LMFDB - Embedded newform 450.6.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.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:	$72.1727189158$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{4081}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1020 $ x^2 - x - 1020
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$32.4414$ of defining polynomial
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000 q^{2} +16.0000 q^{4} -141.648 q^{7} +64.0000 q^{8} +O(q^{10})$	$q+4.00000 q^{2} +16.0000 q^{4} -141.648 q^{7} +64.0000 q^{8} +113.296 q^{11} -61.7038 q^{13} -566.592 q^{14} +256.000 q^{16} +1670.48 q^{17} -662.241 q^{19} +453.185 q^{22} +86.4812 q^{23} -246.815 q^{26} -2266.37 q^{28} -3233.30 q^{29} -3812.24 q^{31} +1024.00 q^{32} +6681.92 q^{34} -10249.6 q^{37} -2648.96 q^{38} -13459.4 q^{41} +4697.57 q^{43} +1812.74 q^{44} +345.925 q^{46} -15264.8 q^{47} +3257.19 q^{49} -987.260 q^{52} +498.632 q^{53} -9065.48 q^{56} -12933.2 q^{58} -15294.3 q^{59} -31811.3 q^{61} -15249.0 q^{62} +4096.00 q^{64} -49274.9 q^{67} +26727.7 q^{68} +21057.7 q^{71} -39447.8 q^{73} -40998.2 q^{74} -10595.8 q^{76} -16048.2 q^{77} +72949.3 q^{79} -53837.8 q^{82} +100897. q^{83} +18790.3 q^{86} +7250.96 q^{88} -146714. q^{89} +8740.22 q^{91} +1383.70 q^{92} -61059.2 q^{94} +43544.9 q^{97} +13028.8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 32 q^{4} + 100 q^{7} + 128 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 + 32 * q^4 + 100 * q^7 + 128 * q^8	$ 2 q + 8 q^{2} + 32 q^{4} + 100 q^{7} + 128 q^{8} - 540 q^{11} - 890 q^{13} + 400 q^{14} + 512 q^{16} - 492 q^{17} + 592 q^{19} - 2160 q^{22} - 3660 q^{23} - 3560 q^{26} + 1600 q^{28} - 5700 q^{29} - 5708 q^{31} + 2048 q^{32} - 1968 q^{34} - 11300 q^{37} + 2368 q^{38} - 15420 q^{41} - 6320 q^{43} - 8640 q^{44} - 14640 q^{46} + 7800 q^{47} + 44844 q^{49} - 14240 q^{52} + 27828 q^{53} + 6400 q^{56} - 22800 q^{58} - 50520 q^{59} - 29126 q^{61} - 22832 q^{62} + 8192 q^{64} - 97400 q^{67} - 7872 q^{68} - 6180 q^{71} - 32900 q^{73} - 45200 q^{74} + 9472 q^{76} - 173916 q^{77} + 7912 q^{79} - 61680 q^{82} + 163464 q^{83} - 25280 q^{86} - 34560 q^{88} - 164640 q^{89} - 191416 q^{91} - 58560 q^{92} + 31200 q^{94} - 52430 q^{97} + 179376 q^{98}+O(q^{100}) $ 2 * q + 8 * q^2 + 32 * q^4 + 100 * q^7 + 128 * q^8 - 540 * q^11 - 890 * q^13 + 400 * q^14 + 512 * q^16 - 492 * q^17 + 592 * q^19 - 2160 * q^22 - 3660 * q^23 - 3560 * q^26 + 1600 * q^28 - 5700 * q^29 - 5708 * q^31 + 2048 * q^32 - 1968 * q^34 - 11300 * q^37 + 2368 * q^38 - 15420 * q^41 - 6320 * q^43 - 8640 * q^44 - 14640 * q^46 + 7800 * q^47 + 44844 * q^49 - 14240 * q^52 + 27828 * q^53 + 6400 * q^56 - 22800 * q^58 - 50520 * q^59 - 29126 * q^61 - 22832 * q^62 + 8192 * q^64 - 97400 * q^67 - 7872 * q^68 - 6180 * q^71 - 32900 * q^73 - 45200 * q^74 + 9472 * q^76 - 173916 * q^77 + 7912 * q^79 - 61680 * q^82 + 163464 * q^83 - 25280 * q^86 - 34560 * q^88 - 164640 * q^89 - 191416 * q^91 - 58560 * q^92 + 31200 * q^94 - 52430 * q^97 + 179376 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−141.648	−1.09261	−0.546306	−	0.837586i	$-0.683966\pi$
$7$	−141.648	−1.09261	−0.546306	+	0.837586i	$0.683966\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	0	0
$11$	113.296	0.282315	0.141157	−	0.989987i	$-0.454918\pi$
$11$	113.296	0.282315	0.141157	+	0.989987i	$0.454918\pi$
$12$	0	0
$13$	−61.7038	−0.101264	−0.0506318	−	0.998717i	$-0.516123\pi$
$13$	−61.7038	−0.101264	−0.0506318	+	0.998717i	$0.516123\pi$
$14$	−566.592	−0.772593
$15$	0	0
$16$	256.000	0.250000
$17$	1670.48	1.40191	0.700954	−	0.713207i	$-0.252758\pi$
$17$	1670.48	1.40191	0.700954	+	0.713207i	$0.252758\pi$
$18$	0	0
$19$	−662.241	−0.420854	−0.210427	−	0.977610i	$-0.567485\pi$
$19$	−662.241	−0.420854	−0.210427	+	0.977610i	$0.567485\pi$
$20$	0	0
$21$	0	0
$22$	453.185	0.199627
$23$	86.4812	0.0340880	0.0170440	−	0.999855i	$-0.494574\pi$
$23$	86.4812	0.0340880	0.0170440	+	0.999855i	$0.494574\pi$
$24$	0	0
$25$	0	0
$26$	−246.815	−0.0716042
$27$	0	0
$28$	−2266.37	−0.546306
$29$	−3233.30	−0.713922	−0.356961	−	0.934119i	$-0.616187\pi$
$29$	−3233.30	−0.713922	−0.356961	+	0.934119i	$0.616187\pi$
$30$	0	0
$31$	−3812.24	−0.712486	−0.356243	−	0.934393i	$-0.615942\pi$
$31$	−3812.24	−0.712486	−0.356243	+	0.934393i	$0.615942\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	6681.92	0.991298
$35$	0	0
$36$	0	0
$37$	−10249.6	−1.23084	−0.615419	−	0.788200i	$-0.711013\pi$
$37$	−10249.6	−1.23084	−0.615419	+	0.788200i	$0.711013\pi$
$38$	−2648.96	−0.297589
$39$	0	0
$40$	0	0
$41$	−13459.4	−1.25045	−0.625227	−	0.780443i	$-0.714993\pi$
$41$	−13459.4	−1.25045	−0.625227	+	0.780443i	$0.714993\pi$
$42$	0	0
$43$	4697.57	0.387438	0.193719	−	0.981057i	$-0.437945\pi$
$43$	4697.57	0.387438	0.193719	+	0.981057i	$0.437945\pi$
$44$	1812.74	0.141157
$45$	0	0
$46$	345.925	0.0241039
$47$	−15264.8	−1.00797	−0.503984	−	0.863713i	$-0.668133\pi$
$47$	−15264.8	−1.00797	−0.503984	+	0.863713i	$0.668133\pi$
$48$	0	0
$49$	3257.19	0.193800
$50$	0	0
$51$	0	0
$52$	−987.260	−0.0506318
$53$	498.632	0.0243832	0.0121916	−	0.999926i	$-0.496119\pi$
$53$	498.632	0.0243832	0.0121916	+	0.999926i	$0.496119\pi$
$54$	0	0
$55$	0	0
$56$	−9065.48	−0.386296
$57$	0	0
$58$	−12933.2	−0.504819
$59$	−15294.3	−0.572005	−0.286002	−	0.958229i	$-0.592327\pi$
$59$	−15294.3	−0.572005	−0.286002	+	0.958229i	$0.592327\pi$
$60$	0	0
$61$	−31811.3	−1.09460	−0.547302	−	0.836935i	$-0.684345\pi$
$61$	−31811.3	−1.09460	−0.547302	+	0.836935i	$0.684345\pi$
$62$	−15249.0	−0.503803
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	−49274.9	−1.34103	−0.670516	−	0.741895i	$-0.733927\pi$
$67$	−49274.9	−1.34103	−0.670516	+	0.741895i	$0.733927\pi$
$68$	26727.7	0.700954
$69$	0	0
$70$	0	0
$71$	21057.7	0.495752	0.247876	−	0.968792i	$-0.420267\pi$
$71$	21057.7	0.495752	0.247876	+	0.968792i	$0.420267\pi$
$72$	0	0
$73$	−39447.8	−0.866394	−0.433197	−	0.901299i	$-0.642615\pi$
$73$	−39447.8	−0.866394	−0.433197	+	0.901299i	$0.642615\pi$
$74$	−40998.2	−0.870333
$75$	0	0
$76$	−10595.8	−0.210427
$77$	−16048.2	−0.308460
$78$	0	0
$79$	72949.3	1.31508	0.657542	−	0.753418i	$-0.271596\pi$
$79$	72949.3	1.31508	0.657542	+	0.753418i	$0.271596\pi$
$80$	0	0
$81$	0	0
$82$	−53837.8	−0.884204
$83$	100897.	1.60762	0.803808	−	0.594889i	$-0.202804\pi$
$83$	100897.	1.60762	0.803808	+	0.594889i	$0.202804\pi$
$84$	0	0
$85$	0	0
$86$	18790.3	0.273960
$87$	0	0
$88$	7250.96	0.0998134
$89$	−146714.	−1.96334	−0.981671	−	0.190585i	$-0.938962\pi$
$89$	−146714.	−1.96334	−0.981671	+	0.190585i	$0.938962\pi$
$90$	0	0
$91$	8740.22	0.110642
$92$	1383.70	0.0170440
$93$	0	0
$94$	−61059.2	−0.712741
$95$	0	0
$96$	0	0
$97$	43544.9	0.469903	0.234951	−	0.972007i	$-0.424507\pi$
$97$	43544.9	0.469903	0.234951	+	0.972007i	$0.424507\pi$
$98$	13028.8	0.137037
$99$	0	0
$100$	0	0
$101$	−49966.3	−0.487386	−0.243693	−	0.969852i	$-0.578359\pi$
$101$	−49966.3	−0.487386	−0.243693	+	0.969852i	$0.578359\pi$
$102$	0	0
$103$	50579.6	0.469766	0.234883	−	0.972024i	$-0.424529\pi$
$103$	50579.6	0.469766	0.234883	+	0.972024i	$0.424529\pi$
$104$	−3949.04	−0.0358021
$105$	0	0
$106$	1994.53	0.0172415
$107$	−191553.	−1.61744	−0.808721	−	0.588193i	$-0.799840\pi$
$107$	−191553.	−1.61744	−0.808721	+	0.588193i	$0.799840\pi$
$108$	0	0
$109$	71531.6	0.576676	0.288338	−	0.957529i	$-0.406897\pi$
$109$	71531.6	0.576676	0.288338	+	0.957529i	$0.406897\pi$
$110$	0	0
$111$	0	0
$112$	−36261.9	−0.273153
$113$	163395.	1.20377	0.601885	−	0.798583i	$-0.294417\pi$
$113$	163395.	1.20377	0.601885	+	0.798583i	$0.294417\pi$
$114$	0	0
$115$	0	0
$116$	−51732.7	−0.356961
$117$	0	0
$118$	−61177.2	−0.404468
$119$	−236621.	−1.53174
$120$	0	0
$121$	−148215.	−0.920298
$122$	−127245.	−0.774002
$123$	0	0
$124$	−60995.8	−0.356243
$125$	0	0
$126$	0	0
$127$	34350.7	0.188984	0.0944922	−	0.995526i	$-0.469877\pi$
$127$	34350.7	0.188984	0.0944922	+	0.995526i	$0.469877\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	0	0
$131$	−349293.	−1.77833	−0.889164	−	0.457588i	$-0.848713\pi$
$131$	−349293.	−1.77833	−0.889164	+	0.457588i	$0.848713\pi$
$132$	0	0
$133$	93805.1	0.459830
$134$	−197100.	−0.948253
$135$	0	0
$136$	106911.	0.495649
$137$	−274334.	−1.24876	−0.624379	−	0.781122i	$-0.714648\pi$
$137$	−274334.	−1.24876	−0.624379	+	0.781122i	$0.714648\pi$
$138$	0	0
$139$	−47842.2	−0.210027	−0.105013	−	0.994471i	$-0.533489\pi$
$139$	−47842.2	−0.210027	−0.105013	+	0.994471i	$0.533489\pi$
$140$	0	0
$141$	0	0
$142$	84230.6	0.350549
$143$	−6990.80	−0.0285882
$144$	0	0
$145$	0	0
$146$	−157791.	−0.612633
$147$	0	0
$148$	−163993.	−0.615419
$149$	−335645.	−1.23855	−0.619276	−	0.785174i	$-0.712574\pi$
$149$	−335645.	−1.23855	−0.619276	+	0.785174i	$0.712574\pi$
$150$	0	0
$151$	336370.	1.20054	0.600268	−	0.799799i	$-0.295061\pi$
$151$	336370.	1.20054	0.600268	+	0.799799i	$0.295061\pi$
$152$	−42383.4	−0.148794
$153$	0	0
$154$	−64192.8	−0.218114
$155$	0	0
$156$	0	0
$157$	381573.	1.23546	0.617729	−	0.786391i	$-0.288053\pi$
$157$	381573.	1.23546	0.617729	+	0.786391i	$0.288053\pi$
$158$	291797.	0.929905
$159$	0	0
$160$	0	0
$161$	−12249.9	−0.0372450
$162$	0	0
$163$	579006.	1.70692	0.853462	−	0.521155i	$-0.174499\pi$
$163$	579006.	1.70692	0.853462	+	0.521155i	$0.174499\pi$
$164$	−215351.	−0.625227
$165$	0	0
$166$	403587.	1.13676
$167$	128108.	0.355455	0.177728	−	0.984080i	$-0.443125\pi$
$167$	128108.	0.355455	0.177728	+	0.984080i	$0.443125\pi$
$168$	0	0
$169$	−367486.	−0.989746
$170$	0	0
$171$	0	0
$172$	75161.2	0.193719
$173$	487187.	1.23760	0.618800	−	0.785548i	$-0.287619\pi$
$173$	487187.	1.23760	0.618800	+	0.785548i	$0.287619\pi$
$174$	0	0
$175$	0	0
$176$	29003.8	0.0705787
$177$	0	0
$178$	−586855.	−1.38829
$179$	−848235.	−1.97872	−0.989359	−	0.145497i	$-0.953522\pi$
$179$	−848235.	−1.97872	−0.989359	+	0.145497i	$0.953522\pi$
$180$	0	0
$181$	−575423.	−1.30554	−0.652771	−	0.757555i	$-0.726394\pi$
$181$	−575423.	−1.30554	−0.652771	+	0.757555i	$0.726394\pi$
$182$	34960.9	0.0782355
$183$	0	0
$184$	5534.79	0.0120519
$185$	0	0
$186$	0	0
$187$	189259.	0.395779
$188$	−244237.	−0.503984
$189$	0	0
$190$	0	0
$191$	18362.7	0.0364211	0.0182105	−	0.999834i	$-0.494203\pi$
$191$	18362.7	0.0364211	0.0182105	+	0.999834i	$0.494203\pi$
$192$	0	0
$193$	349513.	0.675414	0.337707	−	0.941251i	$-0.390349\pi$
$193$	349513.	0.675414	0.337707	+	0.941251i	$0.390349\pi$
$194$	174180.	0.332272
$195$	0	0
$196$	52115.0	0.0968998
$197$	−371791.	−0.682547	−0.341274	−	0.939964i	$-0.610858\pi$
$197$	−371791.	−0.682547	−0.341274	+	0.939964i	$0.610858\pi$
$198$	0	0
$199$	300569.	0.538036	0.269018	−	0.963135i	$-0.413301\pi$
$199$	300569.	0.538036	0.269018	+	0.963135i	$0.413301\pi$
$200$	0	0
$201$	0	0
$202$	−199865.	−0.344634
$203$	457990.	0.780039
$204$	0	0
$205$	0	0
$206$	202318.	0.332175
$207$	0	0
$208$	−15796.2	−0.0253159
$209$	−75029.4	−0.118813
$210$	0	0
$211$	59499.2	0.0920036	0.0460018	−	0.998941i	$-0.485352\pi$
$211$	59499.2	0.0920036	0.0460018	+	0.998941i	$0.485352\pi$
$212$	7978.11	0.0121916
$213$	0	0
$214$	−766210.	−1.14370
$215$	0	0
$216$	0	0
$217$	539997.	0.778470
$218$	286127.	0.407772
$219$	0	0
$220$	0	0
$221$	−103075.	−0.141962
$222$	0	0
$223$	866124.	1.16632	0.583160	−	0.812357i	$-0.301816\pi$
$223$	866124.	1.16632	0.583160	+	0.812357i	$0.301816\pi$
$224$	−145048.	−0.193148
$225$	0	0
$226$	653581.	0.851194
$227$	−737058.	−0.949374	−0.474687	−	0.880155i	$-0.657439\pi$
$227$	−737058.	−0.949374	−0.474687	+	0.880155i	$0.657439\pi$
$228$	0	0
$229$	846501.	1.06669	0.533346	−	0.845897i	$-0.320935\pi$
$229$	846501.	1.06669	0.533346	+	0.845897i	$0.320935\pi$
$230$	0	0
$231$	0	0
$232$	−206931.	−0.252409
$233$	−590760.	−0.712888	−0.356444	−	0.934317i	$-0.616011\pi$
$233$	−590760.	−0.712888	−0.356444	+	0.934317i	$0.616011\pi$
$234$	0	0
$235$	0	0
$236$	−244709.	−0.286002
$237$	0	0
$238$	−946482.	−1.08310
$239$	1.60375e6	1.81611	0.908054	−	0.418853i	$-0.137568\pi$
$239$	1.60375e6	1.81611	0.908054	+	0.418853i	$0.137568\pi$
$240$	0	0
$241$	−347071.	−0.384925	−0.192463	−	0.981304i	$-0.561647\pi$
$241$	−347071.	−0.384925	−0.192463	+	0.981304i	$0.561647\pi$
$242$	−592860.	−0.650749
$243$	0	0
$244$	−508981.	−0.547302
$245$	0	0
$246$	0	0
$247$	40862.7	0.0426172
$248$	−243983.	−0.251902
$249$	0	0
$250$	0	0
$251$	536438.	0.537446	0.268723	−	0.963217i	$-0.413398\pi$
$251$	536438.	0.537446	0.268723	+	0.963217i	$0.413398\pi$
$252$	0	0
$253$	9797.99	0.00962356
$254$	137403.	0.133632
$255$	0	0
$256$	65536.0	0.0625000
$257$	763565.	0.721130	0.360565	−	0.932734i	$-0.382584\pi$
$257$	763565.	0.721130	0.360565	+	0.932734i	$0.382584\pi$
$258$	0	0
$259$	1.45183e6	1.34483
$260$	0	0
$261$	0	0
$262$	−1.39717e6	−1.25747
$263$	−341099.	−0.304082	−0.152041	−	0.988374i	$-0.548585\pi$
$263$	−341099.	−0.304082	−0.152041	+	0.988374i	$0.548585\pi$
$264$	0	0
$265$	0	0
$266$	375221.	0.325149
$267$	0	0
$268$	−788399.	−0.670516
$269$	750163.	0.632084	0.316042	−	0.948745i	$-0.397646\pi$
$269$	750163.	0.632084	0.316042	+	0.948745i	$0.397646\pi$
$270$	0	0
$271$	−1.64462e6	−1.36033	−0.680163	−	0.733061i	$-0.738091\pi$
$271$	−1.64462e6	−1.36033	−0.680163	+	0.733061i	$0.738091\pi$
$272$	427643.	0.350477
$273$	0	0
$274$	−1.09734e6	−0.883005
$275$	0	0
$276$	0	0
$277$	−1.04631e6	−0.819337	−0.409668	−	0.912235i	$-0.634356\pi$
$277$	−1.04631e6	−0.819337	−0.409668	+	0.912235i	$0.634356\pi$
$278$	−191369.	−0.148511
$279$	0	0
$280$	0	0
$281$	−54503.6	−0.0411774	−0.0205887	−	0.999788i	$-0.506554\pi$
$281$	−54503.6	−0.0411774	−0.0205887	+	0.999788i	$0.506554\pi$
$282$	0	0
$283$	−150130.	−0.111429	−0.0557147	−	0.998447i	$-0.517744\pi$
$283$	−150130.	−0.111429	−0.0557147	+	0.998447i	$0.517744\pi$
$284$	336923.	0.247876
$285$	0	0
$286$	−27963.2	−0.0202149
$287$	1.90650e6	1.36626
$288$	0	0
$289$	1.37065e6	0.965344
$290$	0	0
$291$	0	0
$292$	−631164.	−0.433197
$293$	−1.54001e6	−1.04799	−0.523993	−	0.851723i	$-0.675558\pi$
$293$	−1.54001e6	−1.04799	−0.523993	+	0.851723i	$0.675558\pi$
$294$	0	0
$295$	0	0
$296$	−655972.	−0.435167
$297$	0	0
$298$	−1.34258e6	−0.875788
$299$	−5336.21	−0.00345188
$300$	0	0
$301$	−665402.	−0.423319
$302$	1.34548e6	0.848907
$303$	0	0
$304$	−169534.	−0.105214
$305$	0	0
$306$	0	0
$307$	914773.	0.553946	0.276973	−	0.960878i	$-0.410669\pi$
$307$	914773.	0.553946	0.276973	+	0.960878i	$0.410669\pi$
$308$	−256771.	−0.154230
$309$	0	0
$310$	0	0
$311$	1.59808e6	0.936910	0.468455	−	0.883487i	$-0.344811\pi$
$311$	1.59808e6	0.936910	0.468455	+	0.883487i	$0.344811\pi$
$312$	0	0
$313$	2.38412e6	1.37552	0.687760	−	0.725938i	$-0.258594\pi$
$313$	2.38412e6	1.37552	0.687760	+	0.725938i	$0.258594\pi$
$314$	1.52629e6	0.873601
$315$	0	0
$316$	1.16719e6	0.657542
$317$	2.00013e6	1.11792	0.558960	−	0.829194i	$-0.311200\pi$
$317$	2.00013e6	1.11792	0.558960	+	0.829194i	$0.311200\pi$
$318$	0	0
$319$	−366320.	−0.201551
$320$	0	0
$321$	0	0
$322$	−48999.6	−0.0263362
$323$	−1.10626e6	−0.589999
$324$	0	0
$325$	0	0
$326$	2.31602e6	1.20698
$327$	0	0
$328$	−861404.	−0.442102
$329$	2.16223e6	1.10132
$330$	0	0
$331$	−1.48362e6	−0.744306	−0.372153	−	0.928171i	$-0.621380\pi$
$331$	−1.48362e6	−0.744306	−0.372153	+	0.928171i	$0.621380\pi$
$332$	1.61435e6	0.803808
$333$	0	0
$334$	512432.	0.251345
$335$	0	0
$336$	0	0
$337$	−2.42490e6	−1.16311	−0.581553	−	0.813508i	$-0.697555\pi$
$337$	−2.42490e6	−1.16311	−0.581553	+	0.813508i	$0.697555\pi$
$338$	−1.46994e6	−0.699856
$339$	0	0
$340$	0	0
$341$	−431912.	−0.201145
$342$	0	0
$343$	1.91931e6	0.880864
$344$	300645.	0.136980
$345$	0	0
$346$	1.94875e6	0.875116
$347$	1.26681e6	0.564793	0.282397	−	0.959298i	$-0.408871\pi$
$347$	1.26681e6	0.564793	0.282397	+	0.959298i	$0.408871\pi$
$348$	0	0
$349$	2.54257e6	1.11740	0.558701	−	0.829369i	$-0.311300\pi$
$349$	2.54257e6	1.11740	0.558701	+	0.829369i	$0.311300\pi$
$350$	0	0
$351$	0	0
$352$	116015.	0.0499067
$353$	−1.28509e6	−0.548906	−0.274453	−	0.961601i	$-0.588497\pi$
$353$	−1.28509e6	−0.548906	−0.274453	+	0.961601i	$0.588497\pi$
$354$	0	0
$355$	0	0
$356$	−2.34742e6	−0.981671
$357$	0	0
$358$	−3.39294e6	−1.39916
$359$	4.16251e6	1.70459	0.852294	−	0.523063i	$-0.175211\pi$
$359$	4.16251e6	1.70459	0.852294	+	0.523063i	$0.175211\pi$
$360$	0	0
$361$	−2.03754e6	−0.822882
$362$	−2.30169e6	−0.923158
$363$	0	0
$364$	139844.	0.0553209
$365$	0	0
$366$	0	0
$367$	1.13066e6	0.438195	0.219098	−	0.975703i	$-0.429689\pi$
$367$	1.13066e6	0.438195	0.219098	+	0.975703i	$0.429689\pi$
$368$	22139.2	0.00852201
$369$	0	0
$370$	0	0
$371$	−70630.3	−0.0266413
$372$	0	0
$373$	312371.	0.116251	0.0581257	−	0.998309i	$-0.481488\pi$
$373$	312371.	0.116251	0.0581257	+	0.998309i	$0.481488\pi$
$374$	757037.	0.279858
$375$	0	0
$376$	−976948.	−0.356371
$377$	199507.	0.0722943
$378$	0	0
$379$	−2.19990e6	−0.786692	−0.393346	−	0.919390i	$-0.628683\pi$
$379$	−2.19990e6	−0.786692	−0.393346	+	0.919390i	$0.628683\pi$
$380$	0	0
$381$	0	0
$382$	73450.8	0.0257536
$383$	−3.93331e6	−1.37013	−0.685065	−	0.728482i	$-0.740226\pi$
$383$	−3.93331e6	−1.37013	−0.685065	+	0.728482i	$0.740226\pi$
$384$	0	0
$385$	0	0
$386$	1.39805e6	0.477590
$387$	0	0
$388$	696719.	0.234951
$389$	4.20593e6	1.40925	0.704625	−	0.709580i	$-0.251115\pi$
$389$	4.20593e6	1.40925	0.704625	+	0.709580i	$0.251115\pi$
$390$	0	0
$391$	144465.	0.0477883
$392$	208460.	0.0685185
$393$	0	0
$394$	−1.48716e6	−0.482634
$395$	0	0
$396$	0	0
$397$	2.39484e6	0.762606	0.381303	−	0.924450i	$-0.375475\pi$
$397$	2.39484e6	0.762606	0.381303	+	0.924450i	$0.375475\pi$
$398$	1.20228e6	0.380449
$399$	0	0
$400$	0	0
$401$	4.42161e6	1.37315	0.686577	−	0.727057i	$-0.259112\pi$
$401$	4.42161e6	1.37315	0.686577	+	0.727057i	$0.259112\pi$
$402$	0	0
$403$	235230.	0.0721488
$404$	−799460.	−0.243693
$405$	0	0
$406$	1.83196e6	0.551571
$407$	−1.16124e6	−0.347484
$408$	0	0
$409$	−2.61075e6	−0.771717	−0.385858	−	0.922558i	$-0.626095\pi$
$409$	−2.61075e6	−0.771717	−0.385858	+	0.922558i	$0.626095\pi$
$410$	0	0
$411$	0	0
$412$	809273.	0.234883
$413$	2.16641e6	0.624979
$414$	0	0
$415$	0	0
$416$	−63184.7	−0.0179010
$417$	0	0
$418$	−300117.	−0.0840138
$419$	1.01624e6	0.282789	0.141395	−	0.989953i	$-0.454841\pi$
$419$	1.01624e6	0.282789	0.141395	+	0.989953i	$0.454841\pi$
$420$	0	0
$421$	5.96915e6	1.64137	0.820687	−	0.571379i	$-0.193591\pi$
$421$	5.96915e6	1.64137	0.820687	+	0.571379i	$0.193591\pi$
$422$	237997.	0.0650564
$423$	0	0
$424$	31912.4	0.00862076
$425$	0	0
$426$	0	0
$427$	4.50601e6	1.19598
$428$	−3.06484e6	−0.808721
$429$	0	0
$430$	0	0
$431$	−2.97145e6	−0.770504	−0.385252	−	0.922811i	$-0.625885\pi$
$431$	−2.97145e6	−0.770504	−0.385252	+	0.922811i	$0.625885\pi$
$432$	0	0
$433$	4.95237e6	1.26938	0.634692	−	0.772765i	$-0.281127\pi$
$433$	4.95237e6	1.26938	0.634692	+	0.772765i	$0.281127\pi$
$434$	2.15999e6	0.550461
$435$	0	0
$436$	1.14451e6	0.288338
$437$	−57271.3	−0.0143461
$438$	0	0
$439$	−1.12115e6	−0.277652	−0.138826	−	0.990317i	$-0.544333\pi$
$439$	−1.12115e6	−0.277652	−0.138826	+	0.990317i	$0.544333\pi$
$440$	0	0
$441$	0	0
$442$	−412300.	−0.100382
$443$	2.66233e6	0.644544	0.322272	−	0.946647i	$-0.395553\pi$
$443$	2.66233e6	0.644544	0.322272	+	0.946647i	$0.395553\pi$
$444$	0	0
$445$	0	0
$446$	3.46449e6	0.824713
$447$	0	0
$448$	−580191.	−0.136576
$449$	−3.88781e6	−0.910101	−0.455050	−	0.890466i	$-0.650379\pi$
$449$	−3.88781e6	−0.910101	−0.455050	+	0.890466i	$0.650379\pi$
$450$	0	0
$451$	−1.52490e6	−0.353022
$452$	2.61432e6	0.601885
$453$	0	0
$454$	−2.94823e6	−0.671309
$455$	0	0
$456$	0	0
$457$	−2.52630e6	−0.565841	−0.282920	−	0.959143i	$-0.591303\pi$
$457$	−2.52630e6	−0.565841	−0.282920	+	0.959143i	$0.591303\pi$
$458$	3.38600e6	0.754265
$459$	0	0
$460$	0	0
$461$	4.63589e6	1.01597	0.507985	−	0.861366i	$-0.330390\pi$
$461$	4.63589e6	1.01597	0.507985	+	0.861366i	$0.330390\pi$
$462$	0	0
$463$	−3.62380e6	−0.785619	−0.392810	−	0.919620i	$-0.628497\pi$
$463$	−3.62380e6	−0.785619	−0.392810	+	0.919620i	$0.628497\pi$
$464$	−827724.	−0.178480
$465$	0	0
$466$	−2.36304e6	−0.504088
$467$	3.72757e6	0.790922	0.395461	−	0.918483i	$-0.370585\pi$
$467$	3.72757e6	0.790922	0.395461	+	0.918483i	$0.370585\pi$
$468$	0	0
$469$	6.97970e6	1.46523
$470$	0	0
$471$	0	0
$472$	−978835.	−0.202234
$473$	532217.	0.109380
$474$	0	0
$475$	0	0
$476$	−3.78593e6	−0.765870
$477$	0	0
$478$	6.41500e6	1.28418
$479$	−6.88374e6	−1.37084	−0.685418	−	0.728150i	$-0.740381\pi$
$479$	−6.88374e6	−1.37084	−0.685418	+	0.728150i	$0.740381\pi$
$480$	0	0
$481$	632436.	0.124639
$482$	−1.38829e6	−0.272183
$483$	0	0
$484$	−2.37144e6	−0.460149
$485$	0	0
$486$	0	0
$487$	8.58990e6	1.64122	0.820608	−	0.571492i	$-0.193635\pi$
$487$	8.58990e6	1.64122	0.820608	+	0.571492i	$0.193635\pi$
$488$	−2.03593e6	−0.387001
$489$	0	0
$490$	0	0
$491$	−1.46999e6	−0.275176	−0.137588	−	0.990490i	$-0.543935\pi$
$491$	−1.46999e6	−0.275176	−0.137588	+	0.990490i	$0.543935\pi$
$492$	0	0
$493$	−5.40116e6	−1.00085
$494$	163451.	0.0301349
$495$	0	0
$496$	−975934.	−0.178121
$497$	−2.98278e6	−0.541664
$498$	0	0
$499$	−3.13043e6	−0.562798	−0.281399	−	0.959591i	$-0.590798\pi$
$499$	−3.13043e6	−0.562798	−0.281399	+	0.959591i	$0.590798\pi$
$500$	0	0
$501$	0	0
$502$	2.14575e6	0.380032
$503$	2.29993e6	0.405317	0.202658	−	0.979249i	$-0.435042\pi$
$503$	2.29993e6	0.405317	0.202658	+	0.979249i	$0.435042\pi$
$504$	0	0
$505$	0	0
$506$	39192.0	0.00680488
$507$	0	0
$508$	549611.	0.0944922
$509$	−5.16574e6	−0.883767	−0.441884	−	0.897072i	$-0.645690\pi$
$509$	−5.16574e6	−0.883767	−0.441884	+	0.897072i	$0.645690\pi$
$510$	0	0
$511$	5.58770e6	0.946632
$512$	262144.	0.0441942
$513$	0	0
$514$	3.05426e6	0.509916
$515$	0	0
$516$	0	0
$517$	−1.72945e6	−0.284564
$518$	5.80732e6	0.950936
$519$	0	0
$520$	0	0
$521$	−1.12853e7	−1.82146	−0.910729	−	0.413004i	$-0.864480\pi$
$521$	−1.12853e7	−1.82146	−0.910729	+	0.413004i	$0.864480\pi$
$522$	0	0
$523$	4.86022e6	0.776966	0.388483	−	0.921456i	$-0.372999\pi$
$523$	4.86022e6	0.776966	0.388483	+	0.921456i	$0.372999\pi$
$524$	−5.58869e6	−0.889164
$525$	0	0
$526$	−1.36440e6	−0.215019
$527$	−6.36828e6	−0.998839
$528$	0	0
$529$	−6.42886e6	−0.998838
$530$	0	0
$531$	0	0
$532$	1.50088e6	0.229915
$533$	830498.	0.126625
$534$	0	0
$535$	0	0
$536$	−3.15360e6	−0.474126
$537$	0	0
$538$	3.00065e6	0.446951
$539$	369027.	0.0547125
$540$	0	0
$541$	−5.23864e6	−0.769530	−0.384765	−	0.923015i	$-0.625717\pi$
$541$	−5.23864e6	−0.769530	−0.384765	+	0.923015i	$0.625717\pi$
$542$	−6.57849e6	−0.961896
$543$	0	0
$544$	1.71057e6	0.247825
$545$	0	0
$546$	0	0
$547$	−2.88586e6	−0.412389	−0.206194	−	0.978511i	$-0.566108\pi$
$547$	−2.88586e6	−0.412389	−0.206194	+	0.978511i	$0.566108\pi$
$548$	−4.38934e6	−0.624379
$549$	0	0
$550$	0	0
$551$	2.14122e6	0.300457
$552$	0	0
$553$	−1.03331e7	−1.43688
$554$	−4.18525e6	−0.579359
$555$	0	0
$556$	−765476.	−0.105013
$557$	−3.27936e6	−0.447869	−0.223935	−	0.974604i	$-0.571890\pi$
$557$	−3.27936e6	−0.447869	−0.223935	+	0.974604i	$0.571890\pi$
$558$	0	0
$559$	−289858.	−0.0392334
$560$	0	0
$561$	0	0
$562$	−218014.	−0.0291168
$563$	664023.	0.0882901	0.0441451	−	0.999025i	$-0.485944\pi$
$563$	664023.	0.0882901	0.0441451	+	0.999025i	$0.485944\pi$
$564$	0	0
$565$	0	0
$566$	−600518.	−0.0787925
$567$	0	0
$568$	1.34769e6	0.175275
$569$	−3.00851e6	−0.389557	−0.194778	−	0.980847i	$-0.562399\pi$
$569$	−3.00851e6	−0.389557	−0.194778	+	0.980847i	$0.562399\pi$
$570$	0	0
$571$	1.07571e7	1.38072	0.690359	−	0.723467i	$-0.257453\pi$
$571$	1.07571e7	1.38072	0.690359	+	0.723467i	$0.257453\pi$
$572$	−111853.	−0.0142941
$573$	0	0
$574$	7.62602e6	0.966091
$575$	0	0
$576$	0	0
$577$	−1.53859e7	−1.92390	−0.961952	−	0.273220i	$-0.911911\pi$
$577$	−1.53859e7	−1.92390	−0.961952	+	0.273220i	$0.911911\pi$
$578$	5.48260e6	0.682601
$579$	0	0
$580$	0	0
$581$	−1.42918e7	−1.75650
$582$	0	0
$583$	56493.1	0.00688374
$584$	−2.52466e6	−0.306316
$585$	0	0
$586$	−6.16005e6	−0.741038
$587$	−1.16896e7	−1.40025	−0.700125	−	0.714020i	$-0.746872\pi$
$587$	−1.16896e7	−1.40025	−0.700125	+	0.714020i	$0.746872\pi$
$588$	0	0
$589$	2.52462e6	0.299853
$590$	0	0
$591$	0	0
$592$	−2.62389e6	−0.307709
$593$	2.29301e6	0.267774	0.133887	−	0.990997i	$-0.457254\pi$
$593$	2.29301e6	0.267774	0.133887	+	0.990997i	$0.457254\pi$
$594$	0	0
$595$	0	0
$596$	−5.37031e6	−0.619276
$597$	0	0
$598$	−21344.9	−0.00244084
$599$	−1.58740e7	−1.80767	−0.903834	−	0.427884i	$-0.859259\pi$
$599$	−1.58740e7	−1.80767	−0.903834	+	0.427884i	$0.859259\pi$
$600$	0	0
$601$	−1.16506e7	−1.31571	−0.657855	−	0.753144i	$-0.728536\pi$
$601$	−1.16506e7	−1.31571	−0.657855	+	0.753144i	$0.728536\pi$
$602$	−2.66161e6	−0.299332
$603$	0	0
$604$	5.38193e6	0.600268
$605$	0	0
$606$	0	0
$607$	−9.40797e6	−1.03639	−0.518196	−	0.855262i	$-0.673396\pi$
$607$	−9.40797e6	−1.03639	−0.518196	+	0.855262i	$0.673396\pi$
$608$	−678134.	−0.0743972
$609$	0	0
$610$	0	0
$611$	941896.	0.102070
$612$	0	0
$613$	−9.30509e6	−1.00016	−0.500080	−	0.865979i	$-0.666696\pi$
$613$	−9.30509e6	−1.00016	−0.500080	+	0.865979i	$0.666696\pi$
$614$	3.65909e6	0.391699
$615$	0	0
$616$	−1.02708e6	−0.109057
$617$	268346.	0.0283780	0.0141890	−	0.999899i	$-0.495483\pi$
$617$	268346.	0.0283780	0.0141890	+	0.999899i	$0.495483\pi$
$618$	0	0
$619$	−7.97678e6	−0.836760	−0.418380	−	0.908272i	$-0.637402\pi$
$619$	−7.97678e6	−0.836760	−0.418380	+	0.908272i	$0.637402\pi$
$620$	0	0
$621$	0	0
$622$	6.39233e6	0.662496
$623$	2.07817e7	2.14517
$624$	0	0
$625$	0	0
$626$	9.53647e6	0.972640
$627$	0	0
$628$	6.10516e6	0.617729
$629$	−1.71217e7	−1.72552
$630$	0	0
$631$	2.94679e6	0.294629	0.147314	−	0.989090i	$-0.452937\pi$
$631$	2.94679e6	0.294629	0.147314	+	0.989090i	$0.452937\pi$
$632$	4.66876e6	0.464952
$633$	0	0
$634$	8.00053e6	0.790489
$635$	0	0
$636$	0	0
$637$	−200981.	−0.0196248
$638$	−1.46528e6	−0.142518
$639$	0	0
$640$	0	0
$641$	1.28989e7	1.23996	0.619981	−	0.784617i	$-0.287140\pi$
$641$	1.28989e7	1.23996	0.619981	+	0.784617i	$0.287140\pi$
$642$	0	0
$643$	−9.20262e6	−0.877777	−0.438889	−	0.898542i	$-0.644628\pi$
$643$	−9.20262e6	−0.877777	−0.438889	+	0.898542i	$0.644628\pi$
$644$	−195998.	−0.0186225
$645$	0	0
$646$	−4.42504e6	−0.417192
$647$	−177907.	−0.0167083	−0.00835417	−	0.999965i	$-0.502659\pi$
$647$	−177907.	−0.0167083	−0.00835417	+	0.999965i	$0.502659\pi$
$648$	0	0
$649$	−1.73279e6	−0.161485
$650$	0	0
$651$	0	0
$652$	9.26410e6	0.853462
$653$	1.11867e7	1.02664	0.513321	−	0.858197i	$-0.328415\pi$
$653$	1.11867e7	1.02664	0.513321	+	0.858197i	$0.328415\pi$
$654$	0	0
$655$	0	0
$656$	−3.44562e6	−0.312613
$657$	0	0
$658$	8.64893e6	0.778749
$659$	2.34591e6	0.210425	0.105213	−	0.994450i	$-0.466448\pi$
$659$	2.34591e6	0.210425	0.105213	+	0.994450i	$0.466448\pi$
$660$	0	0
$661$	−3.73844e6	−0.332803	−0.166401	−	0.986058i	$-0.553215\pi$
$661$	−3.73844e6	−0.332803	−0.166401	+	0.986058i	$0.553215\pi$
$662$	−5.93446e6	−0.526304
$663$	0	0
$664$	6.45740e6	0.568378
$665$	0	0
$666$	0	0
$667$	−279619.	−0.0243362
$668$	2.04973e6	0.177728
$669$	0	0
$670$	0	0
$671$	−3.60410e6	−0.309023
$672$	0	0
$673$	−6.11773e6	−0.520658	−0.260329	−	0.965520i	$-0.583831\pi$
$673$	−6.11773e6	−0.520658	−0.260329	+	0.965520i	$0.583831\pi$
$674$	−9.69961e6	−0.822440
$675$	0	0
$676$	−5.87977e6	−0.494873
$677$	−6.50221e6	−0.545242	−0.272621	−	0.962121i	$-0.587891\pi$
$677$	−6.50221e6	−0.545242	−0.272621	+	0.962121i	$0.587891\pi$
$678$	0	0
$679$	−6.16805e6	−0.513421
$680$	0	0
$681$	0	0
$682$	−1.72765e6	−0.142231
$683$	−1.37285e7	−1.12609	−0.563044	−	0.826427i	$-0.690370\pi$
$683$	−1.37285e7	−1.12609	−0.563044	+	0.826427i	$0.690370\pi$
$684$	0	0
$685$	0	0
$686$	7.67722e6	0.622865
$687$	0	0
$688$	1.20258e6	0.0968595
$689$	−30767.5	−0.00246913
$690$	0	0
$691$	1.36316e7	1.08606	0.543028	−	0.839715i	$-0.317278\pi$
$691$	1.36316e7	1.08606	0.543028	+	0.839715i	$0.317278\pi$
$692$	7.79499e6	0.618800
$693$	0	0
$694$	5.06726e6	0.399369
$695$	0	0
$696$	0	0
$697$	−2.24837e7	−1.75302
$698$	1.01703e7	0.790123
$699$	0	0
$700$	0	0
$701$	2.08567e6	0.160306	0.0801531	−	0.996783i	$-0.474459\pi$
$701$	2.08567e6	0.160306	0.0801531	+	0.996783i	$0.474459\pi$
$702$	0	0
$703$	6.78767e6	0.518003
$704$	464061.	0.0352894
$705$	0	0
$706$	−5.14037e6	−0.388135
$707$	7.07763e6	0.532524
$708$	0	0
$709$	1.23349e7	0.921554	0.460777	−	0.887516i	$-0.347571\pi$
$709$	1.23349e7	0.921554	0.460777	+	0.887516i	$0.347571\pi$
$710$	0	0
$711$	0	0
$712$	−9.38968e6	−0.694146
$713$	−329687.	−0.0242872
$714$	0	0
$715$	0	0
$716$	−1.35718e7	−0.989359
$717$	0	0
$718$	1.66500e7	1.20533
$719$	−2.83517e6	−0.204530	−0.102265	−	0.994757i	$-0.532609\pi$
$719$	−2.83517e6	−0.204530	−0.102265	+	0.994757i	$0.532609\pi$
$720$	0	0
$721$	−7.16450e6	−0.513272
$722$	−8.15015e6	−0.581865
$723$	0	0
$724$	−9.20677e6	−0.652771
$725$	0	0
$726$	0	0
$727$	1.19423e7	0.838017	0.419009	−	0.907982i	$-0.362378\pi$
$727$	1.19423e7	0.838017	0.419009	+	0.907982i	$0.362378\pi$
$728$	559374.	0.0391178
$729$	0	0
$730$	0	0
$731$	7.84721e6	0.543152
$732$	0	0
$733$	−2.50999e7	−1.72549	−0.862744	−	0.505642i	$-0.831256\pi$
$733$	−2.50999e7	−1.72549	−0.862744	+	0.505642i	$0.831256\pi$
$734$	4.52265e6	0.309851
$735$	0	0
$736$	88556.7	0.00602597
$737$	−5.58267e6	−0.378593
$738$	0	0
$739$	−1.66189e6	−0.111942	−0.0559708	−	0.998432i	$-0.517825\pi$
$739$	−1.66189e6	−0.111942	−0.0559708	+	0.998432i	$0.517825\pi$
$740$	0	0
$741$	0	0
$742$	−282521.	−0.0188383
$743$	988158.	0.0656681	0.0328340	−	0.999461i	$-0.489547\pi$
$743$	988158.	0.0656681	0.0328340	+	0.999461i	$0.489547\pi$
$744$	0	0
$745$	0	0
$746$	1.24948e6	0.0822022
$747$	0	0
$748$	3.02815e6	0.197890
$749$	2.71331e7	1.76724
$750$	0	0
$751$	2.42260e7	1.56741	0.783704	−	0.621135i	$-0.213328\pi$
$751$	2.42260e7	1.56741	0.783704	+	0.621135i	$0.213328\pi$
$752$	−3.90779e6	−0.251992
$753$	0	0
$754$	798026.	0.0511198
$755$	0	0
$756$	0	0
$757$	8.03239e6	0.509454	0.254727	−	0.967013i	$-0.418014\pi$
$757$	8.03239e6	0.509454	0.254727	+	0.967013i	$0.418014\pi$
$758$	−8.79960e6	−0.556275
$759$	0	0
$760$	0	0
$761$	−8.44956e6	−0.528899	−0.264449	−	0.964400i	$-0.585190\pi$
$761$	−8.44956e6	−0.528899	−0.264449	+	0.964400i	$0.585190\pi$
$762$	0	0
$763$	−1.01323e7	−0.630083
$764$	293803.	0.0182105
$765$	0	0
$766$	−1.57333e7	−0.968828
$767$	943716.	0.0579232
$768$	0	0
$769$	−1.68674e7	−1.02857	−0.514285	−	0.857620i	$-0.671943\pi$
$769$	−1.68674e7	−1.02857	−0.514285	+	0.857620i	$0.671943\pi$
$770$	0	0
$771$	0	0
$772$	5.59220e6	0.337707
$773$	−2.37205e7	−1.42782	−0.713912	−	0.700236i	$-0.753078\pi$
$773$	−2.37205e7	−1.42782	−0.713912	+	0.700236i	$0.753078\pi$
$774$	0	0
$775$	0	0
$776$	2.78687e6	0.166136
$777$	0	0
$778$	1.68237e7	0.996490
$779$	8.91339e6	0.526259
$780$	0	0
$781$	2.38575e6	0.139958
$782$	577861.	0.0337914
$783$	0	0
$784$	833840.	0.0484499
$785$	0	0
$786$	0	0
$787$	2.21145e7	1.27274	0.636370	−	0.771384i	$-0.280435\pi$
$787$	2.21145e7	1.27274	0.636370	+	0.771384i	$0.280435\pi$
$788$	−5.94865e6	−0.341274
$789$	0	0
$790$	0	0
$791$	−2.31446e7	−1.31525
$792$	0	0
$793$	1.96288e6	0.110844
$794$	9.57936e6	0.539244
$795$	0	0
$796$	4.80910e6	0.269018
$797$	1.86780e6	0.104156	0.0520779	−	0.998643i	$-0.483416\pi$
$797$	1.86780e6	0.104156	0.0520779	+	0.998643i	$0.483416\pi$
$798$	0	0
$799$	−2.54996e7	−1.41308
$800$	0	0
$801$	0	0
$802$	1.76864e7	0.970967
$803$	−4.46928e6	−0.244596
$804$	0	0
$805$	0	0
$806$	940918.	0.0510169
$807$	0	0
$808$	−3.19784e6	−0.172317
$809$	−203197.	−0.0109156	−0.00545778	−	0.999985i	$-0.501737\pi$
$809$	−203197.	−0.0109156	−0.00545778	+	0.999985i	$0.501737\pi$
$810$	0	0
$811$	2.27341e7	1.21374	0.606870	−	0.794801i	$-0.292425\pi$
$811$	2.27341e7	1.21374	0.606870	+	0.794801i	$0.292425\pi$
$812$	7.32785e6	0.390019
$813$	0	0
$814$	−4.64494e6	−0.245708
$815$	0	0
$816$	0	0
$817$	−3.11092e6	−0.163055
$818$	−1.04430e7	−0.545686
$819$	0	0
$820$	0	0
$821$	2.56056e7	1.32579	0.662897	−	0.748710i	$-0.269327\pi$
$821$	2.56056e7	1.32579	0.662897	+	0.748710i	$0.269327\pi$
$822$	0	0
$823$	7.10878e6	0.365844	0.182922	−	0.983127i	$-0.441444\pi$
$823$	7.10878e6	0.365844	0.182922	+	0.983127i	$0.441444\pi$
$824$	3.23709e6	0.166087
$825$	0	0
$826$	8.66563e6	0.441927
$827$	2.69334e7	1.36939	0.684695	−	0.728830i	$-0.259935\pi$
$827$	2.69334e7	1.36939	0.684695	+	0.728830i	$0.259935\pi$
$828$	0	0
$829$	2.89570e7	1.46341	0.731707	−	0.681619i	$-0.238724\pi$
$829$	2.89570e7	1.46341	0.731707	+	0.681619i	$0.238724\pi$
$830$	0	0
$831$	0	0
$832$	−252739.	−0.0126579
$833$	5.44107e6	0.271689
$834$	0	0
$835$	0	0
$836$	−1.20047e6	−0.0594067
$837$	0	0
$838$	4.06498e6	0.199962
$839$	1.95591e7	0.959279	0.479639	−	0.877466i	$-0.340767\pi$
$839$	1.95591e7	0.959279	0.479639	+	0.877466i	$0.340767\pi$
$840$	0	0
$841$	−1.00569e7	−0.490316
$842$	2.38766e7	1.16063
$843$	0	0
$844$	951987.	0.0460018
$845$	0	0
$846$	0	0
$847$	2.09944e7	1.00553
$848$	127650.	0.00609580
$849$	0	0
$850$	0	0
$851$	−886393.	−0.0419568
$852$	0	0
$853$	−1.74251e7	−0.819981	−0.409990	−	0.912090i	$-0.634468\pi$
$853$	−1.74251e7	−0.819981	−0.409990	+	0.912090i	$0.634468\pi$
$854$	1.80241e7	0.845684
$855$	0	0
$856$	−1.22594e7	−0.571852
$857$	2.03324e7	0.945663	0.472831	−	0.881153i	$-0.343232\pi$
$857$	2.03324e7	0.945663	0.472831	+	0.881153i	$0.343232\pi$
$858$	0	0
$859$	−1.19607e7	−0.553061	−0.276530	−	0.961005i	$-0.589185\pi$
$859$	−1.19607e7	−0.553061	−0.276530	+	0.961005i	$0.589185\pi$
$860$	0	0
$861$	0	0
$862$	−1.18858e7	−0.544829
$863$	−3.87101e7	−1.76928	−0.884640	−	0.466274i	$-0.845596\pi$
$863$	−3.87101e7	−1.76928	−0.884640	+	0.466274i	$0.845596\pi$
$864$	0	0
$865$	0	0
$866$	1.98095e7	0.897590
$867$	0	0
$868$	8.63995e6	0.389235
$869$	8.26488e6	0.371268
$870$	0	0
$871$	3.04045e6	0.135798
$872$	4.57802e6	0.203886
$873$	0	0
$874$	−229085.	−0.0101442
$875$	0	0
$876$	0	0
$877$	−3.76410e7	−1.65258	−0.826289	−	0.563247i	$-0.809552\pi$
$877$	−3.76410e7	−1.65258	−0.826289	+	0.563247i	$0.809552\pi$
$878$	−4.48459e6	−0.196330
$879$	0	0
$880$	0	0
$881$	2.92452e6	0.126945	0.0634723	−	0.997984i	$-0.479783\pi$
$881$	2.92452e6	0.126945	0.0634723	+	0.997984i	$0.479783\pi$
$882$	0	0
$883$	3.90249e7	1.68438	0.842190	−	0.539181i	$-0.181266\pi$
$883$	3.90249e7	1.68438	0.842190	+	0.539181i	$0.181266\pi$
$884$	−1.64920e6	−0.0709811
$885$	0	0
$886$	1.06493e7	0.455761
$887$	1.16063e7	0.495318	0.247659	−	0.968847i	$-0.420339\pi$
$887$	1.16063e7	0.495318	0.247659	+	0.968847i	$0.420339\pi$
$888$	0	0
$889$	−4.86571e6	−0.206487
$890$	0	0
$891$	0	0
$892$	1.38580e7	0.583160
$893$	1.01090e7	0.424208
$894$	0	0
$895$	0	0
$896$	−2.32076e6	−0.0965741
$897$	0	0
$898$	−1.55513e7	−0.643538
$899$	1.23261e7	0.508659
$900$	0	0
$901$	832955.	0.0341830
$902$	−6.09962e6	−0.249624
$903$	0	0
$904$	1.04573e7	0.425597
$905$	0	0
$906$	0	0
$907$	3.80327e7	1.53511	0.767554	−	0.640984i	$-0.221473\pi$
$907$	3.80327e7	1.53511	0.767554	+	0.640984i	$0.221473\pi$
$908$	−1.17929e7	−0.474687
$909$	0	0
$910$	0	0
$911$	4.76995e7	1.90422	0.952112	−	0.305748i	$-0.0989066\pi$
$911$	4.76995e7	1.90422	0.952112	+	0.305748i	$0.0989066\pi$
$912$	0	0
$913$	1.14312e7	0.453854
$914$	−1.01052e7	−0.400110
$915$	0	0
$916$	1.35440e7	0.533346
$917$	4.94767e7	1.94302
$918$	0	0
$919$	−3.06445e7	−1.19692	−0.598459	−	0.801154i	$-0.704220\pi$
$919$	−3.06445e7	−1.19692	−0.598459	+	0.801154i	$0.704220\pi$
$920$	0	0
$921$	0	0
$922$	1.85436e7	0.718400
$923$	−1.29934e6	−0.0502016
$924$	0	0
$925$	0	0
$926$	−1.44952e7	−0.555517
$927$	0	0
$928$	−3.31090e6	−0.126205
$929$	4.20442e7	1.59833	0.799165	−	0.601112i	$-0.205275\pi$
$929$	4.20442e7	1.59833	0.799165	+	0.601112i	$0.205275\pi$
$930$	0	0
$931$	−2.15704e6	−0.0815614
$932$	−9.45216e6	−0.356444
$933$	0	0
$934$	1.49103e7	0.559267
$935$	0	0
$936$	0	0
$937$	−2.09921e7	−0.781101	−0.390551	−	0.920582i	$-0.627715\pi$
$937$	−2.09921e7	−0.781101	−0.390551	+	0.920582i	$0.627715\pi$
$938$	2.79188e7	1.03607
$939$	0	0
$940$	0	0
$941$	1.06080e7	0.390536	0.195268	−	0.980750i	$-0.437442\pi$
$941$	1.06080e7	0.390536	0.195268	+	0.980750i	$0.437442\pi$
$942$	0	0
$943$	−1.16399e6	−0.0426255
$944$	−3.91534e6	−0.143001
$945$	0	0
$946$	2.12887e6	0.0773430
$947$	−2.66948e7	−0.967280	−0.483640	−	0.875267i	$-0.660686\pi$
$947$	−2.66948e7	−0.967280	−0.483640	+	0.875267i	$0.660686\pi$
$948$	0	0
$949$	2.43408e6	0.0877341
$950$	0	0
$951$	0	0
$952$	−1.51437e7	−0.541552
$953$	−2.68015e7	−0.955932	−0.477966	−	0.878378i	$-0.658626\pi$
$953$	−2.68015e7	−0.955932	−0.477966	+	0.878378i	$0.658626\pi$
$954$	0	0
$955$	0	0
$956$	2.56600e7	0.908054
$957$	0	0
$958$	−2.75350e7	−0.969328
$959$	3.88589e7	1.36441
$960$	0	0
$961$	−1.40960e7	−0.492364
$962$	2.52974e6	0.0881331
$963$	0	0
$964$	−5.55314e6	−0.192463
$965$	0	0
$966$	0	0
$967$	3.43270e7	1.18051	0.590255	−	0.807217i	$-0.299027\pi$
$967$	3.43270e7	1.18051	0.590255	+	0.807217i	$0.299027\pi$
$968$	−9.48576e6	−0.325375
$969$	0	0
$970$	0	0
$971$	−1.43606e7	−0.488791	−0.244396	−	0.969676i	$-0.578590\pi$
$971$	−1.43606e7	−0.488791	−0.244396	+	0.969676i	$0.578590\pi$
$972$	0	0
$973$	6.77676e6	0.229477
$974$	3.43596e7	1.16051
$975$	0	0
$976$	−8.14370e6	−0.273651
$977$	−1.16248e7	−0.389627	−0.194813	−	0.980840i	$-0.562410\pi$
$977$	−1.16248e7	−0.389627	−0.194813	+	0.980840i	$0.562410\pi$
$978$	0	0
$979$	−1.66221e7	−0.554280
$980$	0	0
$981$	0	0
$982$	−5.87996e6	−0.194579
$983$	−3.11250e6	−0.102737	−0.0513683	−	0.998680i	$-0.516358\pi$
$983$	−3.11250e6	−0.102737	−0.0513683	+	0.998680i	$0.516358\pi$
$984$	0	0
$985$	0	0
$986$	−2.16046e7	−0.707709
$987$	0	0
$988$	653804.	0.0213086
$989$	406251.	0.0132070
$990$	0	0
$991$	−2.33325e7	−0.754704	−0.377352	−	0.926070i	$-0.623165\pi$
$991$	−2.33325e7	−0.754704	−0.377352	+	0.926070i	$0.623165\pi$
$992$	−3.90373e6	−0.125951
$993$	0	0
$994$	−1.19311e7	−0.383014
$995$	0	0
$996$	0	0
$997$	−5.35497e7	−1.70616	−0.853079	−	0.521782i	$-0.825267\pi$
$997$	−5.35497e7	−1.70616	−0.853079	+	0.521782i	$0.825267\pi$
$998$	−1.25217e7	−0.397958
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.bf.1.1	yes	2
3.2	odd	2		450.6.a.ba.1.1	yes	2
5.2	odd	4		450.6.c.p.199.3		4
5.3	odd	4		450.6.c.p.199.2		4
5.4	even	2		450.6.a.y.1.2	✓	2
15.2	even	4		450.6.c.q.199.1		4
15.8	even	4		450.6.c.q.199.4		4
15.14	odd	2		450.6.a.bd.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.6.a.y.1.2	✓	2	5.4	even	2
450.6.a.ba.1.1	yes	2	3.2	odd	2
450.6.a.bd.1.2	yes	2	15.14	odd	2
450.6.a.bf.1.1	yes	2	1.1	even	1	trivial
450.6.c.p.199.2		4	5.3	odd	4
450.6.c.p.199.3		4	5.2	odd	4
450.6.c.q.199.1		4	15.2	even	4
450.6.c.q.199.4		4	15.8	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +16.0000 q^{4} -141.648 q^{7} +64.0000 q^{8} +O(q^{10})\)	\(q+4.00000 q^{2} +16.0000 q^{4} -141.648 q^{7} +64.0000 q^{8} +113.296 q^{11} -61.7038 q^{13} -566.592 q^{14} +256.000 q^{16} +1670.48 q^{17} -662.241 q^{19} +453.185 q^{22} +86.4812 q^{23} -246.815 q^{26} -2266.37 q^{28} -3233.30 q^{29} -3812.24 q^{31} +1024.00 q^{32} +6681.92 q^{34} -10249.6 q^{37} -2648.96 q^{38} -13459.4 q^{41} +4697.57 q^{43} +1812.74 q^{44} +345.925 q^{46} -15264.8 q^{47} +3257.19 q^{49} -987.260 q^{52} +498.632 q^{53} -9065.48 q^{56} -12933.2 q^{58} -15294.3 q^{59} -31811.3 q^{61} -15249.0 q^{62} +4096.00 q^{64} -49274.9 q^{67} +26727.7 q^{68} +21057.7 q^{71} -39447.8 q^{73} -40998.2 q^{74} -10595.8 q^{76} -16048.2 q^{77} +72949.3 q^{79} -53837.8 q^{82} +100897. q^{83} +18790.3 q^{86} +7250.96 q^{88} -146714. q^{89} +8740.22 q^{91} +1383.70 q^{92} -61059.2 q^{94} +43544.9 q^{97} +13028.8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 32 q^{4} + 100 q^{7} + 128 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 + 32 * q^4 + 100 * q^7 + 128 * q^8	\( 2 q + 8 q^{2} + 32 q^{4} + 100 q^{7} + 128 q^{8} - 540 q^{11} - 890 q^{13} + 400 q^{14} + 512 q^{16} - 492 q^{17} + 592 q^{19} - 2160 q^{22} - 3660 q^{23} - 3560 q^{26} + 1600 q^{28} - 5700 q^{29} - 5708 q^{31} + 2048 q^{32} - 1968 q^{34} - 11300 q^{37} + 2368 q^{38} - 15420 q^{41} - 6320 q^{43} - 8640 q^{44} - 14640 q^{46} + 7800 q^{47} + 44844 q^{49} - 14240 q^{52} + 27828 q^{53} + 6400 q^{56} - 22800 q^{58} - 50520 q^{59} - 29126 q^{61} - 22832 q^{62} + 8192 q^{64} - 97400 q^{67} - 7872 q^{68} - 6180 q^{71} - 32900 q^{73} - 45200 q^{74} + 9472 q^{76} - 173916 q^{77} + 7912 q^{79} - 61680 q^{82} + 163464 q^{83} - 25280 q^{86} - 34560 q^{88} - 164640 q^{89} - 191416 q^{91} - 58560 q^{92} + 31200 q^{94} - 52430 q^{97} + 179376 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 + 32 * q^4 + 100 * q^7 + 128 * q^8 - 540 * q^11 - 890 * q^13 + 400 * q^14 + 512 * q^16 - 492 * q^17 + 592 * q^19 - 2160 * q^22 - 3660 * q^23 - 3560 * q^26 + 1600 * q^28 - 5700 * q^29 - 5708 * q^31 + 2048 * q^32 - 1968 * q^34 - 11300 * q^37 + 2368 * q^38 - 15420 * q^41 - 6320 * q^43 - 8640 * q^44 - 14640 * q^46 + 7800 * q^47 + 44844 * q^49 - 14240 * q^52 + 27828 * q^53 + 6400 * q^56 - 22800 * q^58 - 50520 * q^59 - 29126 * q^61 - 22832 * q^62 + 8192 * q^64 - 97400 * q^67 - 7872 * q^68 - 6180 * q^71 - 32900 * q^73 - 45200 * q^74 + 9472 * q^76 - 173916 * q^77 + 7912 * q^79 - 61680 * q^82 + 163464 * q^83 - 25280 * q^86 - 34560 * q^88 - 164640 * q^89 - 191416 * q^91 - 58560 * q^92 + 31200 * q^94 - 52430 * q^97 + 179376 * q^98

Introduction

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