.520059	−	0.854130i	$-0.325910\pi$
$929$	2.73604e7	1.04012	0.520059	+	0.854130i	$0.325910\pi$
$930$	0	0
$931$	−2.39461e7	−0.905443
$932$	− 1.52982e7i	− 0.576900i
$933$	0	0
$934$	1.57208e7	0.589666
$935$	0	0
$936$	0	0
$937$	1.23707e7i	0.460306i	0.973154	+	0.230153i	$0.0739227\pi$
$937$	1.23707e7i	0.460306i	−0.973154	+	0.230153i	$0.926077\pi$
$938$	7.09801e7i	2.63408i
$939$	0	0
$940$	0	0
$941$	1.77596e7	0.653822	0.326911	−	0.945055i	$-0.393992\pi$
$941$	1.77596e7	0.653822	0.326911	+	0.945055i	$0.393992\pi$
$942$	0	0
$943$	2.95126e7i	1.08076i
$944$	1.73381e7	0.633244
$945$	0	0
$946$	1.22291e7	0.444292
$947$	− 4.50046e7i	− 1.63073i	−0.578949	−	0.815364i	$-0.696537\pi$
$947$	− 4.50046e7i	− 1.63073i	0.578949	−	0.815364i	$-0.303463\pi$
$948$	0	0
$949$	1.13389e7	0.408701
$950$	0	0
$951$	0	0
$952$	− 9.14188e6i	− 0.326921i
$953$	4.10110e7i	1.46274i	0.681979	+	0.731372i	$0.261119\pi$
$953$	4.10110e7i	1.46274i	−0.681979	+	0.731372i	$0.738881\pi$
$954$	0	0
$955$	0	0
$956$	1.69285e7	0.599065
$957$	0	0
$958$	2.45797e7i	0.865293i
$959$	2.73141e7	0.959048
$960$	0	0
$961$	−2.85601e7	−0.997587
$962$	− 6.90001e7i	− 2.40388i
$963$	0	0
$964$	4.67279e7	1.61951
$965$	0	0
$966$	0	0
$967$	− 2.23360e7i	− 0.768137i	−0.923305	−	0.384069i	$-0.874523\pi$
$967$	− 2.23360e7i	− 0.768137i	0.923305	−	0.384069i	$-0.125477\pi$
$968$	− 6.07644e6i	− 0.208430i
$969$	0	0
$970$	0	0
$971$	3.34728e7	1.13932	0.569658	−	0.821882i	$-0.307076\pi$
$971$	3.34728e7	1.13932	0.569658	+	0.821882i	$0.307076\pi$
$972$	0	0
$973$	− 4.30284e7i	− 1.45705i
$974$	2.87942e7	0.972539
$975$	0	0
$976$	−5.52880e6	−0.185783
$977$	8.52076e6i	0.285589i	0.989752	+	0.142795i	$0.0456089\pi$
$977$	8.52076e6i	0.285589i	−0.989752	+	0.142795i	$0.954391\pi$
$978$	0	0
$979$	2.63018e7	0.877058
$980$	0	0
$981$	0	0
$982$	− 1.09343e6i	− 0.0361836i
$983$	2.01155e7i	0.663969i	0.943285	+	0.331985i	$0.107718\pi$
$983$	2.01155e7i	0.663969i	−0.943285	+	0.331985i	$0.892282\pi$
$984$	0	0
$985$	0	0
$986$	5.53502e7	1.81312
$987$	0	0
$988$	5.90015e7i	1.92296i
$989$	1.89652e7	0.616547
$990$	0	0
$991$	3.24195e7	1.04863	0.524315	−	0.851524i	$-0.324321\pi$
$991$	3.24195e7	1.04863	0.524315	+	0.851524i	$0.324321\pi$
$992$	− 2.12764e6i	− 0.0686465i
$993$	0	0
$994$	−3.23801e7	−1.03947
$995$	0	0
$996$	0	0
$997$	− 4.47995e7i	− 1.42737i	−0.700469	−	0.713683i	$-0.747026\pi$
$997$	− 4.47995e7i	− 1.42737i	0.700469	−	0.713683i	$-0.252974\pi$
$998$	− 1.83482e7i	− 0.583133i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.6.b.k.199.1		4
3.2	odd	2		225.6.b.j.199.4		4
5.2	odd	4		45.6.a.f.1.2	yes	2
5.3	odd	4		225.6.a.k.1.1		2
5.4	even	2	inner	225.6.b.k.199.4		4
15.2	even	4		45.6.a.d.1.1	✓	2
15.8	even	4		225.6.a.r.1.2		2
15.14	odd	2		225.6.b.j.199.1		4
20.7	even	4		720.6.a.be.1.1		2
60.47	odd	4		720.6.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.6.a.d.1.1	✓	2	15.2	even	4
45.6.a.f.1.2	yes	2	5.2	odd	4
225.6.a.k.1.1		2	5.3	odd	4
225.6.a.r.1.2		2	15.8	even	4
225.6.b.j.199.1		4	15.14	odd	2
225.6.b.j.199.4		4	3.2	odd	2
225.6.b.k.199.1		4	1.1	even	1	trivial
225.6.b.k.199.4		4	5.4	even	2	inner
720.6.a.y.1.1		2	60.47	odd	4
720.6.a.be.1.1		2	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	225.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.52080i q^{2} -40.6040 q^{4} -160.416i q^{7} +73.3128i q^{8} +279.584 q^{11} -541.664i q^{13} -1366.87 q^{14} -674.644 q^{16} -777.334i q^{17} +2682.66 q^{19} -2382.28i q^{22} -3694.48i q^{23} -4615.41 q^{26} +6513.53i q^{28} -8356.62 q^{29} -262.849 q^{31} +8094.51i q^{32} -6623.51 q^{34} +14949.9i q^{37} -22858.4i q^{38} -7988.28 q^{41} +5133.38i q^{43} -11352.2 q^{44} -31479.9 q^{46} -10567.8i q^{47} -8926.28 q^{49} +21993.7i q^{52} +21069.4i q^{53} +11760.5 q^{56} +71205.1i q^{58} -25699.6 q^{59} +8195.14 q^{61} +2239.68i q^{62} +47383.1 q^{64} -51928.9i q^{67} +31562.9i q^{68} +23689.2 q^{71} +20933.5i q^{73} +127385. q^{74} -108926. q^{76} -44849.7i q^{77} +39279.7 q^{79} +68066.6i q^{82} +35911.1i q^{83} +43740.5 q^{86} +20497.1i q^{88} +94074.7 q^{89} -86891.5 q^{91} +150011. i q^{92} -90046.3 q^{94} +12249.9i q^{97} +76059.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 42 q^{4} + 1600 q^{11} - 3300 q^{14} - 1374 q^{16} + 3024 q^{19} - 12200 q^{26} - 2600 q^{29} - 11648 q^{31} - 5060 q^{34} + 800 q^{41} - 2300 q^{44} - 79680 q^{46} + 2828 q^{49} - 5700 q^{56} - 126400 q^{59}+ \cdots - 121280 q^{94}+O(q^{100}) \) 4 * q - 42 * q^4 + 1600 * q^11 - 3300 * q^14 - 1374 * q^16 + 3024 * q^19 - 12200 * q^26 - 2600 * q^29 - 11648 * q^31 - 5060 * q^34 + 800 * q^41 - 2300 * q^44 - 79680 * q^46 + 2828 * q^49 - 5700 * q^56 - 126400 * q^59 - 98232 * q^61 + 53342 * q^64 + 130400 * q^71 + 271600 * q^74 - 263752 * q^76 + 92576 * q^79 - 57200 * q^86 + 174000 * q^89 - 241600 * q^91 - 121280 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more